Growth Curve Models and Statistical Diagnostics

Statistical diagnostics is one of the most useful techniques in statistical science. The aim of diagnostics is to detect outliers that deviate from the postulated model, to identify influential observations that have large effects on the statistical inference drawn from the postulated model, and to validate the chosen statistical model. The main theme of this book is to comprehensively explore multivariate diagnostic techniques, which are specifically suitable for diagnosing the adequacy of multivariate models, with particular emphasis on the application to growth curve models. The approaches employed are case-deletion and local influence within the likelihood and Bayesian frameworks. We give a brief introduction to statistical diagnostics in Section 1.1 and the associated multivariate techniques in Section 1.2. Section 1.3 is devoted to a brief review of growth curve models as well as model selection criteria with respect to covariance structures. In Section 1.4, the main approaches and results in this book on statistical diagnostics for growth curve models are outlined in a summarized form. Some preparatory materials related to matrix derivatives and matrix-variate distributions are given in Section 1.5 for later use.

In this chapter the fundamental concepts of the growth curve model (GCM) are introduced and several commonly encountered forms of the GCM are described through a variety of practical examples in biology, agriculture, and medical research. Some basic statistical inference of the GCM, such as generalized least square estimate (GLSEs) and the admissibility of estimates on linear combinations of regression coefficients, are discussed in detail. It is shown that the GLSE of the regression coefficient is also the best linear unbiased estimate (BLUE) in the sense of the matrix loss function. In addition, the necessary and sufficient conditions of admissible estimates on linear combinations of regression coefficients are studied. The main theme of this chapter is to demonstrate the use of the GCM in practice and to comprehensively introduce the theory of generalized least square estimation as well. Maximum likelihood estimate (MLE) and restricted maximum likelihood (REML) estimate will be discussed in Chapter 3.

In this chapter maximum likelihood estimates (MLEs) of the parameters in growth curve models are discussed. Also expectations and variancecovariance matrices of the estimates are considered. In general, the MLE of the regression coefficient is different from the generalized least square estimate (GLSE) discussed in Chapter 2, because the former is a nonlinear function of the response variable while the latter is linear. There is indeed a special case, however, in which the MLE is completely identical to the GLSE, making the statistical inferences based on MLEs in growth curve models more easily analytical and tractable. This special case is nothing but Rao’s simple covariance structure (SCS), in which the dispersion component E consists of two orthogonal parts. Many useful covariance structures are included in the SCS as special cases. Among those, two examples are the uniform covariance structure and the random regression coefficient structure, which are commonly encountered in correlation analysis and longitudinal studise. With the assumption of the SCS restricted maximum likelihood (REML) estimate is studied together with the MLE technique in this chapter.

This chapter is devoted to discussion of statistical diagnostics for the growth curve model (GCM), based on the case-deletion method or global influence approach. Under Rao’s simple covariance structure (SCS) discussed in Section 3.2 of Chapter 3 and unstructured covariance (UC), two of the most commonly encountered covariance structures for growth analysis, the multiple individual deletion model (MIDM) and the mean shift regression model (MSRM) are studied, respectively. These can be employed to detect multiple discordant outliers as well as to assess effects of a set of observations on growth regression fittings. Based on the generalized Cook-type distance and the confidence ellipsoid’s volume of the regression coefficient in the GCM, several diagnostic statistics are proposed to measure the influence of a subset of observations. Also, the influence of observations on a linear function of the regression coefficient is measured. For illustration, the practical examples studied in Chapter 2 are reanalyzed using the techniques of discordant outlier detection and influential observation identification.

In this chapter, the local influence approach proposed by Cook (1986) is applied to diagnostics of growth curve models (GCM) with Rao’s simple covariance structure (SCS) and unstructured covariance (UC), respectively. Under these two covariance structures, we study the observed information matrix and the Hessian matrix for the parameters in the GCM; the Hessian matrix serves as a basis for likelihood-based local influence assessment, as pointed out by Cook (1986). As an ancillary result, the Hessian matrix is invariant under a one-to-one measurable transformation on parameters. For illustration, the practical data sets addressed in previous chapters are analyzed by using the local influence approach, which is useful in practice and not overwhelming in its computation.

In this chapter we discuss the influence of a subset of observations on growth curve models from the Bayesian point of view. With a noninformative prior, the posterior distributions of parameters in growth curve models (GCMs) with Rao’s simple covariance structure SCS and unstructured covariance UC are obtained analytically, respectively. A Baysian entropy, namely, Kullback—Leibler divergence (KLD), as mentioned in Subsection 4.1.2 in Chapter 4, is used to measure the change of the posterior distributions when a subset of observations is deleted from the data. Also, the practical data studied in the pervious chapters are reanalyzed using the approaches addressed in this chapter.

This chapter is devoted to the discussion of local influence procedures in growth curve models (GCMs) with Rao’s simple covariance structure (SCS) and unstructured covariance (UC), from the Bayesian point of view. The fundamental idea behind this procedure is to replace likelihood displacement in likelihood-based local influence method (see Subsection 5.1.1 in Chapter 5) with a Bayesian entropy, for example, the Kullback—Leibler divergence (KLD) addressed in Chapter 6. With SCS and UC, the two commonly used covariance structures, Bayesian Hessian matrices of the regression coefficient and the dispersion component in GCMs are studied under an abstract perturbation scheme, which serves as a basis of the Bayesian local influence analysis in these models. Also, some new properties of the Bayesian Hessian matrix are obtained as ancillary results. Similar to likelihood-based local influence analysis addressed in Chapter 5, a covariance-weighted perturbation scheme is employed to demonstrate the use of this procedure. To illustrate, the practical data sets discussed in previous chapters are reanalyzed using Bayesian local influence procedures. This analysis reveals that the Bayesian local in fluence method is a practical diagnostic approach.