Prediction in multilevel generalized linear models

Summary.  We discuss prediction of random effects and of expected responses in multilevel generalized linear models. Prediction of random effects is useful for instance in small area estimation and disease mapping, effectiveness studies and model diagnostics. Prediction of expected responses is useful for planning, model interpretation and diagnostics. For prediction of random effects, we concentrate on empirical Bayes prediction and discuss three different kinds of standard errors; the posterior standard deviation and the marginal prediction error standard deviation (comparative standard errors) and the marginal sampling standard deviation (diagnostic standard error). Analytical expressions are available only for linear models and are provided in an appendix . For other multilevel generalized linear models we present approximations and suggest using parametric bootstrapping to obtain standard errors. We also discuss prediction of expectations of responses or probabilities for a new unit in a hypothetical cluster, or in a new (randomly sampled) cluster or in an existing cluster. The methods are implemented in gllamm and illustrated by applying them to survey data on reading proficiency of children nested in schools. Simulations are used to assess the performance of various predictions and associated standard errors for logistic random-intercept models under a range of conditions.     

Computation aspects of the parameter estimates of linear mixed effects model in multivariate repeated measures set-up

The number of parameters mushrooms in a linear mixed effects (LME) model in the case of multivariate repeated measures data. Computation of these parameters is a real problem with the increase in the number of response variables or with the increase in the number of time points. The problem becomes more intricate and involved with the addition of additional random effects. A multivariate analysis is not possible in a small sample setting. We propose a method to estimate these many parameters in bits and pieces from baby models, by taking a subset of response variables at a time, and finally using these bits and pieces at the end to get the parameter estimates for the mother model, with all variables taken together. Applying this method one can calculate the fixed effects, the best linear unbiased predictions (BLUPs) for the random effects in the model, and also the BLUPs at each time of observation for each response variable, to monitor the effectiveness of the treatment for each subject. The proposed method is illustrated with an example of multiple response variables measured over multiple time points arising from a clinical trial in osteoporosis.     

On asymptotic distribution of prediction in functional linear regression

There has been substantial recent attention on problems involving a functional linear regression model with scalar response. Among them, there have been few works dealing with asymptotic distribution of prediction in functional linear regression models. In recent literature, the centeral limit theorem for prediction has been discussed, but the proof and conditions under which the random bias terms for a fixed predictor converge to zero have been ignored so that the impact of these terms on the convergence of the prediction has not been well understood. Clarifying the proof and conditions under which the bias terms converge to zero, we show that the asymptotic distribution of the prediction is normal. Furthermore, we have derived those results related to other terms that already obtained by others, under milder conditions. Finally, we conduct a simulation study to investigate performance of the asymptotic distribution under various parameter settings.     

Empirical Bayes estimates for correlated hierarchical data with overdispersion

An extension of the generalized linear mixed model was constructed to simultaneously accommodate overdispersion and hierarchies present in longitudinal or clustered data. This so‐called combined model includes conjugate random effects at observation level for overdispersion and normal random effects at subject level to handle correlation, respectively. A variety of data types can be handled in this way, using different members of the exponential family. Both maximum likelihood and Bayesian estimation for covariate effects and variance components were proposed. The focus of this paper is the development of an estimation procedure for the two sets of random effects. These are necessary when making predictions for future responses or their associated probabilities. Such (empirical) Bayes estimates will also be helpful in model diagnosis, both when checking the fit of the model as well as when investigating outlying observations. The proposed procedure is applied to three datasets of different outcome types. Copyright © 2014 John Wiley & Sons, Ltd.     

Nearest Neighbor Adjusted Best Linear Unbiased Prediction

Statistical inference for linear models has classically focused on either estimation or hypothesis testing of linear combinations of fixed effects or of variance components for random effects. A third form of inference—prediction of linear combinations of fixed and random effects—has important advantages over conventional estimators in many applications. None of these approaches will result in accurate inference if the data contain strong, unaccounted for local gradients, such as spatial trends in field-plot data. Nearest neighbor methods to adjust for such trends have been widely discussed in recent literature. So far, however, these methods have been developed exclusively for classical estimation and hypothesis testing. In this article a method of obtaining nearest neighbor adjusted (NNA) predictors, along the lines of “best linear unbiased prediction,” or BLUP, is developed. A simulation study comparing “NNABLUP” to conventional NNA methods and to non-NNA alternatives suggests considerable potential for improved efficiency.     

