Using multiple imputation to estimate cumulative distribution functions in longitudinal data analysis with data missing at random

In longitudinal clinical studies, after randomization at baseline, subjects are followed for a period of time for development of symptoms. The interested inference could be the mean change from baseline to a particular visit in some lab values, the proportion of responders to some threshold category at a particular visit post baseline, or the time to some important event. However, in some applications, the interest may be in estimating the cumulative distribution function (CDF) at a fixed time point post baseline. When the data are fully observed, the CDF can be estimated by the empirical CDF. When patients discontinue prematurely during the course of the study, the empirical CDF cannot be directly used. In this paper, we use multiple imputation as a way to estimate the CDF in longitudinal studies when data are missing at random. The validity of the method is assessed on the basis of the bias and the Kolmogorov–Smirnov distance. The results suggest that multiple imputation yields less bias and less variability than the often used last observation carried forward method. Copyright © 2013 John Wiley & Sons, Ltd.     

A semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data

We propose a flexible semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data. The model can handle either single cycle or, more generally, multiple consecutive cycle data. The approach models the mean of responses by parametric fixed effects and a smooth nonparametric function for the underlying time effects, and the relationship across the bivariate responses by a bivariate Gaussian random field and a joint distribution of random effects. The proposed model not only can model complicated individual profiles, but also allows for more flexible within-subject and between-response correlations. The fixed effects regression coefficients and the nonparametric time functions are estimated using maximum penalized likelihood, where the resulting estimator for the nonparametric time function is a cubic smoothing spline. The smoothing parameters and variance components are estimated simultaneously using restricted maximum likelihood. Simulation results show that the parameter estimates are close to the true values. The fit of the proposed model on a real bivariate longitudinal dataset of pre-menopausal women also performs well, both for a single cycle analysis and for a multiple consecutive cycle analysis. The Canadian Journal of Statistics 48: 471–498; 2020 © 2020 Statistical Society of Canada     

Serial correlation or random subject effects

In longitudinal data analysis with random subject effects, there is often within subject serial correlation and possibly unequally spaced observations. This serial correlation can be partially confounded with the random between subject effects. In real data, it is often not clear whether there is serial correlation, random subject effects or both. Using inference based on the likelihood function, it is not always possible to identify the correct model, especially in small samples. However, it is important that some effort be made to attempt to find a good model rather than just making assumptions. This often means trying models with random coefficients, with serial correlation, and with both. Model selection criteria such as likelihood ratio tests and Akaike's Information Criterion (AIC) can be used. The problem of modelling serial correlation with unequally spaced observations is addressed. A real data example is presented where there is an apparent heterogeneity of variances, possible serial correlation and between subject random effects. In this example, it turns out that the random subject effects explains both the serial correlation and the variance heterogeneity.     

Identifying treatment effects using trimmed means when data are missing not at random

Patients often discontinue from a clinical trial because their health condition is not improving or they cannot tolerate the assigned treatment. Consequently, the observed clinical outcomes in the trial are likely better on average than if every patient had completed the trial. If these differences between trial completers and non-completers cannot be explained by the observed data, then the study outcomes are missing not at random (MNAR). One way to overcome this problem—the trimmed means approach for missing data due to study discontinuation—sets missing values as the worst observed outcome and then trims away a fraction of the distribution from each treatment arm before calculating differences in treatment efficacy (Permutt T, Li F. Trimmed means for symptom trials with dropouts. Pharm Stat. 2017;16(1):20–28). In this paper, we derive sufficient and necessary conditions for when this approach can identify the average population treatment effect. Simulation studies show the trimmed means approach's ability to effectively estimate treatment efficacy when data are MNAR and missingness due to study discontinuation is strongly associated with an unfavorable outcome, but trimmed means fail when data are missing at random. If the reasons for study discontinuation in a clinical trial are known, analysts can improve estimates with a combination of multiple imputation and the trimmed means approach when the assumptions of each hold. We compare the methodology to existing approaches using data from a clinical trial for chronic pain. An R package trim implements the method. When the assumptions are justifiable, using trimmed means can help identify treatment effects notwithstanding MNAR data.     

