Variable order ante-dependence models

Ante-dependence models can be used to model the covariance structure in problems involving repeated measures through time. They are conditional regression models which generalize Gabriel’s constant-order ante-dependence model. Likelihood-based procedures are presented, together with simple expressions for likelihood ratio test statistics in terms of sum of squares from appropriate analysis of covariance. The estimation of the orders is approached as a model selection problem, and penalized likelihood criteria are suggested. Extensions of all procedures discussed here to situations with a monotone pattern of missing data are presented.     

Bayesian longitudinal models for paediatric kidney transplant recipients

Chronic kidney disease is a progressive loss of renal function which results in the inability of the kidneys to properly filter waste from the blood. Renal function is usually estimated by the glomerular filtration rate (eGFR), which decreases with the worsening of the disease. Bayesian longitudinal models with covariates, random effects, serial correlation and measurement error are discussed to analyse the progression of eGFR in first transplanted children taken from a study in València, Spain.     

Leverage analysis for linear mixed models

We consider a generalized leverage matrix useful for the identification of influential units and observations in linear mixed models and show how a decomposition of this matrix may be employed to identify high leverage points for both the marginal fitted values and the random effect component of the conditional fitted values. We illustrate the different uses of the two components of the decomposition with a simulated example as well as with a real data set.     

Covariance structure selection in general mixed models

This article describes a unified approach to variance modeling and inference in the context of a general form of the normal-theory linear mixed model. The primary variance modeling objects are parameterized covari-ance structures, examples being diagonal, compound-symmetry, unstructured, timeseries, and spatial. These structures can enter in two different places in the general mixed model, and the combination of one or both of these places with the variety of structures provides a rich class of variance models. The approach is likelihood-based, and involves the use of both maximum likelihood and restricted maximum likelihood. Two examples provide illustration.     

Least squares estimation of regression parameters in mixed effects models with unmeasured covariates

We consider mixed effects models for longitudinal, repeated measures or clustered data. Unmeasured or omitted covariates in such models may be correlated with the included covanates, and create model violations when not taken into account. Previous research and experience with longitudinal data sets suggest a general form of model which should be considered when omitted covariates are likely, such as in observational studies. We derive the marginal model between the response variable and included covariates, and consider model fitting using the ordinary and weighted least squares methods, which require simple non-iterative computation and no assumptions on the distribution of random covariates or error terms, Asymptotic properties of the least squares estimators are also discussed. The results shed light on the structure of least squares estimators in mixed effects models, and provide large sample procedures for statistical inference and prediction based on the marginal model. We present an example of the relationship between fluid intake and output in very low birth weight infants, where the model is found to have the assumed structure.     

GEEs for repeated categorical responses based on generalized residuals

Clustered or correlated samples of categorical response data arise frequently in many fields of application. The method of generalized estimating equations (GEEs) introduced in Liang and Zeger [Longitudinal data analysis using generalized linear models, Biometrika 73 (1986), pp. 13–22] is often used to analyse this type of data. GEEs give consistent estimates of the regression parameters and their variance based upon the Pearson residuals. Park et al. [Alternative GEE estimation procedures for discrete longitudinal data, Comput. Stat. Data Anal. 28 (1998), pp. 243–256] considered a modification of the GEE approach using the Anscombe residual and the deviance residual. In this work, we propose to extend this idea to a family of generalized residuals. A wide simulation study is conducted for binary and Poisson correlated outcomes and also two numerical illustrations are presented.     

Investigation of mixed model repeated measures analyses and non‐linear random coefficient models in the context of long‐term efficacy data

The longitudinal data from 2 published clinical trials in adult subjects with upper limb spasticity (a randomized placebo‐controlled study [NCT01313299] and its long‐term open‐label extension [NCT01313312]) were combined. Their study designs involved repeat intramuscular injections of abobotulinumtoxinA (Dysport®), and efficacy endpoints were collected accordingly. With the objective of characterizing the pattern of response across cycles, Mixed Model Repeated Measures analyses and Non‐Linear Random Coefficient (NLRC) analyses were performed and their results compared. The Mixed Model Repeated Measures analyses, commonly used in the context of repeated measures with missing dependent data, did not involve any parametric shape for the curve of changes over time. Based on clinical expectations, the NLRC included a negative exponential function of the number of treatment cycles, with its asymptote and rate included as random coefficients in the model. Our analysis focused on 2 specific efficacy parameters reflecting complementary aspects of efficacy in the study population. A simulation study based on a similar study design was also performed to further assess the performance of each method under different patterns of response over time. This highlighted a gain of precision with the NLRC model, and most importantly the need for its assumptions to be verified to avoid potentially biased estimates. These analyses describe a typical situation and the conditions under which non‐linear mixed modeling can provide additional insights on the behavior of efficacy parameters over time. Indeed, the resulting estimates from the negative exponential NLRC can help determine the expected maximal effect and the treatment duration required to reach it.     

