Discovering subpopulation structure with latent class mixed models

The linear mixed model is a well-known method for incorporating heterogeneity (for example, subject-to-subject variation) into a statistical analysis for continuous responses. However heterogeneity cannot always be fully captured by the usual assumptions of normally distributed random effects. Latent class mixed models offer a way of incorporating additional heterogeneity which can be used to uncover distinct subpopulations, to incorporate correlated non-normally distributed outcomes and to classify individuals. The methodology is motivated with examples in health care studies and a detailed illustration is drawn from the Nutritional Prevention of Cancer trials. Latent class models are used with longitudinal data on prostate specific antigen (PSA) as well as incidence of prostate cancer. The models are extended to accommodate prostate cancer as a survival endpoint; this is compared to treating it as a binary endpoint. Four subpopulations are identified which differ both with regard to their PSA trajectories and their incidence rates of prostate cancer.     

Estimation of genetic and environmental factors for binary traits using family data

While the family-based analysis of genetic and environmental contributions to continuous or Gaussian traits is now straightforward using the linear mixed models approach, the corresponding analysis of complex binary traits is still rather limited. In the latter we usually rely on twin studies or pairs of relatives, but these studies often have limited sample size or have difficulties in dealing with the dependence between the pairs. Direct analysis of extended family data can potentially overcome these limitations. In this paper, we will describe various genetic models that can be analysed using an extended family structure. We use the generalized linear mixed model to deal with the family structure and likelihood-based methodology for parameter inference. The method is completely general, accommodating arbitrary family structures and incomplete data. We illustrate the methodology in great detail using the Swedish birth registry data on pre-eclampsia, a hypertensive condition induced by pregnancy. The statistical challenges include the specification of sensible models that contain a relatively large number of variance components compared to standard mixed models. In our illustration the models will account for maternal or foetal genetic effects, environmental effects, or a combination of these and we show how these effects can be readily estimated using family data.     

A semi-parametric Bayesian approach to generalized linear mixed models

The linear mixed effects model with normal errors is a popular model for the analysis of repeated measures and longitudinal data. The generalized linear model is useful for data that have non-normal errors but where the errors are uncorrelated. A descendant of these two models generates a model for correlated data with non-normal errors, called the generalized linear mixed model (GLMM). Frequentist attempts to fit these models generally rely on approximate results and inference relies on asymptotic assumptions. Recent advances in computing technology have made Bayesian approaches to this class of models computationally feasible. Markov chain Monte Carlo methods can be used to obtain ‘exact’ inference for these models, as demonstrated by Zeger and Karim. In the linear or generalized linear mixed model, the random effects are typically taken to have a fully parametric distribution, such as the normal distribution. In this paper, we extend the GLMM by allowing the random effects to have a non-parametric prior distribution. We do this using a Dirichlet process prior for the general distribution of the random effects. The approach easily extends to more general population models. We perform computations for the models using the Gibbs sampler. © 1998 John Wiley & Sons, Ltd.     

Robust non‐linear mixed modelling of longitudinal PSA levels after prostate cancer treatment

The objective of this study was to develop a robust non‐linear mixed model for prostate‐specific antigen (PSA) measurements after a high‐intensity focused ultrasound (HIFU) treatment for prostate cancer. The characteristics of these data are the presence of outlying values and non‐normal random effects. A numerical study proved that parameter estimates can be biased if these characteristics are not taken into account. The intra‐patient variability was described by a Student‐t distribution and Dirichlet process priors were assumed for non‐normal random effects; a process that limited the bias and provided more efficient parameter estimates than a classical mixed model with normal residuals and random effects. It was applied to the determination of the best dynamic PSA criterion for the diagnosis of prostate cancer recurrence, but could be used in studies that rely on PSA data to improve prognosis or compare treatment efficiencies and also with other longitudinal biomarkers that, such as PSA, present outlying values and non‐normal random effects. Copyright © 2010 John Wiley & Sons, Ltd.     

