Dynamic conditionally linear mixed models for longitudinal data

We develop a new class of models, dynamic conditionally linear mixed models, for longitudinal data by decomposing the within-subject covariance matrix using a special Cholesky decomposition. Here 'dynamic' means using past responses as covariates and 'conditional linearity' means that parameters entering the model linearly may be random, but nonlinear parameters are nonrandom. This setup offers several advantages and is surprisingly similar to models obtained from the first-order linearization method applied to nonlinear mixed models. First, it allows for flexible and computationally tractable models that include a wide array of covariance structures; these structures may depend on covariates and hence may differ across subjects. This class of models includes, e.g., all standard linear mixed models, antedependence models, and Vonesh-Carter models. Second, it guarantees the fitted marginal covariance matrix of the data is positive definite. We develop methods for Bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited.     

Quantitative trait locus analysis of longitudinal quantitative trait data in complex pedigrees

There is currently considerable interest in genetic analysis of quantitative traits such as blood pressure and body mass index. Despite the fact that these traits change throughout life they are commonly analyzed only at a single time point. The genetic basis of such traits can be better understood by collecting and effectively analyzing longitudinal data. Analyses of these data are complicated by the need to incorporate information from complex pedigree structures and genetic markers. We propose conducting longitudinal quantitative trait locus (QTL) analyses on such data sets by using a flexible random regression estimation technique. The relationship between genetic effects at different ages is efficiently modeled using covariance functions (CFs). Using simulated data we show that the change in genetic effects over time can be well characterized using CFs and that including parameters to model the change in effect with age can provide substantial increases in power to detect QTL compared with repeated measure or univariate techniques. The asymptotic distributions of the methods used are investigated and methods for overcoming the practical difficulties in fitting CFs are discussed. The CF-based techniques should allow efficient multivariate analyses of many data sets in human and natural population genetics.     

QUANTITATIVE GENETICS OF MORPHOLOGICAL DIFFERENTIATION IN PEROMYSCUS. I. TESTS OF THE HOMOGENEITY OF GENETIC COVARIANCE STRUCTURE AMONG SPECIES AND SUBSPECIES

Phenotypic and additive genetic covariance matrices were estimated for 15 morphometric characters in three species and subspecies of Peromyscus. Univariate and multivariate ANOVAs indicate these groups are highly diverged in all characters, P. leucopus having the largest body size, P. maniculatus bairdii the smallest, and P. maniculatus nebrascensis being intermediate. Comparing the structure of P and G within each taxon revealed significant similarities in all three cases. This proportionality was strong enough to justify using P in the place of G to analyze evolutionary processes using quantitative genetic models when G can not be estimated, as in fossil material. However, the similarity between genetic and phenotypic covariance structures is sufficiently low that estimates of the genetic parameters should be used when possible. The additive genetic covariance matrices were compared to examine the assumption that they remain constant during evolution, an assumption which underlies many applications of quantitative-genetic models. While matrix permutation tests indicated statistically significant proportionality between the genetic covariance structures of the two P. maniculatus subspecies, there is no evidence of significant genetic structural similarity between species. This result suggests that the assumption of constant genetic covariance structure may be valid only within species. (It does not, however, necessarily imply a causal relationship between speciation and heterogeneity of genetic covariance structures.) The low matrix correlation for the two P. maniculatus subspecies' genetic covariance matrices indicates G may not be functionally constant, even within species. The lack of similarity observed here may be due partly to sampling variation.     

Accurate critical constants for the one-sided approximate likelihood ratio test of a normal mean vector when the covariance matrix is estimated

Tang, Gnecco, and Geller (1989, Biometrika 76, 577-583) proposed an approximate likelihood ratio (ALR) test of the null hypothesis that a normal mean vector equals a null vector against the alternative that all of its components are nonnegative with at least one strictly positive. This test is useful for comparing a treatment group with a control group on multiple endpoints, and the data from the two groups are assumed to follow multivariate normal distributions with different mean vectors and a common covariance matrix (the homoscedastic case). Tang et al. derived the test statistic and its null distribution assuming a known covariance matrix. In practice, when the covariance matrix is estimated, the critical constants tabulated by Tang et al. result in a highly liberal test. To deal with this problem, we derive an accurate small-sample approximation to the null distribution of the ALR test statistic by using the moment matching method. The proposed approximation is then extended to the heteroscedastic case. The accuracy of both the approximations is verified by simulations. A real data example is given to illustrate the use of the approximations.     

