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The genetic analysis of quantitative traits includes two major tasks: (1) identifying the location of QTLs affecting a quantitative trait using a genetic linkage map constructed from molecular markers, and (2) estimating the genetic effects of the QTLs on the phenotype. If the genotypes of a putative QTL were known for all individuals, its genomic location could be readily determined using a marker linkage analysis. Furthermore, the genetic effects of the QTL could be precisely estimated and tested by simple t tests or ANOVA. However, it is not possible for the genotypes of QTLs to be directly observed; instead they should be inferred from observed marker and phenotypic information. As was seen in Chapter 8, a marker analysis cannot unambiguously separate the genetic effects of a QTL from the recombination fraction between the markers and QTL. [ABSTRACT FROM AUTHOR]

QTL mapping approaches were developed originally for experimental crosses, such as the backcross, double haploid, RILs, or F2, derived from inbred lines. Because of the homozygosity of inbred lines, the Mendelian (co)segregation of all markers with two alternative alleles in such crosses can be observed directly. In practice, there also is a group of species called outcrossing species, such as forest trees or large animals, in which it is difficult or impossible to generate inbred lines due to long generation intervals and high heterozygosity, although experimental hybrids can be commercially used for their genetic improvement. [ABSTRACT FROM AUTHOR]

Lander and Botstein's interval method has an advantage for mapping QTLs genomewide by scanning for the position of a QTL throughout the genome. But this method can lead to biased estimates of QTL positions and effects when multiple QTLs occur on the same linkage group because it makes use of one single-marker interval at a time and has no mechanism to alleviate the impact of other QTLs outside the interval. For this reason, if a real QTL is located near a marker interval with no QTL, interval mapping may still detect a "ghost" QTL due to the linkage between the real QTL and the interval being tested (Martinez and Curnow 1992). Although a simultaneous search for multiple QTLs on different intervals can overcome this problem, this will bring about new difficulties in parameter estimation and model identifiability. [ABSTRACT FROM AUTHOR]

In the preceding chapter, we described the basic principle for interval mapping of QTLs within the maximum likelihood context, but two fundamental questions should be addressed toward the QTL analysis of complex traits. First, what is the critical threshold of the test statistic that can be used to declare the statistical significance of a QTL? Second, after the significant QTL is determined, how are the estimates of likelihoods and QTL parameters adequately precise to make a scientific inference about QTL position and effect? Statistically, these two questions present different aspects of QTL mapping, but we will describe them here in a single chapter as two important follow-ups of QTL mapping. [ABSTRACT FROM AUTHOR]

As discussed in Chapter 10, interval mapping is powerful for the separation of QTL effects and QTL-marker linkage. Interval mapping based on a regression approach is computationally faster than a maximum likelihood (ML) approach, with comparable results between the two approaches in some particular cases. Kao (2000) discussed analytically and through simulation studies the conditions under which the regressionand ML-based approaches generate different results. Further comparisons by Mayer (2005) between the two approaches were made in terms of power, accuracy of position and effect estimates, and estimation of the residual variance when multiple linked QTLs are involved in the genetic control of a quantitative trait. [ABSTRACT FROM AUTHOR]

In the previous chapter, statistical approaches were described for evaluating associations between markers and phenotypes. Such an association analysis can provide evidence for the genetic control of trait variation but is not very precise because the genetic effects associated with marker genotypes are confounded by the position of a functional QTL and its actual effect. Of course, if markers are so highly dense that they are generated at QTL positions, a simple marker-phenotype association analysis may be useful. The generation of such high-density maps is not possible for a majority of species in practice. Powerful analytical techniques are needed to separate the effects of a QTL from its location. [ABSTRACT FROM AUTHOR]

In the preceding chapters, we described numerous statistical issues related to linkage analysis of molecular markers and the construction of genetic linkage maps. One of the most important aims of these marker analyses is to provide ordered hallmarks on chromosomes with which one can map functional quantitative trait loci (QTLs) determining complex phenotypic variation to particular genomic regions. The genomewide identification of QTLs, their locations and effects, is of fundamental importance for agricultural, evolutionary, and biomedical genetics. [ABSTRACT FROM AUTHOR]

