Shared Parameter Models Research Articles

Shared random parameter models: A legacy of the biostatistics program at the National Heart, Lung, and Blood Institute

Shared random parameter models (SRPMs) were first introduced by researchers at the National Heart Lung and Blood Institute (NHLBI) Biostatistics Branch for analyzing longitudinal data with informative dropout (Wu and Carroll, 1987; Wu and Bailey, 1988; Follmann and Wu, 1995; Albert and Follmann, 2000; Albert et al, 2002). This work was all focused on characterizing the longitudinal data process in the presence of an informative missing data mechanism that is treated as a nuisance. Shared random parameter modeling approaches have also been developed from the perspective of characterizing the relationship between longitudinal data and a subsequent outcome that may be an event time, a dichotomous measurement, or another longitudinal outcome. This article will review the early contributions of the NHLBI biostatisticians on SRPMs for analyzing longitudinal data with dropout and demonstrate how these ideas have, more recently, been applied in these other areas of biostatistics. Rather than focus on technical details or specific analyses, this article presents a conceptual framework for SRPMs within a historical context.

Shared parameter models for joint analysis of longitudinal and survival data with left truncation due to delayed entry - Applications to cystic fibrosis.

Many longitudinal studies observe time to occurrence of a clinical event such as death, while also collecting serial measurements of one or more biomarkers that are predictive of the event, or are surrogate outcomes of interest. Joint modeling can be used to examine the relationship between the biomarker and the event, and also as a way of adjusting analyses of the biomarker for non-ignorable dropout. In settings such as registry studies, an additional complexity is caused when follow-up of subjects is delayed, referred to as left-truncation of follow-up in the survival analysis setting. If not adjusted for, this can cause bias in estimation of parameters of the survival distribution for the clinical event and in parameters of the longitudinal outcome such as the profile or rate of change over time because subjects may die or have the clinical event before follow-up starts. This paper illustrates how a broad class of shared parameter models can be used to jointly model a time to event outcome along with a longitudinal marker using available nonlinear mixed modeling software, when follow-up times are left truncated. Methods are applied to jointly model survival and decline in lung function in cystic fibrosis patients.

An approximate joint model for multiple paired longitudinal outcomes and time-to-event data.

Joint modeling of multivariate paired longitudinal data and time-to-event data presents computational challenges that supersede full likelihood estimation due to the large dimensional random effects vector needed to capture correlation due to clustering with respect to pairs, subjects, and outcomes. We propose an alternative, computationally simpler approach to estimation of complex shared parameter models where missing data is imputed based on the Posterior Predictive Distribution from a Conditional Linear Model (CLM) approximation. Existing methods for complete data are then implemented to obtain estimates of the event time model parameters. Our method is applied to examine the effects of discordant growth in anthropometric measures of longitudinal fetal growth in twin fetuses and the timing of birth. Simulation results are presented to show that our method performs relatively well with moderate measurement errors under certain CLM approximations.

A bivariate model for analyzing recurrent multi-type automobile failures

The failure mechanism in an automobile can be defined as a system of multi-type recurrent failures where failures can occur due to various multi-type failure modes and these failures are repetitive such that more than one failure can occur from each failure mode. In analysing such automobile failures, both the time and type of the failure serve as response variables. However, these two response variables are highly correlated with each other since the timing of failures has an association with the mode of the failure. When there are more than one correlated response variables, the fitting of a multivariate model is more preferable than separate univariate models. Therefore, a bivariate model of time and type of failure becomes appealing for such automobile failure data. When there are multiple failure observations pertaining to a single automobile, such data cannot be treated as independent data because failure instances of a single automobile are correlated with each other while failures among different automobiles can be treated as independent. Therefore, this study proposes a bivariate model consisting time and type of failure as responses adjusted for correlated data. The proposed model was formulated following the approaches of shared parameter models and random effects models for joining the responses and for representing the correlated data respectively. The proposed model is applied to a sample of automobile failures with three types of failure modes and up to five failure recurrences. The parametric distributions that were suitable for the two responses of time to failure and type of failure were Weibull distribution and multinomial distribution respectively. The proposed bivariate model was programmed in SAS Procedure Proc NLMIXED by user programming appropriate likelihood functions. The performance of the bivariate model was compared with separate univariate models fitted for the two responses and it was identified that better performance is secured by the bivariate model. The proposed model can be used to determine the time and type of failure that would occur in the automobiles considered here.

