1CHAPTER ONE 1.0 INTRODUCTION 1. 1 BACKGROUND Health issues have been one of the primary and indeed a paramount concern in man‟s quest for knowledge since time immemorial. Good health is basic to our survival. Adequate food and sound nutrition are essential to good health; they are crucial not only for human survival, but also for prevention of and recovery from illness (Ghana Demographic and Health Survey, GDHS, 2003). This is especially so for newborns. A total of 19 million newborns are born every year in the developing world with low birth weight, which is related to maternal malnutrition. Worldwide, more than 9 million children under 5 years of age die each year. Malnutrition underlies a majority of these under 5 deaths, 70% of which occur in the first year of life. Infant and young child feeding practices directly impact the nutritional status and, ultimately, the child survival of children under 2 years of age. Therefore, improving infant and young child feeding is critical to ensuring their optimal health, nutrition and development (Unicef.org, 2008). Appropriate feeding practices are of fundamental importance for the survival, growth, development, health and nutrition of infants and children. Feeding practices are one of the underlying determinants of child‟s nutritional status which in turn influence the risk of illness and ultimately death (GDHS,2003). Breastfeeding, according to the Oxford Advanced Learners Dictionary, means giving the baby breast milk soon after birth. The baby is put to breast and begins to suck the mother‟s nipple to stimulate the breast to release milk. Also en.wikipedia.org defines 2 breastfeeding as the feeding of an infant or young child with breast milk directly from female human breast ( i.e. via lactation) rather than from a baby bottle or other containers. Breastfeeding is sufficient and beneficial for an infant‟s nutrition in the first six months of life. Exclusive breastfeeding implies feeding the infant only with breast milk without any additional food or drink, not even water (WHO, 2003). According to UNICEF.org.media (2008), high average with optimal breastfeeding practices, especially exclusive breastfeeding for the first six months of life, could have the single largest impact on child survival, with the potential to prevent 1.4 million under five deaths. Yet in 2008 rates of exclusive breastfeeding were only about 37% in developing countries. The World Health Assembly (WHA), in May 2001, recommended exclusive breastfeeding for the first six months of a child‟s life. It emphasized the media role in promoting and protecting exclusive breastfeeding to improve child health. Since Ghana adopted this policy (Exclusive breastfeeding for the first six months), various seminars and programs are being held and information being spread through National and Private Television stations, radios and news papers to promote and encourage the practice. All women are encouraged to breastfeed their infants exclusively for the first six months and then complement the breastfeeding with nutritious food for at least two years. Pre-lacteal feeding (given something other than breast milk in the first three days of life) is generally discouraged since it inhibits breastfeeding and expose the new born infant to illness (GDHS,2003). Despite the recommendation by the WHA, some infants are fed with complementary foods in addition to breast milk. There are other infants, especially those in orphanage homes in Ghana, who do not receive any breast milk. 3 One way of measuring a healthy baby is to look at the weight gain. If the baby gains enough weight during its first year then everything is fine. This study is intended to compare the growth of babies under four forms of feeding practices namely exclusive breastfeeding, breastfeeding with water, breastfeeding with complimentary foods and feeding without breast milk. This would be done by comparing their weights for the first nine months of their life. Children gain weight and grow more rapidly during infancy and childhood than at any other time in life. However, some children fail to gain weight at a normal rate, either because of expected variations related to genes, being born prematurely, or because of under nutrition, which may occur for a variety of reasons (Rebecca, T. K., 2006). Under nutrition is sometimes called a growth deficit or failure to thrive. It is important to recognize and treat children who are not gaining weight normally, because it may be a sign of under nutrition or an underlying medical problem that requires treatment. This, according to Rebecca, T. K. (2006), is because, each year, under nutrition contributes to the deaths of about 5.6 million children younger than 5 years in the developing world. Another 146 million children within this age group are underweight and are at an increased risk of early deaths, illness, disability and underachievement. In least developed countries, 42 percent of children are stunted and 36 percent are underweight. Under nutrition can have complications such as a weakened immune system, shorter than normal height, or difficulties with learning apart from deaths in children. These complications are more common in children who are undernourished for a long period of time (Rebecca, T. K., 2006). Poor weight gain is defined as gaining weight at a slower rate than other children who are the same age and sex. "Normal" ranges for weight and height are based upon the weight and height 4 of thousands of children. In the United States, standard growth charts are published by the Centers for Disease Control and Prevention; these charts are available for boys and girls and are appropriate for all races and nationalities. Weight gain normally follows a predictable course from infancy through adolescence. However, some children do not gain weight normally from birth, while other children gain weight normally for a while, then slow or stop gaining weight. Weight gain usually slows down before the child slows or stops growing in length. A child is said to have poor weight gain if he or she does not grow at the expected rate for their age and sex. Poor weight gain is not a disease, but rather a symptom, which has many possible causes (Rebecca, T. K., 2006). Three factors account for poor weight gain in children. The first factor includes not consuming adequate amount of calories or the right combination of protein, fat, and carbohydrates. The second factor is when the child is not absorbing adequate amount of nutrients, which leads to poor weight gain. The third factor has to do with requiring a higher than normal amount of calories (Rebecca, T. K., 2006). Poor weight gain can occur as a result of a medical problem, a developmental or behavioral problem, and lack of adequate food, a social problem at home, or most frequently, a combination of these problems (Rebecca, T. K., 2006). If an infant or a child slows or stops gaining weight, it is important to try to determine and treat the underlying cause. The first step is a complete medical history and physical examination. Most children will not require blood testing or imaging tests, although testing may be recommended in certain situations. Parents should also mention if they have eliminated foods from the child's diet due to concern about the effects of these foods (e.g. abdominal pain, diarrhea, "hyperactivity"). The provider may also ask about the child's household, including who 5 lives in the child's house, if there have been recent changes or stresses (e.g. divorce, illness, death, new sibling), or if anyone in the house has a medical or psychiatric illness. The provider may also ask about the food supply (e.g. if there have been days when anyone in the family went hungry because there was not enough money for food). Although these questions can be difficult to answer, it is important to be honest (Rebecca, T. K., 2006). 1.1.1 Growth Monitoring In Infants and Children For a relatively small expenditure per child, growth monitoring can greatly strengthen preventive health programmes. Growth is the best general index of the health of an individual child, and regular measurements of growth permit the early detection of malnutrition, frequently association with diarrhea, and other illnesses, when remedial action is relatively easy. Although acute signs of malnutrition are easily noted by health workers, it is often too late, and always more expensive, to help the severely malnourished children (UNICEF, 2006). For early detection of children with growth retardation and high risk of malnutrition and mortality, health workers need special tools and training in growth monitoring. The growth status of children is a measure of the health and well-being of the whole community. Birth weight is of a particular significance in determining the nutritional status of a community, as low birth weight is a good indicator of subsequent illness and death in children (UNICEF, 2006). Various body measurements are used to access growth. Some are easier to use, more accurate and more useful than others. Monitoring the growth of a child usually requires taking the same measurements at regular intervals and seeing how they change. A single measurement only indicates the child‟s size or weight is increasing, staying the same, or declining. Careful repeated measurements and comparisons with previous measurements are necessary because most 6 children will continue to grow a little, unless they are very ill, and it is easy to mistake some growth for adequate growth. Growth measures are usually compared to a reference population. Gathering data to establish a local reference population is a major undertaking. Therefore, western standards are usually used for comparison, such as Tanner and Boston, or more recently, those of the National Centre for Health Statistics (UNICEF, 2006). 1.1.2 Weight-For-Age of Children To obtain weight-for-age, the weight of the child (in kilograms) is compared with that of an ideally healthy child of the same age from a reference population. This is the basis of the weight- for-age, or Gomez classification of nutritional status. A child weighing less than 60 per cent of the reference weight-for-age is considered to be severely malnourished. For these reasons, countries have different ways of assessing the growth of children through weight taking and at different places. For instant, in Indonesia 2.5 million infants and young children are being weighed regularly at the traditional monthly meetings of village women. The results are entered on growth charts kept by the mothers themselves. In Thailand, a programme based on the home use of growth charts by parents in several villages, helped to eliminate completely third degree malnutrition, and reduced second degree malnutrition by 44 per cent during 1981 – 1982, even though no additional food was provided. In Colombia, improvements in weight gain for a majority of children suffering from mild, moderate and severe malnutrition have been achieved in poor communities by nutrition programmes incorporating the “Carnet de Salud” or health card kept at home by mothers. In Jamaica, a systematic programme to improve the health and growth of over 6,000 young children using growth charts, immunization and nutrition education and 7 milk supplements, has resulted in a 40% decline in the prevalence of malnutrition and a 60 percent fall in infant mortality (UNICEF, 1985). In Ghana, mothers and family members have traditionally used variety of indicators and milestones when assessing their children‟s growth. The signs used and local standards children are compared to vary culturally, but are important factors affecting child care practice. These were taken into consideration and used by the community health workers to understand the community‟s perceptions, so that they could be related to the use of growth charts. One quite productive community survey in central Ghana did examine closely the growth related beliefs and practices of local mothers and families in order to suggest ways of applying the information to improve rate monitoring at the local level (Love et al 1985). One key question concerned how mothers know when their children are growing or not growing well. The responses were rich and comprehensive in scope, covering indicators such as appetite, stages of physical and mental development, general health appearance and the feel and look of the skin. Height and weight were also mentioned, as well as variation in mood and activity level (Brownlee, 1990). A wide range of physical changes both for periods of good and poor growth were mentioned. Children growing well cry and move normally. When growth is poor the abdomen becomes distended, bigger than normal, the hair turns brown or grayish, the face puffy or pale, the fontanel sunken or large. Observations were both sensitive and detailed, and often included accurate descriptions of the signs of anemia and dehydration, two major causes of child mortality in Ghana. Size was emphasized, with clothing often mentioned as a useful indicator: “in poor 8 growth, „dresses become loose around the body and clothes bought for a young child are still a good fit many months later‟ ” (Brownlee, 1990). Traditionally “anthropometric” measures were some of the most interesting. It is customary in central Ghana, as in many other cultures, to make strings or beads for a newborn and put them around the waist, wrist and legs. They are intended as decoration but used by many parents to assess growth. “One mother explained that by the time the child had reached the age of 5 months, the bead-strings around the waist should have been changed or adjusted five times.” Other items mentioned included metal bracelets, necklaces and finger rings (Brownlee, 1990). 1.1.3 Interventions to Improve Weight Gain of Infants and Children In 2000, the Center for Diseases Control and prevention (CDC) established the International Micronutrient Malnutrition Prevention and Control Program (IMMPaCt) to support the global effort to eliminate vitamin and mineral deficiencies, or hidden hunger in both developed and developing nations. Through the IMMPaCt program, CDC provides funding and/or technical assistance directly to countries through cooperative and interagency agreements with UNICEF, the World Health Organization (WHO), the U.S. Agency of International Development (USAID), and the Global Alliance for Improved Nutrition (GAIN), and the Micronutrient Initiative (MI). With these partners, CDC has assisted countries in assessing the burden of hidden hunger through national surveys and surveillance systems that allow countries to monitor the coverage and impact of their food fortification and micronutrient supplementation programs. In addition, computer and web-based training tools and regional and national training workshops developed by CDC have strengthened the capacity of countries to assess the burden of poor weight gain through malnutrition, track the effectiveness of interventions strategies through 9 surveillance systems, and plan social marketing and health communication strategies to promote the consumption of vitamin- and mineral-fortified foods. In 2002, in collaboration with the WHO‟s Eastern Mediterranean Regional Office (EMRO), CDC provided funding support and consultation toward a national micronutrient survey to generate baseline data on iron status of adult women and preschool children in order to monitor the impact of the recently initiated national flour fortification program in Jordan. To help improve nutrition worldwide, the CDC IMMPaCt Program helped launch the Flour Fortification Initiative (FFI) in 2002. The Initiative was formalized in 2005. The FFI Leaders Group, a network of government and international agencies, wheat and flour industries, academia, and consumer and civic organizations, was established to promote flour fortification. FFI supports fortification of flour with essential vitamins and minerals, especially folic acid and iron, as one important way to help improve the nutritional status of populations, especially women and children, around the world. In many Asian and African countries commercially produced infant foods are either not commonly used or readily accessible through markets in remote areas. Through the IMMPaCt program, CDC is actively planning pilot interventions in Kenya and Tajikistan to assess the feasibility of alternative approaches to sustainable distribution, through small local markets and house-to-house sales, of easy-to-use, “in-home” fortificants to enrich baby foods. These efforts will require public-private-civic sector partnerships to be nurtured and strengthened over time. 10 1.2 STATEMENT OF THE PROBLEM Weight gain is one of the observable signs of growth especially at the early stages of life. Regular assessment of weight gain among babies has been a way of assessing their health status and growth. Too much of it leads to child obesity while too little is a sign of ill-health. Ghana is much concerned about the health and growth of her children and consequently instituted a monitoring mechanism for babies (both prenatal and postnatal) up to the age of sixty months. This mechanism involves monthly assessment of weight right after birth and compulsory immunization against some diseases (six childhood killer diseases) such as Tuberculosis, Poliomyelitis, Diphtheria, Yellow fever etc at every public or private health center. One could access these weight records when a visit is made to these centers. Weight change, especially among individual children, could be biological (variation related to genes), due to socio-economic status of the mother or due to child‟s nutritional status. Children who show poor weight gain could be a sign of malnutrition. Appropriate feeding practices are of fundamental importance for the growth, survival, development, health and nutrition of infants and children (GDHS, 2003). Parents do not see the need to practice a particular feeding type and feel reluctant to participate in the monthly weighing exercise as they do not know the importance of them. It is in the light of this that the study seeks to examine the relationship between infant feeding practice and weight gain to enable the parents appreciate the feeding practice that is beneficial and results in good health of their babies. 1.3 THE OBJECTIVE(S) OF THE STUDY The main objective is to find out if different forms of feeding practice have effect on 11 weights of babies (growth of babies). To do this the study seeks to: i. Compare the weights of babies under four different forms of feeding practice namely, Exclusive breastfeeding, breastfeeding with water only, breastfeeding with any other food (water inclusive) and feeding without breast milk (non-breastfeeding) . ii. Investigate how weights vary by months for the four modes of feeding practices. iii. Determine which mode(s) of feeding practice differ(s) from others. iv. Find out what happens to the weights for the first three months after introduction of complimentary (other) foods in case of exclusive breastfeeding. v. To compare the rate of weight gain for the age periods 1-3 months, 4-6 months and 7-9 months. 