Partitioned GMM logistic regression models for longitudinal data

Correlation is inherent in longitudinal studies due to the repeated measurements on subjects, as well as due to time-dependent covariates in the study. In the National Longitudinal Study of Adolescent to Adult Health (Add Health), data were repeatedly collected on children in grades 7-12 across four waves. Thus, observations obtained on the same adolescent were correlated, while predictors were correlated with current and future outcomes such as obesity status, among other health issues. Previous methods, such as the generalized method of moments (GMM) approach have been proposed to estimate regression coefficients for time-dependent covariates. However, these approaches combined all valid moment conditions to produce an averaged parameter estimate for each covariate and thus assumed that the effect of each covariate on the response was constant across time. This assumption is not necessarily optimal in applications such as Add Health or health-related data. Thus, we depart from this assumption and instead use the Partitioned GMM approach to estimate multiple coefficients for the data based on different time periods. These extra regression coefficients are obtained using a partitioning of the moment conditions pertaining to each respective relationship. This approach offers a deeper understanding and appreciation into the effect of each covariate on the response. We conduct simulation studies, as well as analyses of obesity in Add Health, rehospitalization in Medicare data, and depression scores in a clinical study. The Partitioned GMM methods exhibit benefits over previously proposed models with improved insight into the nonconstant relationships realized when analyzing longitudinal data.     

Repeated-measures model for the investigation of temporal trends using longitudinal family studies: application to systolic blood pressure

A contemporary path model for the analysis of familial resemblance is extended to incorporate repeated measurements on the entire pedigree over time, in order to assess age-related changes in familiality. The parameters of the model can be defined as arbitrary functions of the ages, age differences, or cohabitation times of the family members at the exact time of measurement. Tracking of the phenotypes is decomposed into a familial and a nonfamilial component, which varies with both the time span between measurements and the ages at measurement. Some of the family members may have data missing on one or more visits, and the visits may be unequally spaced both within and across families. The method incorporates all measurements available from all visits into a single model. The model is applied to longitudinal data on systolic blood pressure in 490 East Boston families measured two times at 3-year intervals. Evidence for some nonfamilial tracking is found. Additionally, significant temporal trends are demonstrated in the familiality as a function of age, t2(A), which appears to be near zero at birth, grow to a maximum of about 40% at around age 30, and then appears to monotonically decrease again. No evidence was found for temporal trends in marital resemblance or residual sibling environmental effects. This model provides an objective method of investigating developmental changes in the correlational structure of families over time using repeated-measures and of estimating continuous changes in familiality with age.     

Sample size and power calculations for correlations between bivariate longitudinal data

The analysis of a baseline predictor with a longitudinally measured outcome is well established and sample size calculations are reasonably well understood. Analysis of bivariate longitudinally measured outcomes is gaining in popularity and methods to address design issues are required. The focus in a random effects model for bivariate longitudinal outcomes is on the correlations that arise between the random effects and between the bivariate residuals. In the bivariate random effects model, we estimate the asymptotic variances of the correlations and we propose power calculations for testing and estimating the correlations. We compare asymptotic variance estimates to variance estimates obtained from simulation studies and compare our proposed power calculations for correlations on bivariate longitudinal data to power calculations for correlations on cross‐sectional data. Copyright © 2010 John Wiley & Sons, Ltd.     

Interim analysis of continuous long-term endpoints in clinical trials with longitudinal outcomes

This paper discusses interim analysis for clinical trials where the primary endpoint is observed at a specific long-term follow-up time, but where repeated measures of the same outcome are also taken at earlier times. Methods are considered for improving the efficiency with which the long-term treatment difference is estimated, making use of information from shorter-term follow-up times. This approach to interim analysis has previously been studied for binary endpoints assessed at two time points during follow-up. Here we adapt and extend this methodology to include continuous endpoints assessed at an arbitrary number of follow-up times, making use of methods for analysing multivariate normal data subject to monotone missingness and unstructured mean and covariance relationships. The magnitude of efficiency gains obtained by using short-term measurements is considered, as well as how these gains depend on the number and timing of the short-term measurements. Sequential analysis of treatment differences is discussed, including the extent to which efficiency gains translate into reductions in the expected duration of a sequentially monitored trial. The methods are illustrated on a data set involving a placebo-controlled comparison of longitudinal cholesterol measurements.     