EFFECTS OF CEREBROVASCULAR INFARCT BURDEN ON COGNITIVE DECLINE IN THE PRESENCE OF DEMENTIA AND DEATH: ACCOUNTING FOR POTENTIALLY INFORMATIVE DROPOUT USING COMPETING RISK SHARED PARAMETER MODELS: THE ARIC STUDY

Large biases may occur in cognitive decline studies due to informative dropout, where participants with the worst cognitive decline also have more missing outcomes due to competing dementia/death/other events. Shared parameter models (SPM) jointly estimate longitudinal submodels with event submodels through a set of “shared parameters” (latent/random effects) to address this selection bias. We compare cerebrovascular infarct relationships with cognitive decline using SPM versus standard models in the Atherosclerosis Risk in Communities (ARIC) study. Infarct burden was assessed using 1.5T MRI in 1,835 stroke-free participants of the ARIC study (V3:1993–1995; mean age 62.8, range 51–73 years; 60% women) and categorized as no-burden (no infarct like lesions (ILL)<3mm nor any infarcts>3mm, n=1580), small-burden (ILL<3mm only, n=48), large-burden (infarcts≥3 mm only, n=174), and double-burden (co-existingILL<3mm plus infarcts≥3 mm (n=30)). Global cognition z-scores over 20 years were available in up to five visits; dementia and death events were available via cohort surveillance. Random-slope linear mixed models (LMM) estimated associations of infarct burden with 10-year cognitive decline under inherent missing at random (MAR) assumptions. SPM approaches used this LMM submodel alongside cause-specific Weibull survival submodels for dementia and death events, sharing the random intercepts and slopes between the longitudinal and event submodels under missing not at random (MNAR) assumptions. The double-burden group experienced triple the dementia (Cox-PHM HR=2.8, [1.5, 5.2]) and double the death rates (HR=2.3, [1.4, 3.7]) compared to the no-burden group. LMM gave 10-year cognitive change estimates for the double-burden group which were counterintuitively positive (change=+0.10 SD) and statistically supported as better (p<0.001) than the no-burden group (change= -0.32 SD). Conversely, SPM gave 10-year cognitive change estimates for the double-burden group which were modestly negative (change=-0.05 SD) and non-supported (p=0.220) as being different from the no-burden group (change=-0.34 SD). SPM estimates of informative relationships connecting the longitudinal (cognitive decline) submodel with the survival submodels (dementia and death) were highly supported (p<0.0001 for both). Including informative competing events such as dementia and deaths in studies of cognitive decline can improve estimates and understanding of the interplay between cognitive decline and cause-specific dropout mechanisms.

Joint modeling of bottle use, daily milk intake from bottles, and daily energy intake in toddlers

ABSTRACTThe current study follows an educational intervention on bottle-weaning to simultaneously evaluate effects of the bottle-weaning intervention on reducing bottle use, daily milk intake from bottles, and daily energy intake in toddlers aged 11–13 months. In this paper, we propose to use shared parameter models and random effects models to model these outcomes jointly. Our joint models consist of two submodels: a two-part submodel for modeling the odds of bottle use and the intensity of daily milk intake from bottles, and a linear mixed submodel for modeling the intensity of daily energy intake. The two submodels are linked by either shared random effects or separate but correlated random effects. We investigate whether the intervention effects, parameter estimates, and model fit differ between shared parameter models and random effects models.

Mixed models approaches for joint modeling of different types of responses

ABSTRACTIn many biomedical studies, one jointly collects longitudinal continuous, binary, and survival outcomes, possibly with some observations missing. Random-effects models, sometimes called shared-parameter models or frailty models, received a lot of attention. In such models, the corresponding variance components can be employed to capture the association between the various sequences. In some cases, random effects are considered common to various sequences, perhaps up to a scaling factor; in others, there are different but correlated random effects. Even though a variety of data types has been considered in the literature, less attention has been devoted to ordinal data. For univariate longitudinal or hierarchical data, the proportional odds mixed model (POMM) is an instance of the generalized linear mixed model (GLMM; Breslow and Clayton, 1993). Ordinal data are conveniently replaced by a parsimonious set of dummies, which in the longitudinal setting leads to a repeated set of dummies. When ordinal longitudinal data are part of a joint model, the complexity increases further. This is the setting considered in this paper. We formulate a random-effects based model that, in addition, allows for overdispersion. Using two case studies, it is shown that the combination of random effects to capture association with further correction for overdispersion can improve the model’s fit considerably and that the resulting models allow to answer research questions that could not be addressed otherwise. Parameters can be estimated in a fairly straightforward way, using the SAS procedure NLMIXED.