1.3.1 Hypotheses In order to achieve the set objective, the following hypotheses are outlined 1. H₀: There is no difference in weight gains under the different feeding practices. (H₀: µ₁ = µ₂ = µ₃ = µ₄ ) H₁ ; There is at least one feeding practice that differ in weight gain from the rest. (H₁: ≠ = for i and j ) 2. H₀: Feeding practices have no effect on weight of babies H₁: Feeding practices affect weight of babies 12 1.4 RESEARCH QUESTIONS Some concerns which this study is attempting to address are (a) Is the monthly increase in weight influenced by the birth weight of a baby? (b) Does the type of feeding practice influence the weight of a baby? (c) Which age period do we have high rate of weight gain? 1.5 SIGNIFICANCE OF THE STUDY A baby must have proper nutrition to grow and thrive. According to Ghana Health Service (GHS) (2003), the most cost effective way to address the pressing public health challenge of malnutrition is to prevent it. That means ensuring that all of the children who are normal weight at birth continue within the normal range, and those who are low birth weight at birth are brought swiftly into a healthy growth range. If the baby weighs too little, it is important to ensure that the nutrition it needs to grow is given. When the baby‟s weight is too much, and the baby is overweight or obese, one may need to consult a doctor for a more balanced and healthy diet. Many researchers including Ghanaian researchers have shown that food insecurity is not the most important factor that contributes to a high prevalence of under nutrition. Instead inappropriate feeding and caring practices of children and mothers, poor environment (e.g. poor sanitation and hygiene), and limited utilization of basic health care services are among the most important determining factors (GHS, 2003). From economic point of view, poor weight gain causes ill-health of babies which is a potential source of drain of parents‟ income. Appropriate feeding method ensures the optimum weight gain for a baby. 13 The research is to encourage parents, especially lactating mothers, to practice appropriate feeding method for their babies. This will help reduce instance of diseases associated with improper feeding practices and ultimately help reduce infant mortality. 1.6 SCOPE OF THE STUDY The study was carried out at three centers in Mampong in the Ashanti Region of Ghana. The centers were Mampong Babies Home, Mampong Technical College of Education Primary School – New Darman, and Akyeremade - near Birth and Death Registry. Basically, the data involving feeding without breast milk was collected from the Babies Home while the data on the other three feeding practices were obtained from the two centers mentioned. The respondents in the research were babies of not less than nine months as the data involves the birth weight and weights for the first nine months after birth. 1.7 THE OUTLINE OF THE STUDY The study is divided into five chapters. Chapter one is the Introduction and it focuses on the background of the study, the statement of the problem, the objectives of the study, the research questions, the significant of the study, the scope of the study and the outline of the study. Chapter two presents the review of literature related and relevant to the study. Chapter three deals with the Methodology which involves the sampling procedure, data sources and collection, and the overview of the analysis and modeling techniques. It also entails the underlying basic theories and concepts and how they are applied. Chapter four presents the Data analysis and the results of the analysis. The last chapter, Chapter five, discusses the key findings of the study which are followed by conclusions and recommendations. 14 CHAPTER TWO 2.0 LITERATURE REVIEW Parents are primarily responsible for their baby‟s health. It is extremely important for them to understand the unique nutritional requirement and feeding practices of infants. A lot of studies have been carried out concerning nutritional requirements and feeding practices of infants. 2.1 FEEDING PRACTICES OF INFANTS The two main ways of feeding infants, which are orthodox, are Breastfeeding and Non- breastfeeding. Breastfeeding has been the basic norm for humans from the genesis of creation while Non-Breastfeeding is artificial to human race. Non-Breastfeeding takes place in most cases when Breastfeeding is medically contraindicated such as health of mother, the baby being unable to breastfeed, absence of mother, personal preferences, beliefs and experiences. Breastfeeding is practiced in three main ways. It can be done exclusively or with water only or with water and other foods. The third form, breastfeeding with water and other foods may be due to a baby being considered at risk for malnutrition, lactation insufficiency, decision or preference of parents etc. Both Non-Breastfeeding and breastfeeding with water and other foods are considered automatic, for example Bottle-feeding, and have the benefit of allowing freedom to leave the baby with others (Wambach and Koehn, 2004). 2.1.1 Breastfeeding Breastfeeding was the rule in ancient times up to recent human history and babies were carried with the mothers and fed as required. With 18 th and 19 th century industrialization in the Western world, mothers in many urban centers began dispensing with 15 breastfeeding due to work requirement in urban Europe. Breastfeeding declined significantly from 1900 to 1960, due to improved sanitation, nutritional technologies, and increasingly negative social attitude towards the practice (Riordan, 1980). By the 1960s the predominant attitude to breastfeeding was that it was something practiced by the uneducated and those lacking temperament of lower classes. The practice was considered old – fashioned and “a little disgusting”, left for those who could not afford infant formula and discouraged by medical practitioners and media of times. In fact, letters and editorials to the women‟s magazine Chatelaine from 1945 to as late as 1995, regarded breastfeeding with a predominately negative attitude. However, since the middle 1960s there has been a steady resurgence in the practice of breastfeeding in Canada and the US, especially among more educated affluent women (Nathoo, 2009). After the addition of solid food, mothers are advised to continue breastfeeding for at least a year. The World Health Organisation recommends nursing for at least two years or more. The World Health Organisation (WHO) and the American Academy of Pediatrics (AAP) emphasize the value of breastfeeding for mothers as well as children. Both recommend exclusive breastfeeding for the first six months of life. The AAP recommends that this be followed by supplemented breastfeeding for at least one year, while WHO recommends that supplemented breastfeeding continues two years or more. While recognizing the superiority of breastfeeding , regulating authorities also work to minimize the risk of artificial feeding (Nathoo, 2009). According to Ip S, Chung M, Raman G, et al (2007) , scientific research such as the studies summarized in a 2007 review for the US Agency for Healthcare Research and Quality (AHRQ) and a 2007 review for the WHO (Horta BL, 2007) have found numerous benefits 16 of breastfeeding for the infant. According to AAP, research shows that breastfeeding provides advantages with regard to general health, growth and development. Not breastfeeding significantly increases risks for a large number of acute and chronic diseases including lower respiratory infection, ear infections, bacteremia, bacterial meningitis, botulism, urinary tract infections, and necrotizing enterocolitis (Lucas, A; Cole, T.J, 1990). There are a number of studies that show a possible protective effect of breast milk feeding against sudden infant death syndrome, insulin – dependent diabetes mellitus, Crohn‟s disease, ulcerative colitis, lymphoma, allergic diseases, digestive diseases and a possible enhancement of cognitive development (PEDIATRICS, Feb., 2005). Under modern health care, human breast milk is considered the healthiest form of milk for babies (Picciano, M., 2001). Breastfeeding promotes health for both mother and infant and helps to prevent diseases (Riordan, JM., 1997). Longer breastfeeding has also been associated with better mental health through childhood and into adolescence (Ody, Wendy H.; Kendall GE.; et al, 2010). Exclusive breastfeeding for the first six months of life has the single largest impact on child survival (Unicef.org. media ,2008). Supplementing breast milk before six months is unnecessary and is strongly discouraged because of the likelihood of contamination, the unaffordability of breast milk substitutes, and the resulting increased risk of diarrheal diseases. The early introduction of liquids and solids reduces milk output because the production and release of milk is influenced by the frequency and intensity of sucking (GDHS, 2003). Despite the recommendations that babies be exclusively breastfed for the first 6 months, less than 40% of infants below this age are exclusively breastfed worldwide. The overwhelming majority of American babies are not exclusively breastfed for this period – 17 in 2005 under 12% of babies were breastfed exclusively for the first 6 months, with 60% of babies of 2 months of age being fed formula and approximately one in four breastfed infants having infant formula feeding within two days of birth (WHO, 2008). Despite the high breastfeeding prevalence (97%) in Ghana, the majority of infants are not fed in compliance with WHO / UNICEF recommendations (World Health Assembly, 2001). The recommendations call for a period of exclusive breastfeeding for six months and the introduction of complementary foods after the age of six months. Fifty-three percent (53%) of children under six months of age are exclusively breastfed in Ghana as at 2003, (GDHS, 2003), which is a slight increase over the proportion reported in the 1998 GDHS (Ghana Statistical Service and MI, 1999). Exclusive breastfeeding drops sharply from 65% at age 2 – 3 months to 39% at age 4 – 5 months. Further, 6% of children aged 2 – 3 months and 32% of children aged 4 – 5 months are receiving complementary foods in addition to breast milk. This indicates that there are many infants who are at risk of being exposed to bacterial contamination and poor quality foods, even if they started out well with early initiation of breastfeeding (GDHS, 2003). 2.1.2 Artificial Feeding Artificial Feeding involves not breastfeeding and supplementing breastfeeding with other foods. Experts agree that breastfeeding is beneficial and have concerns about the effects of artificial formulas. Artificial feeding is associated with more deaths from diarrhea in infants in both developing and developed countries. There are few exceptions, such as when the mother is taking certain drugs or is infected with hum T – lymphotropic virus, or has active untreated tuberculosis. In developed countries with access to infant formula 18 and clean drinking water, maternal HIV infection is an absolute contraindication to breastfeeding (regardless of maternal HIV viral load or antiretroviral treatment) due to the s risk for mother – to – child HIV transmission (Horton, 1996). Infant formula is a manufactured food designed and marketed for feeding to babies and infants under 12 months of age, usually prepared for bottle – feeding or cup – feeding from powder (mixed with water) or liquid (with or without additional water). The US Federal Food, Drug and Cosmetic Act (FFDCA) defines infant formula as “ a food which purports to be or is represented for special dietary use solely as a food for infants by reason of its simulation of human milk or its suitability as a complete or partial substitute for human milk (Wells, J.C.K., 1996). A 2001 WHO report found that infant formula prepared in accordance with applicable Codex Alimentarius Standards was a safe complementary food and a suitable breast milk substitute. In 2003, the WHO and UNICEF published their Global Strategy for infant and young child feeding which restated that “processed – food products for infants and young children should when sold or otherwise distributed, meet applicable standards recommended by the Codex Alimentarius Commission, and also warned that “lack of breastfeeding – and especially lack of exclusive breastfeeding during the first half – year of life – are important risk factors for infant and childhood morbidity and mortality”. In particular, the use of infant formula in less economically developed countries is linked to poorer health outcomes because of the prevalence of unsanitary preparation conditions, including lack of clean water and lack of sanitizing equipment (WHO, 2003). UNICEF(2007), estimates that a formula – fed child living in unhygienic conditions is 19 between 6 and 25 times more likely to die of diarrhea and four times more likely to die of pneumonia than a breastfed child. Throughout history, Schuman (2003) stated, mothers who could not breastfeed their babies either employed a wet nurse or less frequently, prepared food for their babies, a process known as “dry nursing”. Baby food composition varied according to region and economic status (Oliver, 2004). As early as 1846, scientists and nutritionists noted that an increase in medical problems and infant mortality was associated with dry nursing. Improvement of quality of manufactured baby foods gave birth to Raw milk formulas (mixture of cow milk, water, cream, and sugar or honey in specific ratios) to achieve the nutritional balance believed to approximate human milk reformulated in such a way as to accommodate the believed digestive capability of the infant (Fomon, 2001). In 1920s and 1930s, evaporated milk began to be widely commercially available at low prices, and several clinical studies suggested that babies fed with evaporated milk formula thrive as well as breastfed babies. Then came commercial Formulas followed by Generic brand formulas and later Follow – on toddler formulas (Fomon, 2001). The global infant formula market has been estimated at $7.9 billion, with North America and Western Europe accounting for 33% of the market and considered largely saturated, and Asia representing 53% of the market. Infant formula is the largest segment of the baby food market, with the fraction given as between 40% and 70% (Markos, 2005). Leading health organizations (e.g. WHO, US CDC and Department of Health and Human Services) are attempting to reduce the use of infant formula and increase the prevalence of breastfeeding from birth through 12 to 24 months of age through public health awareness 20 campaigns (CDC, 2007). The specific goals and approaches of these breastfeeding promotion programs, and the policy environment surrounding their implementation, vary by country. As a policy basic framework, the International Code of Marketing of Breast-Milk Substitutes, adopted by the WHO‟s World Health Assembly in 1981, requires infant formula companies to preface their product information with statements that breastfeeding is the best way of feeding babies and that a substitute should only be used after consultation with health professionals. The Baby Friendly Hospital Initiative also restricts use by hospitals of free formula or other infant care aids provided by formula companies (WHO, 1981). 2.2 NUTRITIONAL STATUS AND PREVALENCE OF UNDERWEIGHT CHILDREN IN AFRICA On nutritional status of children in Ghana, according to the 2003 GDHS, 30% of children under five are stunted and 11% severely stunted. Weight – for – age results show that 22% of children under five are underweight, with 5% severely underweight. Dinesh et al (2006) studied the nutritional status of under-five children and whether infant feeding practices are associated with the under nutrition in anganwari (AW) areas of urban Allahabad. Under-five-years children and their mothers in selected four anganwari areas of urban Allahabad (UP) participated in the study. Nutritional assessment by WHO criterion (SD- classification) using summary indices of nutritional status: weight-for-age, height-for-age and weight-for-height were done. Normal test of proportions, Chi-square test for testing association of nutritional status with different characteristics and risk analysis using odds ratios with 95% confidence intervals was also done. Among all under five children surveyed, 36.4% underweight (<2SD weight- for -age), 51.6% stunted (<2SD height- for- age), and 10.6% wasted (<2SD 21 weight- for- height). Proportions of underweight (45.5%) and stunting (81.8%) were found maximum among children aged 13-24 months. Wasting was most prevalent (18.2%) among children aged 37-48 months. Initiations of breast-feeding after six hours of birth, deprivation from colostrums and improper complementary feeding were found to be significant (P<0.05) risk factors for underweight. Wasting was not significantly associated (P>0.10) with any infant feeding practice studied. ICDS benefits received by children failed to improve the nutritional status of children. They concluded that delayed initiation of breast-feeding, deprivation from colostrums, and improper weaning are significant risk factors for under nutrition among under- fives. There is need for promotion and protection of optimal infant feeding practices for improving nutrition. According to Dramane and Carolyn (2006), malnutrition of children (0-59 months) is a public health concern in Africa, particularly in the Sahelian countries. In their study to evaluate the trends of the malnutrition status of children under five, from the project monitoring /impact indicators, they determine that, nutritional status of children under-five continues to deteriorate in spite of better agro climatic conditions and agricultural production in many of these countries. Using data collected from the GFSI project, they conducted their analysis base on the following indicators such as Underweight (percent of children of a given age range with weight-for-age z score less than -2 or less than -3 standard deviation) and Stunting (percent of children of a given age range with height-for-age z score less than -2 standard deviation). In the first exploratory phase of the analysis, descriptive statistics and factor analyses (Principal Component Analysis - PCA, Multiple Correspondences Analysis - MCA) were used. The PCA was used for the analysis of underweight children data and for the analysis of the stunted children under-five data, the use of the MCA was chosen. 22 The analysis of anthropometric data on the prevalence of underweight children (under five) over the period from December 2000 to September 2005 shows that this form of malnutrition continues to be a challenge (with rates > 20%) in the study zone. It reaches its peak in June with an average rate varying from 43% to 44% of weighed children. The rate of severely malnourished children (Weight/Age < -3 SD) follows the same trends as that of the rate of global malnutrition. It is largely influenced by the food availability/access at the household level (see figure2.1). The minimum rate of severe malnutrition is observed in September. The maximum recorded rate is 11% in June 2005. That means that in June 2005, 11 children out of 100 in the project zone should have been referred to hospitals as emergency cases due to severe malnutrition problems. Figure 2.1: Evolution Child Malnutrition and Food Availability The rate of underweight children under-five is more important among the sedentary population than the pastoralists. The average rate of global (severe + moderate) underweight children under- 0 2 4 6 8 10 12 D e c e m b e r D e c e m b e r M a r c h J u n e S e p t e m b e r D e c e m b e r M a r c h J u n e S e p t e m b e r D e c e m b e r 2000 2003 2004 2005 % 0 1 2 3 4 5 6 7 8 N b o f m o n t h % Severe Malnutris Children Nb months of food availability 23 five in the pastoral zone is 37.5% (CV=38%) against 36%, 40 %( CV=26%) and 41% (CV=37%) for the children under-five in the zones of river, Irrigated Village Perimeters (IVP), and of the Lakes respectively. However the average rate of severely underweight children is higher among the children of pastoralists (8.84%) than those of sedentary IVP cereal producers (7.60%), and flooded rice growers (6.50%). The rate of severely underweight children in the livestock zone remains lower than the rate of malnourished children living in the villages located around the lakes (9.91%). Fawzi et al (1998) studied the relationship between prolonged breastfeeding and child growth by examining prospectively among 28,753 Sudanese children less than 36 months of age enrolled in a broader cohort study of child health and nutrition. 