Estimating parameters in extended growth curve models with special covariance structures

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.

Equalities of various estimators in the general growth curve model and the restricted growth curve model

The Growth Curve Model (GCM) is a Generalized Multivariate Analysis of Variance (GMANOVA) model especially useful in the analysis of longitudinal data, growth curves as well as other response curves. The model is a natural extension of the classical Multivariate Analysis of Variance (MANOVA) model and among other assumptions, relies on the assumption that the mean for each group can be represented as a polynomial of degree q. The assumption that the mean over time for all groups follows a polynomial of the same degree is not always satisfied, since individuals across the different groups may respond differently. An excellent scenario is when we have clustered longitudinal data, where the response over time can be represented by polynomials of different degrees. In such situations, the natural extension is to use the Extended Growth Curve Model (EGCM), where one can assume different shapes to represent different groups or clusters. In this paper, we formulate hypotheses motivated by real life scenarios involving clustered longitudinal data, and propose tests that are motivated by residuals in the EGCM. We then mathematically derive the tests and evaluate performances using simulations. We provide real data examples as illustrations.

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.

Any given set of asset parameters yields a specific mean–variance optimal tangency portfolio. Yet, when the number of assets is large, there are some general characteristics of optimal portfolios that hold ‘almost surely’. This paper investigates these characteristics. We analytically show that the proportion of assets held short converges to 50% as the number of assets grows. This is a fundamental and robust property of mean–variance optimal portfolios, and it does not depend on the parameter estimation method, the investment horizon, or on a special covariance structure. While it is known that optimal portfolios may all have positive weights in some special situations (e.g. uncorrelated assets), the analysis shows that these cases occupy a zero measure in the parameter space, and therefore should not be expected to be observed empirically. Thus, our analysis offers a general explanation for the empirical finding of many short positions in optimal portfolios.

Three different estimators of the variance parameters in the growth curve model with generalized uniform correlation structure are compared on the basis of mean square error. Since the situation in general depends on specific choice of the structure matrix, we investigate two important special cases.

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.

The COVID-19 pandemic imposed extreme living conditions of social distancing, which triggered negative mental health problems and created challenges in seeking mental health support. Mindfulness-based interventions (MBIs) have been found to enhance wellbeing and mental health by reducing stress and anxiety and improving emotion regulation. Preliminary evidence suggests that online, synchronous MBIs may produce beneficial effects similar to face-to-face programs. However, the effectiveness of such online-MBIs to support mental health in highly stressful times, such as a global pandemic, requires further study. To this end, we investigated the effect of an online 8-week Mindfulness-Based Stress Reduction (MBSR) program on aspects of mental health during the first wave of the COVID-19 pandemic. Participants (N=92) who expressed interest in discounted online-MBSR programs were recruited for the study. The division into experimental and control groups was based on actual enrollment to the courses. Those who enrolled in a program were assigned to the experimental condition and those who decided not to enroll served as controls. Participants were assessed pre-intervention, post-intervention, and 1-month post-intervention for levels of mindfulness, perceived stress, anxiety, emotion regulation, and intolerance of uncertainty. Differences between the groups were tested using the general linear mixed effects model (GLMM) and Individual Growth Curve Models (IGCM) in intent to treat analysis. The findings indicated that, relative to the control group, MBSR improved mindfulness abilities (p <0.001), decreased anxiety (p <0.001), and stress (p <0.001) and increased emotion regulation (p <0.001). These effects were found to persist 1 month after the end of the program, despite the increased governmental public-health restrictions due to COVID-19 at that time. The ability to tolerate uncertainty, a central characteristic of the pandemic, was not found to be affected by the program. A mediation analysis revealed that the effect of the intervention on mental health improvement was partially mediated by the improvement in emotion regulation. Overall, the findings provide positive evidence for the feasibility of an online-MBSR program to support the mental health of individuals from the general population through the mediation of emotion regulation in challenging times, such as a global pandemic.

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.

The coding of time in growth curve models has important implications for the interpretation of the resulting model that are sometimes not transparent. The authors develop a general framework that includes predictors of growth curve components to illustrate how parameter estimates and their standard errors are exactly determined as a function of receding time in growth curve models. Linear and quadratic growth model examples are provided, and the interpretation of estimates given a particular coding of time is illustrated. How and why the precision and statistical power of predictors of lower order growth curve components changes over time is illustrated and discussed. Recommendations include coding time to produce readily interpretable estimates and graphing lower order effects across time with appropriate confidence intervals to help illustrate and understand the growth process.

Multivariate analysis is difficult when there are missing observations in the response vectors. Kleinbaum (1973) proposed a Wald statistic useful in the analysis of incomplete multivariate data. SUBROUTINE C0EF calculates the estimated parameter matrix g in the generalization of the Potthoff-Roy (1964) growth curve model proposed by Kleinbaum (1973). SUBROUTINE WALD calculates the Wald statistic for hypotheses of the form Hn: H 5 D = 0 as proposed by Kleinbaum (1973).

It has been shown in the literature that the data of technological substitutions exhibit a strong correlation across different time periods, that is, a strong serial correlation. Significant improvement in predicting such substitutions has been achieved by incorporating serial correlation and power transformation parameters into four growth curve models. The modified models, or data-based transformed models, however, break down when the number of time points is small. This article proposes a generalized growth curve model for forecasting technological substitutions with concurrent short time series. The model combines the concepts of power transformations and repeated measurements with a common serial covariance structure. Concurrent time series for several cases provide the repeated measurement requirement of the model. Improvement in forecasting accuracy by using this model is demonstrated with a set of telephone switching data.

Kleinbaum (1973) developed a generalized growth curve model for analyzing incomplete longitudinal data. In this paper the small sample properties of several related test statistics are investigated via Monte Carlo techniques. The covariance matrix is estimated by each of three non-iterative methods. The null and non-null distributions of these test statistics are examined.

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.