Joint analysis of longitudinal and survival data measured on nested timescales by using shared parameter models: an application to fecundity data.

We consider the joint modelling, analysis and prediction of a longitudinal binary process and a discrete time-to-event outcome. We consider data from a prospective pregnancy study, which provides day level information regarding the behaviour of couples attempting to conceive. Reproductive epidemiologists are particularly interested in developing a model for individualized predictions of time to pregnancy (TTP). A couple's intercourse behaviour should be an integral part of such a model and is one of the main focuses of the paper. In our motivating data, the intercourse observations are a long series of binary data with a periodic probability of success and the amount of available intercourse data is a function of both the menstrual cycle length and TTP. Moreover, these variables are dependent and observed on different, and nested, timescales (TTP is measured in menstrual cycles whereas intercourse is measured on days within a menstrual cycle) further complicating its analysis. Here, we propose a semiparametric shared parameter model for the joint modelling of the binary longitudinal data (intercourse behaviour) and the discrete survival outcome (TTP). Further, we develop couple-based dynamic predictions for the intercourse profiles, which in turn are used to assess the risk for subfertility (i.e. TTP longer than six menstrual cycles).

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] and a simulation.

Joint modelling of repeated measurements and time-to-event outcomes: flexible model specification and exact likelihood inference.

Random effects or shared parameter models are commonly advocated for the analysis of combined repeated measurement and event history data, including dropout from longitudinal trials. Their use in practical applications has generally been limited by computational cost and complexity, meaning that only simple special cases can be fitted by using readily available software. We propose a new approach that exploits recent distributional results for the extended skew normal family to allow exact likelihood inference for a flexible class of random-effects models. The method uses a discretization of the timescale for the time-to-event outcome, which is often unavoidable in any case when events correspond to dropout. We place no restriction on the times at which repeated measurements are made. An analysis of repeated lung function measurements in a cystic fibrosis cohort is used to illustrate the method.

Joint models for predicting transplant-related mortality from quality of life data

To test whether longitudinally measured health-related quality of life (HRQL) predicts transplant-related mortality (TRM) in pediatric hematopoietic stem cell transplant (HSCT). The predictors of interest were emotional functioning, physical functioning, role functioning, and global HRQL, as rated by the parent about the child up to 6 times over 12 months of follow-up and measured by the Child Health Ratings Inventories. We used joint models, specifically shared parameter models, with time to TRM as the outcome of interest and other causes of mortality as a competing risk, via the JM software package in R. Choosing shared parameter models instead of standard survival models, such as Cox models with time-dependent covariates, enabled us to address measurement error in the HRQL trajectories and appropriately handle missing data. The nonlinear trajectories for each HRQL domain were modeled by random spline functions. The survival submodels were adjusted for baseline patient, family, and transplant characteristics. Hazard ratios per one-half standard deviation difference in emotional, physical, and role functioning, and global HRQL were 0.61 (95 % CI 0.46-0.81; p < 0.001), 0.70 (0.51-0.96; p = 0.03), 0.54 (0.34-0.85; p = 0.007), and 0.57 (0.41-0.79; p < 0.001), respectively. HRQL trajectories were predictive of TRM in pediatric HSCT, even after adjusting the survival outcome for baseline characteristics.

Practical and statistical issues in missing data for longitudinal patient-reported outcomes

Patient-reported outcomes are increasingly used in health research, including randomized controlled trials and observational studies. However, the validity of results in longitudinal studies can crucially hinge on the handling of missing data. This paper considers the issues of missing data at each stage of research. Practical strategies for minimizing missingness through careful study design and conduct are given. Statistical approaches that are commonly used, but should be avoided, are discussed, including how these methods can yield biased and misleading results. Methods that are valid for data which are missing at random are outlined, including maximum likelihood methods, multiple imputation and extensions to generalized estimating equations: weighted generalized estimating equations, generalized estimating equations with multiple imputation, and doubly robust generalized estimating equations. Finally, we discuss the importance of sensitivity analyses, including the role of missing not at random models, such as pattern mixture, selection, and shared parameter models. We demonstrate many of these concepts with data from a randomized controlled clinical trial on renal cancer patients, and show that the results are dependent on missingness assumptions and the statistical approach.