In the preceding chapters, we discussed linkage analysis for a single controlled cross derived from inbred lines or outbred lines. We described a general framework for performing linkage analysis with any type of marker, fully or partially informative, that is qualitatively scored on the presence or absence of a DNA band for an allele. For an outbred cross, the procedure was given to simultaneously estimate the linkage, parental diplotype, and gene order. However, there are many situations in practice in which the strategies for linkage analysis described thus far cannot be appropriately used. In this chapter, we will consider two important issues that have been common or are becoming increasingly common in linkage analysis. The first concerns linkage analysis in a complicated nuclear pedigree, whereas the second deals with linkage analysis based on quantitative marker information. [ABSTRACT FROM AUTHOR]

We have described statistical methods for linkage analysis of different markers in a controlled cross. But these methods can only be appropriate for the markers whose segregation follows the Mendelian ratio (1:1 for the backcross or 1:2:1 for the F2). In a practical molecular experiment, many markers may deviate from Mendelian segregation ratios due to some genetic or biological reason (e.g., differential viability, where viability is defined as an individual's ability to survive). If marker segregation is disturbed by viability effects, we can incorporate such effects into linkage analysis using modified methods. Bailey (1961) discussed several models for analyzing distorted markers in great detail. In this chapter we will describe the methods for estimating the recombination fraction between markers subject to differential viability. [ABSTRACT FROM AUTHOR]

Recombinant inbred lines (RILs) have proven powerful for QTL mapping. They can be derived either by repeated selfing or by repeated sibling (brother-sister) mating from the offspring of an F1 cross between two inbred lines. Because of continuous inbreeding for a sufficiently number of generations (e.g., 7-10), RILs tend to be homozygous for all genes. With such fixed genotypes (ignoring mutations), RILs can be propagated eternally, allowing the replication of identical genotypes on the scale of time and space aimed to address many fundamental biological and genetic issues. Furthermore, RILs accumulate crossovers that occur at each meiosis with every generation, and thus the proportion of recombinant zygotes in RILs (i.e., the probability that two linked loci have different parental alleles) is higher than what it would be in the F2. As a result, RILs that have been increasingly available due to community efforts by geneticists and breeders (Threadgill et al. 2002; Complex Trait Consortium 2004) provide powerful material for high-resolution mapping of QTLs. [ABSTRACT FROM AUTHOR]

Statistical methods for linkage analysis in a backcross or F2 population derived from two inbred lines were described in the previous chapter. An advantage of linkage analysis using these inbred line crosses is that the parental linkage phase between different genetic loci is known and therefore the patterns of marker segregation can be determined and the linkage measured in terms of the recombination fraction tested. However, this inbred line-based analysis is not appropriate for outcrossing species in which it is not possible to generate homozygous lines through successive inbreeding. [ABSTRACT FROM AUTHOR]

Linkage is the tendency for genes to be inherited together because they are located near one another on the same chromosome. Linkage analysis of markers lays a foundation for the construction of a genetic linkage map and the subsequent molecular dissection of quantitative traits using the map. Linkage analysis is based on the cosegregation of adjacent markers and their cotransmission to the next progeny generation. The prerequisite of linkage analysis between any two markers is their known allelic arrangements (i.e., linkage phases) on the homologous chromosome so that parental (or nonrecombinant) vs. nonparental (or recombinant) haplotypes can be readily distinguished. In many domesticated plants and animals, phase-known mapping pedigrees can be established using a segregating population, such as the backcross or F2, derived from two homologous inbred lines. (Recall the definitions in Section 1.5.) Theories for linkage analysis in such segregating pedigrees have been well-developed. [ABSTRACT FROM AUTHOR]

