Multilevel models for survival analysis with random effects

A method for modeling survival data with multilevel clustering is described. The Cox partial likelihood is incorporated into the generalized linear mixed model (GLMM) methodology. Parameter estimation is achieved by maximizing a log likelihood analogous to the likelihood associated with the best linear unbiased prediction (BLUP) at the initial step of estimation and is extended to obtain residual maximum likelihood (REML) estimators of the variance component. Estimating equations for a three-level hierarchical survival model are developed in detail, and such a model is applied to analyze a set of chronic granulomatous disease (CGD) data on recurrent infections as an illustration with both hospital and patient effects being considered as random. Only the latter gives a significant contribution. A simulation study is carried out to evaluate the performance of the REML estimators. Further extension of the estimation procedure to models with an arbitrary number of levels is also discussed.     

REML estimation of variance parameters in nonlinear mixed effects models using the SAEM algorithm

Nonlinear mixed effects models are now widely used in biometrical studies, especially in pharmacokinetic research or for the analysis of growth traits for agricultural and laboratory species. Most of these studies, however, are often based on ML estimation procedures, which are known to be biased downwards. A few REML extensions have been proposed, but only for approximated methods. The aim of this paper is to present a REML implementation for nonlinear mixed effects models within an exact estimation scheme, based on an integration of the fixed effects and a stochastic estimation procedure. This method was implemented via a stochastic EM, namely the SAEM algorithm. The simulation study showed that the proposed REML estimation procedure considerably reduced the bias observed with the ML estimation, as well as the residual mean squared error of the variance parameter estimations, especially in the unbalanced cases. ML and REML based estimators of fixed effects were also compared via simulation. Although the two kinds of estimates were very close in terms of bias and mean square error, predictions of individual profiles were clearly improved when using REML vs. ML. An application of this estimation procedure is presented for the modelling of growth in lines of chicken.     

Analysis of a Generalized Linear Mixed Model: A Case Study and Simulation Results

A class of generalized linear mixed models can be obtained by introducing random effects in the linear predictor of a generalized linear model, e.g. a split plot model for binary data or count data. Maximum likelihood estimation, for normally distributed random effects, involves high-dimensional numerical integration, with severe limitations on the number and structure of the additional random effects. An alternative estimation procedure based on an extension of the iterative re-weighted least squares procedure for generalized linear models will be illustrated on a practical data set involving carcass classification of cattle. The data is analysed as overdispersed binomial proportions with fixed and random effects and associated components of variance on the logit scale. Estimates are obtained with standard software for normal data mixed models. Numerical restrictions pertain to the size of matrices to be inverted. This can be dealt with by absorption techniques familiar from e.g. mixed models in animal breeding. The final model fitted to the classification data includes four components of variance and a multiplicative overdispersion factor. Basically the estimation procedure is a combination of iterated least squares procedures and no full distributional assumptions are needed. A simulation study based on the classification data is presented. This includes a study of procedures for constructing confidence intervals and significance tests for fixed effects and components of variance. The simulation results increase confidence in the usefulness of the estimation procedure.     

Mixed-model analysis of a censored normal distribution with reference to animal breeding

A mixed-model procedure for analysis of censored data assuming a multivariate normal distribution is described. A Bayesian framework is adopted which allows for estimation of fixed effects and variance components and prediction of random effects when records are left-censored. The procedure can be extended to right- and two-tailed censoring. The model employed is a generalized linear model, and the estimation equations resemble those arising in analysis of multivariate normal or categorical data with threshold models. Estimates of variance components are obtained using expressions similar to those employed in the EM algorithm for restricted maximum likelihood (REML) estimation under normality.     

Diallel analysis for sex-linked and maternal effects

Genetic models including sex-linked and maternal effects as well as autosomal gene effects are described. Monte Carlo simulations were conducted to compare efficiencies of estimation by minimum norm quadratic unbiased estimation (MINQUE) and restricted maximum likelihood (REML) methods. MINQUE(1), which has 1 for all prior values, has a similar efficiency to MINQUE(), which requires prior estimates of parameter values. MINQUE(1) has the advantage over REML of unbiased estimation and convenient computation. An adjusted unbiased prediction (AUP) method is developed for predicting random genetic effects. AUP is desirable for its easy computation and unbiasedness of both mean and variance of predictors. The jackknife procedure is appropriate for estimating the sampling variances of estimated variances (or covariances) and of predicted genetic effects. A t-test based on jackknife variances is applicable for detecting significance of variation. Worked examples from mice and silkworm data are given in order to demonstrate variance and covariance estimation and genetic effect prediction.     