Generalized linear mixed joint model for longitudinal and survival outcomes

Longitudinal studies often entail categorical outcomes as primary responses. When dropout occurs, non-ignorability is frequently accounted for through shared parameter models (SPMs). In this context, several extensions from Gaussian to non-Gaussian longitudinal processes have been proposed. In this paper, we formulate an approach for non-Gaussian longitudinal outcomes in the framework of joint models. As an extension of SPMs, based on shared latent effects, we assume that the history of the response up to current time may have an influence on the risk of dropout. This history is represented by the current, expected, value of the response. Since the time a subject spends in the study is continuous, we parametrize the dropout process through a proportional hazard model. The resulting model is referred to as Generalized Linear Mixed Joint Model (GLMJM). To estimate model parameters, we adopt a maximum likelihood approach via the EM algorithm. In this context, the maximization of the observed data log-likelihood requires numerical integration over the random effect posterior distribution, which is usually not straightforward; under the assumption of Gaussian random effects, we compare Gauss-Hermite and Pseudo-Adaptive Gaussian quadrature rules. We investigate in a simulation study the behaviour of parameter estimates in the case of Poisson and Binomial longitudinal responses, and apply the GLMJM to a benchmark dataset.

A structured framework for assessing sensitivity to missing data assumptions in longitudinal clinical trials

The objective of this research was to demonstrate a framework for drawing inference from sensitivity analyses of incomplete longitudinal clinical trial data via a re-analysis of data from a confirmatory clinical trial in depression. A likelihood-based approach that assumed missing at random (MAR) was the primary analysis. Robustness to departure from MAR was assessed by comparing the primary result to those from a series of analyses that employed varying missing not at random (MNAR) assumptions (selection models, pattern mixture models and shared parameter models) and to MAR methods that used inclusive models. The key sensitivity analysis used multiple imputation assuming that after dropout the trajectory of drug-treated patients was that of placebo treated patients with a similar outcome history (placebo multiple imputation). This result was used as the worst reasonable case to define the lower limit of plausible values for the treatment contrast. The endpoint contrast from the primary analysis was - 2.79 (p = .013). In placebo multiple imputation, the result was - 2.17. Results from the other sensitivity analyses ranged from - 2.21 to - 3.87 and were symmetrically distributed around the primary result. Hence, no clear evidence of bias from missing not at random data was found. In the worst reasonable case scenario, the treatment effect was 80% of the magnitude of the primary result. Therefore, it was concluded that a treatment effect existed. The structured sensitivity framework of using a worst reasonable case result based on a controlled imputation approach with transparent and debatable assumptions supplemented a series of plausible alternative models under varying assumptions was useful in this specific situation and holds promise as a generally useful framework.

Evidence Synthesis for Decision Making 2

We set out a generalized linear model framework for the synthesis of data from randomized controlled trials. A common model is described, taking the form of a linear regression for both fixed and random effects synthesis, which can be implemented with normal, binomial, Poisson, and multinomial data. The familiar logistic model for meta-analysis with binomial data is a generalized linear model with a logit link function, which is appropriate for probability outcomes. The same linear regression framework can be applied to continuous outcomes, rate models, competing risks, or ordered category outcomes by using other link functions, such as identity, log, complementary log-log, and probit link functions. The common core model for the linear predictor can be applied to pairwise meta-analysis, indirect comparisons, synthesis of multiarm trials, and mixed treatment comparisons, also known as network meta-analysis, without distinction. We take a Bayesian approach to estimation and provide WinBUGS program code for a Bayesian analysis using Markov chain Monte Carlo simulation. An advantage of this approach is that it is straightforward to extend to shared parameter models where different randomized controlled trials report outcomes in different formats but from a common underlying model. Use of the generalized linear model framework allows us to present a unified account of how models can be compared using the deviance information criterion and how goodness of fit can be assessed using the residual deviance. The approach is illustrated through a range of worked examples for commonly encountered evidence formats.

Generalized shared-parameter models and missingness at random

When data are incomplete, models are often catalogued according to one of the three modelling frameworks to which they belong: selection models (SeM), pattern-mixture models (PMM) and shared-parameter models (SPM). The missing data mechanism is conventionally classified as missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR). Under MCAR, measurement and missingness mechanism are independent, but that is not the case for MAR. The definition of MAR is in SeM terms. Molenberghs et al. (1998) provided a characterization for PMM. Here, MAR is characterized in the SPM framework, using an extended SPM class. A subfamily, satisfying the MAR condition, is studied in detail. Particular implications for non-monotone missingness as well as for longitudinal data subject to dropout are studied. It is indicated how SPM can be constrained such that dropout at a given point in time can depend on current and past, but not on future measurements. Although, a natural requirement, it is less easily imposed in the PMM and SPM frameworks than in the SeM case. Some of the models proposed are illustrated using a clinical trial in toenail dermatophyte onychomycosis.

