A note on a manova model applied to problems in growth curve

Fuzzy Clusterwise Growth Curve Models via Generalized Estimating Equations: An Application to the Antisocial Behavior of Children

The growth curve model has been a useful tool for the analysis of repeated measures data. However, it is designed for an aggregate-sample analysis based on the assumption that the entire sample of respondents are from a single homogenous population. Thus, this method may not be suitable when heterogeneous subgroups exist in the population with qualitatively distinct patterns of trajectories. In this paper, the growth curve model is generalized to a fuzzy clustering framework, which explicitly accounts for such group-level heterogeneity in trajectories of change over time. Moreover, the proposed method estimates parameters based on generalized estimating equations thereby relaxing the assumption of correct specification of the population covariance structure among repeated responses. The performance of the proposed method in recovering parameters and the number of clusters is investigated based on two Monte Carlo analyses involving synthetic data. In addition, the empirical usefulness of the proposed method is illustrated by an application concerning the antisocial behavior of a sample of children.

Non-linear random effects model for multivariate responses with missing data

The use of random-effects models for the analysis of longitudinal data with missing responses has been discussed by several authors. In this paper, we extend the non-linear random-effects model for a single response to the case of multiple responses, allowing for arbitrary patterns of observed and missing data. Parameters for this model are estimated via the EM algorithm and by the first-order approximation available in SAS Proc NLMIXED. The set of equations for this estimation procedure is derived and these are appropriately modified to deal with missing data. The methodology is illustrated with an example using data coming from a study involving 161 pregnant women presenting to a private obstetrics clinic in Santiago, Chile.

Effect Size and Power in Assessing Moderating Effects of Categorical Variables Using Multiple Regression: A 30-Year Review

The authors conducted a 30-year review (1969-1998) of the size of moderating effects of categorical variables as assessed using multiple regression. The median observed effect size (f(2)) is only .002, but 72% of the moderator tests reviewed had power of .80 or greater to detect a targeted effect conventionally defined as small. Results suggest the need to minimize the influence of artifacts that produce a downward bias in the observed effect size and put into question the use of conventional definitions of moderating effect sizes. As long as an effect has a meaningful impact, the authors advise researchers to conduct a power analysis and plan future research designs on the basis of smaller and more realistic targeted effect sizes.

Relating the classical covariance adjustment techniques of multivariate growth curve models to modern univariate mixed effects models

The relationship between the modern univariate mixed model for analyzing longitudinal data, popularized by Laird and Ware (1982, Biometrics 38, 963-974), and its predecessor, the classical multivariate growth curve model, summarized by Grizzle and Allen (1969, Biometrics 25, 357-381), has never been clearly established. Here, the link between the two methodologies is derived, and balanced polynomial and cosinor examples cited in the literature are analyzed with both approaches. Relating the two models demonstrates that classical covariance adjustment for higher-order terms is analogous to including them as random effects in the mixed model. The polynomial example clearly illustrates the relationship between the methodologies and shows their equivalence when all matrices are properly defined. The cosinor example demonstrates how results from each method may differ when the total variance-covariance matrix is positive definite, but that the between-subjects component of that matrix is not so constrained by the growth curve approach. Additionally, advocates of each approach tend to consider different covariance structures. Modern mixed model analysts consider only those terms in a model's expectation (or linear combinations), and preferably the most parsimonious subset, as candidates for random effects. Classical growth curve analysts automatically consider all terms in a model's expectation as random effects and then investigate whether "covariance adjusting" for higher-order terms improves the model. We apply mixed model techniques to cosinor analyses of a large, unbalanced data set to demonstrate the relevance of classical covariance structures that were previously conceived for use only with completely balanced data.

Analysis of drug dissolution data

Drug absorption in the human body depends on the dissolution rate of the drug. Suitable dissolution characteristics are important to ensure that the drug will achieve the desired therapeutic effects. To assess the similarity of dissolution rates of several drug lots, we apply a general growth curve model with different covariance structures. The Box-Cox power transformation and the naive log transformation are applied to a function of the dissolution rate. The predictive sample-reuse, or cross-validation, method is employed in selecting an appropriate model with best predictive accuracy. A testing procedure for examining the similarity among the drug lots is also conducted. A partially Bayesian approach is used for the assessment of dissolution equivalence.

Satisfying a Simplex Structure is Simpler Than it Should Be: A Latent Curve Analysis Revisit

This article is concerned with the utility of the structural equation modeling (SEM) methodology for studying change. Two broad classes of models are considered, individual and group change models. The relationship between the constant rate of change and simplex models, as popular representatives of either class, is examined. Both models are shown to be special cases of the comprehensive latent curve analysis (Meredith & Tisak, 1990). Sensitivity of its models to differences in individual growth curves is demonstrated on data from Rogosa and Willett (1985a). Benefits of studying longitudinal change using SEM and related issues of model choice are discussed.