Now that we have seen the basics of genetics, we turn to an introduction to the statistical methodologies that we will use throughout this book. Most of the statistical inferences that we will make will be based on likelihood analysis, and we will be concerned not only with constructing the appropriate likelihood function for a given model but also with methods for computing and optimizing the likelihood. We start here with some basics and work our way toward likelihood analysis of a genetic linkage model. [ABSTRACT FROM AUTHOR]

There have been enormous advances in the science of genetics. A huge amount of information regarding the precise molecular mechanisms of genetic transmission from parent to offspring is becoming increasingly available. In this chapter, we briefly review basic terminology and principles of genetics from Mendelian, population, quantitative and molecular perspectives at a level appropriate for understanding the research methods to be described in this book. Much of the description for classic Mendelian genetics is adapted from Bailey's (1961) book. To learn more about modern genetics, please look into the more general genetics textbooks that are listed at the end of this chapter. [ABSTRACT FROM AUTHOR]

In this chapter we mention some other issues related to nonparametric inference including: measurement error, inverse problems, nonparametric Bayesian inference, semiparametric inference, correlated errors, classification, sieves, shape-restricted inference, testing, and computation. [ABSTRACT FROM AUTHOR]

This chapter concerns estimating functions that are spatially inhomogeneous, functions r(x) whose smoothness varies substantially with x. For example, Figure 9.1 shows the "blocks" function whose definition is given in Example 9.39. The function is very smooth except for several abrupt jumps. The top right plot shows 100 data points drawn according to Yi = r(xi) + ∈i with ∈i ≈ N(0, 1), and the xis equally spaced. [ABSTRACT FROM AUTHOR]

In this chapter we use orthogonal function methods for nonparametric inference. Specifically, we use an orthogonal basis to convert regression and density estimation into a Normal means problem and then we construct estimates and confidence sets using the theory from Chapter 7. In the regression case, the resulting estimators are linear smoothers and thus are a special case of the estimators described in Section 5.2. We discuss another approach to orthogonal function regression based on wavelets in the next chapter. [ABSTRACT FROM AUTHOR]

Let F be a distribution with probability density f = F′ and let $$ X_1 , \ldots ,X_n \sim F $$ be an IID sample from F. The goal of nonparametric density estimation is to estimate f with as few assumptions about f as possible. We denote the estimator by $$ \widehat{f_n } $$. As with nonparametric regression, the estimator will depend on a smoothing parameter h and choosing h carefully is important. [ABSTRACT FROM AUTHOR]

In this chapter we will discuss the many Normal means problem which unifies some nonparametric problems and will be the basis for the methods in the next two chapters. The material in this chapter is more theoretical than in the rest of the book. If you are not interested in the theoretical details, I recommend reading sections 7.1, 7.2, and 7.3 and then skipping to the next chapter, referring back as needed. If you want more details on this topic, I recommend Johnstone (2003). [ABSTRACT FROM AUTHOR]

In this chapter we will study nonparametric regression, also known as "learning a function" in the jargon of machine learning. We are given n pairs of observations (x1, Y1). . ., (xn, Yn) as in Figures 5.1, 5.2 and 5.3. The response variableY is related to the covariatex by the equations (5.1)$$ Y_i = r(x_i + \in _i ),{\text{ }}\mathbb{E}{\text{(}} \in _i {\text{)}} = {\text{0, }}i = 1, \ldots ,n $$ where r is the regression function. The variable x is also called a feature. We want to estimate (or "learn") the function r under weak assumptions. The estimator of r(x) is denoted by $$ \widehat{r_n }(x) $$. We also refer to $$ \widehat{r_n }(x) $$ as a smoother. At first, we will make the simplifying assumption that the variance $$ \mathbb{V}{\text{(}} \in _i {\text{) = }}\sigma ^{\text{2}} $$ does not depend on x. We will relax this assumption later. [ABSTRACT FROM AUTHOR]

