Normal Random Effects Research Articles

Mixed effect models of longitudinal Alzheimer's disease data: a cautionary note.

Longitudinal studies of cognitive function in Alzheimer's disease (AD) patients are powerful tools to better understand the biology and natural history of the disease, but the attributes of the studies that make them valuable also pose special challenges to analysts. A fundamental problem is the accurate measure of time at which cognitive decline begins. Investigators typically use the date of AD diagnosis or the date of enrollment in an AD study. If the rate of cognitive decline is non-linear, variables associated with the time of diagnosis or enrollment might artificially be associated with the rate of decline. Unlike the mixed effects models typically used to analyse cognitive decline, summary measure analyses do not directly compare the rate of decline with time since decline began, and, therefore, are less sensitive to biased measures of time of decline. We simulated trajectories of cognitive decline using the multivariate normal random effect model and tested the ability of the two analytic techniques to discriminate between true and spurious associations. Our analyses suggest summary measure models are less likely to detect spurious associations generated by biased measures of time at which decline begins, and more likely to detect true associations concealed by biased time measurement.

Maximum Likelihood for Generalized Linear Models with Nested Random Effects via High-Order, Multivariate Laplace Approximation

Nested random effects models are often used to represent similar processes occurring in each of many clusters. Suppose that, given cluster-specific random effects b, the data y are distributed according to f(y|b, Θ), while b follows a density p(b|Θ). Likelihood inference requires maximization of ∫ f(y|b, Θ)p(b|Θdb with respect to Θ. Evaluation of this integral often proves difficult, making likelihood inference difficult to obtain. We propose a multivariate Taylor series approximation of the log of the integrand that can be made as accurate as desired if the integrand and all its partial derivatives with respect to b are continuous in the neighborhood of the posterior mode of b|Θ,y. We then apply a Laplace approximation to the integral and maximize the approximate integrated likelihood via Fisher scoring. We develop computational formulas that implement this approach for two-level generalized linear models with canonical link and multivariate normal random effects. A comparison with approximations based on penalized quasi-likelihood, Gauss—Hermite quadrature, and adaptive Gauss-Hermite quadrature reveals that, for the hierarchical logistic regression model under the simulated conditions, the sixth-order Laplace approach is remarkably accurate and computationally fast.

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.

A Reminder of the Fallibility of the Wald Statistic: Likelihood Explanation

The Wald statistic is one of the most commonly used tools in applied statistics, so it is sobering to read Fears, Benichou, and Gail's recent reminder of its fallibility. What makes their example particularly relevant is the fact that the problem is manifest in a simple normal random effects model on a balanced dataset for a seemingly harmless parameter, while, in practice, one tends to rely on the Wald test in complicated or nonnormal models where there are no exact tests to serve as a gold standard. This article explains the failure of the Wald test via the profile likelihood functions, which graphically look decidedly nonnormal. The methodology used to derive the profile likelihood and the discussion of this example could be instructive for a senior or beginning graduate class in theory of statistics.

Mixed effects models with bivariate and univariate association parameters for longitudinal bivariate binary response data.

When two binary responses are measured for each study subject across time, it may be of interest to model how the bivariate associations and marginal univariate risks involving the two responses change across time. To achieve such a goal, marginal models with bivariate log odds ratio and univariate logit components are extended to include random effects for all components. Specifically, separate normal random effects are specified on the log odds ratio scale for bivariate responses and on the logit scale for univariate responses. Assuming conditional independence given the random effects facilitates the modeling of bivariate associations across time with missing at random incomplete data. We fit the model to a dataset for which such structures are feasible: a longitudinal randomized trial of a cardiovascular educational program where the responses of interest are change in hypertension and hypercholestemia status. The proposed model is compared to a naive bivariate model that assumes independence between time points and univariate mixed effects logit models.

Analysis of correlated data in human biomonitoring studies. The case of high sister chromatid exchange frequency cells

Sister chromatid exchange (SCE) analysis in peripheral blood lymphocytes is a well established technique that aims to evaluate human exposure to toxic agents. The individual mean value of SCE per cell had been the only recommended index to measure the extent of this cytogenetic damage until the early 1980's, when the concept of high frequency cells (HFC) was introduced to increase the sensitivity of the assay. All statistical analyses proposed thus far to handle these data are based on measures which refer to the individual mean values and not to the single cell. Although this approach allows the use of simple statistical methods, part of the information provided by the distribution of SCE per single cell within the individual is lost. Using the appropriate methods developed for the analysis of correlated data, it is possible to exploit all the available information. In particular, the use of random-effects models seems to be very promising for the analysis of clustered binary data such as HFC. Logistic normal random-effects models, which allow modelling of the correlation among cells within individuals, have been applied to data from a large study population to highlight the advantages of using this methodology in human biomonitoring studies. The inclusion of random-effects terms in a regression model could explain a significant amount of variability, and accordingly change point and/or interval estimates of the corresponding coefficients. Examples of coefficients that change across different regression models and their interpretation are discussed in detail. One model that seems particularly appropriate is the random intercepts and random slopes model.

Bayesian marginal influence assessment

Case-influence diagnostics are in wide use in classical linear regression and are common in Bayesian analysis. Most common Bayesian diagnostics assess influence on all parameters; influence on parameter subsets have not been formally developed, in contrast with classically based influence diagnostics. Influence on parameter subsets should be assessed when some parameters are ‘nuisance’ parameters while others are of primary inferential interest. Influence on parameter subsets can be substantially less than influence on the full parameter vector, leading to different conclusions from the influence analysis. We discuss Bayesian influence assessment in normal linear and hierarchical normal random effects models. We give formulae for case deletion influence diagnostics in normal linear regression for joint and marginal posterior distributions using several divergence measures. In more complex models, we describe a nested sampling procedure for computing marginal influence measures when closed-form computations are not available.

Subject-Specific and Population-Averaged Continuation Ratio Logit Models for Multiple Discrete Time Survival Profiles

SUMMARY Subject-specific and population-averaged continuation ratio logit models are presented for multivariate discrete time survival data. The models characterize data from a psychological experiment by using a quadratic polynomial relationship across time that depends on a time-independent condition. A multivariate normal random effects distribution is imposed on intercept, linear and quadratic terms in the subject-specific model, which is fitted by using a combination of Gibbs sampling and buffered stochastic substitution. Variance components that tend towards 0 are addressed in this context. In addition, generalized estimating equations estimates of the parameters in the population-averaged model are compared with analogous estimates for the mixed effects model.

Weighted Normal Plots

Abstract Weighted normal plots are proposed as graphical checks on the normality of random effects in Gaussian linear models. The technique is illustrated using the one-way comparisons model Yi = μ i + ϵ i , where the (μ i , ϵ i ), are independent pairs with μ i and ϵ i , independent N(0, σ2) and N(0, σ2 i ), respectively, for i = 1, …, n. When the variance components σ2 and σ2 i are known, an unweighted normal plot of the standardized Zi = Yi (σ2 + σ2 i )-1/2 provides a check of the overall adequacy of the model. Weighted normal plots involve a modification that gives the ith observation a sample weight of Wi = (σ2 + σ2 i )-1. Under the null hypothesis, the sample size must be larger by a factor of (1 + v/m 2), where m and v are the mean and variance of the weights, to produce a weighted plot with approximately the same sampling variance as an unweighted normal plot. Despite this higher variability, we show that weighted plots are more sensitive than unweighted plots to several departures from the assumed distribution on the random effects, μ i . Several numerical examples are included and the effects of substituting maximum likelihood estimates for the parameters σ2 and σ2 i are considered briefly.