PC program extending the Potthoff-Roy longitudinal data analysis model to allow missing data: Kleinbaum's method

Potthoff and Roy (Biometrika, 51 (1964) 313-326) generalized the multivariate analysis of variance model into a form that is especially useful for the study of longitudinal growth curve data. Applications of this method have, however, been limited by the requirement that each case in the sample be measured at the same set of time points, i.e. there can be no missing data. In this paper we describe, illustrate, and make available a user-friendly, interactive PC program implementing Kleinbaum's (J Mult Anal, 3 (1973) 117-124) extension of the Potthoff-Roy model to allow incomplete measurement sequences. These missing data are permitted to arise either randomly or by design as in mixed longitudinal studies.

Implementation of exact and approximate randomization tests for polynomial growth curves

Two stand-alone, menu-driven PC programs, written in GAUSS386i, which compare groups of growth curves in a completely randomized design using either (a) exact or (b) approximate randomization tests, are described, illustrated, and made available to interested readers. The programs accommodate missing data in the context of studies planned to have common times of measurement, but where some of the measurement sequences are incomplete. The measurement whose growth is being monitored need not have a Gaussian distribution. We consider the hypothesis that the mean growth curves in G groups are the same; and either compute the exact P value (exact test), or estimate, and provide a confidence interval for, the P value (approximate test).

Partial and conditional maximum likelihood for variance-component estimation

SUMMARY

Patterson and Thompson's idea of 'error contrasts' (or restricted maximum likelihood) (1971) was extended to multiple sets of linear contrasts for variance component estimtion. The error contrasts were established in such a way that only errors are retained in the model. The error variance was then estimated by maximizing the likelihood function obtained from the error contrasts. More sets of linear contrasts were then progressively established such that each set of linear contrasts contains only one class of random effects and the errors. A likelihood function was constructed and maximized for each variance of random effects given the error variance held at its estimated value. The likelihood function for estimating the covariance component between two classes of random effects was established such that all other random effects are treated as fixed effects. The likelihood function was then maximized with respect to the covariance given the two variance components fixed at their estimated values. The multidimensional optimization problem in the traditional restricted maximum-likelihood problem was then turned into several one-dimensional optimization problems by using this technique. Inasmuch as the error variance was estimated using a partial likelihood function and the other variance components are estimated using likelihood functions conditional on the estimated error variance, the method is referred to as partial and conditional maximum likelihood (PCML). ZUSAMMENFASSUNG: Partielle und bedingte Maximum Likelihood zur Schätzung von Varianzkomponenten Die Patterson und Thompson Vorstellungen von 'Fehlerkontrasten' (1971) (oder beschränkte maximale Likelihood) wurde auf multiple Gruppen linearer Kontraste für Varianzkomponenten- schätzung ausgedehnt. Die Fehlerkontraste erfolgen in der Form, daß nur Fehler im Modell verbleiben. Die Fehlervarianz wurde dann durch Maximierung der Likelihood Funktion von Fehlerkontrasten geschätzt. Weitere Gruppen linearer Kontraste wurden nacheinander etabliert dergestalt, daß jede Gruppe linearer Kontraste nur eine Klasse zufälliger Wirkungen und die Fehler enthält. Eine Likelihood Funktion wurde konstruiert und für jede Varianz von Zufallsgrößen maximiert unter der Voraussetzung, daß die Fehlervarianz auf ihrem geschätzten Wert verbleibt. Die Likelihood Funktion zur Schätzung der Ko-Varianzkomponenten zwischen zwei Klassen zufälliger Wirkungen wurde in der Form aufgestellt, daß alle anderen Zufallswirkungen als fixe behandelt werden. Die Likelihood Funktion wurde maximiert im Hinblick auf Ko-Varianz bei gegebenen geschätzten Varianzkomponenten. Das multidimensionale Optimierungsproblem der traditionellen restringierten Maximum Likelihood wurde auf diese Weise in ein eindimensionales Optimierungsproblem verwandelt. Nachdem die Fehlervarianz aus der partiellen Likelihood Funktion und die anderen Varianzkomponenten unter Verwendung der bedingten Likelihood Funktionen geschätzt worden waren, wurde die Methode als partielle und bedingte Maximum Likelihood (pcml) bezeichnet.

A random coefficient growth curve analysis of mental development in low-birth-weight infants

In many medical studies, longitudinal data are collected on each of a sample of patients. The objectives of such studies often are: to estimate and test bivariate or multivariate relationships within each of several groups of patients from these repeated measures data; to compare these relationships among groups; and to test for the effects of baseline covariates on the relationships. This paper illustrates the use of statistical methods for growth curve analysis recently proposed by Vonesh and Carter for achieving these goals by relating a measure of preschool cognitive development to age in four race by sex groups of low-birth-weight infants. Significant declines in Bayley's Mental Development Index (MDI) with increasing age were found in all groups. Birth-weight did not significantly influence the rate of decline but did influence the overall level of performance. Even so, in the group most comparable to Bayley's normative population, predicted MDI was near the norm even for extremely low-birth-weight infants (that is, 1000 grams). Although there is some risk of mental deficit associated with prematurity, eventual developmental delays in low-birth-weight infants frequently are acquired with age. The rate of decline in MDI was significantly associated with race and mother's education. Assumptions required for the valid application of these methods are discussed and tested in the setting of this applied problem. The assumptions appeared valid in this application. We conclude with a brief discussion of available alternatives when the assumptions are violated and point to areas for future research.