Developing risk models for multicenter data using standard logistic regression produced suboptimal predictions: A simulation study

Although multicenter data are common, many prediction model studies ignore this during model development. The objective of this study is to evaluate the predictive performance of regression methods for developing clinical risk prediction models using multicenter data, and provide guidelines for practice. We compared the predictive performance of standard logistic regression, generalized estimating equations, random intercept logistic regression, and fixed effects logistic regression. First, we presented a case study on the diagnosis of ovarian cancer. Subsequently, a simulation study investigated the performance of the different models as a function of the amount of clustering, development sample size, distribution of center-specific intercepts, the presence of a center-predictor interaction, and the presence of a dependency between center effects and predictors. The results showed that when sample sizes were sufficiently large, conditional models yielded calibrated predictions, whereas marginal models yielded miscalibrated predictions. Small sample sizes led to overfitting and unreliable predictions. This miscalibration was worse with more heavily clustered data. Calibration of random intercept logistic regression was better than that of standard logistic regression even when center-specific intercepts were not normally distributed, a center-predictor interaction was present, center effects and predictors were dependent, or when the model was applied in a new center. Therefore, to make reliable predictions in a specific center, we recommend random intercept logistic regression.     

Multilevel Mixture Cure Models with Random Effects

This paper extends the multilevel survival model by allowing the existence of cured fraction in the model. Random effects induced by the multilevel clustering structure are specified in the linear predictors in both hazard function and cured probability parts. Adopting the generalized linear mixed model (GLMM) approach to formulate the problem, parameter estimation is achieved by maximizing a best linear unbiased prediction (BLUP) type log‐likelihood at the initial step of estimation, and is then extended to obtain residual maximum likelihood (REML) estimators of the variance component. The proposed multilevel mixture cure model is applied to analyze the (i) child survival study data with multilevel clustering and (ii) chronic granulomatous disease (CGD) data on recurrent infections as illustrations. A simulation study is carried out to evaluate the performance of the REML estimators and assess the accuracy of the standard error estimates.     

Estimating effects of a single gene and polygenes on quantitative traits from a diallel design

A genetic model is developed with additive and dominance effects of a single gene and polygenes as well as general and specific reciprocal effects for the progeny from a diallel mating design. The methods of ANOVA, minimum norm quadratic unbiased estimation (MINQUE), restricted maximum likelihood estimation (REML), and maximum likelihood estimation (ML) are suggested for estimating variance components, and the methods of generalized least squares (GLS) and ordinary least squares (OLS) for fixed effects, while best linear unbiased prediction, linear unbiased prediction (LUP), and adjusted unbiased prediction are suggested for analyzing random effects. Monte Carlo simulations were conducted to evaluate the unbiasedness and efficiency of statistical methods involving two diallel designs with commonly used sample sizes, 6 and 8 parents, with no and missing crosses, respectively. Simulation results show that GLS and OLS are almost equally efficient for estimation of fixed effects, while MINQUE (1) and REML are better estimators of the variance components and LUP is most practical method for prediction of random effects. Data from a Drosophila melanogaster experiment (Gilbert 1985a, Theor appl Genet 69:625–629) were used as a working example to demonstrate the statistical analysis. The new methodology is also applicable to screening candidate gene(s) and to other mating designs with multiple parents, such as nested (NC Design I) and factorial (NC Design II) designs. Moreover, this methodology can serve as a guide to develop new methods for detecting indiscernible major genes and mapping quantitative trait loci based on mixture distribution theory. The computer program for the methods suggested in this article is freely available from the authors.     

A note on type II error under random effects misspecification in generalized linear mixed models

Litière, Alonso, and Molenberghs (2007, Biometrics, 63, 1038-1044) presented the results of simulation studies that they claimed showed that misspecification of the shape of the random effects distribution can produce marked increases in Type II error (decreases in power) of tests based on fits of generalized linear mixed models. However, the article contains a logical fallacy that invalidates this claim. We present logically correct simulation studies that demonstrate little increase in Type II error, consistent with the earlier work that shows little effect due to misspecification.     

基于不同特征挖掘方法结合广义提升回归模型估测安徽省土壤pH

为探讨不同特征挖掘方法与广义提升回归模型相结合在数字土壤制图中的应用,本研究首先使用递归特征消除和过滤式两种特征筛选方法对环境协变量进行筛选,再分别使用原始环境协变量、筛选后的最优变量组合作为自变量,建立基于广义提升回归模型和随机森林模型的安徽省土壤pH预测模型并进行制图。结果表明: 引入两种特征挖掘方法均可有效提高广义提升回归模型和随机森林模型预测土壤pH的精度,并且可以起到降维的作用;相较于随机森林模型,广义提升回归模型的验证集预测精度略低,在训练集中,广义提升回归模型的精度却远高于随机森林模型,模型解释度高,整体效果较好;随机森林模型的主要参数ntree和mtry对于模型的影响程度较低,而不同参数对于广义提升回归模型的预测精度影响较大,不同参数组合模型精度不同,建模前需要进行调参。空间制图结果表明,安徽省土壤pH呈“南酸北碱”趋势。     

