Lipid data from NHLBI veteran twins: Interpreting genetic analyses when model assumptions fail

Analyses were performed on lipid data from the NHLBI Veteran Twin Study. The analyses focused on longitudinal multivariate models, describing how the genetic effects on lipids vary over time. Our pedigree-based model selection approach allows simultaneous estimation of both covariance structure parameters and regression parameters. The analyses reveal strong correlations between additive genetic effects over time, implying that genetic effects on lipids are somewhat constant throughout the life span represented within this sample. Both univariate preliminary analyses and robust fitting applied to the longitudinal models indicate that several assumptions underlying the twin analyses are violated. Although variance component and correlation parameter estimates are not much changed by robust fitting analyses, questions remain about the behavior of parameter estimates in multivariate genetic models under departures from model assumptions. © 1993 Wiley-Liss, Inc.     

A linear mixed model for predicting a binary event from longitudinal data under random effects misspecification

The use of longitudinal data for predicting a subsequent binary event is often the focus of diagnostic studies. This is particularly important in obstetrics, where ultrasound measurements taken during fetal development may be useful for predicting various poor pregnancy outcomes. We propose a modeling framework for predicting a binary event from longitudinal measurements where a shared random effect links the two processes together. Under a Gaussian random effects assumption, the approach is simple to implement with standard statistical software. Using asymptotic and simulation results, we show that estimates of predictive accuracy under a Gaussian random effects distribution are robust to severe misspecification of this distribution. However, under some circumstances, estimates of individual risk may be sensitive to severe random effects misspecification. We illustrate the methodology with data from a longitudinal fetal growth study.     

Comparison of alternative regression models for paired binary data

We used simulated data, derived from real ophthalmologic examples, to evaluate the performance of alternative logistic regression approaches for paired binary data. Approaches considered were: standard logistic regression (ignoring the correlation between fellow eyes, treating individuals classified on the basis of their more impaired eye as the unit of analysis, or considering only right eyes); marginal logistic regression models fitted by the maximum likelihood approach of Lipsitz, Laird and Harrington or the estimating equation approach of Liang and Zeger; and conditional logistic regression models fitted by the maximum likelihood approach of Rosner or the estimating equation approach of Connolly and Liang. Taylor series approximations were used to compare conditional and marginal parameter estimates. Consideration of type I and II error rates found application of standard logistic regression to be inferior to methods that treated the eye as the unit of analysis and accounted for the correlation between fellow eyes. Among these latter approaches, none was uniformly superior to the others across the range of conditions considered.     

Random effects logistic models for analysing efficacy of a longitudinal randomized treatment with non-adherence

We present a random effects logistic approach for estimating the efficacy of treatment for compliers in a randomized trial with treatment non-adherence and longitudinal binary outcomes. We use our approach to analyse a primary care depression intervention trial. The use of a random effects model to estimate efficacy supplements intent-to-treat longitudinal analyses based on random effects logistic models that are commonly used in primary care depression research. Our estimation approach is an extension of Nagelkerke et al.'s instrumental variables approximation for cross-sectional binary outcomes. Our approach is easily implementable with standard random effects logistic regression software. We show through a simulation study that our approach provides reasonably accurate inferences for the setting of the depression trial under model assumptions. We also evaluate the sensitivity of our approach to model assumptions for the depression trial.     

Rank‐preserving regression: a more robust rank regression model against outliers

Mean‐based semi‐parametric regression models such as the popular generalized estimating equations are widely used to improve robustness of inference over parametric models. Unfortunately, such models are quite sensitive to outlying observations. The Wilcoxon‐score‐based rank regression (RR) provides more robust estimates over generalized estimating equations against outliers. However, the RR and its extensions do not sufficiently address missing data arising in longitudinal studies. In this paper, we propose a new approach to address outliers under a different framework based on the functional response models. This functional‐response‐model‐based alternative not only addresses limitations of the RR and its extensions for longitudinal data, but, with its rank‐preserving property, even provides more robust estimates than these alternatives. The proposed approach is illustrated with both real and simulated data. Copyright © 2016 John Wiley & Sons, Ltd.     