81% of children were breast-fed at 12 months, but this prevalence declined to 62% at 18 months and 27% at 24 months. At baseline and at each of three 6-monthly follow-up visits breastfeeding status was assessed and all subjects were weighed and measured. Their results show that undernourished children were more likely to be breastfed for a longer period of time compared with normal children. They found a small difference between breastfed and fully weaned children in the gain in height over the following 6-month period. However, breastfed children were likely to gain significantly less weight, particularly among children who were aged 6-12 months. Similar findings were noted when these associations were examined among children who were normally nourished at the time of breastfeeding assessment. The inverse association between breastfeeding status and weight gain was significantly larger among children of poor or illiterate mothers compared with children of relatively more affluent or literate mothers, respectively. 24 In conclusion, their findings suggest that the inverse association is not causal, and may be explained by poorer complementary feeding among breastfed compared with weaned children. Children from poorer households and whose parents are illiterate are more likely to have less than adequate complementary feeding. The importance of adequate complementary feeding in the second half of infancy needs to be stressed in nutrition education programmes. 2.3 FACTORS AFFECTING THE PREVALENCE OF MALNUTRITION Salah E. et al (2006) states that, malnutrition affects growth, morbidity, mortality, cognitive development, reproduction, and physical work capacity, and it consequently impacts on human performance, health and survival. It is an underlying factor in many diseases for both children and adults, and is particularly prevalent in developing countries, where it affects one out of every 3 preschool-age children. A well-nourished child is one whose weight and height measurements compare very well with the standard normal distribution of heights and weights of healthy children of the same age and sex. Factors that contribute to malnutrition are many and varied. Their study was to evaluate the level of malnutrition and the impact of some socio-economic and demographic factors of households on the nutritional status of children under 3 years of age in Botswana. Factors included: the number of children under 3 years of age in the family, occupation of the parents, marital status, family income, parental education, maternal nutritional knowledge, residence location (urban or rural), gender, and breastfeeding practices. The study was a cross-sectional descriptive survey using a structured questionnaire and measurements of 25 weight and height. Four hundred households and mothers of children under three, representing the 23 Health Regions of Botswana, participated in the study. Reference standards used were those of the National Center for Health Statistics (NCHS). EPI Info software (version 5) was used for data entry and analysis. The results show that the level of wasting, stunting, and underweight in children under three years of age was 5.5 %, 38.7 %, and 15.6 % respectively. Malnutrition was significantly (p < 0.01) higher among boys than among girls. Underweight was less prevalent among children whose parents worked in the agricultural sector than among children whose parents were involved in informal business. Children brought up by single parents suffered from underweight to a significantly (p < 0.01) higher level than children living with both parents. The prevalence of underweight decreased significantly (p<0.01) as family income increased. The higher the level of the mother‟s education, the lower the level of child underweight observed. Breastfeeding was found to reduce the occurrence of underweight among children. The study findings imply that efforts for redressing child undernutrition issues in Botswana should focus on factors associated with development outcomes such as maternal income, maternal education, and the creation of employment or economic engagements that do not compromise important child care practices such as breastfeeding. Martens (2007) studied into the Impact of School Feeding Programme in Ghana with the aim of determining the nutrient intake from school meals and the out-school food consumption of the children and the impact on demand for locally produced foods. Data were collected in 4 primary 26 schools in 4 different districts in Central Region in Ghana from February to April 2007 with the study population of 129 3 rd grade children aged 7 to 16 years. Anthropometric measurements were taken to determine nutritional status. Data collection on nutrient intake from school meals was done using 1-day weighed dietary records and weighing the portion sizes of the selected 3rd grade children. The primary caretakers of the 3rd grade children were interviewed by trained translators using 24hr recalls to determine the out-school food consumption. The demand for locally produced foods per district was determined via the production figures of staple foods gathered from the district Agricultural Extension Service and the information from the weighed dietary records. The food composition data were derived from 3 tables, of which the leading table was: „Table de composition d‟aliments du Mali‟ (Barikmo, 2004) (supplements: „Foods commonly used in Ghana‟ (Eyeson, 1975) and the „NEVO table‟ (NEVO, 2006)). The statistical analysis comprised of descriptive analyses on all data, ANOVA followed by a Tukey test for normally distributed variables, Kruskal-Wallis followed by a Dunnett test for skewed variables, a Spearman correlation test, an independent sample T-test and a paired sample T-test.The total food consumption had a DDS of 6.4 (±1.0), which is on average 1 (±0.8) food group higher than the home food consumption (sig. 0.001). Only 22.6% of the children still received a home lunch. The nutrient intake recommendations for energy (30-45% of RDA) and protein (60-70% of RDA), formulated by the Ghana SFP National Secretariat have been met (31.9% and 67.6% respectively). The iron intake (6.1±5.8) is low compared to the weighted DRI of this age group of 22.9 mg/day (bioavailability of 5%). The demand for locally produced staples is on average 0.07% of the total production of staples in the district. This may potentially increase to 1.11% in the year 2010. 27 The Ghana SFP succeeded in increasing the dietary diversity of the diet of the school children in the selected schools. This may reflect in the nutritional adequacy and nutritional status of the primary school children, bet no internationally agreed upon cut-off points are available to use as a reference. The Ghana SFP meets their recommendations for energy intake and protein intake. Vitamin A intake is probably sufficient, but the iron intake remains low, which raises concern. The impact of the Ghana SFP on the local demand for staple foods at district level seems limited. Maseta et al (2008) conducted a comparative cross-sectional study to compare childcare practices and nutritional status of children aged 6–36 months in Mwembesongo and Mjimpya wards that had long and short experiences respectively with the Child Survival, Protection and Development (CSPD) programme. The purpose of the study was to establish whether the long- term implementation of the CSPD programme had an impact compared to that of a short-term programme. The findings showed that the children from Mwembesongo were exclusively breast- fed for a significantly longer period (50 days) than those in the Mjimpya ward (32 days) and that significantly more mothers (95.7%) in Mwembesongo than in Mjimpya (84.5%) attended growth monitoring programmes. On the other hand, significantly more mothers in Mjimpya (71.5%) compared to those in Mwembesongo (51.8%) breast-fed immediately (less than one hour) after birth. The study revealed that there was no significant difference in children‟s nutritional status (wasting and underweight) between the two wards, except for stunting. More children in Mwembesongo (39.7%) than in Mjimpya (27.5%) were stunted. The findings have demonstrated that financial capacity to support children‟s food and care requirements forms a springboard from which to launch additional efforts for improved nutritional status. Janet and Mabel (1992) used Demographic Health Survey data I Malawi to assess the association between breast-feeding practices, socio-economic and morbidity variables, and the nutritional 28 status of children under the age of five years using multilevel models. About 27% of under-five children in Malawi are underweight, and nearly 50% are stunted. The results of this study suggest that socio-economic factors, morbidity, and inappropriate feeding practices are some of the factors associated with malnutrition in Malawi. High socio-economic status, as measured by urban residence, the presence of modern amenities, and some maternal education, is associated with better nutritional status, whereas morbidity within two weeks before the survey is associated with low weight-for-age Z scores. Breast-feeding is almost universal and is carried on for about 21 months, but the introduction of complementary food starts much too early; only 3% of Malawian children under the age of 4 months are exclusively breastfed. Children aged 12 months or older who were still breastfeeding at the time of the survey were of lower nutritional status than those who had stopped breastfeeding. The analysis also showed a significant intra-family correlation of weight-for-age Z scores of children of the same family of about 39%. In his study to the causes of morbidity and mortality in developing world, Adam (2005) identified under-nutrition and infection as the major causes. These two problems are interrelated. Under-nutrition compromises barrier function, allowing easier access by pathogens, and compromises immune function. Thus malnutrition predisposes to infection. Studies done worldwide by UNICEF indicate that Africa has the worst nutritional status with high levels of underweight, stunting and wasting countrywide (UDHS, 2000 - 2001 and 1995). UNICEF also estimated that 40% of all deaths among children under five years of age were related to malnutrition, with high levels of underweight up to 12%. The general objective of this study was to assess the nutritional status of children under five years of age in Rweibaare Parish, Bushenyi District. A descriptive cross-sectional study using quantitative data collection methods was used to collect data from mothers or caretakers of 215 children under five years. A structured 29 questionnaire was used to collect the data and anthropometric measurements (weight and height) were taken and recorded in a table. The measurements were transformed into the standard anthropometrical measurements and standard indices of physical growth were considered. The data collected was analyzed manually using frequency tables. The results revealed that the rate of malnutrition was high, of the 21 5 children assessed 32% of the children were stunted, 13% underweight and 2.8% wasted. The factors contributing to this were likely to be related with socio-economic status of the mothers which included age of the mother, education level and main occupation. Lack of information about nutrition was also a contributing factor as evidenced by inadequate breastfeeding with children that stopped breast feeding at four months and below having a high likelihood of malnutrition. Chronic malnutrition was a predominant form of malnutrition in this study as seen by malnourished children being mostly stunted (32%) and underweight (13%). Nutrients requirements are high and the diets given are often inadequate because of lack of suitable food, parental poverty, ignorance and the social customs to which they are subjected. One of the most striking problems facing the World Health Organisation is the high morbidity and mortality rates in infants and under fives, (WHO, 2002). Under nutrition is described as the most pervasive human problem especially in less developed countries. It has an adverse effect on the quality of life and social and economic development (Gabir 1995). The problem of under nutrition is most prevalent among vulnerable groups especially in developing countries notably among children, adolescents, pregnant and lactating women and people living in difficult situations, such as the landless, the displaced and the poor (MOH, 2002). In Uganda, as in other developing countries children below five years of age have been the target groups for nutrition programmes due to the high rates of mortality and morbidity in this age group. 30 UNICEF (1998), reported that the infant mortality rate was 97/1,000 live births and the under five mortality rate was 141/1,000 live births in Uganda. Thirty to forty percent of the children under five years of age are malnourished, 38% in the same age group are stunted 5% wasted, and 61% of the population living below the poverty line, (UDHS, 1995). Despite the apparent abundance of food in Uganda, subsistence agriculture, the favorable climate, and enormous natural resources, malnutrition still exists. This is due to regional food imbalances due to ecological differences between the various regions of the country, the under-developed inter- regional food marketing system, poor road network and inadequate food processing and storage facilities. The nutritional status of children is influenced by both diet and frequency of infections. Under-five children have high mortality rates; they suffer from frequent infections such as malaria, measles, intestinal worms and skin diseases, (Jelliffe, 1996). It was therefore, concluded that malnutrition is a significant problem in Rweibaale Parish Bushenyi district. It can also be concluded that malnutrition is caused by many factors hence a need for a multi-sectoral approach if the problem is to be solved. The results could also be an indicator of a need for reproductive health services especially in planning for their families, caring for their children and have a source of income. It is therefore, recommended that another study be done to cover whole District to give a broader picture of nutritional status of children since this study covered only one Parish. Other studies that would use other indicators in addition to what was used (height and weight) such as upper arm circumference and head circumference is necessary. Health workers should be trained and re-oriented to do growth monitoring and promotion of good nutrition as part of integrated management of childhood illnesses. According to Khaled (2009), Major progress has been made over the last 30 years in reducing the prevalence of malnutrition amongst children less than 5 years of age in developing countries. 31 However, approximately 27% of children under the age of 5 in these countries are still malnourished. His work focuses on the childhood malnutrition in one of the biggest developing countries, Egypt. The study examined the association between bio-demographic and socioeconomic determinants and the malnutrition problem in children less than 5 years of age using the 2003 Demographic and Health survey data for Egypt. In the first step, separate geoadditive Gaussian models with the continuous response variables stunting (height-for-age), underweight (weight-for-age) were used, and wasting (weight-for-height) as indicators of nutritional status in the case study. In a second step, based on the results of the first step, we apply the geoadditive Gaussian latent variable model for continuous indicators in which the 3 measurements of the malnutrition status of children are assumed as indicators for the latent variable “nutritional status”. 32 CHAPTER THREE 3.0 METHODOLOGY 3.1 INTRODUCTION This chapter reviews the statistical methods and tools used for the study and their underlying concepts and theories. The rational for adopting the methods and the procedures involved in the analyses would be determined. Paramount among these methods in this regard are univariate and multivariate repeated measures analyses of variance, general linear model ANOVA and general linear mixed model ANOVA. These methods would be considered under longitudinal data analysis. Thorough exploratory data analysis for some descriptive statistics, preliminary and further inferential analyses would be conducted. 3.2 SAMPLING PROCEDURE All nursing mothers presented at the three Weighing Centers at Mampong Metropolis for scheduled clinic visits from Monday 18 th March to Friday 22 nd March were invited to participate in the study. The qualitative strand of the data which involves the feeding practice of the babies was collected using a brief questionnaire. 3.3 SOURCES AND COLLECTION OF DATA Data were collected at three Weighing Centers at Mampong namely, Mampong Babies Home, Mampong Technical College of Education Primary School and Akyeremade (near Birth and Death Registry). These Centers have a monthly weighing programme for children of age one month after birth to sixty months (five years) after birth. Every month children within this age group are brought to the Weighing Centers for weighing. It is from this exercise 33 that the data (birth weights and monthly weights of babies for the first nine months ) was extracted and the form of breastfeeding the baby was undergoing during that period was solicited from the mother. The child‟s weighing card was picked to record the weights and the mother supplied a kind of food the baby was given within the first six months after birth. Also the weighing cards of children within the age group in the orphanage who were supposedly not undergoing the breastfeeding were requested and the weights extracted. A sample of 109 babies altogether was obtained from all the centers. Thus the weight of total of 109 babies were used for the study. 3.4 DATA ANALYSIS The following exploratory data analyses were carried out: 1. Tabulated statistics for sex and feeding practice 2. Tally for discrete variables involving sex and feeding practice 3. Descriptive statistics involving maximum, minimum and mean monthly weights of babies under each form of feeding practice 4. A component line graph of the mean weights of babies under each feeding practice and a plot of weight by time with respect to gender to observe the trend of weight gain 5. Tabulated correlation coefficients of monthly weights of babies At the preliminary inferential analysis stage repeated measures analysis of variance for both univariate (ANOVA) and multivariate (MANOVA) cases were used. Furthermore, General Linear Model was employed for some inferential analysis. Linear mixed models and General linear mixed models were applied for further inferential analysis. 