Statistical models for the analysis of controlled trials on acute migraine

Specific efficacy criteria were defined by the International Headache Society for controlled clinical trials on acute migraine. They are derived from the pain profile and the timing of rescue medication intake. We present a methodology to improve the analysis of such trials. Instead of analysing each endpoint separately, we model the joint distribution and derive success rates in any criteria as predictions. We use cumulative regression models for each response at a time and a multivariate normal copula to model the dependence between responses. Parameters are estimated using maximum likelihood. Benefits of the method include a reduction in the number of tests performed and an increase in their power. The method is well suited to dose–response trials from which predictions can be used to select doses and optimize the design of subsequent trials. More generally, our method permits a very flexible modelling of longitudinal series of ordinal data. Copyright © 2003 John Wiley & Sons, Ltd.     

Goodness-of-fit measures of R 2 for repeated measures mixed effect models

Linear mixed effects model (LMEM) is efficient in modeling repeated measures longitudinal data. However, little research has been done in developing goodness-of-fit measures that can evaluate the models, particularly those that can be interpreted in an absolute sense without referencing a null model. This paper proposes three coefficient of determination (R ²) as goodness-of-fit measures for LMEM with repeated measures longitudinal data. Theorems are presented describing the properties of R ² and relationships between the R ² statistics. A simulation study was conducted to evaluate and compare the R ² along with other criteria from literature. Finally, we applied the proposed R ² to a real virologic response data of an HIV-patient cohort. We conclude that our proposed R ² statistics have more advantages than other goodness-of-fit measures in the literature, in terms of robustness to sample size, intuitive interpretation, well-defined range, and unnecessary to determine a null model.     

Latent variable models with mixed continuous and polytomous data

Owing to the nature of the problems and the design of questionnaires, discrete polytomous data are very common in behavioural, medical and social research. Analysing the relationships between the manifest and the latent variables based on mixed polytomous and continuous data has proven to be difficult. A general structural equation model is investigated for these mixed outcomes. Maximum likelihood (ML) estimates of the unknown thresholds and the structural parameters in the covariance structure are obtained. A Monte Carlo–EM algorithm is implemented to produce the ML estimates. It is shown that closed form solutions can be obtained for the M-step, and estimates of the latent variables are produced as a by-product of the analysis. The method is illustrated with a real example.     

Diagnosing explainable heterogeneity of variance in random‐effects models

Data‐analytic tools for models other than the normal linear regression model are relatively rare. Here we develop plots and diagnostic statistics for nonconstant variance for the random‐effects model (REM). REMs for longitudinal data include both within‐ and between‐subject variances. A basic assumption is that the two variance terms are constant across subjects. However, we often find that these variances are functions of covariates, and the data set has what we call explainable heterogeneity, which needs to be allowed for in the model. We characterize several types of heterogeneity of variance in REMs and develop three diagnostic tests using the score statistic: one for each of the two variance terms, and the third for a form of multivariate nonconstant variance. For each test we present an adjusted residual plot which can identify cases that are unusually influential on the outcome of the test.     

Applying skovgaard's modified directed likelihood statistic to mixed linear models

The introduction of software to calculate maximum likelihood estimates for mixed linear models has made likelihood estimation a practical alternative to methods based on sums of squares. Likelihood based tests and confidence intervals, however, may be misleading in problems with small sample sizes. This paper discusses an adjusted version of the directed log-likelihood statistic for mixed models that is highly accurate for testing one parameter hypotheses. Indroduced by Skovgaard (1996, Journal of the Bernoulli Society,2,145-165), we show in mixed models that the statistic has a simple conpact from that may be obtained from standard software. Simulation studies indicate that this statistic is more accurate than many of the specialized procedure that have been advocated.     

Bivariate Longitudinal Model for Detecting Prescribing Change in Two Drugs Simultaneously with Correlated Errors

Bivariate responses of repeated measures data are usually analysed as two separate responses in the literature by several authors. The two responses usually tend to be related in some way and analysing this data jointly presents an opportunity to account for the joint movement, which may impact on the conclusions reached compared to analysing the responses separately. In this paper, a bivariate regression model with random effects (linear mixed model) is used to detect a change if any in the prescribing habits in the UK at the general practice (family medicine) level due to an educational intervention given repeated measures data before and after the intervention and a control group. The message was to increase the prescribing of one drug while simultaneously decreasing the prescribing of another. The effects of modelling a bivariate auto-regressive process are evaluated.     

Selection of terms in random coefficient regression models

The selection of suitable terms in random coefficient regression models is a challenging problem to practitioners. Although many techniques, ranging from those with a theoretical flavour to those with an exploratory spirit, have been proposed for such purposes, no particular one may be considered as a paradigm. In fact, many authors advocate that they should be used in a complementary way. We consider exploratory methods based on fitting standard regression models to the individual response profiles or to the rows of the sample within-units covariance matrix (for balanced data) that may serve as additional tools in the process of selecting an appropriate model. We evaluate the performance of the proposal via a simulation study and consider applications to two examples in the field of Biostatistics.     