A comparison of generalized linear mixed model procedures with estimating equations for variance and covariance parameter estimation in longitudinal studies and group randomized trials.

Response data in longitudinal studies and group randomized trials are gathered on units that belong to clusters, within which data are usually positively correlated. Therefore, estimates and confidence intervals for intraclass correlation or variance components are helpful when designing a longitudinal study or group randomized trial. Data simulated from both study designs are used to investigate the estimation of variance and covariance parameters from the following procedures: for continuous outcomes, restricted maximum likelihood (REML) and estimating equations (EE); for binary outcomes, restricted pseudo-likelihood (REPL) and estimating equations (EE). We evaluate these procedures to see which provide valid and precise estimates as well as correct standard errors for the intraclass correlation coefficient or variance components. REML seems the better choice for estimating terms related to correlation for models with normal outcomes, especially in group randomized trial situations. Results for REML and EE are mixed when outcomes are continuous and non-normal. With binary outcomes neither REPL nor EE provides satisfactory estimation or inference in longitudinal study situations, while REPL is preferable for group randomized trials.     

Robust inference for multivariate survival data

Multivariate survival data arise when an individual records multiple survival events or when individuals recording single survival events are grouped into clusters. In this paper we propose a new method for the analysis of multivariate survival data. The technique is a synthesis of the Poisson regression formulation for univariate censored survival analysis and the generalized estimating equation approach for obtaining valid variance estimates for generalized linear models in the presence of clustering. When the survival data are clustered, combining the methods provides not only valid estimates for the variances of regression parameters but also estimates of the dependence between survival times. The approach entails specifying parametric models for the marginal hazards and a dependence structure, but does not require specification of the joint multivariate survival distribution. Properties of the methodology are investigated by simulation and through an illustrative example.     

Regression models for clustered binary responses: implications of ignoring the intracluster correlation in an analysis of perinatal mortality in twin gestations

PURPOSE: Dependent binary responses, such as health outcomes in twin pairs or siblings, frequently arise in perinatal epidemiologic research. This gives rise to correlated data, which must be taken into account during analysis to avoid erroneous statistical and biological inferences. METHODS: An analysis of perinatal mortality (fetal deaths plus deaths within the first 28 days) in twins in relation to cluster-varying (those that are unique to each fetus within a twin pregnancy such as birthweight) and cluster-constant (those that are identical for both twins within a sibship such as maternal smoking status) risk factors is presented. Marginal (ordinary logistic regression [OLR] and logistic regression using generalized estimating equations [GEE]) and cluster-specific (conditional and random-intercept logistic regression models) regression models are fit and their results contrasted. The United States "matched multiple data" file of twin births (1995-1997), which includes 285,226 twins from 142,613 pregnancies, was used to examine the implications of ignoring of clustering on regression inferences. RESULTS: The OLR models provide variance estimates for cluster constant covariates that ranged from 7% to 71% smaller than those from GEE-based models. This underestimation is even more pronounced for some cluster-varying covariates, ranging from 21% to 198%. CONCLUSIONS: Ignoring the cluster dependency is likely to affect the precision of covariate effects and consequently interpretation of results. With widespread availability of appropriate software, statistical methods for taking the intracluster dependency into account are easily implemented and necessary.     

Meta‐analysis of binary data: which within study variance estimate to use?