34 Computer software packages, Excel, Minitab and SAS were used to analyze the data. 3.5 MODEL FORMULATION The response variable was the weight of the child with the explanatory variables as the four types of feeding practices of the child. Other variables that might affect the weight increase such as the child‟s birth weight and the gender of the baby were considered under the explanatory variables. Each child was assessed for the first nine months after birth. Since the outcomes were repeated measurements over time on the same subjects (weight of the same babies when they were 1 month, 2 months, …. , 9 months) the data is a longitudinal data. This called for analyzing the data based on repeated measures analysis and longitudinal data analysis 3.5.1 Repeated Measures Analysis Repeated measurement analysis sometimes known as longitudinal studies are used in many fields of study and the need to analyze this unique data is growing increasingly. Sometimes a distinction is drawn between longitudinal designs (where subjects are followed for extended periods of time) and repeated measures designs (where the measurements are collected over a relatively short period) (Ware, 1985). Repeated measures designs come in several forms such as split-plot, change over, sources of variability, and longitudinal studies. Split plot designs are mostly used in agricultural experiments where the field is split into multiple plots. Such plots are treated with different fertilizer, and crops are randomly assigned to the subplots within each fertilized plot (Kotz & Johnson, 1988). Each fertilized plot will produce several types of crops and a measure of each will be collected, yielding repeated measures for each plot. Change over designs may be used when testing two types of drugs. With this type of design, selected people 35 are usually put into two groups with half given drug A and the others given drug B. Then drug A and drug B are switched and the experiment is rerun. This design is a repeated measures experiment because every person will have two measures, one for drug A and one for drug B (Kotz& Johnson, 1988). Source of variability studies may include taking several randomly selected items from a manufacturing process and allowing several people to test each one, possibly over several days (Kotz& Johnson, 1988). Longitudinal studies address situations such as the growth of chicks, where weights of each chick may be measured every few days. Usually, in longitudinal designs, the observational units are sampled over an extended period of time. However, there exists a subset of longitudinal designs where time is not an independent variable (Ware, 1985). In this case, the interest is studies that are not broken into intervals of time but rather are categorized according to variables composed of concepts, items, or locations in space (Kotz& Johnson, 1988). 3.5.2 Model 1: (Where Every Subject Is Tested At Every Level) In the need to test the efficacy of a drug in a clinical experiment, two methods of experimentation are available. First, an experimenter may collect observations on N I J = × people, randomly assign th i of them to one of th j treatment groups, and give each treatment group a different dose of the drug. Under this design, exactly one reading per person is taken. Alternately, an experimenter could collect observations on th i person and test it at each of the th j levels of the factor. This second method is categorized as repeated measures because the same set of people is used in multiple tests. The repeated measures experimental design is beneficial as it eliminates the variation due to subjects, which can significantly reduce the mean square error, ultimately making detecting real differences between the treatment doses easier (Montgomery, 1984). 36 In this Model, we will simulate an experiment with J treatments and I people. Each person is tested once per treatment, which helps minimize the amount of variation due to the subjects. This sort of study is used in many fields; one such example is in the field of psychology where several subjects were asked to read a set of sentences (Lorch& Myers, 1990). Each sentence was timed, so each subject provided multiple readings. In this particular model, we have I subjects with indices 1, 2, 3, , i I = in our experiment with th j repeated measures per person with indices 1, 2, 3, , j J = person is timed as they read J sentences, and a time is recorded for each sentence. Let N I J = × denote the total number of observations. In Model 1, ij y is the measurement of the time it took the th i person to read the th j sentence. In this example, 1 X will be a dummy coded variable for which sentence is being read. But in other experiments, 1 X could just as easily be a continuous variable, such as the dose of a drug taken for a study. Other independent variables may exist as well, such as the age of the person, gender, or IQ score, denoted , , … , however, only one independent variable is used in these simulations. β is a vector of the coefficients, and ε is the vector of errors. Let Σ be the covariance matrix of the errors. The model will be of the form: y X| c = + …………………………………………………………. 3.1 where X is a matrix of the independent variables, | is the vector of the coefficients, and c is the vector of errors In matrix notation we have: 37 11 12 1 21 j ij y y y y y y ( ( ( ( ( = ( ( ( ( ( ( ¸ ¸ , 11 1 12 2 1 11 1 1 1 1 1 1 1 k k j kj k j kj x x x x X x x x x x x ( ( ( ( ( = ( ( ( ( ( ( ¸ ¸ , 0 1 k | | | | ( ( ( = ( ( ¸ ¸ , 11 12 1 21 j ij c c c c c c ( ( ( ( ( = ( ( ( ( ( ( ¸ ¸ 11 11 12 11 1 11 21 11 11 11 12 12 12 1 12 21 12 12 11 1 12 1 1 1 21 1 1 1 cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( j ij j ij j j j j j ij j c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c E = 1 21 12 21 1 21 21 21 21 11 12 1 21 , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) cov( , ) j ij ij ij j ij ij ij ij c c c c c c c c c c c c c c c c c c c ( ( ( ( ( ( ( ( ( ( ( ¸ ¸ It is also noted that these matrices were created for a regression analysis. If RM ANOVA is utilized, which requires categorical variables, the X matrix must be adjusted. In this model, the X matrix used for RM ANOVA would have two categorical variables, which would be dummy coded with J - 1 levels for the treatments and I - 1 dummy coded variables for the subjects and result in rank(X) = (J - 1) + (I - 1) + 1. Figure 3.1 shows how data from such a model would look if graphed, where each observational unit is given its own symbol. For example, if there was a study on a blood pressure drug the graph could look as following, where a patient (denoted +,×,○, or ♦ ) has three blood pressure readings taken, one at each dosage of a drug (maybe 1, 2, and 3mgs). 38 Figure 3.1 Graphical Representation of Model 1 Crowder and Hand (1990) describe many inefficient methods of analyzing this experiment type. These methods include performing multiple t-tests on different groupings of the factors and observational units. This analysis tends to be invalid because the involvement of multiple hypothesis tests increase the chance of a Type I error (Bergh, 1995). Others suggest performing multiple regressions on these crossover design experiments, allotting one regression for each observational unit. Some suggest performing an area under the curve analysis, but this examination only inspects one very specific feature of the regression curve. Similarly, other aspects like time to the peak, half- life of the curve, or distance back to the baseline are not accounted for (Crowder & Hand, 1990). 39 Crowder and Hand (1990) suggest the best way to analyze data is to have one all-inclusive ANOVA or regression model, so as not to have multiple statistical tests. 3.5.3 Repeated Measures Analysis of Variance (RM ANOVA) In Model 1, each measurement on a subject is taken at each of the levels of treatment. Therefore, if, as before, an experimenter wanted to test a drug‟s effectiveness he or she could simply test each person at, for example, J = 3 dosage levels. All subjects must be tested at all treatment levels for repeated measures ANOVA (RM ANOVA) to be a valid test (Montgomery, 1985). The null hypothesis is that effectiveness is statistically the same at each treatment level (Misangyi et al., 2006). If the F value is large enough, then the null hypothesis will be rejected, and the treatments will be considered statistically distinct. The calculations for the F values as shown in Table 3.1 is given as MSTR F MSE = ………………………………………………………………. 3.2 MSBS F MSE = ………………………………………………………………………………………………….3.3 MSTR is mean square treatment, MSBS is mean square between subject and MSE is mean square error. The interest is in the first F value in the above equations because that is used to test the null hypothesis that the treatments levels are the same. This analysis is identical to randomized complete block design with subjects as the blocks (Montgomery, 1985). 40 Table 3.1 ANOVA with One Within-Subject Factor Sources df SS MS F Between Subject i-1 SSBS MSBS MSBE/MSE Within subject i(j-1) SSWS MSWS Treatment J-1 SSTR MSTR MSTR/MSE Error (j-1)(i-1) SSE MSE Total IJ-1 SST In a standard ANOVA setting, without repeated measures, all of the error terms are assumed independent and normally distributed, with equal variance. Also, in ANOVA, the independent variable is considered as a factor variable that contains levels. In RM ANOVA, since the variation broken into two parts, one due to the observational units and one due to the factor levels, the assumptions must be modified. The assumptions of independence and constant variance no longer necessarily hold, and this assumption is change to one of sphericity, or circularity, which restricts the variances and correlations of the measurements (Bergh, 1995). Compound symmetry, a condition where all variances of the factor levels are equal and all covariances between each pair of factor levels are equal (O‟Brien & Kaiser, 1985), can be examined in place of the less restrictive sphericity assumption because, if it does not hold, the sphericity assumption usually will not hold. First, the variance of each level of the factor must be identical so as not to violate the assumptions. For example, we may test children at ages 2, 4, 6, and 8; with sphericity it is assumed that the correlation between observations taken at ages 2 and 4, and the correlation between observations taken at ages 4 and 6 would be the same. Usually, it 41 would not be the case that the correlation between observations taken at ages 2 and 8 is the same as the correlation between observations taken at ages 2 and 4, but compound symmetry requires this supposition (O‟Brien & Kaiser, 1985). The sphericity assumption generally will be violated when individuals are being tested over a time range or more than two measurements are taken: two characteristics of most longitudinal designs (O‟Brien & Kaiser, 1985). In this way, the sphericity assumption is almost always violated in the repeated measures situation making RM ANOVA (O‟Brien & Kaiser, 1985) and may lead to an increase in Type I errors if the degrees of freedom are not adjusted (Misangyi et al., 2006). 3.5.4 Univariate Repeated Measure Model In the univariate repeated measures ANOVA the correlation is assumed to come from the individual specific random effects; this is due to the fact that each subject is assumed to have an underlying level of response that persists over time and influences all measurements on that subject. The times of measurement are treated as a within-subject factor and the effect of time is assumed to be the same for all subjects. The response for the i th subject is assumed to be related to discrete covariates and is assumed to be different from the population mean μ. Repeated measures ANOVA can be expressed ij j i ij e V y + + + = t µ , ¹ ´ ¦ = = t j s i , 2 , 1 . 2 , 1 …………………………………… 3.4 where ( ) j ij ij V X Y E + = = µ ' The parameter j v is the effect of time. The parameter ( ) 2 , 0 t o t N i ~ is the random subject effect that gives the between-subject variance, and ( ) 2 , 0 e ij N e o ~ is a within-subject measurement error and it gives the within-subjects variance. The covariance matrix of the ANOVA has a compound 42 symmetry structure, where the variance and covariance are homogeneous across time and equal to 2 2 2 e o o o t t + 3.5.5 Multivariate Analysis of Variance (MANOVA) Multivariate analysis of variate (MANOVA) is an extension of the univariate analysis of variate (ANOVA) to accommodate two or more dependent variables. MANOVA is a technique that measures the differences for two or more metric dependent variables base on the set of categorical variables acting as independent variables. Like ANOVA, MANOVA is concerned with differences between groups or experimental treatments. MANOVA is termed multivariate procedure because it is used to assess group difference across multiple metric dependent variables simultaneously. In MANOVA, each treatment is observed on two or more dependent variables. The concept of multivariate analysis of variance was introduced more than seventy (70) years ago by Wilks (1932). However, it was not until the development of appropriate test statistics with table distributions and more recent wide spread availability of computer programs to compute the statistics that it became a practical tool for researchers. 3.5.6 Multivariate Procedure for Assessing Group Difference As statistical inference procedure, both the univariate techniques (t test and ANOVA) and their multivariate extensions (Hotelling 2 T and MANOVA) are use to assess the statistical significance of difference between groups. In the t test ANOVA the null hypothesis tested is the equality of single dependent variable means across groups. In multivariate techniques, the null 43 hypothesis tested is the equality of vectors of means on multiple dependent variables across groups. The differences between the hypotheses tested in both cases are shown below: n o H µ µ µ µ = = = = 3 2 1 : ………………………………………… (3.5) Null hypothesis o H = all the group means are equal, that is they come from the same population. This is for the univariate analysis of variance case. For the multivariate analysis of variance (MANOVA), the null hypothesis o H is stated as follows: ( ( ( ( ( ( ¸ ( ¸ = ( ( ( ( ( ( ¸ ( ¸ = ( ( ( ( ( ( ¸ ( ¸ pk k k p p o H µ µ µ µ µ µ µ µ µ 2 1 2 22 12 1 21 11 ........ : ……………………………………. (3.6) Unlike the ANOVA, a variate is tested for equality in MANOVA and the researchers actually have two variates, one for the dependent variables and another for the independent variables. The dependent variable variate is of more interest because the metric-dependent measures can be combined in a linear combination as in multiple regression and discriminant analysis. The uniqueness of MANOVA is that the variate optimally combines the multiple dependent measures into a single value that maximizes the differences across groups. 44 3.5.7 The Two-group MANOVA Assume that we have two groups of multivariate observations, as in the example above, and that there are 1 n observations in group 1 and 2 n observations in group 2. Hence we observe the vectors p ÷ ij y , 1 2 1, 2, , , 1, 2, , j n i n = = , or equivalently expressed, we observe the ( ) p n n × + 2 1 matrix . Formally, the statistical model that underlies the MANOVA is "just" an ANOVA model for each variable together with an error structure that allows for covariance between response variables: ij i ij y e µ = + …………………………………………. (3.7) For 1 1, 2, i n = and 2 1, 2, j n = where, if the number of variables equals three: 3 = p | | | . | \ | = 31 21 11 1 µ µ µ µ and | | | . | \ | = 32 22 12 2 µ µ µ µ ………………………………………… (3.8) are the two expected group means. As in univariate ANOVA, we interested in an overall statistical significance test for the hypothesis of no group difference at all 0 1 2 1 1 2 : : H H µ µ µ µ = = Note, that this hypothesis expresses that the groups are equal in all the response variables, and that the alternative expresses that in at least one variable there is a group difference. So this is a composite hypothesis jointly in all variables. Computationally, the analysis is handled by computing the group and overall mean: 45 1 2 1 1 2 2 1 1 1 1 1 1 , n n j j j j y y y y n n = = = = ¿ ¿ …………………………….. (3.9) ¿ = + = 2 1 2 1 1 i i i y n n n y …………………………………. (3.10) the (weighted) between group variance-covariance structure of these averages: ( )( ) ¿ = ÷ ÷ = 2 1 i T i i i Between y y y y n SS and the within-group covariances of the group-corrected data: ( )( ) ¿¿ = = ÷ ÷ = 2 1 1 i n j T i ij i ij Within i y y y y SS From a statistical perspective, the two group centroids are estimates of the unknown group centroids 1 µ and 2 µ and the within group error variance-covariance matrix Eis estimated as 1 2 1 2 w within S SS n n E = = + ÷ ………………………………………………. (3.11) in analogy with the univariate case. From the example above it is clear that the key information about the group differences is given by the distance between the two centroids. This difference can be measured statistically by the Mahalanobis distance: 2 1 1 2 1 2 ( ) ( ) T w MD y y S y y ÷ = ÷ ÷ …………………………………. (3.12) Recall, how this distance measure down weighs directions in which there is a high residual variability and up weighs directions of low residual variability (allowing for covariances between 46 each variable). In the case of two groups the 2 MD related to the multivariate generalization of the student's t-test, Hotelling's 2 T as: 2 2 1 2 1 2 n n T MD n n = + ………………………………………………..(3.13) and the 2 T can be transformed into an exact F-ratio as follows: ( ) 2 1 2 1 2 1 2 n n p F T n n p + ÷ ÷ = + ÷ ………………………………………………………………………..(3.14) which has an F-distribution with p and 1 2 1 ÷ ÷ + p n n degrees of freedom (under the null hypothesis). Multivariate confidence ellipsoids for group mean and/or mean difference can be found from the F-distribution and the inverse within group covariance matrix 1 ÷ w S . 