Nonparametric Bayesian functional two-part random effects model for longitudinal semicontinuous data analysis.

Two-part models are an attractive approach for analysing longitudinal semicontinuous data consisting of a mixture of true zeros and continuously distributed positive values. When the population-averaged (marginal) covariate effects are of interest, two-part models that provide straightforward interpretation of the marginal effects are desirable. Presently, the only available approaches for fitting two-part marginal models to longitudinal semicontinuous data are computationally difficult to implement. Therefore, there exists a need to develop two-part marginal models that can be easily implemented in practice. We propose a fully likelihood-based two-part marginal model that satisfies this need by using the bridge distribution for the random effect in the binary part of an underlying two-part mixed model; and its maximum likelihood estimation can be routinely implemented via standard statistical software such as the SAS NLMIXED procedure. We illustrate the usage of this new model by investigating the marginal effects of pre-specified genetic markers on physical functioning, as measured by the Health Assessment Questionnaire, in a cohort of psoriatic arthritis patients from the University of Toronto Psoriatic Arthritis Clinic. An added benefit of our proposed marginal model when compared to a two-part mixed model is the robustness in regression parameter estimation when departure from the true random effects structure occurs. This is demonstrated through simulation.

Two-part mixed effects models are often used for analyzing longitudinal data with many zeros. Typically, these models are formulated with binary and continuous components separately with random effects that are correlated between the two components. Researchers have developed maximum-likelihood and Bayesian approaches for fitting these models that often require using particular software packages or very specialized software. We propose an imputation approach that will allow practitioners to separately use standard linear and generalized linear mixed models to estimate the fixed effects for two-part mixed effects models with complex random effects structures. An approximation to the conditional distribution of positive measurements given an individual's pattern of non-zero measurements is proposed that can be easily estimated and then imputed from. We show that for a wide range of parameter values, the imputation approach results in nearly unbiased estimation and can be implemented with standard software. We illustrate the proposed imputation approach for the analysis of longitudinal clinical trial data with many zeros.

Longitudinal time use data afford the opportunity to study within- and between-individual differences, but can present challenges in data analysis. Often the response set includes a large number of zeros representing those who did not engage in the target behavior. Coupled with this is a continuous measure of time use for those who did engage. The latter is strictly positive and skewed to the right if relatively few individuals engage in the behavior to a greater extent. Data analysis is further complicated for repeated measures, because within-individual responses are typically correlated, and some respondents may have missing data. This combination of zeros and positive responses is characteristic of a type of semicontinuous data in which the response is equal to a discrete value and is otherwise continuous. Two-part models have been successfully applied to cross-sectional time use data when the research goals distinguish between a respondent's likelihood to engage in a behavior and the time spent conditional on any time being spent, as these models allow different covariates to relate to each distinct aspect of a behavior. Two-part mixed-effects models extend two-part models for analysis of longitudinal semicontinuous data to simultaneously address longitudinal decisions to engage in a behavior and time spent conditional on any time spent. Heterogeneity between and within individuals can be studied in unique ways. This paper presents applications of these models to daily diary data to study individual differences in time spent relaxing or engaged in leisure activities for an adult sample.

The analysis of longitudinal data with non-ignorable missingness remains an active area in biostatistics research. This article discusses various random effects and latent process models which have been proposed for analyzing longitudinal binary data subject to both non-ignorable intermittent missing data and dropout. These models account for non-ignorable missingness by introducing random effects or a latent process which is shared between the response model and the model for the missing-data mechanism. We discuss various random effects and latent processes approaches and compare these approaches with analyses from an opiate clinical trial data set, which had high proportion of intermittent missingness and dropout. We also compare these random effect and latent process approaches with other methods for accounting for non-ignorable missingness using this data set.