Real longitudinal data analysis for real people: Building a good enough mixed model

Mixed effects models have become very popular, especially for the analysis of longitudinal data. One challenge is how to build a good enough mixed effects model. In this paper, we suggest a systematic strategy for addressing this challenge and introduce easily implemented practical advice to build mixed effects models. A general discussion of the scientific strategies motivates the recommended five‐step procedure for model fitting. The need to model both the mean structure (the fixed effects) and the covariance structure (the random effects and residual error) creates the fundamental flexibility and complexity. Some very practical recommendations help to conquer the complexity. Centering, scaling, and full‐rank coding of all the predictor variables radically improve the chances of convergence, computing speed, and numerical accuracy. Applying computational and assumption diagnostics from univariate linear models to mixed model data greatly helps to detect and solve the related computational problems. Applying computational and assumption diagnostics from the univariate linear models to the mixed model data can radically improve the chances of convergence, computing speed, and numerical accuracy. The approach helps to fit more general covariance models, a crucial step in selecting a credible covariance model needed for defensible inference. A detailed demonstration of the recommended strategy is based on data from a published study of a randomized trial of a multicomponent intervention to prevent young adolescents' alcohol use. The discussion highlights a need for additional covariance and inference tools for mixed models. The discussion also highlights the need for improving how scientists and statisticians teach and review the process of finding a good enough mixed model. Copyright © 2009 John Wiley & Sons, Ltd.     

A note on genetic variance components in mixed models

Burton et al. ([1999] Genet. Epidemiol. 17:118-140) proposed a series of generalized linear mixed models for pedigree data that account for residual correlation between related individuals. These models may be fitted using Markov chain Monte Carlo methods, but the posterior mean for small variance components can exhibit marked positive bias. Burton et al. ([1999] Genet. Epidemiol. 17:118-140) suggested that this problem could be overcome by allowing the variance components to take negative values. We examine this idea in depth, and show that it can be interpreted as a computational device for locating the posterior mode without necessarily implying that the original random effects structure is incorrect. We illustrate the application of this technique to mixed models for familial data.     

A latent factor linear mixed model for high‐dimensional longitudinal data analysis

High‐dimensional longitudinal data involving latent variables such as depression and anxiety that cannot be quantified directly are often encountered in biomedical and social sciences. Multiple responses are used to characterize these latent quantities, and repeated measures are collected to capture their trends over time. Furthermore, substantive research questions may concern issues such as interrelated trends among latent variables that can only be addressed by modeling them jointly. Although statistical analysis of univariate longitudinal data has been well developed, methods for modeling multivariate high‐dimensional longitudinal data are still under development. In this paper, we propose a latent factor linear mixed model (LFLMM) for analyzing this type of data. This model is a combination of the factor analysis and multivariate linear mixed models. Under this modeling framework, we reduced the high‐dimensional responses to low‐dimensional latent factors by the factor analysis model, and then we used the multivariate linear mixed model to study the longitudinal trends of these latent factors. We developed an expectation–maximization algorithm to estimate the model. We used simulation studies to investigate the computational properties of the expectation–maximization algorithm and compare the LFLMM model with other approaches for high‐dimensional longitudinal data analysis. We used a real data example to illustrate the practical usefulness of the model. Copyright © 2013 John Wiley & Sons, Ltd.     

Joint modelling of longitudinal and multi‐state processes: application to clinical progressions in prostate cancer

Joint modelling of longitudinal and survival data is increasingly used in clinical trials on cancer. In prostate cancer for example, these models permit to account for the link between longitudinal measures of prostate‐specific antigen (PSA) and time of clinical recurrence when studying the risk of relapse. In practice, multiple types of relapse may occur successively. Distinguishing these transitions between health states would allow to evaluate, for example, how PSA trajectory and classical covariates impact the risk of dying after a distant recurrence post‐radiotherapy, or to predict the risk of one specific type of clinical recurrence post‐radiotherapy, from the PSA history. In this context, we present a joint model for a longitudinal process and a multi‐state process, which is divided into two sub‐models: a linear mixed sub‐model for longitudinal data and a multi‐state sub‐model with proportional hazards for transition times, both linked by a function of shared random effects. Parameters of this joint multi‐state model are estimated within the maximum likelihood framework using an EM algorithm coupled with a quasi‐Newton algorithm in case of slow convergence. It is implemented under R, by combining and extending mstate and JM packages. The estimation program is validated by simulations and applied on pooled data from two cohorts of men with localized prostate cancer. Thanks to the classical covariates available at baseline and the repeated PSA measurements, we are able to assess the biomarker's trajectory, define the risks of transitions between health states and quantify the impact of the PSA dynamics on each transition intensity. Copyright © 2016 John Wiley & Sons, Ltd.     