3.5.8 The Multiple group one-way MANOVA Assume that we have I groups of multivariate observations with i n observations in group i, 1, 2. , i I = . Hence we observe the vectors p ÷ , ij y 1, 2, , i i n = , 1, 2, , j I = . The model is expressed exactly as for the two-group case: ( ) , var ij i ij ij y e e µ = + = ¿ …………………………………. (3.15) For 1, 2. , i I = and 1, 2, , i j n = The hypothesis of no group difference at all is given as 0 1 : I H µ µ = = ……………………………………… (3.16) 47 with the alternative that for at least on variable there is at least two groups that differ. Again, this is a composite hypothesis jointly in all variables, even to a higher degree than for the two-group case, since it is composite in as well the group structure as in the variables. Computationally, the analysis is handled exactly as above. First the averages are computed as; Treatment mean 1 1 1 1 1 n ij j y y n = = ¿ And the total mean as 1 1 I i i i y n y n = = ¿ (3.17) where I n n n n , 2 1 + + = . Then the between-group information is found: 1 ( )( ) I T Between i i i i SS n y y y y = = ÷ ÷ ¿ (3.18) and the within-group covariance of the group-corrected data: ( )( ) 1 1 i n I T Within ij i ij i i j SS y y y y = = = ÷ ÷ ¿¿ (3.19) And as before the group means are estimates of the unknown group mean i µ and the within group error variance-covariance matrix Eis estimated as 48 1 W Within S SS n I E = = ÷ (3.20) In the univariate hierarchy of methodology the F-test is introduced as a necessary tool when going from two to more than two groups. This is because there is no direct generalization of the t-statistic (a difference between two groups) to more than two groups, since e.g. three groups require two distances to summarize the group difference information. So instead of comparing the (two) groups directly they are all compared with the overall mean/centroids of the data, that is, the between group variation. In the univariate case, the F-statistic has the form (assuming for a moment that the SS's are univariate) / / Between Between Within Within SS df F SS df = (3.21) In the multivariate case the measure of group differences in the light of the within-group variability is given by the matrix 1 Within Between SS SS ÷ × (3.22) In fact the unique decomposition of the total variability often expressed in the univariate case also holds for multivariate: Total Between Within SS SS SS = + (3.23) Where Total SS ( )( ) 1 1 i n I T Total ij ij i j SS y y y y = = = ÷ ÷ ¿¿ (3.24) 49 The exercise of doing a statistical hypothesis test for the hypothesis of no group differences is to evaluate the degree of extremeness of this statistic compared to what would be expected under the null hypothesis. Due to the multivariate nature, extremeness can be defined in various ways. Four of these are usually standard output from statistical software: 1 , 1 K k k k Pillai sTrace ì ì = = + ¿ 1 , K k k Hotelling sTrace ì = = ¿ 1 1 ' 1 K k k Wilk s ì = A = + [ max max ' arg 1 Roy s L est Root ì ì = + Where K ì ì ì , , 2 1 are the eigenvalues of the Between within SS SS × ÷1 matrix. The number of eigen values K is equal to the minimum of the dimension of the variables space and the number of groups minus 1, ( ) 1 , min ÷ = I p K . Being familiar with PCA a way to express this would be: Perform a PCA on the 2 / 1 ÷ w S pre-processed group average data and compare the explained variances of each component with what comes out of data where the groups are truly equal. Note that eigenvalues are the variations of the principal directions of the variables that are corresponding to the covariance matrix Between SS . In other words the first (largest) eigenvalue of this matrix express the largest possible variation (between group means) of a new combined variable (the first principal component). The largest eigenvalue of Between within SS SS × ÷1 expressed 50 the largest possible variation between group means taking the within-covariance (point scatter shape) into account. The theoretical null hypothesis distribution of these statistics is in principle known and described in the literature. However, often they are substituted by approximate F-ratios. We give here no formulas for these approximations. However in the two group case all of these statistics are equivalent and can be exactly equivalently expressed by the F-ratio and/or Hotelling's 2 T . In Sharma (1995), Olson (1974) is referred to as showing that the Pillai's Trace was the most robust and had adequate power to detect true differences under different conditions, and hence Pillai's Trace is recommended to test multivariate significance. 3.6 LONGITUDINAL DATA ANALYSIS CONCEPTS The objectives of longitudinal data analysis are to examine and compare responses over time. The defining feature of a longitudinal data model is its ability to study changes over time within subjects and changes over time between groups. For example, longitudinal models can estimate individual-level (subject-specific) regression parameters and population-level regression parameters (Patetta et. al, 2002). Longitudinal data sets differ from time series data sets because longitudinal data usually consists of a large number of a short series of time points. However, time series data sets usually consist of a single, long series of time points (Diggle, Liang, and Zeger 1994). For example, the monthly average of the Dow Jones Industrials Index for several years is a time series data set, and the efficacy of a drug treatment over time for several patients is a longitudinal data set. To illustrate the ability of longitudinal models to study changes over time, consider cross- sectional studies in which a single outcome is measured for each subject. In the figure 3.2 below 51 where each point represents one subject, blood pressure appears to be positively related to age. However, you can reach no conclusions regarding blood pressure changes over time within subjects. Blood Pressure Age Figure 3.2 Cross-Sectional Analysis Now expand the cross-sectional study of baseline data to a longitudinal study with repeated measurements over time. The baseline data still shows a positive relationship between blood pressure and age. However, now you can distinguish changes over time within subjects from differences among subjects at their baseline or initial starting values. Cross-sectional models cannot make this distinction (Diggle, Liang, and Zeger 1994). Blood 3 Pressure 1 4 2 Age Figure 3.3 Longitudinal analysis 52 An example of a longitudinal study is the Baltimore Longitudinal Study of Aging (Shock, et al. 1984). This is a multidisciplinary observational study where participants return approximately every two years for three days of biomedical and psychological examinations. One objective of the study is to look for markers, which can detect prostate cancer at an early stage. One marker with this potential is the prostate-specific antigen (PSA), which is an enzyme produced by both normal and cancerous prostate cells and its level is related to the volume of prostate tissue. However, an elevated PSA level is not necessarily an indicator of prostate cancer because patients with benign prostatic hyperplasia can also have increased PSA levels. Therefore, researchers have hypothesized that the rate of change in the PSA level might be a more accurate method of detecting prostate cancer in the early stages of the disease. A longitudinal model can address this hypothesis (Patetta et. al, 2002). Another example of a longitudinal study is the Indonesian children‟s health study (Sommer 1982). In this study over 3000 children had quarterly medically exams for up to six visits to assess whether they suffered from respiratory or diarrheal infection and xerophthalmia. One objective of the study is to determine whether children who have a vitamin A deficiency are at increased risk of respiratory infection. 3.6.1 Notation Let it Y be the response for the th i subject ( 1, 2,..., ) i N = at the th t occasion where ( 1, 2,..., ) i t n = . The total number of subjects is equal to N i i n ¿ , 1 i i y n = × is the vector of responses, and 1 i X p = × is the covariate vector for subject i at time t. The matrix of covariates is i i X n p = × for subject i and will usually include an intercept. For example in a 14 repeated measures data with four factors, if i = subject, 1, ,14 t = (number of repeated measures), 14 1 i y = × , 4 1 it x = × and 53 14 4 i X = × . If the fixed effects are age and gender because they do not change throughout the duration of the study and the within individual effects are time and body weight, the data set will be a balanced with time. This meaning that all subjects will be measured at a common set of occasions and there will be no missing data. 3.6.2 Advantages and Disadvantages There are several advantages and disadvantages to using longitudinal studies. Some of the advantages are: subjects serving as their own controls which mean the direct study of change can be measured; fewer subjects are required because the measurements are being repeated, between-subject variation is excluded from the error, and longitudinal data can separate aging effects from cohort effects. Some of the disadvantages are: the dependence of the measurements which must be accounted for in the analysis, models are not as well developed; the risk of attrition, carry-over effects, and the improvement or the decline could be caused by treatment or fatigue. 3.6.3 Problems with Model Assuming Independence Special methods of statistical analysis are needed for longitudinal data because the set of measurements on one subject tends to be correlated, measurements on the same subject close in time tend to be more highly correlated than measurements far apart in time, and the variances of longitudinal data often change with time. These potential patterns of correlation and variation may combine to produce a complicated covariance structure. This covariance structure must be taken into account to draw valid statistical inferences. Therefore, standard regression and ANOVA models may produce invalid results because two of the parametric assumptions (independent observations and equal variances) may not be valid (Patetta et. al, 2002). 54 3.6.4 Variance – Covariance Matrix for continuous outcome To illustrate the differences between longitudinal data models and other types of models, consider the variance-covariance matrix of the response variable below. Subject X Y 1 4 10 2 2 7 3 6 12 4 8 11 If you have four observations, you would have a 4-by-4 variance-covariance matrix with the variances on the main diagonal and the covariances (a measure of how two observations vary together) on the off diagonals. In linear regression with continuous responses, the assumptions are that the responses have equal variances and are independent. Therefore, the variances along the diagonal are equal and the covariances along the off-diagonals are 0 (Patetta et. al, 2002). Longitudinal data Subject X 1 4 10 6 6 2 2 7 5 3 3 6 12 9 8 4 8 11 14 16 With longitudinal data (as above), there are now multiple measurements taken on each subject. You not only can examine the differences between subjects, but you can also examine the change 0 0 0 0 0 0 0 0 0 0 0 0 55 within subjects across time. There are still only four subjects and the response is continuous. How does this change your variance-covariance matrix? 3.6.5 Variance-Covariance Matrix for Longitudinal Data Subject Time X Y 1 1 4 10 1 2 4 6 1 3 4 6 2 1 2 7 2 2 2 5 2 3 2 3 3 1 6 12 3 2 6 9 3 3 6 8 4 1 8 11 4 2 8 14 4 3 8 16 For longitudinal data models fit in the MIXED procedure, the number of observations is not the number of subjects but rather the number of measurements taken on all the subjects. Because there are three repeated measurements on each subject, you now have 12 observations and a 12- by-12 variance-covariance matrix (as shown above). For a simple longitudinal model, the matrix is now a block-diagonal matrix in which the observations within each block (the block corresponds to a subject) are assumed to be correlated and the observations outside of the blocks 0 0 0 0 0 0 0 0 0 0 0 0 56 are assumed to be independent. In other words, the subjects are still assumed to be independent of each other and the measurements within each subject are assumed to be correlated. a). Effect on Time-Independent Predictor Variables If the observations are positively correlated, which often occurs with longitudinal data, then the variances of the time-independent predictor variables (variables that estimate the group effect (or between-subject effect) such as gender, race, treatment, and so on) are underestimated if the data is analyzed as though the observations are independent. In other words, the Type I error rate (rejecting the null hypothesis when it is true, in other words, a false positive) is inflated for these variables (Dunlop 1994). b). Effect on Time-Dependent Predictor Variables For time-dependent predictor variables (variables that measure the time effect (or within-subject effect) such as how the measurements change over time), ignoring positive correlation leads to a variance estimate that is too large. In other words, the Type II error rate (failing to reject the null hypothesis when it is false, in other words, a false negative) is inflated for these variables (Dunlop 1994). Because the variances of the group effects will be underestimated and the variance of the time effects will be overestimated if positive correlation is ignored, it is evident that correlated outcomes must be addressed to obtain valid analyses. 3.7 STATISTICAL METHODS FOR CONTINUOUS RESPONSES The following are statistical methods for continuous responses -Univariate ANOVA in PROC GLM -Multivariate ANOVA in PROC GLM -Linear mixed model in PROC MIXED 57 The GLM procedure enables users to perform univariate and multivariate ANOVA of repeated measures data and the MIXED procedure enables users to fit linear mixed models. 3.7.1 Univariate ANOVA using PROC GLM Univariate ANOVA for repeated measures data, also called split-plot in time ANOVA, has a data structure where the number of observations equals the number of measurements taken on all subjects. The assumptions are that the measurements have equal variances at all time points and that the correlation between measurements on the same subject is the same regardless of the time lag between measurements. If these assumptions are valid, then univariate ANOVA for repeated measures data is an optimal method of analysis (Littell, Henry, and Ammerman 1998). However, these assumptions are frequently unrealistic with longitudinal data. Measurements close in time are often more highly correlated than measures far apart in time. Therefore, the inferences from univariate ANOVA for repeated measures data may not be valid The condition required for the validity of the univariate ANOVA tests is called the Huynh-Feldt (H-F) condition, which is equivalent to differences between each pair of measurements having equal variances. While the H-F condition is mathematically more general than equal variances and correlations, the differences are minor from a practical point of view (Patetta et. al, 2002). 3.7.2 Multivariate ANOVA using PROC GLM Multivariate ANOVA has a data structure where the rows are subjects and the columns are time points in the study. For example, if there were five time points in the study, then there would be five response variables. For balanced longitudinal data where all subjects have the same number of repeated measurements taken at time points that are the same for all subjects, multivariate ANOVA has some useful features. However, for observational longitudinal studies, this data structure has major limitations. First, it is unreasonable to assume that observational data will be 58 balanced. Second, because PROC GLM uses complete case analysis, if a subject has one missing measurement, then PROC GLM deletes the entire subject (Patetta et. al, 2002). Multivariate ANOVA uses a covariance structure that assumes that the correlation between every pair of measurements within a subject is unique. For example, if there were five time points, then there would be 15 (((1+5)*5)/2) unique parameters that are estimated. For observational data with many time points, a simpler covariance structure would probably be sufficient to characterize the data. This is a limitation for multivariate ANOVA because if the covariance structure (and thus your model) is more complex than the within-subject correlation structure, then you sacrifice power and efficiency in your statistical tests. In other words, if a simpler covariance structure can be used to model the within-subject correlations, then the inferences in the longitudinal study can be severely compromised if multivariate ANOVA is used (Littell, Freund, and Spector 2001). 3.7.3 Complete Case Analysis In complete case analysis, only those observations without any missing values are used in the analysis. Because each time point represents a separate column in multivariate ANOVA, if one measurement is missing then PROC GLM deletes the entire subject. Therefore, a great deal of valid data may be discarded. Another drawback of the complete case analysis method is that severe bias in the inference can result if the missing data is not missing completely at random (MCAR). A simple check of MCAR is to divide the subjects into 2 groups: those with a complete set of measurements and those with missing measurements. If the MCAR assumption holds, then both groups (with their measurements) should be random samples of the same population of measurements. In other 59 words, the probability of missing is independent of the observed measurements and the measurements that would have been available had they not been missing (Patetta et. al, 2002). The MCAR assumption is much more restrictive than the missing at random assumption (MAR). If the measurements are MAR, then the subjects with missing measurements are a random sample of the population of measurements that would have been available if they were not missing, but they do not have to have the same distributional characteristics as the observations with complete measurements. In other words, the probability of missing is independent of the measurements that would have been available had they not been missing but does not have to be independent of the observed measurements (Little and Rubin 1987). 3.7.4 Linear Mixed Model using PROC MIXED The linear mixed model allows a very flexible approach to modeling longitudinal data. The data structure is similar to univariate ANOVA where the number of observations equals the number of measurements for all the subjects. This means the data does not have to be balanced. An advantage of fitting linear mixed models is that PROC MIXED uses all the available data in the analysis. This method differs from complete case analysis in which any observation with a missing value is dropped from the analysis. The method PROC MIXED uses, called a likelihood- based ignorable analysis, leads to a valid analysis when the missing data is MAR (which is a less restrictive assumption than MCAR). When the probability of missing is informative (probability of missing is dependent on the measurements that would have been available if they were not missing), then the ignorable analysis is not valid and more complex modeling is required (Verbeke and Molenberghs 1997). PROC MIXED offers a wide variety of covariance structures. This enables the user to directly address the within-subject correlation structure and incorporate it into a statistical model. By 60 selecting a parsimonious covariance model that adequately accounts for within-subject correlations, the user can avoid the problems associated with univariate and multivariate ANOVA using PROC GLM (Littell, Freund, and Spector 2001). 3.8 MODEL-BUILDING STRATEGIES IN LONGITUDINAL DATA ANALYSIS The first step in any model building process is to conduct a thorough exploratory data analysis. For longitudinal data this involves plotting the individual measurements over time and fitting a smoothing spline over time. Plotting different groups over time and illustrating cross-sectional and longitudinal relationships are also important steps in exploratory data analysis. The second step is to fit a complex mean model in PROC MIXED and output the ordinary least squares residuals. These residuals can be used to create a sample variogram, and the pattern in the sample variogram can be helpful in selecting a covariance structure (Patetta et. al, 2002). The third step is to fit the linear mixed model in PROC MIXED using the selected covariance structure. Eliminating unnecessary terms and fitting a parsimonious model are important steps in the model building process. After a candidate model is selected, the final steps of the model building process are to evaluate model assumptions and to identify potential outliers. 3.8.1 Exploratory data analysis. The first step in any model-building process is exploratory data analysis. In this step you create graphs that expose the patterns relevant to the scientific question. The recommendations below, given by Diggle, Liang, and Zeger (1994), are used to produce the graphs. Recommendations - Graph as much of the relevant raw data as possible. - Highlight aggregate patterns of potential scientific interest. - Identify both cross-sectional and longitudinal patterns. - Identify unusual individuals or observations. 61 3.8.2 Individual Profiles A scatter plot of the response with time is a useful graph. Connecting the repeated measurements for each subject over time shows you whether there is a discernible pattern common to most subjects. These individual profiles can also provide some information on between-subject variability. For example, the figure 3.4 below is a graph of weight over time for several subjects. These individual profiles illustrate several important patterns (Diggle, Liang, and Zeger 1994). 