Comparison of complex modeling strategies for prediction of a binary outcome based on a few,highly correlated predictors

Motivated by a clinical prediction problem, a simulation study was performed to compare different approaches for building risk prediction models. Robust prediction models for hospital survival in patients with acute heart failure were to be derived from three highly correlated blood parameters measured up to four times, with predictive ability having explicit priority over interpretability. Methods that relied only on the original predictors were compared with methods using an expanded predictor space including transformations and interactions. Predictors were simulated as transformations and combinations of multivariate normal variables which were fitted to the partly skewed and bimodally distributed original data in such a way that the simulated data mimicked the original covariate structure. Different penalized versions of logistic regression as well as random forests and generalized additive models were investigated using classical logistic regression as a benchmark. Their performance was assessed based on measures of predictive accuracy, model discrimination, and model calibration. Three different scenarios using different subsets of the original data with different numbers of observations and events per variable were investigated. In the investigated setting, where a risk prediction model should be based on a small set of highly correlated and interconnected predictors, Elastic Net and also Ridge logistic regression showed good performance compared to their competitors, while other methods did not lead to substantial improvements or even performed worse than standard logistic regression. Our work demonstrates how simulation studies that mimic relevant features of a specific data set can support the choice of a good modeling strategy.     

Testing random effects in linear mixed‐effects models with serially correlated errors

In linear mixed‐effects models, random effects are used to capture the heterogeneity and variability between individuals due to unmeasured covariates or unknown biological differences. Testing for the need of random effects is a nonstandard problem because it requires testing on the boundary of parameter space where the asymptotic chi‐squared distribution of the classical tests such as likelihood ratio and score tests is incorrect. In the literature several tests have been proposed to overcome this difficulty, however all of these tests rely on the restrictive assumption of i.i.d. measurement errors. The presence of correlated errors, which often happens in practice, makes testing random effects much more difficult. In this paper, we propose a permutation test for random effects in the presence of serially correlated errors. The proposed test not only avoids issues with the boundary of parameter space, but also can be used for testing multiple random effects and any subset of them. Our permutation procedure includes the permutation procedure in Drikvandi, Verbeke, Khodadadi, and Partovi Nia (2013) as a special case when errors are i.i.d., though the test statistics are different. We use simulations and a real data analysis to evaluate the performance of the proposed permutation test. We have found that random slopes for linear and quadratic time effects may not be significant when measurement errors are serially correlated.     

MIXED MODEL APPROACHES FOR ESTIMATING GENETIC VARIANCES AND COVARIANCES

The limitations of methods for analysis of variance(ANOVA)in estimating genetic variances are discussed. Among the three methods(maximum likelihood ML, restricted maximum likelihood REML, and minimum norm quadratic unbiased estimation MINQUE)for mixed linear models, MINQUE method is presented with formulae for estimating variance components and covariances components and for predicting genetic effects. Several genetic models, which cannot be appropriately analyzed by ANOVA methods, are introduced in forms of mixed linear models. Genetic models with independent random effects can be analyzed by MINQUE(1)method whieh is a MINQUE method with all prior values setting 1. MINQUE(1)method can give unbiased estimation for variance components and covariance components, and linear unbiased prediction (LUP) for genetic effects. There are more complicate genetic models for plant seeds which involve correlated random effects. MINQUE(0/1)method, which is a MINQUE method with all prior covariances setting 0 and all prior variances setting 1, is suitable for estimating variance and covariance components in these models. Mixed model approaches have advantage over ANOVA methods for the capacity of analyzing unbalanced data and complicated models. Some problems about estimation and hypothesis test by MINQUE method are discussed.     

Modelling the growth curve of Maine-Anjou beef cattle using heteroskedastic random coefficients models

A heteroskedastic random coefficients model was described for analyzing weight performances between the 100th and the 650th days of age of Maine-Anjou beef cattle. This model contained both fixed effects, random linear regression and heterogeneous variance components. The objective of this study was to analyze the difference of growth curves between animals born as twin and single bull calves. The method was based on log-linear models for residual and individual variances expressed as functions of explanatory variables. An expectation-maximization (EM) algorithm was proposed for calculating restricted maximum likelihood (REML) estimates of the residual and individual components of variances and covariances. Likelihood ratio tests were used to assess hypotheses about parameters of this model. Growth of Maine-Anjou cattle was described by a third order regression on age for a mean growth curve, two correlated random effects for the individual variability and independent errors. Three sources of heterogeneity of residual variances were detected. The difference of weight performance between bulls born as single and twin bull calves was estimated to be equal to about 15 kg for the growth period considered.     