On nonlinear random effects models for repeated measurements

Linear random effects models for longitudinal data discussed by Laird and Ware (1982), Jennrich and Schluchter (1986), Lange and Laird (1989), and others are extended in a straight forward manner to nonlinear random effects models. This results in a simple computational approach which accommodates patterned covariance matrices and data insufficient for fitting each subject separately. The technique is demonstrated with an interesting medical data set, and a short, simple SAS PROC IML program based on the EM algorithm is presented.     

Bayesian analysis of panel data using an MTAR model

Bayesian analysis of panel data using a class of momentum threshold autoregressive (MTAR) models is considered. Posterior estimation of parameters of the MTAR models is done by using a simple Markov Chain Monte Carlo (MCMC) algorithm. Selection of appropriate differenced variables, test for asymmetry and unit roots are recast as model selections and a simple way of computing posterior probabilities of the candidate models is proposed. The proposed method is applied to the yearly unemployment rates of 51 US states and the results show strong evidence of stationarity and asymmetry.     

A Bayesian approach for analysing longitudinal nominal outcomes using random coefficients transitional generalized logit model: an application to the labour force survey data

A random-effects transition model is proposed to model the economic activity status of household members. This model is introduced to take into account two kinds of correlations; one due to the longitudinal nature of the study, which will be considered using a transition parameter, and the other due to the existing correlation between responses of members of the same household which is taken into account by introducing random coefficients into the model. The results are presented based on the homogeneous (all parameters are not changed by time) and non-homogeneous Markov models with random coefficients. A Bayesian approach via the Gibbs sampling is used to perform parameter estimation. Results of using random-effects transition model are compared, using deviance information criterion, with those of three other models which exclude random effects and/or transition effects. It is shown that the full model gains more precision due to the consideration of all aspects of the process which generated the data. To illustrate the utility of the proposed model, a longitudinal data set which is extracted from the Iranian Labour Force Survey is analysed to explore the simultaneous effect of some covariates on the current economic activity as a nominal response. Also, some sensitivity analyses are performed to assess the robustness of the posterior estimation of the transition parameters to the perturbations of the prior parameters.     

A skew-normal random effects model for longitudinal ordinal categorical responses with missing data

Missing values are common in longitudinal data studies. The missing data mechanism is termed non-ignorable (NI) if the probability of missingness depends on the non-response (missing) observations. This paper presents a model for the ordinal categorical longitudinal data with NI non-monotone missing values. We assumed two separate models for the response and missing procedure. The response is modeled as ordinal logistic, whereas the logistic binary model is considered for the missing process. We employ these models in the context of so-called shared-parameter models, where the outcome and missing data models are connected by a common set of random effects. It is commonly assumed that the random effect follows the normal distribution in longitudinal data with or without missing data. This can be extremely restrictive in practice, and it may result in misleading statistical inferences. In this paper, we instead adopt a more flexible alternative distribution which is called the skew-normal distribution. The methodology is illustrated through an application to Schizophrenia Collaborative Study data [19 D. Hedeker, Generalized linear mixed models, in Encyclopedia of Statistics in Behavioral Science, B. Everitt and D. Howell, eds., John Wiley, London, 2005, pp. 729–738. [Google Scholar]] and a simulation.     

Functional data analysis: estimation of the relative error in functional regression under random left-truncation model

In this paper, we investigate the relationship between a functional random covariable and a scalar response which is subject to left-truncation by another random variable. Precisely, we use the mean squared relative error as a loss function to construct a nonparametric estimator of the regression operator of these functional truncated data. Under some standard assumptions in functional data analysis, we establish the almost sure consistency, with rates, of the constructed estimator as well as its asymptotic normality. Then, a simulation study, on finite-sized samples, was carried out in order to show the efficiency of our estimation procedure and to highlight its superiority over the classical kernel estimation, for different levels of simulated truncated data.     