Pseudo-likelihood methods for longitudinal binary data with non-ignorable missing responses and covariates

In this paper we consider longitudinal studies in which the outcome to be measured over time is binary, and the covariates of interest are categorical. In longitudinal studies it is common for the outcomes and any time-varying covariates to be missing due to missed study visits, resulting in non-monotone patterns of missingness. Moreover, the reasons for missed visits may be related to the specific values of the response and/or covariates that should have been obtained, i.e. missingness is non-ignorable. With non-monotone non-ignorable missing response and covariate data, a full likelihood approach is quite complicated, and maximum likelihood estimation can be computationally prohibitive when there are many occasions of follow-up. Furthermore, the full likelihood must be correctly specified to obtain consistent parameter estimates. We propose a pseudo-likelihood method for jointly estimating the covariate effects on the marginal probabilities of the outcomes and the parameters of the missing data mechanism. The pseudo-likelihood requires specification of the marginal distributions of the missingness indicator, outcome, and possibly missing covariates at each occasions, but avoids making assumptions about the joint distribution of the data at two or more occasions. Thus, the proposed method can be considered semi-parametric. The proposed method is an extension of the pseudo-likelihood approach in Troxel et al. to handle binary responses and possibly missing time-varying covariates. The method is illustrated using data from the Six Cities study, a longitudinal study of the health effects of air pollution.     

A Bayesian approach to joint analysis of longitudinal measurements and competing risks failure time data

In this paper, we develop a Bayesian method for joint analysis of longitudinal measurements and competing risks failure time data. The model allows one to analyze the longitudinal outcome with nonignorable missing data induced by multiple types of events, to analyze survival data with dependent censoring for the key event, and to draw inferences on multiple endpoints simultaneously. Compared with the likelihood approach, the Bayesian method has several advantages. It is computationally more tractable for high‐dimensional random effects. It is also convenient to draw inference. Moreover, it provides a means to incorporate prior information that may help to improve estimation accuracy. An illustration is given using a clinical trial data of scleroderma lung disease. The performance of our method is evaluated by simulation studies. Copyright © 2009 John Wiley & Sons, Ltd.     

A three‐step estimation procedure using local polynomial smoothing for inconsistently sampled longitudinal data

Parametric mixed‐effects models are useful in longitudinal data analysis when the sampling frequencies of a response variable and the associated covariates are the same. We propose a three‐step estimation procedure using local polynomial smoothing and demonstrate with data where the variables to be assessed are repeatedly sampled with different frequencies within the same time frame. We first insert pseudo data for the less frequently sampled variable based on the observed measurements to create a new dataset. Then standard simple linear regressions are fitted at each time point to obtain raw estimates of the association between dependent and independent variables. Last, local polynomial smoothing is applied to smooth the raw estimates. Rather than use a kernel function to assign weights, only analytical weights that reflect the importance of each raw estimate are used. The standard errors of the raw estimates and the distance between the pseudo data and the observed data are considered as the measure of the importance of the raw estimates. We applied the proposed method to a weight loss clinical trial, and it efficiently estimated the correlation between the inconsistently sampled longitudinal data. Our approach was also evaluated via simulations. The results showed that the proposed method works better when the residual variances of the standard linear regressions are small and the within‐subjects correlations are high. Also, using analytic weights instead of kernel function during local polynomial smoothing is important when raw estimates have extreme values, or the association between the dependent and independent variable is nonlinear. Copyright © 2016 John Wiley & Sons, Ltd.     