Bayesian state space models for inferring and predicting temporal gene expression profiles

Prediction of gene dynamic behavior is a challenging and important problem in genomic research while estimating the temporal correlations and non-stationarity are the keys in this process. Unfortunately, most existing techniques used for the inclusion of the temporal correlations treat the time course as evenly distributed time intervals and use stationary models with time-invariant settings. This is an assumption that is often violated in microarray time course data since the time course expression data are at unequal time points, where the difference in sampling times varies from minutes to days. Furthermore, the unevenly spaced short time courses with sudden changes make the prediction of genetic dynamics difficult. In this paper, we develop two types of Bayesian state space models to tackle this challenge for inferring and predicting the gene expression profiles associated with diseases. In the univariate time-varying Bayesian state space models we treat both the stochastic transition matrix and the observation matrix time-variant with linear setting and point out that this can easily be extended to nonlinear setting. In the multivariate Bayesian state space model we include temporal correlation structures in the covariance matrix estimations. In both models, the unevenly spaced short time courses with unseen time points are treated as hidden state variables. Bayesian approaches with various prior and hyper-prior models with MCMC algorithms are used to estimate the model parameters and hidden variables. We apply our models to multiple tissue polygenetic affymetrix data sets. Results show that the predictions of the genomic dynamic behavior can be well captured by the proposed models.     

Analysis of longitudinal data with unmeasured confounders.

Confounding in longitudinal or clustered data creates special problems and opportunities because the relationship between the confounder and covariate of interest may differ across and within individuals or clusters. A well-known example of such confounding in longitudinal data is the presence of cohort and period effects in models of aging in epidemiologic research. We first formulate a data-generating model with confounding and derive the distribution of the response variable unconditional on the confounder. We then examine the properties of the regression coefficient for some analytic approaches when the confounder is omitted from the fitted model. The expected value of the regression coefficient differs in across- and within-individual regression. In the multivariate case, within- and between-individual information is combined and weighted according to the assumed covariance structure. We assume compound symmetry in the fitted covariance matrix and derive the variance, bias, and mean squared error of the slope estimate as a function of the fitted within-individual correlation. We find that even in this simplest multivariate case, the trade-off between bias and variance depends on a large number of parameters. It is generally preferable to fit correlations somewhat above the true correlation to minimize the effect of between-individual confounders or cohort effects. Period effects can lead to situations where it is advantageous to fit correlations that are below the true correlation. The results highlight the trade-offs inherent in the choice of method for analysis of longitudinal data, and show that an appropriate choice can be made only after determining whether within- or between-individual confounding is the major concern.     

Bayesian discrimination with longitudinal data

The motivation for the methodological development is a double-blind clinical trial designed to estimate the effect of regular injection of growth hormone, with the purpose of identifying growth hormone abusers in sport. The data formed part of a multicentre investigation jointly sponsored by the European Union and the International Olympic Committee. The data are such that for each individual there is a matrix of marker variables by time point (nominally 8 markers at each of 7 time points). Data arise out of a double-blind trial in which individuals are given growth hormone at one of two dose levels or placebo daily for 28 days. Monitoring by means of blood samples is at 0, 21, 28, 30, 33, 42 and 84 days. We give a new method of Bayesian discrimination for multivariate longitudinal data. This involves a Kronecker product covariance structure for the time by measurements (markers) data on each individual. This structure is estimated by an empirical Bayes approach, using an ECM algorithm, within a Bayesian Gaussian discrimination model. In future one may have markers for an individual at one or more time points. The method gives probabilities that an individual is on placebo or on one of the two dose regimes.     

Local estimation of age-dependent variance components from longitudinal twin data

In the study of longitudinal twin and family data, interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function. Various parametric models for covariance structures have been proposed but, e.g., in studies of children where growth spurts occur at various ages, it is difficult to a priori determine an appropriate parametric model for the covariance structure. In particular, there is a general lack of the visualization procedures, such as lowess, that are invaluable in the initial stages of constructing a parametric model for a mean function. Here we use kernel smoothing to modify a cross-sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data. The methods are proposed to be exploratory as an aid to parametric modeling rather than inferential, although approximate asymptotic standard errors are derived in the Appendix.     