To estimate a curve—such as a probability density function f or a regression function r—we must smooth the data in some way. The rest of the book is devoted to smoothing methods. In this chapter we discuss some general issues related to smoothing. There are mainly two types of problems we will study. The first is density estimation in which we have a sample X1. . ., Xn from a distribution F with density f, written (4.1)$$ X_1 \ldots ,X_n \sim f, $$ and we want to estimate the probability density function f. The second is regression in which we have a pairs (x1, Y1). . ., (xn, Yn) where (4.2)$$ Y_i = r(x_i ) + \in _i $$ where $$ \mathbb{E}{\text{(}} \in _i {\text{) = 0}} $$, and we want to estimate the regression function r. We begin with some examples all of which are discussed in more detail in the chapters that follow. [ABSTRACT FROM AUTHOR]

The bootstrap and the jackknife are nonparametric methods for computing standard errors and confidence intervals. The jackknife is less computationally expensive, but the bootstrap has some statistical advantages. [ABSTRACT FROM AUTHOR]

The first problem we consider is estimating the cdf. By itself, this is not a very interesting problem. However, it is the first step towards solving more important problems such as estimating statistical functionals. [ABSTRACT FROM AUTHOR]

In this chapter we briefly describe the types of problems with which we will be concerned. Then we define some notation and review some basic concepts from probability theory and statistical inference. [ABSTRACT FROM AUTHOR]

In most of the multivariate linear models analyzed to this point, we formulated relationships between one observed independent set of variables and a set of random dependent variables where the covariance structure of the dependent set involved an unknown and unstructured covariance matrix Σ. Two exceptions to this paradigm included mixed models where Σ was formulated to have structure which contained components of variance and the exploratory factor analysis (EFA) model where Σ depended upon the unknown regression (pattern) coefficients which related unobserved (latent) factors to observed dependent variables. [ABSTRACT FROM AUTHOR]

Discriminant analysis is used to evaluate group separation and to develop rules for assigning observations to groups. Cluster analysis is concerned with group identification. The goal of cluster analysis is to partition a set of observations into a distinct number of unknown groups or clusters in such a manner that all observations within a group are similar, while observations in different groups are not similar. If data are represented as an n x p matrix Y = [yij] where $$ \mathop Y\limits_{n \times p} = \left[ \begin{gathered} y'_1 \\ y'_i \\ \vdots \\ y'_n \\ \end{gathered} \right] $$ the goal of cluster analysis is to develop a classification scheme that will partition the rows of Y into k distinct groups (clusters). The rows of the matrix usually represent items or objects. To uncover the groupings in the data, a measure of nearness, also called a proximity measure needs to be defined. Two natural measures of nearness are the degree of distance or "dissimilarity" and the degree of association or "similarity" between groups. The choice of the proximity measure depends on the subject matter, scale of measurement (nominal, ordinal, interval, ratio), and type of variables (continuous, categorical) being analyzed. In many applications of cluster analysis, one begins with a proximity matrix rather than a data matrix. Given the proximity matrix of order (n x n) say, the entries may represent dissimilarities [drs] or similarities [srs] between the rth and sth objects. Cluster analysis is a tool for classifying objects into groups and is not concerned with the geometric representation of the objects in a low-dimensional space. To explore the dimensionality of the space, one may use multidimensional scaling. [ABSTRACT FROM AUTHOR]

In the previous chapter on discriminant analysis we generated linear combinations of variables called discriminant functions. Applying the discriminant functions to multivariate data obtained from several independent groups resulted in discriminant scores in a smaller dimensional discriminant space. The uncorrelated scores were used to evaluate group differences (separation) and to classify (allocate) observations into groups. In this chapter we investigate three statistical procedures that commonly involve a single group. These procedures were designed to reduce the dimensionality of the data space in order to discover, visualize, and interpret dependences among sets of variables, or to help stabilize the measurements for additional statistical analysis such as regression analysis (Chapter 4) or cluster analysis (Chapter 9). [ABSTRACT FROM AUTHOR]