1. All of the subjects are gaining weight. 2. The subjects that are the heaviest at the beginning of the study tend to be the heaviest throughout the study. 3. The variability of the measurements is smaller at the beginning of the study compared to the end of the study. Weight Time Figure 3.4 Individual Profiles 3.8.3 Group Profiles Besides plotting the response over time, it is also useful to include different subgroups on the same graph to illustrate the relationship between the response and an explanatory variable over 62 time. For example, in the slide below it appears that both males and females have decreasing blood pressures over time. However, the slope for the males seems to be more pronounced than the slope for the females, which may indicate an interaction between gender and time. Blood Pressure Males Females Time Figure 3.5 Group Profile 3.9 GENERAL LINEAR MODEL The General Linear Model (GLM) is written as y = Χβ +ε Where y is the matrix of observed responses; Χ is the design matrix of predictor variables; β is the matrix of regression parameters; ε is the vector of random errors. The GLM incorporates a number of different statistical models; ANOVA, ANCOVA, MANOVA, MANCOVA, ordinary linear regression, t-test and F-test. The general linear model is a generalization of multiple linear regression model to the case of more than one dependent variable. If y, β, and ε were column vectors, the matrix equation above would represent multiple linear regression. 63 Hypothesis test with the GLM can be made in two ways: multivariate or as several independent univariate tests. In multivariate test, the columns of Y are tested together whereas in univariate tests the columns of Y are tested independently as multiple univariate tests with the same design matrix. The standard linear regression model, which is used in the GLM procedure, models the mean of the response variable by using the regression parameters. The random errors are assumed to be independent and normally distributed with a common variance. If these parametric assumptions are valid, then the estimated regression parameters are the best linear unbiased estimates (Patetta et. al, 2002). The general linear mixed model is an extension of the general linear model. The standard linear regression model, which is used in the GLM procedure, models the mean of the response variable by using the regression parameters. The random errors are assumed to be independent and normally distributed with a common variance. If these parametric assumptions are valid, then the estimated regression parameters are the best linear unbiased estimates (BLUE). 3.10 GENERAL LINEAR MIXED MODEL General Linear Mixed Model is written as y = Χβ + Ζγ +ε Where Ζ is the design matrix of random variables; γ is the vector of random-effect parameters; ε is no longer required to be independent and homogeneous. The general linear mixed model extends the general linear model by the addition of random effect parameters and by allowing a more flexible specification of the covariance matrix of the random errors. For example, general linear mixed models allow for both correlated error terms and error terms with heterogeneous variances. The matrix Z can contain continuous or dummy 64 predictor variables, just like the matrix X. The name mixed model indicates that the model contains both fixed-effect parameters and random-effect parameters (Patetta et. al, 2002). 3.10.1 Fixed and Random Effects A represents all possible levels of the variable Drug B represents all levels in which C inferences are to be made Figure 3.6 Fixed Effects A Drug B -Fixed Effects C 7 - Levels represent only a random sample Clinic 18 of a larger set of potential levels 23 -Interest is in drawing inferences that are valid 41 for the complete population of levels Figure 3.7 Fixed and Random Effects 65 Variable effects are either fixed or random depending on how the levels of the variables that appear in the study are selected. For example, the figure 3.6 above represents a clinical trial analyzing the effectiveness of three drugs. If the three drugs are the only candidates for the clinical trial and the conclusions of the clinical trial are restricted to just those three drugs, then the effect of the variable drug is a fixed effect. However, suppose the clinical trial was performed in four clinics and the four clinics are a sample from a larger population of clinics. The conclusions of the clinical trials are not only restricted to the four clinics but rather to the population of clinics. The appropriate model in this study is a general linear mixed model with drug as a fixed-effect variable and clinic as a random-effect variable (figure 3.7) (Patetta et. al, 2002). 3.11 MODEL ASSUMPTIONS IN PROC MIXED i. Random effects and error terms are normally distributed with means of 0. ii. Random effects and error terms are independent of each other. iii. The relationship between the response variable and predictor variables is linear iv. Variance-covariance matrices for random effects and error terms exhibit structures available in PROC MIXED. Even with these assumptions, the general linear mixed model is an extremely flexible model. By appropriately defining the model matrices for fixed and random effects, and the covariance structures for the random effects and the error terms, you can perform numerous mixed model analyses. 3.12 LINEAR MIXED MODELS FOR LONGITUDINAL DATA y = Χβ + Ζγ +ε where γ represents -parameters that are allowed to vary over subjects 66 - subject-specific regression coefficients that reflect the natural heterogeneity in the population. β represents - parameters that are assumed to be the same for all subjects - factors that affect subject-specific regression coefficients and ε - represents within-subject variation - has a covariance matrix that is block diagonal with each block corresponding to a subject. The general linear mixed model can easily be fitted to longitudinal data. The model assumes that the vector of repeated measurements on each subject follows a linear regression model where some of the regression parameters are population-specific (fixed-effects), but other parameters are subject-specific (random-effects). The subject-specific regression coefficients reflect how the response evolves over time for each subject. These subject-specific models can be very flexible, but in practice polynomials involving time will often suffice. However, extensions of this flexibility, such as fractional polynomial models or extended spline functions, can be considered as well (Verbeke and Molenberghs 2000). How the subject-specific regression coefficients relate to known covariates is reflected in the fixed-effect parameter estimates. These parameter estimates represent the population average. A model with the assumption that all variability in subject specific intercepts and slopes is attributed to the fixed-effect variables can be obtained by omitting the random-effect variables. Omitting only the random variables reflecting subject-specific slopes generates a random intercepts model (Patetta et. al, 2002). In the model proposed by Diggle, Liang, and Zeger (1994), it is assumed that the error terms have a constant variance and can be decomposed as 67 = () + () where () is the measurement error reflecting the variation added by the measurement process () is the error associated with the serial correlation in which times closer together are more correlated than times further apart i denotes the subject. If you assume that the measurement errors have an independent covariance structure ( Ι), then you should only concern yourself with covariance structures that reflect the serial correlation. 3.13 SELECTING THE APPROPRIATE COVARIANCE STRUCTURE When finding reasonable estimates for covariance structure of the random errors (denoted by R), if you choose a structure that is too - simple, you risk increasing the Type I error rate - complex, you sacrifice power and efficiency. The validity of the statistical inference of the general linear mixed model depends upon the covariance structure you select for R. Therefore, a large amount of time spent on building the model is spent on choosing a reasonable covariance structure for R. Because the covariance structure models the variability not explained by the fixed effects, selecting the appropriate mean model is critical. For models dealing with data collected in an experiment, a saturated model is usually recommended. However, for models dealing with observational data, saturated models are not feasible. Therefore, it is important to include all the important main effects and interactions (Patetta et. al, 2002). After a candidate mean model is selected, fitting the model using ordinary least squares regression and examining the residuals may help determine the appropriate covariance structure. 68 For example, plotting the residuals by subject over time may indicate whether random intercepts (between-subject variability at baseline) and random slopes (between-subject variability over time) are needed. You can also use the information criteria (such as the AIC and BIC) produced by PROC MIXED as a tool to help you select the most appropriate covariance structure. The smaller the information criteria value, the better the model. Examining a plot of the autocorrelation function of the residuals may be useful for equally spaced data that is roughly stationary. (The residuals have constant mean and variance and the correlations depend only on the length of the time interval.) An alternative function that describes the association among repeated measurements and is easily estimated with irregular observation times is the variogram (Diggle 1990). 3.13.1 Information Criteria In a simulation study conducted by Guerin and Stroup (2000), several information criteria were compared in terms of their ability to choose the right covariance structure. In terms of Type I error control, assuming the Kenward-Roger (KR) adjustment is used, Guerin and Stroup showed that it is better to err in the direction of a more complex covariance structure. More complex covariance structures tend to have inflated Type I error rates only if you fail to use the KR adjustment, while excessively simple covariance structures have inflated Type I error rates that the degrees of freedom adjustment cannot correct. However, because complex covariance structures reduce power, erring too far in the direction of complexity is also not recommended. Guerin and Stroup believe the AIC is the most desirable compromise in practice. However, if the sample size is relatively small, the finite-sample corrected version of AIC, called AICC, may be the most desirable. 69 3.13.2 Variance Component The simplest covariance structure is the independent or variance component model, where the within-subject error correlation is zero. This is the default structure for both the RANDOM and REPEATED statements. For the between-subject errors, the simple covariance structure may be a reasonable assumption. However, for the within-subject errors, the simple covariance structure may be a reasonable choice if the repeated measurements occurred at long enough intervals so that the correlation is effectively zero relative to other variation. 0 0 0 0 0 0 0 0 0 0 0 0 3.13.3 Compound Symmetry The covariance structure with the simplest correlation model is the compound symmetry structure. It assumes that the correlation (ρ) is constant regardless of the distance between the time points. This is the assumption that univariate ANOVA makes, but it is usually not a reasonable choice in longitudinal data analysis. However, this covariance structure may be reasonable when the repeated measurements are not obtained over time. 1 ρ ρ ρ ρ 1 ρ ρ ρ ρ 1 ρ ρ ρ ρ 1 70 3.13.4 Unstructured Covariance The unstructured covariance structure is parameterized directly in terms of variances and covariances where the observations for each pair of times have their own unique correlations. The variances are constrained to be nonnegative and the covariances are unconstrained. This is the covariance structure used in multivariate ANOVA. The correlation coefficient for row 1 column 2 is = √( ) There are two potential problems with using the unstructured covariance. First, it requires the estimation of a large number of variance and covariance parameters. This can lead to severe computational problems, especially with unbalanced data. Second, it does not exploit the existence of trends in variances and covariances over time, and this can result in erratic patterns of standard error estimates (Littell, et. al. 1998). If a simpler covariance structure is a reasonable alternative, then the unstructured covariance structure wastes a great deal of information, which would adversely affect efficiency and power. 71 3.13.5 First-Order Autoregressive (AR1) Covariance The first-order autoregressive covariance structure takes into account a common trend in longitudinal data; the correlation between observations is a function of the number of time points apart. In this structure, the correlation between adjacent observations is ρ, regardless of whether the pair of observations is the 1st and 2 nd pair, the 2nd and 3rd pair, and so on. The correlation is for any pair of observations 2 units apart, and for any pair of observations d units apart. Notice that the AR (1) model only requires estimates for just two parameters, and ρ , whereas the unstructured models require estimates for( (1+T )*T)/2 parameters (where T is the number of time points). The assumption in the AR (1) model is that the longitudinal data is equally spaced (Littell, et. al. 1996). This means that the distance between time 1 and 2 is the same as time 2 and 3, time 3 and 4, and so on. The AR (1) structure also assumes that the correlation structure does not change appreciably over time (Littell, et. al. 2002). 1 ρ ρ 1 ρ ρ 1 ρ ρ 1 3.13.6 Toeplitz Covariance The Toeplitz covariance structure is similar to the AR (1) covariance structure in that the pairs of observations separated by a common distance share the same correlation. However, observations d units apart have correlation instead of . The Toeplitz structure requires the estimation of 72 T parameters instead of just two parameters. You can also specify a banded Toeplitz structure in which you specify the number of time points apart the measurements are still correlated. For example, a TOEP (3) (Toeplitz with 3 bands) structure would indicate that measurements are correlated if they are three or fewer time points apart. If they are four or more time points apart, the correlation is zero. As with the AR (1) structure, the Toeplitz structure assumes that the observations are equally spaced and the correlation structure does not change appreciably over time (Littell, et. al. 2002). 1 1 1 1 3.13.7 Spatial Power Covariance structures that allow for unequal spacing are the spatial covariance structures. These structures are mainly used in geostatistical models, but they are very useful for unequally spaced longitudinal measurements where the correlations decline as a function of time. The connection between geostatistics and longitudinal data is that the unequally spaced data can be viewed as a spatial process in one dimension (Littell, et. al. 1996). The spatial power structure provides a direct generalization of the AR (1) structure for equally spaced data. In fact, the AR (1) structure is more efficient when you have equal spacing. Only two parameters are estimated (σ 2 and ρ). 73 1.0 1.0 1.0 1.0 3.14 OUTPUT FROM PROC MIXED USING SAS 1. The Model Information table shows the name of the data set, the dependent variable, the covariance structure used in the model, the subject effect, the estimation method to compute the parameters for the covariance structure, and the method to compute the degrees of freedom. The default estimation method is REML. The METHOD= option can be used in the PROC MIXED statement to specify other estimation methods. There are four methods for handling the residual variance in the model. The profile method concentrates the residual variance out of the optimization problem, whereas the fit method retains the residual variance as a parameter in the optimization. The factor method keeps the residual fixed, and none is displayed when a residual variance is not a part of the model. The NOPROFILE option in the PROC MIXED statement changes the method based on the chosen covariance structure. The fixed effects standard error method describes the method used to compute the approximate standard errors for the fixed-effects parameter estimates and related functions of them. The default method can be changed using the EMPIRICAL option in the PROC MIXED statement. This option requests robust standard errors obtained from using the sandwich estimator, which has shown to be consistent as long as the mean model is correctly specified. However, if there are any missing observations, the EMPIRICAL option only provides valid inferences for the 74 fixed effects under the MCAR assumption. The EMPIRICAL option is not used here because it cannot be used with the Kenward-Roger degrees of freedom calculation. 2. The Dimensions table lists the sizes of the relevant matrices. This table can be useful in determining CPU time and memory requirements. 3. The Iteration History table describes the optimization of the residual log likelihood. The minimization is performed using a ridge-stabilized Newton-Raphson algorithm, and the rows of the table describe the iterations that this algorithm takes in order to minimize the objective function. 4. The Covariance Parameter Estimates table shows the parameter estimates for the compound symmetry covariance structure. 5. The Fit Statistics table provides information you can use to select the most appropriate covariance structure. Akaike‟s Information Criterion (AIC) (Akaike 1974) penalizes the –2 residual log likelihood by twice the number of covariance parameters in the model. The smaller the value, the better the model. The finite sample version of the AIC (AICC) is also included. The Schwarz‟s Bayesian Information Criterion (BIC) (Schwarz 1978) also penalizes the –2 residual log likelihood, but the penalty is more severe. Therefore, BIC tends to choose less complex models than AIC or AICC. 6. The Null Model Likelihood Ratio Test table shows a test that determines whether it is necessary to model the covariance structure of the data at all. The test statistic is –2 times the log likelihood from the null model (model with an independent covariance structure) minus –2 times the log likelihood from the fitted model. The p-value can be used to assess the significance of the model fit. 75 7. The Type 3 Tests of Fixed Effects table shows the hypothesis tests for the significance of each of the fixed effects. A p-value is computed from an F distribution with the numerator and denominator degrees of freedom. You can use the HTYPE= option in the MODEL statement to obtain tables of Type I (sequential) tests and TYPE II (adjusted) tests in addition to or instead of the table of TYPE III (partial) tests. You can also use the CHISQ option to obtain Wald chi- square tests of the fixed effects (Patetta et. al, 2002). 