Prostate cancer segregation analyses using 4390 families from UK and Australian population‐based studies

Familial aggregation of prostate cancer is likely to be due to multiple susceptibility loci, perhaps acting in conjunction with shared lifestyle risk factors. Models that assume a single mode of inheritance may be unrealistic. We analyzed genetic models of susceptibility to prostate cancer using segregation analysis of occurrence in families ascertained through population‐based series totaling 4390 incident cases. We investigated major gene models (dominant, recessive, general, X‐linked), polygenic models, and mixed models of susceptibility using the pedigree analysis software MENDEL. The hypergeometric model was used to approximate polygenic inheritance. The best‐fitting model for the familial aggregation of prostate cancer was the mixed recessive model. The frequency of the susceptibility allele in the population was estimated to be 0.15 (95% confidence interval (CI) 0.11–0.20), with a relative risk for homozygote carriers of 94 (95% CI 46–192), and a polygenic standard deviation of 2.01 (95% CI 1.72–2.34). These analyses suggest that one or more genes having a strong recessively inherited effect on risk, as well as a number of genes with variants having small multiplicative effects on risk, may account for the genetic susceptibility to prostate cancer. The recessive component would predict the observed higher familial risk for siblings of cases than for fathers, but this could also be due to other factors such as shared lifestyle by siblings, targeted screening effects, and/or non‐additive effects of one or more genes. Genet. Epidemiol. 34:42–50, 2010. © 2009 Wiley‐Liss, Inc.     

Extended information criterion (EIC) approach for linear mixed effects models under restricted maximum likelihood (REML) estimation

In clinical data analysis, the restricted maximum likelihood (REML) method has been commonly used for estimating variance components in the linear mixed effects model. Under the REML estimation, however, it is not straightforward to compare several linear mixed effects models with different mean and covariance structures. In particular, few approaches have been proposed for the comparison of linear mixed effects models with different mean structures under the REML estimation. We propose an approach using extended information criterion (EIC), which is a bootstrap-based extension of AIC, for comparing linear mixed effects models with different mean and covariance structures under the REML estimation. We present simulation studies and applications to two actual clinical data sets.     

An autoregressive linear mixed effects model for the analysis of longitudinal data which show profiles approaching asymptotes

In longitudinal data, a continuous response sometimes shows a profile approaching an asymptote. For such data, we propose a new class of models, autoregressive linear mixed effects models in which the current response is regressed on the previous response, fixed effects, and random effects. Asymptotes can shift depending on treatment groups, individuals, and so on, and can be modelled by fixed and random effects. We also propose error structures that are useful in practice. The estimation methods of linear mixed effects models can be used as long as there is no intermittent missing.     

Concordance correlation coefficient applied to discrete data

In any field in which decisions are subject to measurements, interchangeability between the methods used to obtain these measurements is essential. To consider methods as interchangeable, a certain degree of agreement is needed between the measurements they provide. The concordance correlation coefficient is an index that assesses the strength of agreement and it has been widely applied in situations in which measurements are made on a continuous scale. Recently the concordance correlation coefficient has been defined as a specific intraclass correlation coefficient estimated by the variance components of a Normal-Normal mixed linear model. Although this coefficient was defined for the continuous scale case, it may also be used with a discrete scale. In this case the data are often transformed and normalized, and the concordance correlation is applied. This study discusses the expression of the concordance correlation coefficient for discrete Poisson data by means of the Poisson-Normal generalized linear mixed model. The behaviour of the concordance correlation coefficient estimate is assessed by means of a simulation study, in which the estimates were compared using four models: three Normal-Normal mixed models with raw data, log-transformed data and square-root transformed data, and the Poisson-Normal generalized linear mixed model. An example is provided in which two different methods are used to measure CD34+ cells.     

