Vernon T. Farewell

Bias in 2-part mixed models for longitudinal semicontinuous data

Semicontinuous data in the form of a mixture of zeros and continuously distributed positive values frequently arise in biomedical research. Two-part mixed models with correlated random effects are an attractive approach to characterize the complex structure of longitudinal semicontinuous data. In practice, however, an independence assumption about random effects in these models may often be made for convenience and computational feasibility. In this article, we show that bias can be induced for regression coefficients when random effects are truly correlated but misspecified as independent in a 2-part mixed model. Paralleling work on bias under nonignorable missingness within a shared parameter model, we derive and investigate the asymptotic bias in selected settings for misspecified 2-part mixed models. The performance of these models in practice is further evaluated using Monte Carlo simulations. Additionally, the potential bias is investigated when artificial zeros, due to left censoring from some detection or measuring limit, are incorporated. To illustrate, we fit different 2-part mixed models to the data from the University of Toronto Psoriatic Arthritis Clinic, the aim being to examine whether there are differential effects of disease activity and damage on physical functioning as measured by the health assessment questionnaire scores over the course of psoriatic arthritis. Some practical issues on variance component estimation revealed through this data analysis are considered.

Validation of the Toronto Psoriatic Arthritis Screen Version 2 (ToPAS 2)

Objective. We previously developed and performed an initial validation of a screening questionnaire, the Toronto Psoriatic Arthritis Screen (ToPAS), for psoriatic arthritis (PsA). In our original analysis, we found that the index constructed appeared to discriminate well between those with a confirmed diagnosis of PsA and those without PsA in various clinical settings. However, it was suggested that ToPAS would benefit from additional refinement to the questions and the scoring system, because items pertaining to axial involvement were not included in our original index. Subsequently, a second version of ToPAS was developed, ToPAS 2, which incorporated the suggested refinements. We aimed to validate ToPAS 2 as a screening instrument for PsA. Methods. ToPAS 2 was administered to 3 “diagnostic” groups of individuals — patients with PsA, patients with psoriasis, and healthy controls, and the data collected were analyzed. Results. It was found that the new version of ToPAS, ToPAS 2, again performed well, with the axial domain now featuring in the new scoring system. The constructed index, ToPAS2_cap, had an overall area under the receiver-operation curve of 0.910, with overall values of sensitivity and specificity, at a cutpoint of 8 (or 7), of 87.2% (92.0%) and 82.7% (77.2%), respectively. Conclusion. ToPAS 2 shows much promise as a screening instrument for identifying PsA both in people with psoriasis and in individuals from the general population. Its performance against other proposed screening instruments for PsA should be evaluated in other clinics and for other study designs.

A case-study in the clinical epidemiology of psoriatic arthritis: multistate models and causal arguments

In psoriatic arthritis, permanent joint damage characterizes disease progression and represents a major debilitating aspect of the disease. Understanding the process of joint damage will assist in the treatment and disease management of patients. Multistate models provide a means to examine patterns of disease, such as symmetric joint damage. Additionally, the link between damage and the dynamic course of disease activity (represented by joint swelling and stress pain) at both the individual joint level and otherwise can be represented within a correlated multistate model framework. Correlation is reflected through the use of random effects for progressive models and robust variance estimation for non-progressive models. Such analyses, undertaken with data from a large psoriatic arthritis cohort, are discussed and the extent to which they permit causal reasoning is considered. For this, emphasis is given to the use of the Bradford Hill criteria for causation in observational studies and the concept of local (in)dependence to capture the dynamic nature of the relationships.

Likelihood estimation for a longitudinal negative binomial regression model with missing outcomes

Joint damage in psoriatic arthritis can be measured by clinical and radiological methods, the former being done more frequently during longitudinal follow-up of patients. Motivated by the need to compare findings based on the different methods with different observation patterns, we consider longitudinal data where the outcome variable is a cumulative total of counts that can be unobserved when other, informative, explanatory variables are recorded. We demonstrate how to calculate the likelihood for such data when it is assumed that the increment in the cumulative total follows a discrete distribution with a location parameter that depends on a linear function of explanatory variables. An approach to the incorporation of informative observation is suggested. We present analyses based on an observational database from a psoriatic arthritis clinic. Although the use of the new statistical methodology has relatively little effect in this example, simulation studies indicate that the method can provide substantial improvements in bias and coverage in some situations where there is an important time varying explanatory variable.