Short notes on maximum likelihood inference for control‐based pattern‐mixture models

A likelihood‐based analytical approach has been proposed for the control‐based pattern‐mixture model and its extension. In this note, we derive equivalent but simpler analytical expressions for the treatment effect and its variance for these control‐based pattern mixture models. Our formulae are easier to use and interpret. An application of our formulae to an antidepressant trial is provided, in which the likelihood‐based analysis is compared with the multiple imputation approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Repeated‐measures models in the analysis of QT interval

Because of the recent regulatory emphasis on issues related to drug‐induced cardiac repolarization that can potentially lead to sudden death, QT interval analysis has received much attention in the clinical trial literature. The analysis of QT data is complicated by the fact that the QT interval is correlated with heart rate and other prognostic factors. Several attempts have been made in the literature to derive an optimal method for correcting the QT interval for heart rate; however the QT correction formulae obtained are not universal because of substantial variability observed across different patient populations. It is demonstrated in this paper that the widely used fixed QT correction formulae do not provide an adequate fit to QT and RR data and bias estimates of treatment effect. It is also shown that QT correction formulae derived from baseline data in clinical trials are likely to lead to Type I error rate inflation. This paper develops a QT interval analysis framework based on repeated‐measures models accomodating the correlation between QT interval and heart rate and the correlation among QT measurements collected over time. The proposed method of QT analysis controls the Type I error rate and is at least as powerful as traditional QT correction methods with respect to detecting drug‐related QT interval prolongation. Copyright © 2003 John Wiley & Sons, Ltd.     

Logistic regression and other discrete data models for serially correlated observations

Summary We consider the analysis of discrete serially correlated data in the presence of time dependent covariates. If the interest is to relate the covariates to the marginal distribution of the data, Markov chains are an obvious tool to consider, but their use is complicated by the fact that they are expressed in terms of transitional rather than marginal probabilities. We show how to parametrize the transition matrix in a suitable way so that interpretation is as desired. The focus is on binary and Poisson data, but the methodology can be adopted also with other discrete data distributions.     

Mixed models for data from thorough QT studies: part 1. assessment of marginal QT prolongation

We investigate mixed models for repeated measures data from cross-over studies in general, but in particular for data from thorough QT studies. We extend both the conventional random effects model and the saturated covariance model for univariate cross-over data to repeated measures cross-over (RMC) data; the resulting models we call the RMC model and Saturated model, respectively. Furthermore, we consider a random effects model for repeated measures cross-over data previously proposed in the literature. We assess the standard errors of point estimates and the coverage properties of confidence intervals for treatment contrasts under the various models. Our findings suggest: (i) Point estimates of treatment contrasts from all models considered are similar; (ii) Confidence intervals for treatment contrasts under the random effects model previously proposed in the literature do not have adequate coverage properties; the model therefore cannot be recommended for analysis of marginal QT prolongation; (iii) The RMC model and the Saturated model have similar precision and coverage properties; both models are suitable for assessment of marginal QT prolongation; and (iv) The Akaike Information Criterion (AIC) is not a reliable criterion for selecting a covariance model for RMC data in the following sense: the model with the smallest AIC is not necessarily associated with the highest precision for the treatment contrasts, even if the model with the smallest AIC value is also the most parsimonious model.     

Using regression mixture models with non-normal data: examining an ordered polytomous approach

Mild to moderate skew in errors can substantially impact regression mixture model results; one approach for overcoming this includes transforming the outcome into an ordered categorical variable and using a polytomous regression mixture model. This is effective for retaining differential effects in the population; however, bias in parameter estimates and model fit warrant further examination of this approach at higher levels of skew. The current study used Monte Carlo simulations; 3000 observations were drawn from each of two subpopulations differing in the effect of X on Y. Five hundred simulations were performed in each of the 10 scenarios varying in levels of skew in one or both classes. Model comparison criteria supported the accurate two-class model, preserving the differential effects, while parameter estimates were notably biased. The appropriate number of effects can be captured with this approach but we suggest caution when interpreting the magnitude of the effects.     

Score tests for heterogeneity and overdispersion in zero‐inflated Poisson and binomial regression models

Hall (2000) has described zero‐inflated Poisson and binomial regression models that include random effects to account for excess zeros and additional sources of heterogeneity in the data. The authors of the present paper propose a general score test for the null hypothesis that variance components associated with these random effects are zero. For a zero‐inflated Poisson model with random intercept, the new test reduces to an alternative to the overdispersion test of Ridout, Demério & Hinde (2001). The authors also examine their general test in the special case of the zero‐inflated binomial model with random intercept and propose an overdispersion test in that context which is based on a beta‐binomial alternative.