Estimation of group means in generalized linear mixed models

In this study, we investigate the concept of the mean response for a treatment group mean as well as its estimation and prediction for generalized linear models with a subject‐wise random effect. Generalized linear models are commonly used to analyze categorical data. The model‐based mean for a treatment group usually estimates the response at the mean covariate. However, the mean response for the treatment group for studied population is at least equally important in the context of clinical trials. New methods were proposed to estimate such a mean response in generalized linear models; however, this has only been done when there are no random effects in the model. We suggest that, in a generalized linear mixed model (GLMM), there are at least two possible definitions of a treatment group mean response that can serve as estimation/prediction targets. The estimation of these treatment group means is important for healthcare professionals to be able to understand the absolute benefit vs risk. For both of these treatment group means, we propose a new set of methods that suggests how to estimate/predict both of them in a GLMMs with a univariate subject‐wise random effect. Our methods also suggest an easy way of constructing corresponding confidence and prediction intervals for both possible treatment group means. Simulations show that proposed confidence and prediction intervals provide correct empirical coverage probability under most circumstances. Proposed methods have also been applied to analyze hypoglycemia data from diabetes clinical trials.     

Design and analysis of computer experiments when the output is highly correlated over the input space

To build a predictor, the output of a deterministic computer model or “code” is often treated as a realization of a stochastic process indexed by the code's input variables. The authors consider an asymptotic form of the Gaussian correlation function for the stochastic process where the correlation tends to unity. They show that the limiting best linear unbiased predictor involves Lagrange interpolating polynomials; linear model terms are implicitly included. The authors then develop optimal designs based on minimizing the limiting integrated mean squared error of prediction. They show through several examples that these designs lead to good prediction accuracy.     

Bayesian model selection in linear mixed effects models with autoregressive(p) errors using mixture priors

In this article, we apply the Bayesian approach to the linear mixed effect models with autoregressive(p) random errors under mixture priors obtained with the Markov chain Monte Carlo (MCMC) method. The mixture structure of a point mass and continuous distribution can help to select the variables in fixed and random effects models from the posterior sample generated using the MCMC method. Bayesian prediction of future observations is also one of the major concerns. To get the best model, we consider the commonly used highest posterior probability model and the median posterior probability model. As a result, both criteria tend to be needed to choose the best model from the entire simulation study. In terms of predictive accuracy, a real example confirms that the proposed method provides accurate results.     

Numerical Comparison Between Different Empirical Prediction Intervals Under the Fay-Herriot Model

Recently, an empirical best linear unbiased predictor is widely used as a practical approach to small area inference. It is also of interest to construct empirical prediction intervals. However, we do not know which method should be used from among the several existing prediction intervals. In this article, we first obtain an empirical prediction interval by using the residual maximum likelihood method for estimating unknown model variance parameters. Then we compare the later with other intervals with the residual maximum likelihood method. Additionally, some different parametric bootstrap methods for constructing empirical prediction intervals are also compared in a simulation study.     

Impact of missing data on type 1 error rates in non‐inferiority trials

In this paper, a simulation study is conducted to systematically investigate the impact of different types of missing data on six different statistical analyses: four different likelihood‐based linear mixed effects models and analysis of covariance (ANCOVA) using two different data sets, in non‐inferiority trial settings for the analysis of longitudinal continuous data. ANCOVA is valid when the missing data are completely at random. Likelihood‐based linear mixed effects model approaches are valid when the missing data are at random. Pattern‐mixture model (PMM) was developed to incorporate non‐random missing mechanism. Our simulations suggest that two linear mixed effects models using unstructured covariance matrix for within‐subject correlation with no random effects or first‐order autoregressive covariance matrix for within‐subject correlation with random coefficient effects provide well control of type 1 error (T1E) rate when the missing data are completely at random or at random. ANCOVA using last observation carried forward imputed data set is the worst method in terms of bias and T1E rate. PMM does not show much improvement on controlling T1E rate compared with other linear mixed effects models when the missing data are not at random but is markedly inferior when the missing data are at random. Copyright © 2009 John Wiley & Sons, Ltd.     