Tutorial in biostatistics: spline smoothing with linear mixed models

The semi-parametric regression achieved via penalized spline smoothing can be expressed in a linear mixed models framework. This allows such models to be fitted using standard mixed models software routines with which many biostatisticians are familiar. Moreover, the analysis of complex correlated data structures that are a hallmark of biostatistics, and which are typically analysed using mixed models, can now incorporate directly smoothing of the relationship between an outcome and covariates. In this paper we provide an introduction to both linear mixed models and penalized spline smoothing, and describe the connection between the two. This is illustrated with three examples, the first using birth data from the U.K., the second relating mammographic density to age in a study of female twin-pairs and the third modelling the relationship between age and bronchial hyperresponsiveness in families. The models are fitted in R (a clone of S-plus) and using Markov chain Monte Carlo (MCMC) implemented in the package WinBUGS.     

A latent class mixed model for analysing biomarker trajectories with irregularly scheduled observations

This paper considers a latent class model to uncover subpopulation structure for both biomarker trajectories and the probability of disease outcome in highly unbalanced longitudinal data. A specific pattern of trajectories can be viewed as a latent class in a finite mixture where membership in latent classes is modelled with a polychotomous logistic regression. The biomarker trajectories within a latent class are described by a linear mixed model with possibly time-dependent covariates and the probabilities of disease outcome are estimated via a class specific model. Thus the method characterizes biomarker trajectory patterns to unveil the relationship between trajectories and outcomes of disease. The coefficients for the model are estimated via a generalized EM (GEM) algorithm, a natural tool to use when latent classes and random coefficients are present. Standard errors of the coefficients are calculated using a parametric bootstrap. The model fitting procedure is illustrated with data from the Nutritional Prevention of Cancer trials; we use prostate specific antigen (PSA) as the biomarker for prostate cancer and the goal is to examine trajectories of PSA serial readings in individual subjects in connection with incidence of prostate cancer.     

Autoregressive and cross‐lagged model for bivariate non‐commensurate outcomes

Autoregressive and cross‐lagged models have been widely used to understand the relationship between bivariate commensurate outcomes in social and behavioral sciences, but not much work has been carried out in modeling bivariate non‐commensurate (e.g., mixed binary and continuous) outcomes simultaneously. We develop a likelihood‐based methodology combining ordinary autoregressive and cross‐lagged models with a shared subject‐specific random effect in the mixed‐model framework to model two correlated longitudinal non‐commensurate outcomes. The estimates of the cross‐lagged and the autoregressive effects from our model are shown to be consistent with smaller mean‐squared error than the estimates from the univariate generalized linear models. Inclusion of the subject‐specific random effects in the proposed model accounts for between‐subject variability arising from the omitted and/or unobservable, but possibly explanatory, subject‐level predictors. Our model is not restricted to the case with equal number of events per subject, and it can be extended to different types of bivariate outcomes. We apply our model to an ecological momentary assessment study with complex dependence and sampling data structures. Specifically, we study the dependence between the condom use and sexual satisfaction based on the data reported in a longitudinal study of sexually transmitted infections. We find negative cross‐lagged effect between these two outcomes and positive autoregressive effect within each outcome. Copyright © 2017 John Wiley & Sons, Ltd.     

Regression analysis using dependent Polya trees

Many commonly used models for linear regression analysis force overly simplistic shape and scale constraints on the residual structure of data. We propose a semiparametric Bayesian model for regression analysis that produces data‐driven inference by using a new type of dependent Polya tree prior to model arbitrary residual distributions that are allowed to evolve across increasing levels of an ordinal covariate (e.g., time, in repeated measurement studies). By modeling residual distributions at consecutive covariate levels or time points using separate, but dependent Polya tree priors, distributional information is pooled while allowing for broad pliability to accommodate many types of changing residual distributions. We can use the proposed dependent residual structure in a wide range of regression settings, including fixed‐effects and mixed‐effects linear and nonlinear models for cross‐sectional, prospective, and repeated measurement data. A simulation study illustrates the flexibility of our novel semiparametric regression model to accurately capture evolving residual distributions. In an application to immune development data on immunoglobulin G antibodies in children, our new model outperforms several contemporary semiparametric regression models based on a predictive model selection criterion. Copyright © 2013 John Wiley & Sons, Ltd.     

