Some Tests for the Extended Growth Curve Model and Applications in the Analysis of Clustered Longitudinal Data

In this paper, we study the existence of the uniformly minimum risk equivariant (UMRE) estimators of parameters in a class of normal linear models, which include the normal variance components model, the growth curve model, the extended growth curve model, and the seemingly unrelated regression equations model, and so on. The necessary and sufficient conditions are given for the existence of UMRE estimators of the estimable linear functions of regression coefficients, the covariance matrix V and (tr V ) α , where α > 0 is known, in the models under an affine group of transformations for quadratic losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model, the conclusions given in literature for estimating regression coefficients can be derived by applying the general results in this paper, and the sufficient conditions for non-existence of UMRE estimators of V and tr( V ) are expanded to be necessary and sufficient conditions. In addition, the necessary and sufficient conditions that there exist UMRE estimators of parameters in the variance components model are obtained for the first time.

To prevent biased estimates of intraindividual growth and interindividual variability when working with clustered longitudinal data (e.g., repeated measures nested within students; students nested within schools), individual dependency should be considered. A Monte Carlo study was conducted to examine to what extent two model-based approaches (multilevel latent growth curve model - MLGCM, and maximum model - MM) and one design-based approach (design-based latent growth curve model - D-LGCM) could produce unbiased and efficient parameter estimates of intraindividual growth and interindividual variability given clustered longitudinal data. The solutions of a single-level latent growth curve model (SLGCM) were also provided to demonstrate the consequences of ignoring individual dependency. Design factors considered in the present simulation study were as follows: number of clusters (NC = 10, 30, 50, 100, 150, 200, and 500) and cluster size (CS = 5, 10, and 20). According to our results, when intraindividual growth is of interest, researchers are free to implement MLGCM, MM, or D-LGCM. With regard to interindividual variability, MLGCM and MM were capable of producing accurate parameter estimates and SEs. However, when D-LGCM and SLGCM were applied, parameter estimates of interindividual variability were not comprised exclusively of the variability in individual (e.g., students) growth but instead were the combined variability of individual and cluster (e.g., school) growth, which cannot be interpreted. The take-home message is that D-LGCM does not qualify as an alternative approach to analyzing clustered longitudinal data if interindividual variability is of interest.

SYNOPTIC ABSTRACTThis paper deals with profile analysis in two extended growth curve models. The first is a growth curve model with parallel mean profiles, which has a random effects covariance structure based on a single response variable; the second is a multivariate growth curve model with parallel mean profiles, which has a multivariate random effects covariance structure based on several response variables. For testing “no condition variation” and “level” hypotheses concerning parallel mean profiles of several groups, we obtain the Wald's criteria (Trans. Amer. Math. Soc. 54 (1943) 426–482) and their asymptotic non-null distributions. A numerical example is also given.

The paper contains an overview of recent results in the extended growth curve model, both with fixed and random effects. Explicit results are shown mainly for the model with two components/profiles. A variance components estimability criterion is proposed.

An extended growth curve model with fixed and random effects is considered. Under the assumption of multivariate normality, the maximum likelihood estimators of the fixed effects and the dispersion matrix are determined in a model with random nuisance parameters, both without any assumption on the covariance structure and under the assumption of compound symmetry. For this purpose, rules for differentiation of symmetric matrices are applied. Furthermore, when the experiments are designed in balanced complete blocks, particular symmetric matrices appear in the likelihood equations, allowing closed-form expressions for the estimators. It is also shown that the vector of sufficient statistics for the fixed effects extended growth curve model is also sufficient for the model with random nuisance parameters. The presented results are illustrated using a real data example.

Estimating parameters in extended growth curve models with special covariance structures

This paper presents the conditions for robustness to the nonnormality on\\break three test statistics for a general multivariate linear hypothesis, which were proposed under the normal assumption in a generalized multivariate analysis of variance (GMANOVA) model. The proposed conditions require the cumulants of an unknown population's distribution to vanish in the second terms of the asymptotic expansions for both the mean and variance of the test statistics. With the proposed conditions, the test statistic can be investigated for robustness to nonnormality of the population's distribution. When the conditions are satisfied, the Bartlett correction and the modified Bartlett correction in the normal case improve the quality of the chi-square approximation even under nonnormality.

Score limitation at the top of a scale is commonly termed “ceiling effect.” Ceiling effects can lead to serious artifactual parameter estimates in most data analysis. This study examines the consequences of ceiling effects in longitudinal data analysis and investigates several methods of dealing with ceiling effects through Monte Carlo simulations and empirical data analyses. Data were simulated based on a latent growth curve model with T = 5 occasions. The proportion of the ceiling data [10%–40%] was manipulated by using different thresholds, and estimated parameters were examined for R = 500 replications. The results showed that ceiling effects led to incorrect model selection and biased parameter estimation (shape of the curve and magnitude of the changes) when regular growth curve models were applied. The Tobit growth curve model, instead, performed very well in dealing with ceiling effects in longitudinal data analysis. The Tobit growth curve model was then applied in an empirical cognitive aging study and the results were discussed.