Functional Mixed Effects Model for Small Area Estimation

Functional data analysis has become an important area of research because of its ability of handling high‐dimensional and complex data structures. However, the development is limited in the context of linear mixed effect models and, in particular, for small area estimation. The linear mixed effect models are the backbone of small area estimation. In this article, we consider area‐level data and fit a varying coefficient linear mixed effect model where the varying coefficients are semiparametrically modelled via B‐splines. We propose a method of estimating the fixed effect parameters and consider prediction of random effects that can be implemented using a standard software. For measuring prediction uncertainties, we derive an analytical expression for the mean squared errors and propose a method of estimating the mean squared errors. The procedure is illustrated via a real data example, and operating characteristics of the method are judged using finite sample simulation studies.     

Prediction of transplant-free survival in idiopathic pulmonary fibrosis patients using joint models for event times and mixed multivariate longitudinal data

We implement a joint model for mixed multivariate longitudinal measurements, applied to the prediction of time until lung transplant or death in idiopathic pulmonary fibrosis. Specifically, we formulate a unified Bayesian joint model for the mixed longitudinal responses and time-to-event outcomes. For the longitudinal model of continuous and binary responses, we investigate multivariate generalized linear mixed models using shared random effects. Longitudinal and time-to-event data are assumed to be independent conditional on available covariates and shared parameters. A Markov chain Monte Carlo algorithm, implemented in OpenBUGS, is used for parameter estimation. To illustrate practical considerations in choosing a final model, we fit 37 different candidate models using all possible combinations of random effects and employ a deviance information criterion to select a best-fitting model. We demonstrate the prediction of future event probabilities within a fixed time interval for patients utilizing baseline data, post-baseline longitudinal responses, and the time-to-event outcome. The performance of our joint model is also evaluated in simulation studies.     

Transformation and Smoothing in Sample Survey Data

Abstract. We consider model‐based prediction of a finite population total when a monotone transformation of the survey variable makes it appropriate to assume additive, homoscedastic errors. As the transformation to achieve this does not necessarily simultaneously produce an easily parameterized mean function, we assume only that the mean is a smooth function of the auxiliary variable and estimate it non‐parametrically. The back transformation of predictions obtained on the transformed scale introduces bias which we remove using smearing. We obtain an asymptotic expansion for the prediction error which shows that prediction bias is asymptotically negligible and the prediction mean‐squared error (MSE) using a non‐parametric model remains in the same order as when a parametric model is adopted. The expansion also shows the effect of smearing on the prediction MSE and can be used to compute the asymptotic prediction MSE. We propose a model‐based bootstrap estimate of the prediction MSE. The predictor produces competitive results in terms of bias and prediction MSE in a simulation study, and performs well on a population constructed from an Australian farm survey.     

Optimal Prediction in Stochastic Regression Models with Application to the Analysis of Repeated Surveys

Several results relating to the optimal prediction of regression coefficients and random variables under a general linear model with stochastic coefficients are presented. These results are then applied to the analysis of repeated sample surveys over time. In particular, if the finite population can be modelled by a superpopulation model, a fully efficient method for the analysis of repeated surveys is proposed.     

Optimal predictive design augmentation for spatial generalised linear mixed models

A typical model for geostatistical data when the observations are counts is the spatial generalised linear mixed model. We present a criterion for optimal sampling design under this framework which aims to minimise the error in the prediction of the underlying spatial random effects. The proposed criterion is derived by performing an asymptotic expansion to the conditional prediction variance. We argue that the mean of the spatial process needs to be taken into account in the construction of the predictive design, which we demonstrate through a simulation study where we compare the proposed criterion against the widely used space-filling design. Furthermore, our results are applied to the Norway precipitation data and the rhizoctonia disease data.     