A likelihood-based two-part marginal model for longitudinal semicontinuous data

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.

A corrected formulation for marginal inference derived from two-part mixed models for longitudinal semi-continuous data

For semi-continuous data which are a mixture of true zeros and continuously distributed positive values, the use of two-part mixed models provides a convenient modelling framework. However, deriving population-averaged (marginal) effects from such models is not always straightforward. Su et al. presented a model that provided convenient estimation of marginal effects for the logistic component of the two-part model but the specification of marginal effects for the continuous part of the model presented in that paper was based on an incorrect formulation. We present a corrected formulation and additionally explore the use of the two-part model for inferences on the overall marginal mean, which may be of more practical relevance in our application and more generally.

Tracing studies and analysis of the effect of loss to follow-up on mortality estimation from patient registry data

Before patient registries are used for studies of the long-term mortality that is associated with chronic medical conditions, the potential bias resulting from patients who become lost to follow-up must be investigated. A study design, used for a systemic lupus erythematosus patient registry, is described. The design involves tracing patients who are defined as lost to follow-up according to specific criteria. This provides supplementary information on the mortality experience of patients who are lost to (regular) follow-up. Some methods of analysis are described, based on comparing the mortality experience of patients when under regular follow-up with the experience of patients after they are deemed to be lost to follow-up. The effect of loss to follow-up, death reporting and visits to the clinic on estimation procedures is illustrated and recommendations are made for patient registries which are to be used in mortality studies. Copyright 2003 Royal Statistical Society.

Exploring the existence of a stayer population with mover–stayer counting process models: application to joint damage in psoriatic arthritis

Summary Many psoriatic arthritis patients do not progress to permanent joint damage in any of the 28 hand joints, even under prolonged follow‐up. This has led several researchers to fit models that estimate the proportion of stayers (those who do not have the propensity to experience the event of interest) and to characterize the rate of developing damaged joints in the movers (those who have the propensity to experience the event of interest). However, when fitted to the same data, the paper demonstrates that the choice of model for the movers can lead to widely varying conclusions on a stayer population, thus implying that, if interest lies in a stayer population, a single analysis should not generally be adopted. The aim of the paper is to provide greater understanding regarding estimation of a stayer population by comparing the inferences, performance and features of multiple fitted models to real and simulated data sets. The models for the movers are based on Poisson processes with patient level random effects and/or dynamic covariates, which are used to induce within‐patient correlation, and observation level random effects are used to account for time varying unobserved heterogeneity. The gamma, inverse Gaussian and compound Poisson distributions are considered for the random effects.

Pragmatic Analysis of Longitudinal Data on Disease Activity in Systemic Lupus Erythematosus

The multiple clinical manifestations of systemic lupus erythematosus are a challenge to rheumatologists managing patients with the disease, and there is a need to better understand predictors of disease activity in order to improve and standardize therapy and prevent the development of permanent damage. In this article, we present the analysis of a clinical database for patients with lupus. The database consists of data collected on 440 patients over a period of 10 years. The analysis is based on logistic regression methodology with outcomes defined at the times of clinic visits and develops an approach to examine the interrelationships between disease activity in different systems. The usefulness of separate logistic regressions with dynamic covariates for the analysis of multinomial panel data is illustrated and the efficiency of the approach relative to modeling disease activity in continuous time is investigated.

The use of variance components for the assessment of outcome measures in rheumatology

There is current interest in the development of new or improved outcome measures for rheumatological diseases. In the early stages of development, attention is usually directed to how well the measure distinguishes between patients and whether different observers attach similar values of the measure to the same patient. An approach, based on variance components, to the assessment of outcome measures is presented. The need to assess different aspects of variation associated with a measure is stressed. The terms ‘observer reliability’ and ‘agreement’ are frequently used in the evaluation of measurement instruments, and are often used interchangeably. In this paper, we use the terms to refer to different concepts assessing different aspects of variation. They are likely to correspond well in heterogeneous populations, but not in homogeneous populations where reliability will generally be low but agreement may well be high. Results from a real patient exercise, designed to study a set of tools for assessing myositis outcomes, are used to illustrate the approach that examines both reliability and agreement, and the need to evaluate both is demonstrated. A new measure of agreement, based on the ratio of standard deviations, is presented and inference procedures are discussed. To facilitate the interpretation of the combination of measures of reliability and agreement, a classification system is proposed that provides a summary of the performance of the tools. The approach is demonstrated for discrete ordinal and continuous outcomes.