Concordance correlation coefficients estimated by generalized estimating equations and variance components for longitudinal repeated measurements

The concordance correlation coefficient (CCC) is a commonly accepted measure of agreement between two observers for continuous responses. This paper proposes a generalized estimating equations (GEE) approach allowing dependency between repeated measurements over time to assess intra‐agreement for each observer and inter‐ and total agreement among multiple observers simultaneously. Furthermore, the indices of intra‐, inter‐, and total agreement through variance components (VC) from an extended three‐way linear mixed model (LMM) are also developed with consideration of the correlation structure of longitudinal repeated measurements. Simulation studies are conducted to compare the performance of the GEE and VC approaches for repeated measurements from longitudinal data. An application of optometric conformity study is used for illustration. In conclusion, the GEE approach allowing flexibility in model assumptions and correlation structures of repeated measurements gives satisfactory results with small mean square errors and nominal 95% coverage rates for large data sets, and when the assumption of the relationship between variances and covariances for the extended three‐way LMM holds, the VC approach performs outstandingly well for all sample sizes. Copyright © 2017 John Wiley & Sons, Ltd.     

Selection models and pattern-mixture models to analyse longitudinal quality of life data subject to drop-out

Longitudinally observed quality of life data with large amounts of drop-out are analysed. First we used the selection modelling framework, frequently used with incomplete studies. An alternative method consists of using pattern-mixture models. These are also straightforward to implement, but result in a different set of parameters for the measurement and drop-out mechanisms. Since selection models and pattern-mixture models are based upon different factorizations of the joint distribution of measurement and drop-out mechanisms, comparing both models concerning, for example, treatment effect, is a useful form of a sensitivity analysis.     

An approach to joint analysis of longitudinal measurements and competing risks failure time data

Joint analysis of longitudinal measurements and survival data has received much attention in recent years. However, previous work has primarily focused on a single failure type for the event time. In this paper we consider joint modelling of repeated measurements and competing risks failure time data to allow for more than one distinct failure type in the survival endpoint which occurs frequently in clinical trials. Our model uses latent random variables and common covariates to link together the sub-models for the longitudinal measurements and competing risks failure time data, respectively. An EM-based algorithm is derived to obtain the parameter estimates, and a profile likelihood method is proposed to estimate their standard errors. Our method enables one to make joint inference on multiple outcomes which is often necessary in analyses of clinical trials. Furthermore, joint analysis has several advantages compared with separate analysis of either the longitudinal data or competing risks survival data. By modelling the event time, the analysis of longitudinal measurements is adjusted to allow for non-ignorable missing data due to informative dropout, which cannot be appropriately handled by the standard linear mixed effects models alone. In addition, the joint model utilizes information from both outcomes, and could be substantially more efficient than the separate analysis of the competing risk survival data as shown in our simulation study. The performance of our method is evaluated and compared with separate analyses using both simulated data and a clinical trial for the scleroderma lung disease.     

Missing phenotype data imputation in pedigree data analysis

Mapping complex traits or phenotypes with small genetic effects, whose phenotypes may be modulated by temporal trends in families are challenging. Detailed and accurate data must be available on families, whether or not the data were collected over time. Missing data complicate matters in pedigree analysis, especially in the case of a longitudinal pedigree analysis. Because most analytical methods developed for the analysis of longitudinal pedigree data require no missing data, the researcher is left with the option of dropping those cases (individuals) with missing data from the analysis or imputing values for the missing data. We present the use of data augmentation within Bayesian polygenic and longitudinal polygenic models to produce k complete datasets. The data augmentation, or imputation step of the Markov chain Monte Carlo, takes into account the observed familial information and the observed subject information available at other time points. These k complete datasets can then be used to fit single time point or longitudinal pedigree models. By producing a set of k complete datasets and thus k sets of parameter estimates, the total variance associated with an estimate can be partitioned into a within-imputation and a between-imputation component. The method is illustrated using the Genetic Analysis Workshop simulated data.     