We applied a mixed effects model to investigate between- and within-study variation in improvement rates of 180 schizophrenia outcome studies. The between-study variation was explained by the fixed study characteristics and an additional random study effect. Both rate difference and logit models were used. For a binary proportion outcome p(i) with sample size n(i) in the ith study, (circumflexp(i)(1-circumflexp(i))n)(-1) is the usual estimate of the within-study variance sigma(i)(2) in the logit model, where circumflexpi) is the sample mean of the binary outcome for subjects in study i. This estimate can be highly correlated with logit(circumflexp(i)). We used (macronp(i)(1-macronp)n(i))(-1) as an alternative estimate of sigma(i)(2), where macronp is the weighted mean of circumflexp(i)'s. We estimated regression coefficients (beta) of the fixed effects and the variance (tau(2)) of the random study effect using a quasi-likelihood estimating equations approach. Using the schizophrenia meta-analysis data, we demonstrated how the choice of the estimate of sigma(2)(i) affects the resulting estimates of beta and tau(2). We also conducted a simulation study to evaluate the performance of the two estimates of sigma(2)(i) in different conditions, where the conditions vary by number of studies and study size. Using the schizophrenia meta-analysis data, the estimates of beta and tau(2) were quite different when different estimates of sigma(2)(i) were used in the logit model. The simulation study showed that the estimates of beta and tau(2) were less biased, and the 95 per cent CI coverage was closer to 95 per cent when the estimate of sigma(2)(i) was (macronp(1-macronp)n(i))(-1) rather than (circumflexp(i)(1-circumflexp)n(i))(-1). Finally, we showed that a simple regression analysis is not appropriate unless tau(2) is much larger than sigma(2)(i), or a robust variance is used.     

Estimating correlation between alternative measures of disease progression in a longitudinal study. Modification of Diet in Renal Disease Study

A primary outcome variable in longitudinal studies is often the rate of change of a continuous measurement over time. Examples include the one-second forced expiratory volume (FEV1) in pulmonary studies or glomerular filtration rate (GFR) in renal studies. An individual patient's least-squares estimate of slope obtained from a linear regression is an imprecise measure of the true slope for that patient, and correlations involving the estimated slopes will be biased due to this measurement error. This paper presents methods for estimating the true correlation between these imprecise slope estimates, or between slope estimates and other variables measured with error. In addition to providing a simple consistent estimator of the correlation, we show how the maximum likelihood estimate of the correlation coefficient and a 100(1 - alpha) per cent confidence interval can be obtained. An example estimating the correlation between GFR and inverse serum creatinine slopes in patients with chronic renal disease is given.     

Estimation of relative risk and prevalence ratio

Relative risks (RRs) and prevalence ratios (PRs) are measures of association that are more intuitively interpretable than odds ratios (ORs). Many health science studies report OR estimates, however, even when their designs permit and study questions target RRs and/or PRs. This is, partially, attributable to the popularity and technical advantage (i.e. no restriction on the parameter space) of logistic regression for estimating ORs. To improve this practice, several biostatistical approaches for estimating RR/PR, adjusting for potential confounders, have been proposed. In this paper, we consider two RR/PR estimating methods: (1) the modification of log‐binomial regression with the COPY method; and (2) an inverse‐probability‐of‐treatment‐weighted (IPTW) log‐binomial regression we newly propose. For the COPY method, we rigorously establish the existence and uniqueness of the maximum‐likelihood estimator, provided certain degeneracies in the data do not occur. Moreover, the global maximum of the COPY‐modified likelihood is shown to occur at an interior point of the restricted parameter space. This result explains why the COPY method avoids convergence problems of log‐binomial models frequently. For the IPTW estimator, we show that its simple procedure results in standardized estimates of RR/PR, and discuss its potential challenges, extensions, and an improvement through propensity‐score‐based grouping of observations. Furthermore, we compare the performances of four RR/PR estimation methods, including the COPY method and IPTW log‐binomial regression, on simulated data. We demonstrate a lack of robustness of the COPY method against misspecification of the true relationship between binary outcome and explanatory variables, and show robustness of the IPTW approach in this regard. Copyright © 2010 John Wiley & Sons, Ltd.     

Functional regression analysis using an F test for longitudinal data with large numbers of repeated measures

Longitudinal data sets from certain fields of biomedical research often consist of several variables repeatedly measured on each subject yielding a large number of observations. This characteristic complicates the use of traditional longitudinal modelling strategies, which were primarily developed for studies with a relatively small number of repeated measures per subject. An innovative way to model such 'wide' data is to apply functional regression analysis, an emerging statistical approach in which observations of the same subject are viewed as a sample from a functional space. Shen and Faraway introduced an F test for linear models with functional responses. This paper illustrates how to apply this F test and functional regression analysis to the setting of longitudinal data. A smoking cessation study for methadone-maintained tobacco smokers is analysed for demonstration. In estimating the treatment effects, the functional regression analysis provides meaningful clinical interpretations, and the functional F test provides consistent results supported by a mixed-effects linear regression model. A simulation study is also conducted under the condition of the smoking data to investigate the statistical power for the F test, Wilks' likelihood ratio test, and the linear mixed-effects model using AIC.     