Longitudinal structural mixed models for the analysis of surgical trials with noncompliance

Patient noncompliance complicates the analysis of many randomized trials seeking to evaluate the effect of surgical intervention as compared with a nonsurgical treatment. If selection for treatment depends on intermediate patient characteristics or outcomes, then 'as-treated' analyses may be biased for the estimation of causal effects. Therefore, the selection mechanism for treatment and/or compliance should be carefully considered when conducting analysis of surgical trials. We compare the performance of alternative methods when endogenous processes lead to patient crossover. We adopt an underlying longitudinal structural mixed model that is a natural example of a structural nested model. Likelihood-based methods are not typically used in this context; however, we show that standard linear mixed models will be valid under selection mechanisms that depend only on past covariate and outcome history. If there are underlying patient characteristics that influence selection, then likelihood methods can be extended via maximization of the joint likelihood of exposure and outcomes. Semi-parametric causal estimation methods such as marginal structural models, g-estimation, and instrumental variable approaches can also be valid, and we both review and evaluate their implementation in this setting. The assumptions required for valid estimation vary across approaches; thus, the choice of methods for analysis should be driven by which outcome and selection assumptions are plausible.     

Models of population-based analyses for data collected from large extended families

Large studies of extended families usually collect valuable phenotypic data that may have scientific value for purposes other than testing genetic hypotheses if the families were not selected in a biased manner. These purposes include assessing population-based associations of diseases with risk factors/covariates and estimating population characteristics such as disease prevalence and incidence. Relatedness among participants however, violates the traditional assumption of independent observations in these classic analyses. The commonly used adjustment method for relatedness in population-based analyses is to use marginal models, in which clusters (families) are assumed to be independent (unrelated) with a simple and identical covariance (family) structure such as those called independent, exchangeable and unstructured covariance structures. However, using these simple covariance structures may not be optimally appropriate for outcomes collected from large extended families, and may under- or over-estimate the variances of estimators and thus lead to uncertainty in inferences. Moreover, the assumption that families are unrelated with an identical family structure in a marginal model may not be satisfied for family studies with large extended families. The aim of this paper is to propose models incorporating marginal models approaches with a covariance structure for assessing population-based associations of diseases with their risk factors/covariates and estimating population characteristics for epidemiological studies while adjusting for the complicated relatedness among outcomes (continuous/categorical, normally/non-normally distributed) collected from large extended families. We also discuss theoretical issues of the proposed models and show that the proposed models and covariance structure are appropriate for and capable of achieving the aim.     

A review of some extensions to generalized linear models.

Although generalized linear models are reasonably well known, they are not as widely used in medical statistics as might be appropriate, with the exception of logistic, log-linear, and some survival models. At the same time, the generalized linear modelling methodology is decidedly outdated in that more powerful methods, involving wider classes of distributions, non-linear regression, censoring and dependence among responses, are required. Limitations of the generalized linear modelling approach include the need for the iterated weighted least squares (IWLS) procedure for estimation and deviances for inferences; these restrict the class of models that can be used and do not allow direct comparisons among models from different distributions. Powerful non-linear optimization routines are now available and comparisons can more fruitfully be made using the complete likelihood function. The link function is an artefact, necessary for IWLS to function with linear models, but that disappears once the class is extended to truly non-linear models. Restricting comparisons of responses under different treatments to differences in means can be extremely misleading if the shape of the distribution is changing. This may involve changes in dispersion, or of other shape-related parameters such as the skewness in a stable distribution, with the treatments or covariates. Any exact likelihood function, defined as the probability of the observed data, takes into account the fact that all observable data are interval censored, thus directly encompassing the various types of censoring possible with duration-type data. In most situations this can now be as easily used as the traditional approximate likelihood based on densities. Finally, methods are required for incorporating dependencies among responses in models including conditioning on previous history and on random effects. One important procedure for constructing such likelihoods is based on Kalman filtering.     

Effects of covariance model assumptions on hypothesis tests for repeated measurements: analysis of ovarian hormone data and pituitary-pteryomaxillary distance data

In the analysis of repeated measurements, multivariate methods which account for the correlations among the observations from the same experimental unit are widely used. Two commonly-used multivariate methods are the unstructured multivariate approach and the mixed model approach. The unstructured multivariate approach uses MANOVA types of models and does not require assumptions on the covariance structure. The mixed model approach uses multivariate linear models with random effects and requires covariance structure assumptions. In this paper, we describe the characteristics of tests based on these two methods of analysis and investigate the performance of these tests. We focus particularly on tests for group effects and parallelism of response profiles.