The context of this thesis explores efficiency measures in higher education (HE) in England. Measures of efficiency serve as a crucial link between the economic sustainability of the HE sector and the policymaking establishment. Given that the idea of efficient allocation of resources, in a period of tighter budget constraints, curtailed government funding and increasing competition for a greater share for research funding and number of students has such a powerful influence, the concept of efficiency becomes meaningful serving as a basis for decisions to improve resource allocation. Understanding the nature of efficiency aims to put in place a simpler and more efficient HE and research system in England that encourages competition and choice, enhances quality, and ensures greater accountability and value for money. The thesis unfolds two main disciplines of technical and cost efficiency in HE in England. Therefore the research objectives discussed are driven upon that conceptualization of efficiency and provide further insights into first, the effects of merger activity on efficiency, and second on whether permanent or transient (in) efficiency dominates the English HE sector. Those topics are key aspects and critically important for both policy change and ongoing institutional and structural reform and as thus are explored in the lines of this thesis. Regarding the first research objective on the potential effect of mergers on Higher Education Institutions’ (HEIs) efficiency, in a first stage analysis efficiency scores of English universities are derived for a 17-year period using the frontier estimation method data envelopment analysis (DEA). A second stage analysis explores the effect of merger and other factors on efficiency. We find that mean efficiency for the sector has varied around 60 percent to 70 percent, but that the efficiency levels of the vast majority of individual HEIs are not significantly different from each other. Merged HEIs have efficiency which is 5 percentage points higher post-merger than non-merging HEIs holding all else constant; moreover the efficiency impact of merger comes within 2 years of the merger taking place. Of the other factors included in the second stage analysis, pre-1992 universities have lower efficiency than other types of institution. In addition, having a higher proportion of income from government sources is an incentive to greater efficiency. Finally, a sensitivity analysis was conducted which exposed the post-merger efficiency results to a different method assessment as a validation test of the proposed policy implications. The sensitivity analysis resulted in confirming the main findings of efficiency improvements in the units received the treatment of merger. Turning to the second research objective, a common weakness in most of the models dealing with efficiency is their deficiency to account for unobserved heterogeneity that finally lead to biased efficiency estimates. Most of the cost efficiency frontier models, focused either on the transient or on the persistent part of cost inefficiency, confounding firm effects (that are not part of inefficiency) with persistent inefficiency or blending persistent inefficiency with latent heterogeneity. However a decomposition of the two parts, persistent (long –term) and transient/residual (short-term) inefficiencies, provides an in-depth analysis of whether short term practises or more long term structural changes within colleges and universities affect the degree of cost efficiency in the English HE sector. This distinction seems to be further appealing to the policy makers as a regulatory asset that aims at improving the efficiency of the sector through incentive reforms. Hence, more recent developments in panel data allow a further appealing distinction in the cost efficiency of HEIs in which unobserved firm effects (firm heterogeneity) can be disentangled from time invariant and time varying inefficiency. Hence the purpose of this thesis is partly to assess the level of persistent and transient inefficiency in the English HE sector from 2008/09 to 2013/14 by using a four-way error component model (persistent and transient inefficiency, random firm effects and noise) and so as to retain the apparatus of statistical inference stemming from a generalised true random effects (GTRE) model based on maximum simulated likelihood (MSL) techniques. In order to provide evidence that the aforementioned method ameliorates the predicted power of the model we offer a comparative study through the fundamental models applied in the literature so far. Consequently, statistical inference will be attempted by countering the efficiency estimates of a GRTE model with a random effects (RE) model proposed by Pitt and Lee (1981), informative on the persistent part, and a true random effects model (TRE) proposed by Greene (2005a, 2005b), enlightening the transient part. Finally, omitted variables bias will be controlled by implementing a MGTRE model, minimizing the resulting heterogeneity bias. The comparison reinforces the validity of the GTRE model since it captures every single component of inefficacy while heterogeneity is controlled. For the English HE inefficiency is considered as persistent since short-run efficiency estimates are proven to be higher than the long-run. This gives further rise for more comprehensive and structural changes rather than simple mechanisms for short-term cost savings.

Longitudinal designs for biomarker traits are very appealing for a number of reasons. First, the collection of repeated biomarker measurements over time allows investigators to characterize both the overall and individual trajectories in these measurements. This helps investigators to understand overall effects as well as how individuals vary around these average effects. Distinguishing individual heterogeneity from average trend is impossible without longitudinal data. This distinction is important for understanding individual between-subject in its own right, as well as in developing predictors of disease outcome from longitudinal marker data. There are many statistical issues that are unique to longitudinal data analysis ranging from correctly specifying realistic models which characterize average changes over time and individual variation, accounting for observations that are taken at unequally spaced observation times which may differ between individuals, to accounting for missing data due to occasionally missed measurements or loss to follow-up. When the reason for missing data is not related to the underlying observation process, most longitudinal data analysis methods can be routinely applied. However, complex modelling is necessary when the reason for missing an observation or dropout is directly related to the underlying outcome process [1–2]. Statistical techniques that appropriately analyze continuous and discrete longitudinal data have been discussed by many authors. Diggle, et al. [3] provides a good summary of these techniques for the practitioner, while Fitzmaurice et al. [4] discusses recent advances from a more technical perspective. The three papers in this section present innovative approaches for solving important methodological problems in biomarker studies. Most standard longitudinal data analysis techniques assume that the measurement process is independent of the outcome process. However, when the outcome is related to the underlying observation process, failure to properly account for the observation process can result in bias [5]. Schildcrout et al.[6] proposed an approach for regression modelling for longitudinal binary data where the outcome is measured with probabilities determined by an auxiliary variable that is related, but not identical to the binary outcome. Modelling multivariate longitudinal biomarker data is an increasingly common issue with the advent of an increasing number of important biomarker techniques (e.g., multiple cytokines in a multiplex assay or multiple reproductive hormones that describe complex biological phenomenon). Although various authors have proposed techniques for analyzing multivariate longitudinal data [7–8 and others], little work has focused on estimating multivariate linear mixed models under known constraints in the relationship between the multivariate outcomes. Roy et al [9] proposes such an approach for analyzing multivariate reproductive hormones over the menstrual cycle, where established relationships have solid biological justification. Finally, the collection of longitudinal biomarker data as disease predictors is becoming an increasingly important design in diagnostic medicine. Pfeiffer et al. [10] proposes flexible non-parametric methodology for combining biomarkers that exploits the longitudinal structure in formulating a score that is used for predicting a binary outcome.