Given two or more groups or populations and a set of associated variables one often wants to locate a subset of the variables and associated functions of the subset that leads to maximum separation among the centroids of the groups. The exploratory multivariate procedure of determining variables and a reduced set of functions called discriminants or discriminant functions is called discriminant analysis. Discriminants that are linear functions of the variables are called linear discriminant functions (LDF). The number of functions required to maintain maximum separation for a subset of the original variables is called the rank or dimensionality of the separation. The goals of a discriminant analysis are to construct a set of discriminants that may be used to describe or characterize group separation based upon a reduced set of variables, to analyze the contribution of the original variables to the separation, and to evaluate the degree of separation. Fisher (1936) developed the technique to create a linear discriminant function to establish maximum separation among three species of iris flowers based upon four measurements. In personnel management one may want to discriminate among groups of professionals based upon a skills inventory. In medicine one may want to discriminate among persons who are at high risk or low risk for a specific disease. In a community, the mayor may want to evaluate how far apart several interest groups are on specific issues and to characterize the groups. In industry, one may want to determine when processes are in-control and out-of-control. [ABSTRACT FROM AUTHOR]

In our discussion of the MR model the parameter matrix B contained only fixed parameters and the analysis of univariate fixed effect designs for orthogonal and nonorthogonal MANOVA and MANCOVA designs extended in a natural manner to the multivariate case. The MANOVA model also provided an alternative for the analysis of repeated measurement designs, if there are no covariates changing with time, and allowed the within-subject covariance matrix to have a general structure. The SUR model was introduced to permit the analysis of repeated measures data with changing covariates. The GMANOVA model was also used to analyze repeated measurements, however, exact tests exist only when there is complete data and the within-subject covariance matrix is unstructured. [ABSTRACT FROM AUTHOR]

In the MR regression model, the design matrix X is common to each variable. This limitation does not permit one to associate different design matrices with each dependent variable which in many regression problems may lead to overfitting some variables. To correct this problem, we must be able to model each dependent variable separately within a common, overall model. Using the vec (•) operator on the columns of the data matrix Y, Zellner (1962, 1963) formulated the seemingly unrelated regression (SUR) model as p correlated regression models. Srivastava (1966, 1967) called the design the multiple-design multi-variate (MDM) model. Hecker (1987) formulates the model using the vec (•) operator on the rows of Ynxp and calls the model the completely general MANOVA (CGMANOVA) model. In this chapter, we review the general theory of the SUR model and show how the model may be used to estimate parameters and test hypothesis in complex design situations including the generalized MANOVA (GMANOVA) model developed by Potthoff and Roy (1964), repeated measurement designs with changing covariates, and mixed MANOVA-GMANOVA designs. In addition, goodness of fit tests, tests for nonadditivity, and sum of profile models are discussed. Finally, the multivariate SUR (MSUR) is reviewed. [ABSTRACT FROM AUTHOR]

In Chapter 3, Section 3.6 we introduced the basic theory for estimating the nonrandom, fixed parameter matrix Bqxp for the multivariate (linear) regression (MR) model Ynxp = XnxqBqxp + Enxp and for testing linear hypotheses of the general form CBM = 0. For this model it was assumed that the design matrix X contains fixed nonrandom variables measured without measurement error, the matrix Ynxp contains random variables with or without measurement error, the E (Y) is related to X by a linear function of the parameters in B, and that each row of Y has a MVN distribution. [ABSTRACT FROM AUTHOR]

In this chapter, the multivariate normal distribution, the estimation of its parameters, and the algebra of expectations for vector- and matrix-valued random variables are reviewed. Distributions commonly encountered in multivariate data analysis, the linear model, and the evaluation of multivariate normality and covariance matrices are also reviewed. Finally, tests of locations for one and two groups are discussed. The purpose of this chapter is to familiarize students with multivariate sampling theory, evaluating model assumptions, and analyzing multivariate data for one- and two-group inference problems. [ABSTRACT FROM AUTHOR]