Variance function in regression analysis of longitudinal data using the generalized estimating equation approach

Longitudinal or clustered response data arise in many applications such as biostatistics, epidemiology and environmental studies. The repeated responses cannot in general be assumed to be independent. One method of analysing such data is by using the generalized estimating equations (GEE) approach. The current GEE method for estimating regression effects in longitudinal data focuses on the modelling of the working correlation matrix assuming a known variance function. However, correct choice of the correlation structure may not necessarily improve estimation efficiency for the regression parameters if the variance function is misspecified [Wang YG, Lin X. Effects of variance-function misspecification in analysis of longitudinal data. Biometrics. 2005;61:413–421]. In this connection two problems arise: finding a correct variance function and estimating the parameters of the chosen variance function. In this paper, we study the problem of estimating the parameters of the variance function assuming that the form of the variance function is known and then the effect of a misspecified variance function on the estimates of the regression parameters. We propose a GEE approach to estimate the parameters of the variance function. This estimation approach borrows the idea of Davidian and Carroll [Variance function estimation. J Amer Statist Assoc. 1987;82:1079–1091] by solving a nonlinear regression problem where residuals are regarded as the responses and the variance function is regarded as the regression function. A limited simulation study shows that the proposed method performs at least as well as the modified pseudo-likelihood approach developed by Wang and Zhao [A modified pseudolikelihood approach for analysis of longitudinal data. Biometrics. 2007;63:681–689]. Both these methods perform better than the GEE approach.     

Likelihood-based approach for analysis of longitudinal nominal data using marginalized random effects models

Likelihood-based marginalized models using random effects have become popular for analyzing longitudinal categorical data. These models permit direct interpretation of marginal mean parameters and characterize the serial dependence of longitudinal outcomes using random effects [12,22]. In this paper, we propose model that expands the use of previous models to accommodate longitudinal nominal data. Random effects using a new covariance matrix with a Kronecker product composition are used to explain serial and categorical dependence. The Quasi-Newton algorithm is developed for estimation. These proposed methods are illustrated with a real data set and compared with other standard methods.     

Likelihood inference for a step-stress partially accelerated life test model with Type-I progressively hybrid censored data from Weibull distribution

Recently, progressively hybrid censoring schemes have become quite popular in life testing and reliability studies. In this article, the point and interval maximum-likelihood estimations of Weibull distribution parameters and the acceleration factor are considered. The estimation process is performed under Type-I progressively hybrid censored data for a step-stress partially accelerated test model. The biases and mean square errors of the maximum-likelihood estimators are computed to assess their performances in the presence of censoring developed in this article through a Monte Carlo simulation study.     

Bayesian estimation of varying-coefficient models with missing data,with application to the Singapore Longitudinal Aging Study

Motivated by the Singapore Longitudinal Aging Study (SLAS), we propose a Bayesian approach for the estimation of semiparametric varying-coefficient models for longitudinal continuous and cross-sectional binary responses. These models have proved to be more flexible than simple parametric regression models. Our development is a new contribution towards their Bayesian solution, which eases computational complexity. We also consider adapting all kinds of familiar statistical strategies to address the missing data issue in the SLAS. Our simulation results indicate that a Bayesian imputation (BI) approach performs better than complete-case (CC) and available-case (AC) approaches, especially under small sample designs, and may provide more useful results in practice. In the real data analysis for the SLAS, the results for longitudinal outcomes from BI are similar to AC analysis, differing from those with CC analysis.     

Robust estimation and model identification for longitudinal data varying-coefficient model

It is well known that M-estimation is a widely used method for robust statistical inference and the varying coefficient models have been widely applied in many scientific areas. In this paper, we consider M-estimation and model identification of bivariate varying coefficient models for longitudinal data. We make use of bivariate tensor-product B-splines as an approximation of the function and consider M-type regression splines by minimizing the objective convex function. Mean and median regressions are included in this class. Moreover, with a double smoothly clipped absolute deviation (SCAD) penalization, we study the problem of simultaneous structure identification and estimation. Under approximate conditions, we show that the proposed procedure possesses the oracle property in the sense that it is as efficient as the estimator when the true model is known prior to statistical analysis. Simulation studies are carried out to demonstrate the methodological power of the proposed methods with finite samples. The proposed methodology is illustrated with an analysis of a real data example.     