A Sensitivity Analysis for Shared‐Parameter Models for Incomplete Longitudinal Outcomes

All models for incomplete data either explicitly make assumptions about aspects of the distribution of the unobserved outcomes, given the observed ones, or at least implicitly imply such. One consequence is that there routinely exist a whole class of models, coinciding in their description of the observed portion of the data but differing with respect to their "predictions" of what is unobserved. Within such a class, there always is a single model corresponding to so-called random missingness, in the sense that the mechanism governing missingness depends on covariates and observed outcomes, but given these not further on unobserved outcomes. We employ these results in the context of so-called shared-parameter models where outcome and missingness models are connected by means of common latent variables or random effects, to devise a sensitivity analysis framework. Precisely, the impact of varying unverifiable assumptions about unobserved measurements on parameters of interest is studied. Apart from analytic considerations, the proposed methodology is applied to assess treatment effect in data from a clinical trial in toenail dermatophyte onychomycosis. While our focus is on longitudinal outcomes with incomplete outcome data, the ideas developed in this paper are of use whenever a shared-parameter model could be considered.

Nonignorable Models for Intermittently Missing Categorical Longitudinal Responses

A class of nonignorable models is presented for handling nonmonotone missingness in categorical longitudinal responses. This class of models includes the traditional selection models and shared parameter models. This allows us to perform a broader than usual sensitivity analysis. In particular, instead of considering variations to a chosen nonignorable model, we study sensitivity between different missing data frameworks. An appealing feature of the developed class is that parameters with a marginal interpretation are obtained, while algebraically simple models are considered. Specifically, marginalized mixed-effects models (Heagerty, 1999, Biometrics 55, 688-698) are used for the longitudinal process that model separately the marginal mean and the correlation structure. For the correlation structure, random effects are introduced and their distribution is modeled either parametrically or non-parametrically to avoid potential misspecifications.

Cognition and Quality of Life After Chemotherapy Plus Radiotherapy (RT) vs. RT for Pure and Mixed Anaplastic Oligodendrogliomas: Radiation Therapy Oncology Group Trial 9402

Radiation Therapy Oncology Group 9402 compared procarbazine, lomustine, and vincristine (PCV) chemotherapy plus radiation therapy (PCV + RT) vs. RT alone for anaplastic oligodendroglioma. Here we report longitudinal changes in cognition and quality of life, effects of patient factors and treatments on cognition, quality of life and survival, and prognostic implications of cognition and quality of life. Cognition was assessed by Mini Mental Status Examination (MMSE) and quality of life by Brain-Quality of Life (B-QOL). Scores were analyzed for survivors and within 5 years of death. Shared parameter models evaluated MMSE/B-QOL with survival. For survivors, MMSE and B-QOL scores were similar longitudinally and between treatments. For those who died, MMSE scores remained stable initially, whereas B-QOL slowly declined; both declined rapidly in the last year of life and similarly between arms. In the aggregate, scores decreased over time (p = 0.0413 for MMSE; p = 0.0016 for B-QOL) and were superior with age <50 years (p < 0.001 for MMSE; p = 0.0554 for B-QOL) and Karnofsky Performance Score (KPS) 80-100 (p < 0.001). Younger age and higher KPS were associated with longer survival. After adjusting for patient factors and drop-out, survival was longer after PCV + RT (HR = 0.66, 95% CI = 0.49-0.9, p = 0.0084; HR = 0.74, 95% CI = 0.54-1.01, p = 0.0592) in models with MMSE and B-QOL. In addition, there were no differences in MMSE and B-QOL scores between arms (p = 0.4752 and p = 0.2767, respectively); higher scores predicted longer survival. MMSE and B-QOL scores held steady in the upper range in both arms for survivors. Younger, fitter patients had better MMSE and B-QOL and longer survival.

Last Observation Carried Forward Versus Mixed Models in the Analysis of Psychiatric Clinical Trials

Last Observation Carried Forward Versus Mixed Models in the Analysis of Psychiatric Clinical Trials