In this chapter, we review the fundamental operations of vectors and matrices useful in statistics. The purpose of the chapter is to introduce basic concepts and formulas essential to the understanding of data representation, data manipulation, model building, and model evaluation in applied multivariate analysis. The field of mathematics that deals with vectors and matrices is called linear algebra; numerous texts have been written about the applications of linear algebra and calculus in statistics. In particular, books by Carroll and Green (1997), Dhrymes (2000), Graybill (1983), Harville (1997), Khuri (1993), Magnus and Neudecker (1999), Schott (1997), and Searle (1982) show how vectors and matrices are useful in applied statistics. Because the results in this chapter are to provide the reader with the basic knowledge of vector spaces and matrix algebra, results are presented without proof. [ABSTRACT FROM AUTHOR]

In this book we present applied multivariate data analysis methods for making inferences regarding the mean and covariance structure of several variables, for modeling relationships among variables, and for exploring data patterns that may exist in one or more dimensions of the data. The methods presented in the book usually involve analysis of data consisting of n observations on p variables and one or more groups. As with univariate data analysis, we assume that the data are a random sample from the population of interest and we usually assume that the underlying probability distribution of the population is the multivariate normal (MVN) distribution. The purpose of this book is to provide students with a broad overview of methods useful in applied multivariate analysis. The presentation integrates theory and practice covering both formal linear multivariate models and exploratory data analysis techniques. [ABSTRACT FROM AUTHOR]

Various methods of constructing confidence sets are introduced in this chapter, along with studies of properties of confidence sets. Throughout this chapter X = (X1, …,Xn) denotes a sample from a population P ∈ P; θ = θ(P) denotes a functional from P to Θ ⊂ Rk for a fixed integer k; and C(X) denotes a confidence set for θ, a set in ßΘ (the class of Borel sets on Θ) depending only on X. We adopt the basic concepts of confidence sets introduced in §2.4.3. In particular, infP∈PP(θ ∈ C(X)) is the confidence coefficient of C(X) and, if the confidence coefficient of C(X) is ≥ 1- α for fixed α ∈ (0, 1), then we say that C(X) has significance level 1-α or C(X) is a level 1 - α confidence set. [ABSTRACT FROM AUTHOR]

Unbiased or asymptotically unbiased estimation plays an important role in point estimation theory. Unbiasedness of point estimators is defined in §2.3.2. In this chapter, we discuss in detail how to derive unbiased estimators and, more importantly, how to find the best unbiased estimators in various situations. Although an unbiased estimator (even the best unbiased estimator if it exists) is not necessarily better than a slightly biased estimator in terms of their mse's (see Exercise 63 in §2.6), unbiased estimators can be used as "building blocks" for the construction of better estimators. Furthermore, one may give up the exact unbiasedness, but cannot give up asymptotic unbiasedness since it is necessary for consistency (see §2.5.2). Properties and the construction of asymptotically unbiased estimators are studied in the last part of this chapter. [ABSTRACT FROM AUTHOR]

A general theory of testing hypotheses is presented in this chapter. Let X be a sample from a population P in P, a family of populations. Based on the observed X, we test a given hypothesis H0 : P ∈ P0 versus H1 : P ∈ P1, where P0 and P1 are two disjoint subsets of P and P0∪P1 = P. Notational conventions and basic concepts (such as two types of errors, significance levels, and sizes) given in Example 2.20 and §2.4.2 are used in this chapter. [ABSTRACT FROM AUTHOR]

Estimation methods studied in this chapter are useful for nonparametric models as well as for parametric models in which the parametric model assumptions might be violated (so that robust estimators are required) or the number of unknown parameters is exceptionally large. Some such methods have been introduced in Chapter 3; for example, the methods that produce UMVUE's in nonparametric models, the U- and V-statistics, the LSE's and BLUE's, the Horvitz-Thompson estimators, and the sample (central) moments. [ABSTRACT FROM AUTHOR]

In this chapter, we consider point estimation methods in parametric models. One such method, the moment method, has been introduced in §3.5.2. It is assumed in this chapter that the sample X is from a population in a parametric family P={Pθ : θ ∈ Θ}, where Θ ⊂ Rk for a fixed integer k ≥ 1. [ABSTRACT FROM AUTHOR]

This chapter discusses some fundamental concepts of mathematical statistics. These concepts are essential for the material in later chapters. [ABSTRACT FROM AUTHOR]