3.15 EVALUATING COVARIANCE STRUCTURE 3.15.1 Importance of covariance structure Covariance structures -model all the variability in the data, which cannot be explained by the fixed effects - represent the background variability that the fixed effects are tested against - must be carefully selected to obtain valid inferences for the parameters of the fixed effects. Obtaining valid inferences in a mixed model is much more complex than in a general linear model. For example, inferences are obtained in the GLM procedure by testing the fixed effects against the error variance (residual variance). However, in PROC MIXED the inferences are obtained by testing the fixed effects against the appropriate background variability, which is modeled by the covariance structure. This background variability may consist of several sources of error, so selecting the appropriate covariance structure is not a trivial task. 3.15.2 Sources of Error Longitudinal models usually have three sources of random variation. The between subject variability is represented by the random effects. The within-subject variability is represented by the serial correlation. The correlation between the measurements within subject usually depends on the time interval between the measurements and decreases as the length of the interval 76 increases. A common assumption is that the serial effect is a population phenomenon independent of the subject. Finally, there is potentially also measurement error in the measurement process. The covariance structure that is appropriate for your model is directly related to which component of variability is the dominant component. For example, if the serial correlation among the measurements is minimal, then the random effects will probably account for most of the variability in the data and the remaining error components will have a very simple covariance structure. Diggle, Liang, and Zeger (1994) believe that in most applications, the serial correlation is very often dominated by the combination of random effects and measurement error. Furthermore, Chi and Reinsel (1989) found that models with random effects and serial correlation might sometimes over parameterize the covariance structure because the random effects are often able to represent the serial correlations among the measurements. They conclude that methods for determining the best combination of serial correlation components and random effects are an important topic that deserves further consideration. However, suppose the autocorrelation among the measurements is relatively large, and the between-subject variability not explained by the fixed effects is relatively small. Then choosing the appropriate serial correlation function in the covariance structure becomes important. 77 CHAPTER FOUR DATA ANALYSIS AND RESULTS 4.1 INTRODUCTION This chapter describes the data obtained in easily understandable terms. The chapter examines the data collected, the detailed analysis of the data and the presentation of results. Table 4.1 presents the snapshot of the data used for the analysis. It contains data on one hundred and nine (109) babies of age nine months old. The data was a primary data from three weighing centers in the Mampong metropolis. Table 4.1 Snapshot of the Data Collected for Analysis Respondent Sex Feeding Practice BW W1 W2 W3 W4 W5 W6 W7 W8 W9 1 1 1 3.2 5.2 6.5 7.3 7.9 7.8 8.2 8.5 9.1 9.1 2 2 2 2.6 3.9 4.8 5.5 6.2 6.9 7.6 7.6 7.8 8.0 3 2 2 3.0 4.9 5.6 6.8 7.7 8.5 9.6 10.2 10.5 10.9 4 2 1 3.3 5.0 6.4 7.2 8.0 8.8 9.7 10.5 10.5 10.4 5 2 2 3.0 5.0 6.5 6.9 7.5 8.3 9.3 10.0 10.0 11.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 1 4 2.4 3.1 3.3 3.6 3.8 4.3 5.0 5.6 5.6 6.1 109 2 4 2.7 3.5 3.8 4.2 4.2 4.6 5.8 7.0 6.9 7.0 78 The table consists of 13 columns. Column 1 represents the respondent‟s (baby‟s) serial number, column 2 the sex of the baby (1 for female and 2 for male), column 3 the feeding practice of the baby, column 4 the birth weight of the baby and the successive 9 columns (i.e. column 5 to 13) represents the weight of the baby for 1-9 months (denoted W1, W2, . . ., W9) respectively. The feeding practices in column 3 are in four forms namely exclusive breastfeeding coded 1, breastfeeding with water only coded 2, breastfeeding with water and any other food coded 3 and feeding without breast milk coded 4. (These are shown in Appendix A). 4.2 EXPLORATORY DATA ANALYSIS 4.2.1 Descriptive Statistics The birth weight of each of the babies was recorded and their monthly weights were also recorded repeatedly over nine months. The sex and feeding practice of the baby were recorded to be used to determine if sex, birth weight and feeding practices were the factors that affect the weight gain of the babies. For the random sample of 109 babies obtained for the study, 52 representing 47.71% were females while 57 representing 52.29% were females. Thirty two (32) of the babies, representing 29.36%, were under exclusive breastfeeding, 30 representing 27.52% were under breastfeeding with water only, 26 representing 23.85% were under breastfeeding with water and any other food while 21 representing 19.27% were feeding without breast milk. Statistics on sex and feeding practice indicates that, for the females, 16 were under exclusive breastfeeding, 16 were under breastfeeding with water only, 13 were under breastfeeding with water and any other food and 7 were under feeding without breast milk. For the males, 16 were 79 under exclusive breastfeeding, 14 were under breastfeeding with water only, 13 were under breastfeeding with water and any other food and 14 were under feeding without breast milk. Table 4.2 presents the mean weights of the babies under the four different forms of feeding practice as well as their maximum and minimum weights. Table 4.2 Summary statistics for feeding practices Month Measure FEEDING PRACTICE Exclusive Breastfeeding Breastfeeding with water Breastfeeding with water and any other food Feeding without breast Milk Mean Maximum Minimum 4.688 6.5 3.2 4.63 6.1 3.1 4.777 5.9 3.7 3.662 5.0 3.0 Mean Maximum Minimum 5.672 7.5 4.0 5.477 8.0 3.7 5.665 7.4 4.2 4.405 6.0 2.9 Mean Maximum Minimum 6.409 8.1 4.4 6.323 9.7 4.9 6.177 7.5 5.1 4.929 6.6 3.3 Mean Maximum Minimum 6.969 9.5 4.4 6.83 10.0 5.2 6.719 8.4 5.3 5.548 7.8 3.7 80 Mean Maximum Minimum 7.369 10.2 4.3 7.307 10.8 5.3 7.146 9.2 5.5 6.143 8.8 4.1 Mean Maximum Minimum 7.763 10.5 4.6 7.700 11.4 5.4 7.554 10.1 5.6 6.724 8.8 4.1 Mean Maximum Minimum 8.081 10.8 4.7 8.117 11.9 5.9 7.742 10.0 5.6 7.267 9.8 5.1 Mean Maximum Minimum 8.397 11.5 5.0 8.443 12.2 6.2 8.054 9.8 5.8 7.648 9.9 5.6 Mean Maximum Minimum 8.603 11.7 5.0 8.760 11.8 6.0 8.408 10.3 6.0 8.024 9.7 6.1 The pattern in the table shows a general increasing trend of weights for all the forms of feeding practice. From the table, the mean weights of babies are higher for the exclusive breastfeeding from the second month to the sixth month than the other three modes of feeding. Breastfeeding with water only became higher than all the other three modes of feeding from the 7 th to the 9 th month. Weight for non-breastfeeding babies was lowest throughout the period. 81 4.2.2 Mean Weight of Babies under each feeding practice A composite line graph for the four modes of feeding practices for the babies using mean Fig 4.1 Mean weights of Babies by modes of feeding weights of the babies is plotted in figure 4.1. From the graph it is observed that weights increase by months for all modes of feeding. Weights of exclusive breastfeeding are higher after one month and remain consistently higher than others up to about the seventh month. Feeding without breast milk is the lowest for the whole study period. Breastfeeding with water only is the WEIGHT 2 3 4 5 6 7 8 9 10 11 12 13 MONTH 1 2 3 4 5 6 7 8 9 PRACTICES Breastfeeding+water only Exclusive Feeding without breastmilk breastfeeding+water+other foods 82 second highest after the exclusive breastfeeding followed by breastfeeding with any other food throughout the period. 4.2.3 Weight Gain Table 4.3 represents the mean weights for all the babies for each month for the nine months. Table 4.3 Mean weight of babies Month Mean weight 1 2 3 4 5 6 7 8 9 4.495 5.372 6.045 6.597 7.062 7.495 7.853 8.183 8.488 The mean weight gain for the first three months is 1.55kg (6.045-4.495), fourth to sixth month is 0.898kg (7.495-6.597) and for seventh to ninth month is 0.635kg (8.488-7.853). This suggests that the weight gain is higher in the first three months of life. For various groups of feeding practices the mean weight gains were respectively 1.721kgs (6.409-4.688), 1.639kg (6.323-4.630), 1.4kg (6.177-4.777) and 1.267kg (4.929-3.662) for 83 exclusive breastfeeding, breastfeeding with water only, breastfeeding with water and any other food and feeding without breast milk for the first three months. The mean weight gains for the next three months, from the 4 th to the 6 th months, were respectively, as in the same order, 0.794kg (7.763-6.969), 0.87kg (7.7-6.83), 0.835kg (7.554-6.719) and 1.176kg (6.724-5.548). After the introduction of complimentary foods in the case of exclusive breastfeeding (from seventh to ninth month) the weight of babies under breastfeeding with water only became highest of all the different modes of feeding. This is shown in column four of Table 4.2 as 8.117, 8.443 and 8.760 for the 7 th , 8 th and 9 th months respectively. The mean weight gain, however, for this period were 0.522kg (8.603-8.081) for exclusive breastfeeding, 0.643kg (8.760-8.117) for breastfeeding with water only, 0.666kg (8.408-7.742) for breastfeeding with water and any other food, and 0.757kg (8.024-7.267) for feeding without breast milk. The result is the reverse order of the mean weight gain by various feeding practices as compared to the first three months. This information suggests that after the introduction of complimentary foods in the case of exclusive breastfeeding, the babies‟ rate of gaining weight slows down as compared to the other forms of feeding practices. For the first six months, exclusive breastfeeding babies gain 3.08kgs (7.763- 4.688), breastfeeding with water only gain 3.07kgs (7.7-4.63), breastfeeding with water and other foods gain 2.78kgs (7.554-4.777) and non-breastfeeding babies gain 3.06kgs (6.724-3.662). A plot of weight by time with respect to gender indicates that males in general have higher rate of weight gain than females. This is especially so for babies of age one to nine months. Figure 4.2 indicates this result. The plot shows the graph of males being consistently higher than that of 84 females, suggesting that males are generally heavier than females at the infant stage of life. Figure 4.2 Plot of Weight by Month with respect to Gender. 4.2.4 Correlation between weights of Babies General Linear models are employed in situations where the responses are correlated rather than independent random variables. This occur, for instance, if they are repeated measurements on the same subject or measurement on a group of related subjects obtained, for example, from clustered sampling (Dobson, 2002). Table 4.4 below is drawn to show the correlation between the monthly weights. WEIGHT 2 3 4 5 6 7 8 9 10 11 12 13 MONTH 1 2 3 4 5 6 7 8 9 GENDER Female Male 85 Table 4.4 Correlation Coefficient for the weights of the babies Weight for month Weight for month 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 0.845 0.783 0.739 0.679 0.610 0.556 0.539 0.482 0.892 0.861 0.814 0.772 0.717 0.677 0.631 0.922 0.869 0.801 0.748 0.700 0.648 0.953 0.884 0.843 0.805 0.759 0.941 0.900 0.865 0.819 0.963 0.917 0.894 0.956 0.932 0.964 Generally, there are strong positive correlations between the weights for any two months. The strength of the correlation, however, declines with increasing gap between the months. For instance correlation coefficient between month 1 and month 2 is 0.845; month 1 and month 3 is 0.783 and continues to decline to 0.482 for month 1 and month 9. The P-value for each of the correlation is 0.00 which signifies that there is significant correlation between any two monthly weights. 86 4.3 INFERENTIAL STATISTICS 4.3.1 Introduction Taken cognizance of the seeming disparities in the weights of the babies, this section seeks to delve into what accounts for the phenomenon and to clarify any perception. The tests were done with the level of significance of 5% (CI = 95%). The results from the various analyses performed were obtained using the Minitab and SAS software. 4.3.2 One-way ANOVA for mean weights of three monthly periods The nine months were divided into three age periods, each consisting of three months, which are the first three months, 4-6 months and 7-9 months. Table 4.5 shows results of a one-way analysis of variance of General Linear Model where age group (1-3 months, 4-6 months, 7-9 months) is the factor with mean weight as the response variable. Table 4.5 One-way ANOVA for mean weights of three monthly periods. Source of variation DF SS MS F P Age 2 393.17 196.58 96.49 0.000 Error 93 189.47 2.04 Total 95 582.64 In the table 4.5, the results show the age as been significant with the P-value of 0.000 at ɑ = 0.05 level of significant. This means that the assertion that the weight gains by age groups are the same over the time period would be rejected and confirms the previous suggestion that in the 87 first three month the weight gain is higher. There are differences in weight gains by babies for different months. 4.3.3 One-way ANOVA for Weights Using the means of the birth weights and each of the nine months, a one-way analysis of variance was conducted to find out if there exist differences. Table 4.6 shows the results from the output of the analysis as produced by Minitab. Table 4.6 One-way ANOVA: Birth weight and nine monthly weights The P-value of 0.00 is less than ɑ=0.05 which implies that differences of the mean weights were statistically significant. Thus there exists difference in mean weights at 5% significant level. 4.3.4 ANOVA for Birth weight Considering feeding practice and sex as factors and Birth weight as response variable, an analysis of variance conducted using Minitab gave results as shown in Table 4.7 below. Source DF SS MS F P Factor Error 9 1080 2964.68 1347.84 329.41 1.25 263.96 0.00 Total 1089 4312.52 88 Table 4.7 ANOVA for Birth weight Source DF SS MS F P Feeding Sex Error 3 1 104 0.5809 0.0014 22.6917 0.1936 0.0014 0.89 0.01 0.45 0.936 Total 108 Since the P-value = 0.45 ˃ ɑ = 0.05, the test is not significant. Also P-value = 0.936 ˃ ɑ = 0.05, the test is not significant. This indicates that birth weights are the same for the different feeding practices and sexes. Naturally, birth weights can never be influenced by feeding practice. 4.3.5 ANOVA for each of the nine months using Adjusted SS for Tests. Using weight as response variable, feeding practice and sex as factors, an analysis of variance was conducted for each of the nine months, using Adjusted Sum of Squares for tests. The results of this from Minitab output are shown in Table 4.8. There is indication from the output that there were significant differences in weights of the babies under the four forms of feeding practices for the first six months of life. For each of the six months the P-value is less than an alpha level of 0.05. However, for the 7 th , 8 th and 9 th months, the weights were found to be statistically not significantly different for the various feeding practices since the P-value were either greater or equal to 0.05 (0.05, 0.08, 0. 17 respectively). 89 Table 4.8 ANOVA for each of the nine months weight MONTH SOURCE DF Seq SS Adj SS Adj MS F P 1 FEEDING 3 18.3740 19.5497 6.5166 14.46 0.000 SEX 1 1.6422 1.6422 1.6422 3.64 0.059 ERROR 104 46.8715 46.8715 0.4507 TOTAL 108 66.8877 2 FEEDING 3 25.0907 26.1206 8.7069 12.13 0.000 SEX 1 1.2028 1.2028 1.2028 1.68 0.198 ERROR 104 74.6639 74.6639 0.7179 TOTAL 108 100.9574 3 FEEDING 3 33.1999 35.2999 11.7666 13.27 0.000 SEX 1 2.8572 2.8572 2.8572 3.22 0.076 ERROR 104 92.2327 92.2327 0.8869 TOTAL 108 128.2897 4 FEEDING 3 29.565 30.952 10.317 9.44 0.000 . SEX 1 1.667 1.667 1.667 1.53 0.220 ERROR 104 113.657 113.657 1.093 TOTAL 108 144.889 90 5 FEEDING 3 22.732 24.514 8.171 6.20 0.001 SEX 1 2.717 2.717 2.717 2.06 0.154 ERROR 104 137.066 137.066 1.318 TOTAL 108 162.516 6 FEEDING 3 16.130 18.003 6.001 4.31 0.007 SEX 1 3.655 3.655 3.655 2.63 0.108 ERROR 104 144.642 144.642 1.391 TOTAL 108 164.428 7 FEEDING 3 11.291 13.208 4.403 2.77 0.045 SEX 1 5.213 5.213 5.213 3.29 0.073 ERROR 104 165.047 165.047 1.587 TOTAL 108 181.551 8 FEEDING 3 9.950 11.582 3.861 2.35 0.077 SEX 1 4.442 4.442 4.442 2.70 0.103 ERROR 104 170.818 170.818 1.642 TOTAL 108 185.210 9 FEEDING 3 7.336 8.719 2.906 1.69 0.173 SEX 1 3.965 3.965 3.965 2.31 0.132 ERROR 104 178.533 178.533 1.717 TOTAL 108 189.834 Considering the sex, there were no significant differences in weights for all the nine months. Thus, differences in weights of babies are not dependent on sex of the baby. 91 4.3.6 Repeated Measures Analysis of Variance for Gender As Tests for Hypotheses for Between Subjects Effects, the comparison test between male and female weight gain at the first nine months of life was performed using Univariate Repeated Measures Analysis of Variance. Table 4.9a below shows the results as for the dependent variable (Weight) versus the independent variable (gender) using SAS. Table 4.9a Analysis of Variance for Gender (Between Subject Effects) (Test of Hypotheses for Between Subjects Effects) Source DF Type 111 SS Mean Square F-Value Pr ˃ F Gender 1 10.616209 10.616209 1.07 0.3038 Error 1.07 1063.992438 9.943855 The results show that the between subject effect is not significant. Thus gender is not statistically significant in terms of weight gain. This is indicated by P-value of 0.3038 being greater than ɑ = 0.05 level of significance. This means that we fail to reject the hypothesis that the mean weight gain for the nine months is the same. Sex is therefore not an influential factor on growth at the infant level as indicated earlier using the Minitab. Univariate tests of Hypotheses for Within Subjects effects produced the results given in Table 4.9b using SAS. 92 Table 4.9b Univariate Analysis of Variance for within Subject effects (Univariate Tests of Hypotheses for Within Subjects Effects) Adj. Pr > F Source DF Type111 SS Mean Square F-Value Pr ˃ F G-G H-F Sex 7 884.6088 126.3727 522.75 ˂ 0.0001 ˂ 0.0001 ˂ 0.0001 Sex*Sex 7 1.9994 0.2856 1.18 0.3108 0.3119 0.3126 Error (sex) 749 181.0679 0.2417 The results in table 4.9b present insignificant test for the sex by sex interaction with P – value of 0.3108 at 5% level of significance. It is concluded therefore that the effect of interaction of sexes is not significant on weight gain by the children. It must be noted here that since sex is composed of two mutually exclusive sets (male and female), there is no possibility of interaction of sexes to produce any effect. 4.3.7 Univariate Analysis of Variance by Feeding Practice Table 4.10a and 4.10b shows the Repeated Measures ANOVA test of Hypotheses for Between Subjects Effects and for within subject effects respectively. 93 Table 4.10a Tests of Hypotheses for Between Subjects Effect. (ANOVA for Feeding Practice) Source DF Type 111 SS Mean Square F-value Pr ˃ F Practice 3 140.0670 46.6890 5.25 0.0021 Error 105 934.5417 8.9004 The Between Subjects effects test in table 4.9a produces a P-value of 0.0021. This does not support the hypothesis of equal mean effects. Hence the test for feeding practice is significant indicating that the weight of babies is influenced by the feeding practice. Table 4.10b Univariate Analysis of Variance for within Subject Effects (Univariate Tests of Hypotheses for Within Subjects Effects) Adj. Pr > F Source DF Type111 SS Mean Square F-Value Pr ˃ F G-G H-F Practice 7 888.4970 126.9281 555.84 ˂ 0.0001 ˂ 0.0001 ˂0.0001 Prac.*Prac. 21 15.2276 0.7251 3.18 ˂ 0.0001 0.0030 0.0025 Error (Prac.) 735 167.8398 0.2284 In the within subject effect test above (Table 4.8b), the interaction of feeding practices is shown to be significant at ɑ = 0.05 significance level with P-value of 0.0001. 