Network meta‐analysis of longitudinal data using fractional polynomials

Network meta‐analysis of randomized controlled trials (RCTs) are often based on one treatment effect measure per study. However, many studies report data at multiple time points. Furthermore, not all studies measure the outcomes at the same time points. As an alternative to a network meta‐analysis based on a synthesis of the results at one time point, a network meta‐analysis method is presented that allows for the simultaneous analysis of outcomes at multiple time points. The development of outcomes over time of interventions compared in an RCT is modeled with fractional polynomials, and the differences between the parameters of these polynomials within a trial are synthesized across studies with a Bayesian network meta‐analysis. The proposed models are illustrated with an analysis of RCTs evaluating interventions for osteoarthritis of the knee. Fixed and random effects second order fractional polynomials were applied to the case study. Network meta‐analysis with models that represent the treatment effects in terms of several parameters using fractional polynomials can be considered a useful addition to models for network meta‐analysis of repeated measures previously proposed. When RCTs report treatment effects at multiple follow‐up times, these models can be used to synthesize the results even if reporting times differ across the studies. Copyright © 2015 John Wiley & Sons, Ltd.     

Regression analysis of longitudinal data with informative observation times and application to medical cost data

Longitudinal data analysis is one of the most discussed and applied areas in statistics and a great deal of literature has been developed for it. However, most of the existing literature focus on the situation where observation times are fixed or can be treated as fixed constants. This paper considers the situation where these observation times may be random variables and more importantly, they may be related to the underlying longitudinal variable or process of interest. Furthermore, covariate effects may be time-varying. For the analysis, a joint modeling approach is proposed and in particular, for estimation of time-varying regression parameters, an estimating equation-based procedure is developed. Both asymptotic and finite sample properties of the proposed estimates are established. The methodology is applied to an acute myeloid leukemia trial that motivated this study.     

Inference of nested variance components in a longitudinal myopia intervention trial

This paper was motivated by a double-blind randomized clinical trial of myopia intervention. In addition to the primary goal of comparing treatment effects, we are concerned with the modelling of correlation that may come from two possible sources, one among the longitudinal observations and the other between measurements taken from both eyes per subject. The data are nested repeated measurements. We suggest three models for analysis. Each one expresses the correlation differently in various covariance structures. We articulate their differences and describe the implementations in estimation using commercial statistical software. The computer output can be further utilized to perform model selection with Schwarz criterion. Simulation studies are conducted to evaluate the performance under each model. Data of the myopia intervention trial are reanalysed with these models for illustration. The results indicate that atropine is more effective in reducing the progression rate, the rates are homogeneous across subjects, and, among the suggested models, the one with independent random effects of two eyes fits best. We conclude that model selection is a crucial step before making inference with estimates; otherwise the correlation may be attributed incorrectly to a different mechanism. The same conclusion applies to other variance components as well.     

Multiple outputation for the analysis of longitudinal data subject to irregular observation

Observational cohort studies often feature longitudinal data subject to irregular observation. Moreover, the timings of observations may be associated with the underlying disease process and must thus be accounted for when analysing the data. This paper suggests that multiple outputation, which consists of repeatedly discarding excess observations, may be a helpful way of approaching the problem. Multiple outputation was designed for clustered data where observations within a cluster are exchangeable; an adaptation for longitudinal data subject to irregular observation is proposed. We show how multiple outputation can be used to expand the range of models that can be fitted to irregular longitudinal data. Copyright © 2015 John Wiley & Sons, Ltd.     