A mixed effects model for analyzing area under the curve of longitudinally measured biomarkers with missing data

A simple approach for analyzing longitudinally measured biomarkers is to calculate summary measures such as the area under the curve (AUC) for each individual and then compare the mean AUC between treatment groups using methods such as t test. This two-step approach is difficult to implement when there are missing data since the AUC cannot be directly calculated for individuals with missing measurements. Simple methods for dealing with missing data include the complete case analysis and imputation. A recent study showed that the estimated mean AUC difference between treatment groups based on the linear mixed model (LMM), rather than on individually calculated AUCs by simple imputation, has negligible bias under random missing assumptions and only small bias when missing is not at random. However, this model assumes the outcome to be normally distributed, which is often violated in biomarker data. In this paper, we propose to use a LMM on log-transformed biomarkers, based on which statistical inference for the ratio, rather than difference, of AUC between treatment groups is provided. The proposed method can not only handle the potential baseline imbalance in a randomized trail but also circumvent the estimation of the nuisance variance parameters in the log-normal model. The proposed model is applied to a recently completed large randomized trial studying the effect of nicotine reduction on biomarker exposure of smokers.     

Fitting Bayesian multiple random effects models

Bayesian random effects models may be fitted using Gibbs sampling, but the Gibbs sampler can be slow mixing due to what might be regarded as lack of model identifiability. This slow mixing substantially increases the number of iterations required during Gibbs sampling. We present an analysis of data on immunity after Rubella vaccinations which results in a slow-mixing Gibbs sampler. We show that this problem of slow mixing can be resolved by transforming the random effects and then, if desired, expressing their joint prior distribution as a sequence of univariate conditional distributions. The resulting analysis shows that the decline in antibodies after Rubella vaccination is relatively shallow compared to the decline in antibodies which has been shown after Hepatitis B vaccination.     

Longitudinal data analysis for generalized linear models with follow‐up dependent on outcome‐related variables

In longitudinal studies, observation times are often irregular and subject‐specific. Frequently they are related to the outcome measure or other variables that are associated with the outcome measure but undesirable to condition upon in the model for outcome. Regression analyses that are unadjusted for outcome‐dependent follow‐up then yield biased estimates. The authors propose a class of inverse‐intensity rate‐ratio weighted estimators in generalized linear models that adjust for outcome‐dependent follow‐up. The estimators, based on estimating equations, are very simple and easily computed; they can be used under mixtures of continuous and discrete observation times. The predictors of observation times can be past observed outcomes, cumulative values of outcome‐model covariates and other factors associated with the outcome. The authors validate their approach through simulations and they illustrate it using data from a supported housing program from the US federal government.     

Longitudinal data analysis of mean passage time among malnutrition states: an application of Markov chains

Clinical prognosis of patients can be best described from a longitudinal study and a Markov regression model is an appropriate way of analyzing the prognosis of disease when the outcomes are serially dependent. Mean first passage time (MFPT) is a method to estimate the average number of transitions between the states of a Markov chain. The present study used the secondary data from a longitudinal study which was done during 1982–1986. This study was to illustrate the MFPT among the states of malnutrition, which were classified as Normal, Mild/Moderate and Severe among children aged 5–7 years, in South India. The 95% confidence interval (CI) for the MFPT was calculated using Monte Carlo simulation. Markov regression models were used to test for the association of state transitions across the risk factors. The average time taken for an underweight child to transit from Severe state of malnutrition to become Normal was nearly 2.73 (95% CI 2.60–2.86) years and 3.41 (95% CI 3.25–3.58) years in Rural area and 2.31(95% CI 2.20–2.42) in Urban area. The significant difference between the MFPT for some risk factors are useful to plan interventions. It will especially be useful to find the impact of duration among school-going children on their cognitive disorders.     

Longitudinal data analysis of animal growth via multivariate dynamic models

We propose models to analyze animal growth data with the aim of estimating and predicting quantities of biological and economical interest such as the maturing rate and asymptotic weight. It is also studied the effect of environmental factors of relevant influence in the growth process. The models considered in this paper are based on an extension and specialization of the dynamic hierarchical model (Gamerman & Migon, 1993) to a non–linear growth curve setting, where some of the growth curve parameters are considered exchangeable among the units. The inference for these models are approximate conjugate analysis based on Taylor series expansions and linear Bayes procedures