Linear Transformations of Linear Mixed-Effects Models

A number of articles have discussed the way lower order polynomial and interaction terms should be handled in linear regression models. Only if all lower order terms are included in the model will the regression model be invariant with respect to coding transformations of the variables. If lower order terms are omitted, the regression model will not be well formulated. In this paper, we extend this work to examine the implications of the ordering of variables in the linear mixed-effects model. We demonstrate how linear transformations of the variables affect the model and tests of significance of fixed effects in the model. We show how the transformations modify the random effects in the model, as well as their covariance matrix and the value of the restricted log-likelihood. We suggest a variable selection strategy for the linear mixed-effects model.     

An RKHS framework for functional data analysis

Linear combinations of random variables play a crucial role in multivariate analysis. Two extension of this concept are considered for functional data and shown to coincide using the Loève–Parzen reproducing kernel Hilbert space representation of a stochastic process. This theory is then used to provide an extension of the multivariate concept of canonical correlation. A solution to the regression problem of best linear unbiased prediction is obtained from this abstract canonical correlation formulation. The classical identities of Lawley and Rao that lead to canonical factor analysis are also generalized to the functional data setting. Finally, the relationship between Fisher's linear discriminant analysis and canonical correlation analysis for random vectors is extended to include situations with function-valued random elements. This allows for classification using the canonical Y scores and related distance measures.     

Predicting random effects with an expanded finite population mixed model

Prediction of random effects is an important problem with expanding applications. In the simplest context, the problem corresponds to prediction of the latent value (the mean) of a realized cluster selected via two-stage sampling. Recently, Stanek and Singer [Predicting random effects from finite population clustered samples with response error. J. Amer. Statist. Assoc. 99, 119–130] developed best linear unbiased predictors (BLUP) under a finite population mixed model that outperform BLUPs from mixed models and superpopulation models. Their setup, however, does not allow for unequally sized clusters. To overcome this drawback, we consider an expanded finite population mixed model based on a larger set of random variables that span a higher dimensional space than those typically applied to such problems. We show that BLUPs for linear combinations of the realized cluster means derived under such a model have considerably smaller mean squared error (MSE) than those obtained from mixed models, superpopulation models, and finite population mixed models. We motivate our general approach by an example developed for two-stage cluster sampling and show that it faithfully captures the stochastic aspects of sampling in the problem. We also consider simulation studies to illustrate the increased accuracy of the BLUP obtained under the expanded finite population mixed model.     

Small-area estimation by combining time-series and cross-sectional data

A model involving autocorrelated random effects and sampling errors is proposed for small-area estimation, using both time-series and cross-sectional data. The sampling errors are assumed to have a known block-diagonal covariance matrix. This model is an extension of a well-known model, due to Fay and Herriot (1979), for cross-sectional data. A two-stage estimator of a small-area mean for the current period is obtained under the proposed model with known autocorrelation, by first deriving the best linear unbiased prediction estimator assuming known variance components, and then replacing them with their consistent estimators. Extending the approach of Prasad and Rao (1986, 1990) for the Fay-Herriot model, an estimator of mean squared error (MSE) of the two-stage estimator, correct to a second-order approximation for a small or moderate number of time points, T, and a large number of small areas, m, is obtained. The case of unknown autocorrelation is also considered. Limited simulation results on the efficiency of two-stage estimators and the accuracy of the proposed estimator of MSE are présentés.     

Estimation and Prediction in the Presence of Spatial Confounding for Spatial Linear Models

In studies that produce data with spatial structure, it is common that covariates of interest vary spatially in addition to the error. Because of this, the error and covariate are often correlated. When this occurs, it is difficult to distinguish the covariate effect from residual spatial variation. In an i.i.d. normal error setting, it is well known that this type of correlation produces biased coefficient estimates, but predictions remain unbiased. In a spatial setting, recent studies have shown that coefficient estimates remain biased, but spatial prediction has not been addressed. The purpose of this paper is to provide a more detailed study of coefficient estimation from spatial models when covariate and error are correlated and then begin a formal study regarding spatial prediction. This is carried out by investigating properties of the generalized least squares estimator and the best linear unbiased predictor when a spatial random effect and a covariate are jointly modelled. Under this setup, we demonstrate that the mean squared prediction error is possibly reduced when covariate and error are correlated.