Valid inference can be drawn from a random-effects model for repeated measures that are incomplete if whether the data are missing or not, known as missingness, is independent of the missing data. Data that are missing completely at random or missing at random are two data types for which missingness is ignorable. Given ignorable missingness, statistical inference can proceed without addressing the source of the missing data in the model. If the missingness is not ignorable, however, recommendations are to fit multiple models that represent different plausible explanations of the missing data. A popular choice in methods for evaluating nonignorable missingness is a random-effects pattern-mixture model that extends a random-effects model to include one or more between-subjects variables that represent fixed patterns of missing data. Generally straightforward to implement, a fixed pattern-mixture model is one among several options for assessing nonignorable missingness, and when it is used as the sole model to address nonignorable missingness, understanding the impact of missingness is greatly limited. This paper considers alternatives to a fixed pattern-mixture model for nonignorable missingness that are generally straightforward to fit and encourage researchers to give greater attention to the possible impact of nonignorable missingness in longitudinal data analysis. Patterns of both monotonic and non-monotonic (intermittently) missing data are addressed. Empirical longitudinal psychiatric data are used to illustrate the models. A small Monte Carlo data simulation study is presented to help illustrate the utility of such methods.Supplementary InformationThe online version contains supplementary material available at 10.3758/s13428-023-02128-y.

In clinical trials a fixed effects research model assumes that the patients selected for a specific treatment have the same true quantitative effect and that the differences observed are residual error. If, however, we have reasons to believe that certain patients respond differently from others, then the spread in the data is caused not only by the residual error but also by between-patient differences. The latter situation requires a random effects model. To explain random effects models in analysis of variance and to give examples of studies qualifying for them. If in a particular study the data are believed to be different from one assessing doctor to the other, and if we have no prior theory that 1 or 2 assessing doctors produced the highest scores, but rather expect there may be heterogeneity in the population of doctors at large, then a random effects model will be appropriate. For that purpose between-doctor variability is compared to within-doctor variability. If the data of 2 separate studies of the same new treatment are analyzed simultaneously, it will be safe to consider an interaction effect between the study number and treatment efficacy. If the interaction is significant, a random effects model with the study number as random variable, will be adequate. For that purpose the treatment effect is tested against the interaction effect. In a multicenter study the data are at risk of interaction between centers and treatment efficacy. If this interaction is significant, a random effects model with the health center as random variable, will be adequate. The treatment effect is tested not against residual but against the interaction. If in a crossover study a treatment difference is not observed, this may be due to random subgroup effects. A post-hoc random effects model, with patients effect as random variable, testing the treatment effect against treatments x patients interaction, will be appropriate. Random effects research models enable the assessment of an entire sample of data for subgroup differences without need to split the data into subgroups. Clinical investigators, in general, are hardly aware of this possibility and, therefore, wrongly assess random effects as fixed effects leading to a biased interpretation of the data.

This paper is motivated from a neurophysiological study of muscle fatigue, in which biomedical researchers are interested in understanding the time-dependent relationships of handgrip force and electromyography measures. A varying coefficient model is appealing here to investigate the dynamic pattern in the longitudinal data. The response variable in the study is continuous but bounded on the standard unit interval (0, 1) over time, while the longitudinal covariates are contaminated with measurement errors. We propose a generalization of varying coefficient models for the longitudinal proportional data with errors-in-covariates. We describe two estimation methods with penalized splines, which are formalized under a Bayesian inferential perspective. The first method is an adaptation of the popular regression calibration approach. The second method is based on a joint likelihood under the hierarchical Bayesian model. A simulation study is conducted to evaluate the efficacy of the proposed methods under different scenarios. The analysis of the neurophysiological data is presented to demonstrate the use of the methods.