Mathematical statistics relies on probability theory, which in turn is based on measure theory. The present chapter provides some principal concepts and notational conventions of probability theory, and some important results that are useful tools in statistics. A more complete account of probability theory can be found in a standard textbook, for example, Billingsley (1986), Chung (1974), or Loève (1977). The reader is assumed to be familiar with set operations and set functions (mappings) in advanced calculus. [ABSTRACT FROM AUTHOR]

Most of the methods considered in previous chapters require distributional assumptions on either the joint distribution of a subject's repeated measurements or on the marginal distributions at each time point. For example, Chapters 3, 4, 5, and 6 assume that the vectors of repeated measurements have a multivariate normal joint distribution, either with an arbitrary covariance structure (Chapters 3 and 4) or with some type of structured covariance matrix (Chapters 5 and 6). In Chapter 7, the underlying model for the weighted least squares (WLS) approach is the multinomial distribution. Although the distributional assumptions are much weaker for the methods discussed in Chapter 9, one still must make some basic assumptions concerning the marginal distributions at each time point. [ABSTRACT FROM AUTHOR]

In many applications in which the response variable of interest has a continuous distribution, the normal-theory methods described in Chapters 3-6 may be inappropriate. In addition, in situations where the response variable is categorical, the WLS (Chapter 7) and randomization model (Chapter 8) approaches are not always applicable. For example, although the WLS methodology is often a useful approach to the analysis of repeated binary and ordered categorical outcome variables, it can only accommodate categorical explanatory variables. In addition, the WLS methodology requires a sufficiently large sample size for the marginal response functions at each time point within each subpopulation from the multiway cross-classification of the explanatory variables to have an approximately multivariate normal distribution. The randomization model approach is useful only in one-sample problems. Thus, neither of these methodologies for categorical outcomes can be used in the general repeated measurements setting. [ABSTRACT FROM AUTHOR]

In many settings in which the response variable is categorical, the WLS approach described in Chapter 7 cannot be used. For example, the sample size may be too small, the number of time points may be too large, and/or the response variable may have too many possible values. In addition, the methods described in Chapters 3-6 for analyzing continuous response variables are also often inapplicable. For example, the distribution of the outcome variable may be markedly nonnormal. This chapter describes an alternative methodology based on the randomization model and the multiple hypergeometric distribution. [ABSTRACT FROM AUTHOR]

The methods described in previous chapters are useful in the analysis of continuous, normally distributed outcome variables. This chapter introduces and discusses the use of weighted least squares methods for the analysis of repeated categorical outcomes. [ABSTRACT FROM AUTHOR]

Chapters 3, 4, and 5 have considered the situation in which a normally distributed outcome variable is measured repeatedly from each subject or experimental unit. The analysis of such data must account for the dependence among a subject's multiple measurements. "Classical" methodology is based either on multivariate normal models with general covariance structure (Chapters 3 and 4) or on univariate repeated measures ANOVA (Chapter 5). In practice, however, repeated measurements studies are characterized by: variation among experimental units with respect to the number andtiming of observations;missing data;•time-dependent covariates. [ABSTRACT FROM AUTHOR]

The unstructured multivariate approach described in Chapters 3 and 4 makes no assumptions concerning the covariance structure of the vector of repeated measurements from each experimental unit. Although not making covariance structure assumptions may be advantageous, these approaches can result in low power to detect differences of interest. This is due to the fact that a large number of degrees of freedom must be used to estimate the covariance parameters. [ABSTRACT FROM AUTHOR]

Chapter 3 described methods for the analysis of one-sample and two-sample problems based on Hotelling's T2 statistic. This chapter extends this methodology to the situation in which there are more than two groups of experimental units. [ABSTRACT FROM AUTHOR]

The univariate methods described in Chapter 2 reduce the vector of repeated measurements from each experimental unit to a single number. Although this permits the use of simple analysis methods, the resulting loss of information may not be desirable. [ABSTRACT FROM AUTHOR]