Quantile causal mediation analysis allowing longitudinal data

Mediation analysis has mostly been conducted with mean regression models. With this approach modeling means, formulae for direct and indirect effects are based on changes in means, which may not capture effects that occur in units at the tails of mediator and outcome distributions. Individuals with extreme values of medical endpoints are often more susceptible to disease and can be missed if one investigates mean changes only. We derive the controlled direct and indirect effects of an exposure along percentiles of the mediator and outcome using quantile regression models and a causal framework. The quantile regression models can accommodate an exposure‐mediator interaction and random intercepts to allow for longitudinal mediator and outcome. Because DNA methylation acts as a complex “switch” to control gene expression and fibrinogen is a cardiovascular factor, individuals with extreme levels of these markers may be more susceptible to air pollution. We therefore apply this methodology to environmental data to estimate the effect of air pollution, as measured by particle number, on fibrinogen levels through a change in interferon‐gamma (IFN‐γ) methylation. We estimate the controlled direct effect of air pollution on the qth percentile of fibrinogen and its indirect effect through a change in the pth percentile of IFN‐γ methylation. We found evidence of a direct effect of particle number on the upper tail of the fibrinogen distribution. We observed a suggestive indirect effect of particle number on the upper tail of the fibrinogen distribution through a change in the lower percentiles of the IFN‐γ methylation distribution.     

Sample size and power determination in joint modeling of longitudinal and survival data

Owing to the rapid development of biomarkers in clinical trials, joint modeling of longitudinal and survival data has gained its popularity in the recent years because it reduces bias and provides improvements of efficiency in the assessment of treatment effects and other prognostic factors. Although much effort has been put into inferential methods in joint modeling, such as estimation and hypothesis testing, design aspects have not been formally considered. Statistical design, such as sample size and power calculations, is a crucial first step in clinical trials. In this paper, we derive a closed-form sample size formula for estimating the effect of the longitudinal process in joint modeling, and extend Schoenfeld's sample size formula to the joint modeling setting for estimating the overall treatment effect. The sample size formula we develop is quite general, allowing for p-degree polynomial trajectories. The robustness of our model is demonstrated in simulation studies with linear and quadratic trajectories. We discuss the impact of the within-subject variability on power and data collection strategies, such as spacing and frequency of repeated measurements, in order to maximize the power. When the within-subject variability is large, different data collection strategies can influence the power of the study in a significant way. Optimal frequency of repeated measurements also depends on the nature of the trajectory with higher polynomial trajectories and larger measurement error requiring more frequent measurements.     

Testing model fit in longitudinal data analysis against alternatives with omitted covariates

Several types of common model misspecifications can be re-formulated as problems of omitted covariates. These include situations with unmeasured confounders, measurement errors in observed covariates and informative censoring. Longitudinal data present special opportunities for detecting omitted covariates that are related to the observed ones differently across time than across individuals. This situation arises with period and cohort effects, as well as with usual formulations of classical measurement error in observed covariates. In this article we focus on testing for the existence of omitted covariates in longitudinal data analysis when models are fit by generalized estimation equations. When omitted covariates are present, specification of the correct link function conditionally on only observed covariates under the alternative usually involves complicated numerical integration. We propose a quasi-score test statistic that avoids the need to fit such alternative models. The statistic is asymptotically chi-square distributed under the null hypothesis of no omitted covariates with degrees of freedom determined by the assumed alternative structure. We study the significance level and the power of the quasi-score test in linear and logistic regression models. The test is then applied to an analysis of excessive daytime sleepiness.     

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.     

Smooth bootstrap methods for analysis of longitudinal data

In analysis of longitudinal data, the variance matrix of the parameter estimates is usually estimated by the 'sandwich' method, in which the variance for each subject is estimated by its residual products. We propose smooth bootstrap methods by perturbing the estimating functions to obtain 'bootstrapped' realizations of the parameter estimates for statistical inference. Our extensive simulation studies indicate that the variance estimators by our proposed methods can not only correct the bias of the sandwich estimator but also improve the confidence interval coverage. We applied the proposed method to a data set from a clinical trial of antibiotics for leprosy.