94 4.3.8 Repeated Measures Multivariate Analysis of Variance (MANOVA) The factors, feeding practices and sex are considered for multivariate analysis using repeated measures analysis. Tables 4.11 (a, b and c) show the output for extreme measures for the factors. There are seven columns in each of the tables. The effects of the factors are in the first column, the extreme measures for each factor (statistic) in column two, the values of the extreme measures in column three and F values in column four. Numerator degrees of freedom, Denominator degrees of freedom and the P-value are in columns five, six and seven respectively. Table 4.11 Multivariate Repeated Measures Analysis on Weight Gain (a) MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no SEX Effect Effect Statistic Value F Num DF Den DF Pr ˃ F Sex Wilks‟ Lambda 0.0930 140.74 7 101 ˂ 0.0001 Pillai‟s Trace 0.9070 140.74 7 101 ˂ 0.0001 Hotelling-Lawley Trace 9.7545 140.74 7 101 ˂ 0.0001 Roy‟s Greatest Root 9.7545 140.74 7 101 ˂ 0.0001 95 (b) MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no PRACTICES Effect Effect Statistic Value F Num DF Den DF Pr > F Practice Wilks‟ Lambda 0.0880 146.70 7 99 ˂ 0.0001 Pillai‟s Trace 0.9121 146.70 7 99 ˂ 0.0001 Hotelling-Lawley Trace 10.3729 146.70 7 99 ˂ 0.0001 Roy‟s Greatest Root 10.3729 146.70 7 99 ˂ 0.0001 (c) MANOVA Test Criteria and F Approximations for the Hypothesis of no PRACTICES*PRACTICES Effects. Effect Statistic Value F Num DF Den DF Pr > F Prac*Prac Wilks‟ Lambda 0.7320 1.56 21 284.82 0.0591 Pillai‟s Trace 0.2908 1.55 21 303 0.0606 Hotelling-Lawley Trace 0.3355 1.57 21 200.45 0.0607 Roy‟s Greatest Root 0.1983 2.86 7 101 0.0091 The results in the tables show that both sex and feeding practice are influential factors in determining the weight gain of children of age 1 – 9 months. This is as a result of p-value for all the extreme measures being 0.0001 at 5% significance level. 96 4.4 FURTHER INFERENTIAL ANALYSIS Table 4.12 represents covariance structures investigated in selecting an appropriate covariance structure of the random errors. Table 4.12 Covariance Structures Investigated COVARIANCE STRUCTURE AIC AICC BIC COMPOUND SYMMETRY 2062.8 2062.8 2068.2 UNSTRUCTURED 1271.3 1275.7 1392.4 FIRST ORDER AUTOREGRESSIVE 1407.4 1401.4 1406.7 FIRST ORDER ANTE-DEPENDENCE 1257.6 1258.2 1303.3 HETEROGENEOUS FIRST ORDER AUTOREGRESSIVE 1350.0 1353.2 1379.7 HETEROGENEOUS COMPOUND SYMMETRY 1963.8 1964.1 1990.7 HETEROGENEOUS TOEPLITZ 1333.7 1334.3 1379.4 FIRST ORDER AUTOREGRESSIVE MOVING AVERAGE 1394.8 1394.9 1402.9 HUYNH-FELDT 1974.5 1974.7 2001.4 TOEPLITZ 1373.5 1373.7 1397.7 VARIANCE COMPONENT 2876.3 2876.3 2879.0 From the table the smallest value is 1257.6 which is in the row of the First Order Ante- Dependence corresponding to the AIC column. Thus the best covariance structure to be used is the Ante-Dependence. 97 4.4.0 Longitudinal Model with Ante-dependence Covariance Structure (Mixed Procedure) 4.4.1 Model Information The Model Information table, Table 4.13, shows the name of the data set, the dependent variable, the covariance structure used in the model, the subject effect, the estimation method to compute the parameters for the covariance structures (the default estimation method is REML), the residual variance method, the fixed effects standard error method and the method to compute the degrees of freedom. Table 4.13 Model information Data Set SASUSER.SEK Dependent Variable WEIGHT Covariance Structure Ante-dependence Subject Effect RESPONDENTS Estimation Method REML Residual Variance Method None Fixed Effects SE Method Model-Based Degrees of Freedom Method Satterthwaite 4.4.2 Dimensions The Dimensions table below, Table 4.14, lists the sizes of the relevant matrices. The table is useful in determining CPU time and memory requirements. 98 Table 4.14 Dimensions Covariance Parameters 17 Columns in X 9 Columns in Z 0 Subjects 109 Max Obs Per Subject 9 Number of Observations Read 986 Number of Observations Used 981 Number of Observations Not Used 5 4.4.3 Iteration History The Iteration History table (Table 4.15) describes the optimization of the residual log likelihood. The minimization is performed using a ridge-stabilized Newton-Raphson algorithm, and the rows of the table describe the iterations that this algorithm takes in order to minimize the objective function. 99 Table 4.15 Iteration History Iteration Evaluations -2 Res Log Like Criterion 0 1 2874.32193062 1 2 1238.12934187 0.04301972 2 1 1230.70751929 0.02156881 3 1 1224.01108546 0.00134184 4 1 1223.58269601 0.00009122 5 1 1223.55544908 0.00000076 6 1 1223.55523153 0.00000000 *Convergence criteria met 4.4.4 Covariance Parameter Estimates The covariance Parameter Estimates table (Table 4.16 in Appendix C) shows the parameter estimates for the Ante-dependence covariance structure. 4.4.5 Fit Statistics The Fit Statistics table provides information to be used to select the most appropriate covariance structure. Akaike‟s Information Criterion (AIC)(Akaike,1974) penalizes -2 residual log likelihood by twice the number of covariance parameters in the model. The smaller the value, the better the model. The finite-sample version of the AIC (AICC) is also included. The Schwarz‟s Bayesian Information Criterion (BIC)(Schwarz,1978) also penalizes the -2 residual log likelihood. In The Fit Statistics table below (Table 4.17) AIC is the best option. 100 Table 4.17 Fit Statistics -2 Res Log Likelihood 1223.6 AIC (smaller is better) 1257.6 AICC (smaller is better) 1258.2 BIC (smaller is better) 1303.3 4.4.6 Null Model Likelihood Ratio Test The Null Model Likelihood Ratio Test shows a test that determines whether it is necessary to model the covariance structure of the data at all. The test statistic is -2 times the log likelihood from the null model (model with an independent covariance structure) minus -2 times the log likelihood from the fitted model. The p-value is used to assess the significance of the model fit. Therefore for the p-value of 0.0001, the model is considered significant at 5% level of significant. This is shown in table 4.18. Table 4.18 Null Model Likelihood Ratio Test DF Chi-Square Pr ˃ ChiSq 16 1650.77 ˂ 0.0001 4.4.7 Solution for Fixed Effects Table 4.19 contains the solution for Fixed Effects. It contains the estimates, standard errors, the degrees of freedom, the t-value and the p-values for each of the fixed effects listed in column one. From the table, the initial weight (birth weight), monthly weight gains, and the gender are shown to be significant. This is so as the p-value for each of the effects mentioned is 0.0001. 101 Feeding without breast milk has p-value 0.023 which shows that the practice is significant. Exclusive breastfeeding and Breastfeeding with water only are found not to be significant. Table 4.19 Solution for Fixed Effects Effect Gender Practices Estimate Std. Error DF t-value Pr ˃ |t| Intercept 1.8824 0.3830 108 4.91 ˂ 0.0001 Birth weight 0.7969 0.1151 108 6.92 ˂ 0.0001 Month 0.4275 0.01241 157 34.43 ˂ 0.0001 Gender Female -0.2450 0.1063 108 -2.31 0.0230 Gender Male 0 . . . . Practice BFW -0.03157 0.1483 108 -0.21 0.8318 Practice Exclusive -0.09133 0.1449 108 -0.63 0.5298 Practice NBF -1.0019 0.1630 108 -6.15 ˂ 0.0001 Practice BWFO 0 . . . . 4.4.8 Type 3 Tests of Fixed Effects The Type 3 Test of Fixed Effects table (Table 4.20) shows the hypothesis test for the significance of each of the fixed effects. A p-value is computed from an F distribution with the numerator and denominator degrees of freedom. The p-values are given as 0.0001 for each of the following fixed effects: Birth weight, monthly weight gain and Feeding Practices. The p-value for Gender is 0.0230. Thus these effects are considered significant under this test. 102 Table 4.20 Type 3 Test of Fixed Effects Effect Num DF Den DF F-Value Pr ˃ F Birth weight 1 108 47.92 ˂ 0.0001 Month 1 157 1185.55 ˂ 0.0001 Gender 1 108 5.32 0.0230 Practices 3 108 16.95 ˂ 0.0001 103 CHAPTER FIVE 5.0 DISCUSSIONS, CONCLUSIONS AND RECOMMENDATIONS 5.1 DISCUSSIONS OF RESULTS The study used a random sample of 109 babies. The weights of these babies were recorded repeatedly for nine months. The birth weight and the feeding practice of each of the babies were also recorded. The study was intended to find out whether the monthly weight of babies recorded at the weighing centers were significantly different and if birth weight, age, sex and feeding practice could contribute to weight changes. At the exploratory analysis stage it was found out that there was generally an increasing trend of weights for all the babies. The descriptive statistics of the mean monthly weights of the babies suggested that those under exclusive breastfeeding had higher weights than the others for the first six months after birth. The babies under breastfeeding with water only and breastfeeding with water and any other food were second and third respectively in terms of higher weights. The non-breastfed babies had the lowest mean monthly weights throughout the study period. Considering the mean monthly weight gain for the three age groups, the babies within ages 0-3 months had 2.977kg while those within 4-6 and 7-9 had 1.45kg and 0.993kg respectively. This suggested that the weight gain was higher in the first three months of life and decreased through the next age groups. For the four groups of feeding practices the weight gains were respectively 3.312kg, 3.34kg, 3.015kg and 1.948kg for exclusive breastfeeding, breastfeeding with water only, breastfeeding with water and any other food and feeding without breast milk for the first three months of life. The mean weight gains for the 4 th , 5 th , and the 6 th months were 1.354kg for exclusive breastfeeding, 1.377kg for breastfeeding with water only, 1.377kg for breastfeeding 104 with water and any other food and 1.795kg for feeding without breast milk. The weight gains for the first six months after birth confirmed the assertion that exclusive breastfeeding is the best form of feeding for babies. The weight gains were 4.7kg for exclusive breastfeeding, 4.7kg for breastfeeding with water only, 4.4kg for breastfeeding with water and any other food and 3.7kg for non-breastfeeding. There was seemingly a drastic decrease in weight gain for babies under exclusive breastfeeding from the 7 th to the 9 th month which was just after introduction of complimentary food to the babies although there was general decreasing trend of weight gain. The weight gains as in the same sequence were 0.84kg, 1.06kg, 0.85kg and 1.3kg. A composite line graph of mean weights against age in months in figure 4.1 suggested such a shocking revelation of non-breastfed babies having high rate of weight gain at the latter stages of the study period. A Plot of weight by time with respect to gender indicated that males have high rate than females (fig 4.2) as the graph for male was consistently higher. A table of correlation coefficient gave strong positive correlations between any two different months. The strength of the correlation, however, declines from the closer months to the months wider apart. This prompted the use of general linear model for further analysis. An analysis of variance conducted using General linear model method revealed that age is significant in determining the weight of babies under nine months old. Thus the older the baby the heavier the baby as exposed by the increasing trend in weight by month is confirmed. At 5% significant level the one-way ANOVA revealed that there exist differences in the mean weights of babies. There was also an indication that there were significant differences in weights of the babies under the four forms of feeding practice for the first six months. For the 7 th , 8 th and 105 9 th months, the analysis indicated that there were not significant differences in weights under the feeding practices. Contrary to the suggested assertion that male babies weigh more than female as revealed by the plot in fig 4.2, the analysis by the general linear model showed that sex does not influence weight gain. For all of the nine months, sex was found to be not statistically significant at 5% level. This called for further investigation. Using Repeated Measures Multivariate Analysis of variance (MANOVA), sex was found to be significant for all the statistics; Wilks‟ Lambda, Pillai‟s Trace, Hotelling-Lawley Trace and Roy‟s Greatest Root at 5% level. Similarly, the MANOVA test criteria and exact F statistics for the hypothesis of no feeding practices effect confirmed again that there are differences in weights under the feeding practices. Thus, feeding practice affects the weight gain. There were no interaction effects of sexes and feeding practices on the weights. Since the measurement were repeated measurements over time on the same subject the data is a longitudinal data requiring longitudinal data analysis. Furthermore, the plot of mean monthly weight of the feeding practices against the time (age in months) illustrated longitudinal relationship in the exploratory analysis. The observations were positively correlated, which often occurs with longitudinal data, and meant that the variances of the time-independent predictor variables are underestimated if the data is analyzed as though the observations are independent. In other words the Type I error rate is inflated for these variables. For time-dependent predictor variables also, ignoring positive correlation leads to a variable estimate that is too large. In other words, the Type II error rate is inflated for these variables. 106 Special methods of statistical analysis are needed for longitudinal data such as this because the set of measurements on one subject tends to be correlated; measurements on the same subject close in time tend to be more highly correlated than measurements far apart in time and the variances of longitudinal data often change with time. These potential patterns of correlation and variation may combine to produce a complicated covariance structure. This covariance structure must be taken into account to draw valid statistical inferences. Therefore, standard regression and ANOVA models may produce invalid results because two of the parametric assumptions (independent observation and equal variances) may not be valid as stated and documented in chapter 3 (methodology). The linear mixed model allows a very flexible approach to modeling longitudinal data. The data structure is similar to univariate ANOVA where the number of observations equals the number of measurements for all the subjects. This means the data does not have to be balanced. An advantage of fitting linear mixed model is that PROC MIXED uses all the available data in the analysis. PROC MIXED offers a wide variety of covariance structures. This enables the user to directly address the within-subject correlation structure and incorporate it into a statistical model. By selecting a parsimonious covariance model that adequately accounts for within-subject correlations, the user can avoid the problems associated with univariate and multivariate ANOVA using PROC GLM (Little, Freund and Spector, 2001). Using the information criteria in selection, it was realized that 1 st order Ante-Dependence was the most appropriate covariance structure for the data for the study. In Model Information SASUSER SEX was the data set, Weight was the dependent variable, RESPONDENTS were the 107 Subject Effect, Estimation Method employed was REML, Residual variance method was none, the Fixed Effects Standard Error Method was Model-Based and the Degrees of Freedom Method was Satterthwaite. The Dimensions for the matrices were 17 covariance parameters, 9 columns in X matrix with 0 columns in Z matrix and maximum observations per subject were 9. There were 109 subjects, 986 Number of observations read, 981 Number of observations used and 5 Number of observations not used. The iteration history indicated that convergence criteria were met, Covariance Parameter estimates were given as; respondents are the subjects; variables (1), variables (2), …,variable (9); Rho (1), Rho (2), …, Rho (9) are the covariance parameters with their corresponding estimates listed respectively. AIC was the appropriate Fit statistics used to select the most appropriate covariance structure which was the Ante-dependence. The model validated the graph of the mean monthly weights in the exploratory analysis section and other assertion such as relationship between feeding practice and weight gain, initial weight (or birth weight) and weight gain and monthly weight gains. Thus, it confirmed that feeding practices affect weight of babies, birth weight affect weight of babies and monthly weight gains are different. Sex was found to be influential in weight of babies which is the biggest change. This illustrates how changing the covariance structure can change the inferences made from model. Using an output generated by General Linear Model, and judging from the contributions of each of the feeding practices to the model, the exclusive breastfeeding is found to be very significant, 108 followed by breastfeeding with water only with non-breastfeeding and breastfeeding with water and any other food being not significant for the first six months. Furthermore, considering the solution for fixed effects, using BFWO as the base, both EBF and BFW are found not to be significant while NBF is found to be significant. 5.2 CONCLUSIONS From the review of the related research works, proper nutrition helps give every child the best start in life. Malnutrition of children (0-59 months) is a public health concern in Africa, particularly in the Sahelian countries. Breastfeeding has been found to reduce the occurrence of the pressing public health challenge of malnutrition. From the analyses, it is concluded that the type of feeding practice of a baby affect weight gain significantly. It is also concluded that the mean weight gain is higher in the first three months of life compare to that of 4 th – 6 th and 7 th – 9 th months. The birth weight of a baby has a significant influence on weight gain at the early stage of growth. The Exclusive Breastfeeding babies‟ rate of weight gain is highest for the first six months but slows down drastically and becomes the second lowest after the introduction of complimentary foods. The non-breastfed babies have rate of weight gain to be the lowest throughout the study period. The study has therefore confirmed that Exclusive Breastfeeding is the best form of feeding practice for a baby for the first six months after birth. Comparatively, there is no clear difference in weight gain between female and male babies as there was inconsistency in the results provided by various methods of analysis. 109 5.3 RECOMMENDATIONS 1. There is need for promotion and protection of optimal infant feeding practice for improving nutrition since infant feeding practices constitute a major component of child caring practices. 2. Nursing mothers should be encouraged to frequently report at the weighing centers to facilitate the early detection of ill-health in babies and other early childhood diseases. 3. The ministry of health should open more weighing centers and put up mobile teams to go round the communities to solve the unfriendly overcrowding situations in the existing ones that deter mothers from participating in the exercise. 4. Educational programs on Exclusive Breastfeeding for the first six months of life should be intensified, especially by radios in order to reach the remote areas of the country. 5. Parents or guardians who have babies who do not have the benefit of breastfeeding should consider timely admission to Babies Homes as urgent and important. 6. 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