Random effects models are commonly used to analyze longitudinal categorical data. Marginalized random effects models are a class of models that permit direct estimation of marginal mean parameters and characterize serial correlation for longitudinal categorical data via random effects (Heagerty, 1999). Marginally specified logistic-normal models for longitudinal binary data. Biometrics 55, 688-698; Lee and Daniels, 2008. Marginalized models for longitudinal ordinal data with application to quality of life studies. Statistics in Medicine 27, 4359-4380). In this paper, we propose a Kronecker product (KP) covariance structure to capture the correlation between processes at a given time and the correlation within a process over time (serial correlation) for bivariate longitudinal ordinal data. For the latter, we consider a more general class of models than standard (first-order) autoregressive correlation models, by re-parameterizing the correlation matrix using partial autocorrelations (Daniels and Pourahmadi, 2009). Modeling covariance matrices via partial autocorrelations. Journal of Multivariate Analysis 100, 2352-2363). We assess the reasonableness of the KP structure with a score test. A maximum marginal likelihood estimation method is proposed utilizing a quasi-Newton algorithm with quasi-Monte Carlo integration of the random effects. We examine the effects of demographic factors on metabolic syndrome and C-reactive protein using the proposed models.

Despite the long-standing discussion on fixed effects (FE) and random effects (RE) models, how and under what conditions both methods can eliminate unmeasured confounding bias has not yet been widely understood in practice. Using a simple pretest-posttest design in a linear setting, this paper translates the conventional algebraic formalization of FE and RE models into causal graphs and provides intuitively accessible graphical explanations about their data-generating and bias-removing processes. The proposed causal graphs highlight that FE and RE models consider different data-generating models. RE models presume a data-generating model that is identical to a randomized controlled trial, while FE models allow for unobserved time-invariant treatment-outcome confounding. Augmenting regular causal graphs that describe data-generating processes by adding the computational structures of FE and RE estimators, the paper visualizes how FE estimators (gain score and deviation score estimators) and RE estimators (quasi-deviation score estimators) offset unmeasured confounding bias. In contrast to standard regression or matching estimators that reduce confounding bias by blocking non-causal paths via conditioning, FE and RE estimators offset confounding bias by deliberately creating new non-causal paths and associations of opposite sign. Though FE and RE estimators are similar in their bias-offsetting mechanisms, the augmented graphs reveal their subtle differences that can result in different biases in observational studies.

We discuss the performance of non-parametric maximum likelihood (NPML) estimators for the distribution of a univariate random effect in the analysis of longitudinal data. For continuous data, we analyse generated and real data sets, and compare the NPML method to those that assume a Gaussian random effects distribution and to ordinary least squares. For binary outcomes we use generated data to study the moderate and large-sample performance of the NPML compared with a method based on a Gaussian random effect distribution in logistic regression. We find that estimated fixed effects are compatible for all approaches, but that appropriate standard errors for the NPML require adjusting the likelihood-based standard errors. We conclude that the non-parametric approach provides an attractive alternative to Gaussian-based methods, though additional evaluations are necessary before it can be recommended for general use.

Analysis of binary longitudinal data with time-varying effects

The linear mixed-effects model (LMM) is widely used in the analysis of clustered or longitudinal data. This paper aims to address analytic challenges arising from estimation and selection in the application of the LMM to high-dimensional longitudinal data. We develop a doubly regularized approach in the LMM to simultaneously select fixed and random effects. On the theoretical front, we establish large sample properties for the proposed method under the high-dimensional setting, allowing both numbers of fixed effects and random effects to be much larger than the sample size. We present new regularity conditions for the diverging rates, under which the proposed method achieves both estimation and selection consistency. In addition, we propose a new algorithm that solves the related optimization problem effectively so that its computational cost is comparable with that of the Newton-Raphson algorithm for maximum likelihood estimator in the LMM. Through simulation studies we assess performances of the proposed regularized LMM in both aspects of variable selection and estimation. We also illustrate the proposed method by two data analysis examples.

ObjectivesCatastrophic health expenditure (CHE) means that the medical spending of a household exceeds a certain level of capacity to pay. Previous studies of CHE have focused on benefits supported by...