A marginalized conditional linear model for longitudinal binary data when informative dropout occurs in continuous time

Within the pattern-mixture modeling framework for informative dropout, conditional linear models (CLMs) are a useful approach to deal with dropout that can occur at any point in continuous time (not just at observation times). However, in contrast with selection models, inferences about marginal covariate effects in CLMs are not readily available if nonidentity links are used in the mean structures. In this article, we propose a CLM for long series of longitudinal binary data with marginal covariate effects directly specified. The association between the binary responses and the dropout time is taken into account by modeling the conditional mean of the binary response as well as the dependence between the binary responses given the dropout time. Specifically, parameters in both the conditional mean and dependence models are assumed to be linear or quadratic functions of the dropout time; and the continuous dropout time distribution is left completely unspecified. Inference is fully Bayesian. We illustrate the proposed model using data from a longitudinal study of depression in HIV-infected women, where the strategy of sensitivity analysis based on the extrapolation method is also demonstrated.     

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.     

Bayesian analysis of matrix normal graphical models

We present Bayesian analyses of matrix-variate normal data with conditional independencies induced by graphical model structuring of the characterizing covariance matrix parameters. This framework of matrix normal graphical models includes prior specifications, posterior computation using Markov chain Monte Carlo methods, evaluation of graphical model uncertainty and model structure search. Extensions to matrix-variate time series embed matrix normal graphs in dynamic models. Examples highlight questions of graphical model uncertainty, search and comparison in matrix data contexts. These models may be applied in a number of areas of multivariate analysis, time series and also spatial modelling.     

Analysis of infant growth curves using multivariate adaptive splines

In this paper, we study the effect of cocaine use by a pregnant woman on the growth of her infant after birth. Using a data set from a retrospective study, we found that cocaine use was a marginally significant contributor to the infant growth as measured by bodyweight. From a statistical point of view, the data represent a common, though complex, structure that has received little attention in the statistical literature. To analyze these data, we adopt and further enhance an approach developed recently called MASAL (multivariate adaptive splines for the analysis of longitudinal data). In addition to the fitting of growth curves, we demonstrate particularly how to explore and estimate the underlying covariance structures for the longitudinal data that were collected from a rather irregular schedule.     

Stability of exploratory multivariate data modeling in longitudinal data

Exploratory data-driven multivariate analysis provides a means of investigating underlying structure in complex data. To explore the stability of multivariate data modeling, we have applied a common method of multivariate modeling (factor analysis) to the Genetic Analysis Workshop 13 (GAW13) Framingham Heart Study data. Given the longitudinal nature of the data, multivariate models were generated independently for a number of different time points (corresponding to cross-sectional clinic visits for the two cohorts), and compared. In addition, each multivariate model was used to generate factor scores, which were then used as a quantitative trait in variance component-based linkage analysis to investigate the stability of linkage signals over time. We found surprisingly good correlation between factor models (i.e., predicted factor structures), maximum LOD scores, and locations of maximum LOD scores (0.81< rho <0.94 for factor scores; rho >0.99 for peak locations; and 0.67< rho <0.93 for peak LOD scores). Furthermore, the regions implicated by linkage analysis with these factor scores have also been observed in other studies, further validating our exploratory modeling.     

Random effects modeling of multiple binomial responses using the multivariate binomial logit-normal distribution

The multivariate binomial logit-normal distribution is a mixture distribution for which, (i) conditional on a set of success probabilities and sample size indices, a vector of counts is independent binomial variates, and (ii) the vector of logits of the parameters has a multivariate normal distribution. We use this distribution to model multivariate binomial-type responses using a vector of random effects. The vector of logits of parameters has a mean that is a linear function of explanatory variables and has an unspecified or partly specified covariance matrix. The model generalizes and provides greater flexibility than the univariate model that uses a normal random effect to account for positive correlations in clustered data. The multivariate model is useful when different elements of the response vector refer to different characteristics, each of which may naturally have its own random effect. It is also useful for repeated binary measurement of a single response when there is a nonexchangeable association structure, such as one often expects with longitudinal data or when negative association exists for at least one pair of responses. We apply the model to an influenza study with repeated responses in which some pairs are negatively associated and to a developmental toxicity study with continuation-ratio logits applied to an ordinal response with clustered observations.     

The genetic variance but not the genetic covariance of life‐history traits changes towards the north in a time‐constrained insect

Seasonal time constraints are usually stronger at higher than lower latitudes and can exert strong selection on life‐history traits and the correlations among these traits. To predict the response of life‐history traits to environmental change along a latitudinal gradient, information must be obtained about genetic variance in traits and also genetic correlation between traits, that is the genetic variance‐covariance matrix, G . Here, we estimated G for key life‐history traits in an obligate univoltine damselfly that faces seasonal time constraints. We exposed populations to simulated native temperatures and photoperiods and common garden environmental conditions in a laboratory set‐up. Despite differences in genetic variance in these traits between populations (lower variance at northern latitudes), there was no evidence for latitude‐specific covariance of the life‐history traits. At simulated native conditions, all populations showed strong genetic and phenotypic correlations between traits that shaped growth and development. The variance–covariance matrix changed considerably when populations were exposed to common garden conditions compared with the simulated natural conditions, showing the importance of environmentally induced changes in multivariate genetic structure. Our results highlight the importance of estimating variance–covariance matrixes in environments that mimic selection pressures and not only trait variances or mean trait values in common garden conditions for understanding the trait evolution across populations and environments.     