Longitudinal designs for biomarker traits are very appealing for a number of reasons. First, the collection of repeated biomarker measurements over time allows investigators to characterize both the overall and individual trajectories in these measurements. This helps investigators to understand overall effects as well as how individuals vary around these average effects. Distinguishing individual heterogeneity from average trend is impossible without longitudinal data. This distinction is important for understanding individual between-subject in its own right, as well as in developing predictors of disease outcome from longitudinal marker data. There are many statistical issues that are unique to longitudinal data analysis ranging from correctly specifying realistic models which characterize average changes over time and individual variation, accounting for observations that are taken at unequally spaced observation times which may differ between individuals, to accounting for missing data due to occasionally missed measurements or loss to follow-up. When the reason for missing data is not related to the underlying observation process, most longitudinal data analysis methods can be routinely applied. However, complex modelling is necessary when the reason for missing an observation or dropout is directly related to the underlying outcome process [1–2]. Statistical techniques that appropriately analyze continuous and discrete longitudinal data have been discussed by many authors. Diggle, et al. [3] provides a good summary of these techniques for the practitioner, while Fitzmaurice et al. [4] discusses recent advances from a more technical perspective. The three papers in this section present innovative approaches for solving important methodological problems in biomarker studies. Most standard longitudinal data analysis techniques assume that the measurement process is independent of the outcome process. However, when the outcome is related to the underlying observation process, failure to properly account for the observation process can result in bias [5]. Schildcrout et al.[6] proposed an approach for regression modelling for longitudinal binary data where the outcome is measured with probabilities determined by an auxiliary variable that is related, but not identical to the binary outcome. Modelling multivariate longitudinal biomarker data is an increasingly common issue with the advent of an increasing number of important biomarker techniques (e.g., multiple cytokines in a multiplex assay or multiple reproductive hormones that describe complex biological phenomenon). Although various authors have proposed techniques for analyzing multivariate longitudinal data [7–8 and others], little work has focused on estimating multivariate linear mixed models under known constraints in the relationship between the multivariate outcomes. Roy et al [9] proposes such an approach for analyzing multivariate reproductive hormones over the menstrual cycle, where established relationships have solid biological justification. Finally, the collection of longitudinal biomarker data as disease predictors is becoming an increasingly important design in diagnostic medicine. Pfeiffer et al. [10] proposes flexible non-parametric methodology for combining biomarkers that exploits the longitudinal structure in formulating a score that is used for predicting a binary outcome.

Analysis of binary longitudinal data with time-varying effects

To introduce growth curve modeling for longitudinal headache research. Longitudinal data play an important role in the study of headache-related outcomes by allowing researchers to test hypotheses about change over time. However, headache researchers are often unfamiliar with the flexibility and power that growth curves can offer in analyzing longitudinal data. The goals of this paper are to introduce growth curve models within the multilevel modeling framework for analyzing longitudinal headache-related data and to show how these models can be applied in practice. Longitudinal data for the empirical example came from publicly available data from Wave I to Wave IV of the National Longitudinal Study of Adolescent to Adult Health. In total, 5608 individuals were included in the study and multilevel models were fit to examine, for individuals with and without adolescent migraine, longitudinal changes in depression from age 13 to 27 years old. Findings showed that individuals varied in their longitudinal depression trajectories. A cubic time trend best approximated the data with depression increasing through adolescence, decreasing during young adulthood, and then beginning to increase again in adulthood. Further, results also indicated that individuals with adolescent migraine had higher levels of depression throughout the age span compared those without adolescent migraine, but the shape of change did not differ across the groups. Growth curve models offer a flexible alternative to traditional statistical methods and can rigorously evaluate a wide array of headache-related hypotheses.