Application of robust estimating equations to the analysis of quantitative longitudinal data.

A model fit by general estimating equations (GEE) has been used extensively for the analysis of longitudinal data in medical studies. To some extent, GEE tries to minimize a quadratic form of the residuals, and therefore is not robust in the sense that it, like least squares estimates, is sensitive to heavy-tailed distributions, contaminated distributions and extreme values. This paper describes the family of truncated robust estimating equations and its properties for the analysis of quantitative longitudinal data. Like GEE, the robust estimating equations aim to assess the covariate effects in the generalized linear model in the complete population of observations, but in a manner that is more robust to the influence of aberrant observations. A simulation study has been conducted to compare the finite-sample performance of GEE and the robust estimating equations under a variety of error distributions and data structures. It shows that the parameter estimates based on GEE and the robust estimating equations are approximately unbiased and the type I errors of Wald tests do not tend to be inflated. GEE is slightly more efficient with pure normal data, but the efficiency of GEE declines much more quickly than the robust estimating equations when the data become contaminated or have heavy tails, which makes the robust estimating equations advantageous with non-normal data. Both GEE and the robust estimating equations are applied to a longitudinal analysis of renal function in the Diabetes Control and Complications Trial (DCCT). For this application, GEE seems to be sensitive to the working correlation specification in that different working correlation structures may lead to different conclusions about the effect of intensive diabetes treatment. On the other hand, the robust estimating equations consistently conclude that the treatment effect is highly significant no matter which working correlation structure is used. The DCCT Research Group also demonstrated a significant effect using a mixed-effects longitudinal model.     

The analysis of binary longitudinal data with time-dependent covariates

We consider longitudinal studies with binary outcomes that are measured repeatedly on subjects over time. The goal of our analysis was to fit a logistic model that relates the expected value of the outcomes with explanatory variables that are measured on each subject. However, additional care must be taken to adjust for the association between the repeated measurements on each subject. We propose a new maximum likelihood method for covariates that may be fixed or time varying. We also implement and make comparisons with two other approaches: generalized estimating equations, which may be more robust to misspecification of the true correlation structure, and alternating logistic regression, which models association via odds ratios that are subject to less restrictive constraints than are correlations. The proposed estimation procedure will yield consistent and asymptotically normal estimates of the regression and correlation parameters if the correlation on consecutive measurements on a subject is correctly specified. Simulations demonstrate that our approach can yield improved efficiency in estimation of the regression parameter; for equally spaced and complete data, the gains in efficiency were greatest for the parameter associated with a time-by-group interaction term and for stronger values of the correlation. For unequally spaced data and with dropout according to a missing-at-random mechanism, MARK1ML with correctly specified consecutive correlations yielded substantial improvements in terms of both bias and efficiency. We present an analysis to demonstrate application of the methods we consider. We also offer an R function for easy implementation of our approach.     

Binary variable multiple‐model multiple imputation to address missing data mechanism uncertainty: application to a smoking cessation trial

The true missing data mechanism is never known in practice. We present a method for generating multiple imputations for binary variables, which formally incorporates missing data mechanism uncertainty. Imputations are generated from a distribution of imputation models rather than a single model, with the distribution reflecting subjective notions of missing data mechanism uncertainty. Parameter estimates and standard errors are obtained using rules for nested multiple imputation. Using simulation, we investigate the impact of missing data mechanism uncertainty on post‐imputation inferences and show that incorporating this uncertainty can increase the coverage of parameter estimates. We apply our method to a longitudinal smoking cessation trial where nonignorably missing data were a concern. Our method provides a simple approach for formalizing subjective notions regarding nonresponse and can be implemented using existing imputation software. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust inference for variance components models in families ascertained through probands: I. Conditioning on proband's phenotype