Generalized inferences on the overall treatment effect in meta-analysis with normally distributed outcomes

This paper focuses on inferences about the overall treatment effect in meta-analysis with normally distributed responses based on the concepts of generalized inference. A refined generalized pivotal quantity based on t distribution is presented and simulation study shows that it can provide confidence intervals with satisfactory coverage probabilities and perform hypothesis testing with satisfactory type-I error control at very small sample sizes.     

Hierarchical Bayesian modeling of random and residual variance–covariance matrices in bivariate mixed effects models

Bivariate mixed effects models are often used to jointly infer upon covariance matrices for both random effects ( u ) and residuals ( e ) between two different phenotypes in order to investigate the architecture of their relationship. However, these (co)variances themselves may additionally depend upon covariates as well as additional sets of exchangeable random effects that facilitate borrowing of strength across a large number of clusters. We propose a hierarchical Bayesian extension of the classical bivariate mixed effects model by embedding additional levels of mixed effects modeling of reparameterizations of u‐ level and e ‐level (co)variances between two traits. These parameters are based upon a recently popularized square‐root‐free Cholesky decomposition and are readily interpretable, each conveniently facilitating a generalized linear model characterization. Using Markov Chain Monte Carlo methods, we validate our model based on a simulation study and apply it to a joint analysis of milk yield and calving interval phenotypes in Michigan dairy cows. This analysis indicates that the e ‐level relationship between the two traits is highly heterogeneous across herds and depends upon systematic herd management factors.     

Confidence Intervals for Relative Risks in Disease Mapping

Several analysis of the geographic variation of mortality rates in space have been proposed in the literature. Poisson models allowing the incorporation of random effects to model extra‐variability are widely used. The typical modelling approach uses normal random effects to accommodate local spatial autocorrelation. When spatial autocorrelation is absent but overdispersion persists, a discrete mixture model is an alternative approach. However, a technique for identifying regions which have significant high or low risk in any given area has not been developed yet when using the discrete mixture model. Taking into account the importance that this information provides to the epidemiologists to formulate hypothesis related to the potential risk factors affecting the population, different procedures for obtaining confidence intervals for relative risks are derived in this paper. These methods are the standard information‐based method and other four, all based on bootstrap techniques, namely the asymptotic‐bootstrap, the percentile‐bootstrap, the BC‐bootstrap and the modified information‐based method. All of them are compared empirically by their application to mortality data due to cardiovascular diseases in women from Navarra, Spain, during the period 1988–1994. In the small area example considered here, we find that the information‐based method is sensible at estimating standard errors of the component means in the discrete mixture model but it is not appropriate for providing standard errors of the estimated relative risks and hence, for constructing confidence intervals for the relative risk associated to each region. Therefore, the bootstrap‐based methods are recommended for this matter. More specifically, the BC method seems to provide better coverage probabilities in the case studied, according to a small scale simulation study that has been carried out using a scenario as encountered in the analysis of the real data.     

Empirical Bayes estimation of random effects parameters in mixed effects logistic regression models

We extend an approach for estimating random effects parameters under a random intercept and slope logistic regression model to include standard errors, thereby including confidence intervals. The procedure entails numerical integration to yield posterior empirical Bayes (EB) estimates of random effects parameters and their corresponding posterior standard errors. We incorporate an adjustment of the standard error due to Kass and Steffey (KS; 1989, Journal of the American Statistical Association 84, 717-726) to account for the variability in estimating the variance component of the random effects distribution. In assessing health care providers with respect to adult pneumonia mortality, comparisons are made with the penalized quasi-likelihood (PQL) approximation approach of Breslow and Clayton (1993, Journal of the American Statistical Association 88, 9-25) and a Bayesian approach. To make comparisons with an EB method previously reported in the literature, we apply these approaches to crossover trials data previously analyzed with the estimating equations EB approach of Waclawiw and Liang (1994, Statistics in Medicine 13, 541-551). We also perform simulations to compare the proposed KS and PQL approaches. These two approaches lead to EB estimates of random effects parameters with similar asymptotic bias. However, for many clusters with small cluster size, the proposed KS approach does better than the PQL procedures in terms of coverage of nominal 95% confidence intervals for random effects estimates. For large cluster sizes and a few clusters, the PQL approach performs better than the KS adjustment. These simulation results agree somewhat with those of the data analyses.     

Standard errors for EM estimates in generalized linear models with random effects

A procedure is derived for computing standard errors of EM estimates in generalized linear models with random effects. Quadrature formulas are used to approximate the integrals in the EM algorithm, where two different approaches are pursued, i.e., Gauss-Hermite quadrature in the case of Gaussian random effects and nonparametric maximum likelihood estimation for an unspecified random effect distribution. An approximation of the expected Fisher information matrix is derived from an expansion of the EM estimating equations. This allows for inferential arguments based on EM estimates, as demonstrated by an example and simulations.