Intuition suggests that altering the covariance structure of a parametric model for repeated-measures data alters the variances of the model's estimated mean parameters. The purpose of this article is to sharpen such intuition for a family of growth-curve models with differing numbers of random effects for the individual sampling units and with a fixed structure on the mean. For every member of this family, the maximum likelihood (ML) estimator of the fixed effects is identical to the ordinary least squares (OLS) estimator. In addition, simple closed-form ML and restricted maximum likelihood estimators for the variance and covariance parameters exist for every member. As a consequence, closed-form expressions for the estimated variance-covariance matrix of the OLS estimator of the fixed effects also exist for the entire family. We derive explicit relationships between the variance and covariance parameter estimators from different members of the family and thereby extend some familiar results. For example, it is well known that for balanced and complete longitudinal designs the compound symmetry assumption for the covariance structure of the serial observations (i.e., assuming one random effect for each sampling unit) yields a more precise estimate of the slope of the population growth curve than of its intercept. It is also well known that for such designs the diagonal covariance structure assumption of OLS regression (i.e., no random effects for the units) yields a more precise estimate of the intercept than does the compound symmetry assumption, and a less precise estimate of the slope. We extend such relationships as these to growth-curve models whose covariance structures are of increasing linear complexity (i.e., assuming two or more random effects for each of the sampling units). We find that, in general, the variance of the OLS estimator depends strongly on assumed covariance structure. We also find that, specifically, the linear growth-curve assumption (i.e., two random effects for each sampling unit) is a conservative assumption, in that such will not give misleadingly small variance estimates for both intercept and slope even if a more complex covariance structure actually holds. We illustrate these points with a data example. In the conclusion, we extend our results by showing how they may apply to more general families of balanced growth-curve models that allow wider ranges of covariance structures and designs for the random effects, such as when different population subgroups require different covariance structures. We also address several computational issues involved in transformations of growth-curve models to canonical form.

Intuition suggests that altering the covariance structure of a parametric model for repeated-measures data alters the variances of the model's estimated mean parameters. The purpose of this article is to sharpen such intuition for a family of growth-curve models with differing numbers of random effects for the individual sampling units and with a fixed structure on the mean. For every member of this family, the maximum likelihood (ML) estimator of the fixed effects is identical to the ordinary least squares (OLS) estimator. In addition, simple closed-form ML and restricted maximum likelihood estimators for the variance and covariance parameters exist for every member. As a consequence, closed-form expressions for the estimated variance-covariance matrix of the OLS estimator of the fixed effects also exist for the entire family. We derive explicit relationships between the variance and covariance parameter estimators from different members of the family and thereby extend some familiar results. For example, it is well known that for balanced and complete longitudinal designs the compound symmetry assumption for the covariance structure of the serial observations (i.e., assuming one random effect for each sampling unit) yields a more precise estimate of the slope of the population growth curve than of its intercept. It is also well known that for such designs the diagonal covariance structure assumption of OLS regression (i.e., no random effects for the units) yields a more precise estimate of the intercept than does the compound symmetry assumption, and a less precise estimate of the slope. We extend such relationships as these to growth-curve models whose covariance structures are of increasing linear complexity (i.e., assuming two or more random effects for each of the sampling units). We find that, in general, the variance of the OLS estimator depends strongly on assumed covariance structure. We also find that, specifically, the linear growth-curve assumption (i.e., two random effects for each sampling unit) is a conservative assumption, in that such will not give misleadingly small variance estimates for both intercept and slope even if a more complex covariance structure actually holds. We illustrate these points with a data example. In the conclusion, we extend our results by showing how they may apply to more general families of balanced growth-curve models that allow wider ranges of covariance structures and designs for the random effects, such as when different population subgroups require different covariance structures. We also address several computational issues involved in transformations of growth-curve models to canonical form.

Growth curve models (GCMs) play an important role in longitudinal data analysis. However, a very crucial assumption is that the grouping design matrix is assumed to be known, which may not be true in practice. In this paper, we propose a novel mixture model which can handle the problem caused by the unknown grouping matrix. In the meantime, it approximates the true distribution of the response matrix using a mixture of normal distributions. The new model is studied under the serial covariance structure (SCS), which accounts for serial correlation among different observations within the same individual. The maximum likelihood estimation for the proposed model is considered by the use of EM (expectation maximization) algorithm, along with clustering individuals. A data-driven method is used to find a appropriate model. Simulation studies and real data analysis show that the proposed procedure works well in parameter estimation and individual clustering, simultaneously.

When several continuous outcome measures of interest are collected across time in experimental studies, the use of standard statistical procedures, such as multivariate analysis of variance or growth curve modeling, can be properly used to assess treatment effects. However, when data consist of mixed responses (e.g., continuous and ordered categorical [ordinal] responses), traditional modeling approaches are no longer appropriate. The purpose of this article is to illustrate the use of a more suitable modeling procedure when mixed responses are collected in longitudinal intervention studies. Problems with traditional analyses of such data are discussed, as are potential advantages provided by the proposed modeling approach. The application of the multiple-domain latent growth modeling approach with mixed responses is illustrated for experimental designs with data from the SeniorWISE study (McDougall et al., 2010). This multisite randomized trial assessed memory functioning of 265 elderly adults across a 26-month period after receiving either a memory or health promotion training program. The latent growth models illustrated allow one to examine treatment effects on the growth of multiple mixed outcomes while incorporating associations among multiple responses, which allows for better missing data treatment, greater power, and more accurate control of Type I error. The interpretation of parameters of interest and treatment effects is discussed using the SeniorWISE data. Multiple-domain latent growth modeling with mixed responses is a flexible statistical modeling tool that can have substantial benefits for applied researchers. As such, the use of this modeling approach is expected to increase.