A modified pseudolikelihood approach for analysis of longitudinal data

We consider the analysis of longitudinal data when the covariance function is modeled by additional parameters to the mean parameters. In general, inconsistent estimators of the covariance (variance/correlation) parameters will be produced when the "working" correlation matrix is misspecified, which may result in great loss of efficiency of the mean parameter estimators (albeit the consistency is preserved). We consider using different "working" correlation models for the variance and the mean parameters. In particular, we find that an independence working model should be used for estimating the variance parameters to ensure their consistency in case the correlation structure is misspecified. The designated "working" correlation matrices should be used for estimating the mean and the correlation parameters to attain high efficiency for estimating the mean parameters. Simulation studies indicate that the proposed algorithm performs very well. We also applied different estimation procedures to a data set from a clinical trial for illustration.     

The use of linear mixed models to estimate variance components from data on twin pairs by maximum likelihood.

It is shown that maximum likelihood estimation of variance components from twin data can be parameterized in the framework of linear mixed models. Standard statistical packages can be used to analyze univariate or multivariate data for simple models such as the ACE and CE models. Furthermore, specialized variance component estimation software that can handle pedigree data and user-defined covariance structures can be used to analyze multivariate data for simple and complex models, including those where dominance and/or QTL effects are fitted. The linear mixed model framework is particularly useful for analyzing multiple traits in extended (twin) families with a large number of random effects.     

Selection criterion of work matrix as a function of limiting estimates of the covariance matrix of correlated data in GEE

The modeling of generalized estimating equations used in the analysis of longitudinal data whether in continuous or discrete variables, necessarily requires the prior specification of a correlation matrix in its iterative process in order to obtain the estimates of the regression parameters. Such a matrix is called working correlation matrix and its incorrect specification produces less efficient estimates for the model parameters. Due to this fact, this study aims to propose a selection criterion of working correlation matrix based on the covariance matrix estimates of correlated responses resulting from the limiting values of the association parameter estimates. For validation of the criterion, we used simulation studies considering normal and binary correlated responses. Compared to some criteria in the literature, it was concluded that the proposed criterion resulted in a better performance when the correlation structure for exchangeable working correlation matrix was considered as true structure in the simulated samples and for large samples, the proposed criterion showed similar behavior to the other criteria, resulting in higher success rates.     

The stability of P in coral reef fishes

The constancy of phenotypic variation and covariation is an assumption that underlies most recent investigations of past selective regimes and attempts to predict future responses to selection. Few studies have tested this assumption of constancy despite good reasons to expect that the pattern of phenotypic variation and covariation may vary in space and time. We compared phenotypic variance-covariance matrices (P) estimated for populations of six species of distantly related coral reef fishes sampled at two locations on Australia's Great Barrier Reef separated by more than 1000 km. The intraspecific similarity between these matrices was estimated using two methods: matrix correlation and common principal component analysis. Although there was no evidence of equality between pairs of P, both statistical approaches indicated a high degree of similarity in morphology between the two populations for each species. In general, the hierarchical decomposition of the variance-covariance structure of these populations indicated that all principal components of phenotypic variance-covariance were shared but that they differed in the degree of variation associated with each of these components. The consistency of this pattern is remarkable given the diversity of morphologies and life histories encompassed by these species. Although some phenotypic instability was indicated, these results were consistent with a generally conserved pattern of multivariate selection between populations.     

Estimation of Growth Norms from Incomplete Longitudinal Data

Longitudinal studies are rarely complete due to attrition, mistimed visits and observations missing at random. When the data are missing at random it is possible to estimate the primary location parameters of interest by constructing a modification of Zellner's (1962) seemingly unrelated regression estimator. Such a procedure is developed in this paper and is applied to a longitudinal study of coronary risk factors in children. The method consists of two stages in which the covariance matrix is estimated at the first stage. Using the estimated covariance matrix a generalized least squares estimator of the regression parameter vector is then determined at the second stage. Limitations of the procedure are also discussed.