A robust approach for estimating standard errors of variance components by using quantitative phenotypes from families ascertained through a proband with an extreme phenotypic value is presented. Estimators that use the multivariate normal distribution as a "working likelihood" are obtained by computing conditional ln-likelihoods, conditional first and second derivatives in a Newton-Raphson approach. Robust estimates of standard errors about the estimators are also provided. Tests of hypotheses are based on a modification of the score test, which allows the assumption of multivariate normality to be relaxed. Conditional goodness-of-fit statistics are proposed that can be used to examine the fit of separate pedigrees to the overall model. This robust approach for estimating the standard errors for variance components by conditioning on the proband's phenotype will allow general inferences to be made from the analysis of families ascertained through probands with extreme or unusual phenotypes and should be most appropriate for studying many physiological traits that may be intrinsically nonnormal.     

A refined method for multivariate meta‐analysis and meta‐regression

Making inferences about the average treatment effect using the random effects model for meta‐analysis is problematic in the common situation where there is a small number of studies. This is because estimates of the between‐study variance are not precise enough to accurately apply the conventional methods for testing and deriving a confidence interval for the average effect. We have found that a refined method for univariate meta‐analysis, which applies a scaling factor to the estimated effects’ standard error, provides more accurate inference. We explain how to extend this method to the multivariate scenario and show that our proposal for refined multivariate meta‐analysis and meta‐regression can provide more accurate inferences than the more conventional approach. We explain how our proposed approach can be implemented using standard output from multivariate meta‐analysis software packages and apply our methodology to two real examples. © 2013 The Authors. Statistics in Medicine published by John Wiley & Sons, Ltd.     

Comparison of adaptive treatment strategies based on longitudinal outcomes in sequential multiple assignment randomized trials

In sequential multiple assignment randomized trials, longitudinal outcomes may be the most important outcomes of interest because this type of trials is usually conducted in areas of chronic diseases or conditions. We propose to use a weighted generalized estimating equation (GEE) approach to analyzing data from such type of trials for comparing two adaptive treatment strategies based on generalized linear models. Although the randomization probabilities are known, we consider estimated weights in which the randomization probabilities are replaced by their empirical estimates and prove that the resulting weighted GEE estimator is more efficient than the estimators with true weights. The variance of the weighted GEE estimator is estimated by an empirical sandwich estimator. The time variable in the model can be linear, piecewise linear, or more complicated forms. This provides more flexibility that is important because, in the adaptive treatment setting, the treatment changes over time and, hence, a single linear trend over the whole period of study may not be practical. Simulation results show that the weighted GEE estimators of regression coefficients are consistent regardless of the specification of the correlation structure of the longitudinal outcomes. The weighted GEE method is then applied in analyzing data from the Clinical Antipsychotic Trials of Intervention Effectiveness. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of matched case-control data with incomplete strata: applying longitudinal approaches

BACKGROUND: Matched case-control data have a structure that is similar to longitudinal data with correlated outcomes, except for a retrospective sampling scheme. In conditional logistic regression analysis, sets that are incomplete due to missing covariates and sets with identical values of the covariates do not contribute to the estimation; both situations may cause a loss in efficiency. These problems are more severe when sample sizes are small. We evaluated retrospective models for longitudinal data as alternatives in analyzing matched case-control data. METHODS: We conducted simulations to compare the properties of matched case-control data analyses using conditional likelihood and a commonly used longitudinal approach generalized estimating equation (GEE). We simulated scenarios for one-to-one and one-to-two matching designs, each with various sizes of matching strata, with complete and incomplete strata, and with dichotomous and normal exposures. RESULTS AND CONCLUSIONS: The simulations show that the estimates by conditional likelihood and GEE methods are consistent, and a proper coverage was reached for both binary and continuous exposures. The estimates produced by conditional likelihood have greater standard errors than those obtained by GEE. These relative efficiency losses are more substantial when data contain incomplete matched sets and when the data have small sizes of matching strata; these can be improved by including more controls in the strata. These losses of efficiency also increase as the magnitude of the association increases.