Pseudo maximum likelihood approach for the analysis of multivariate left‐censored longitudinal data

The linear mixed effects model based on a full likelihood is one of the few methods available to model longitudinal data subject to left censoring. However, a full likelihood approach is complicated algebraically because of the large dimension of the numeric computations, and maximum likelihood estimation can be computationally prohibitive when the data are heavily censored. Moreover, for mixed models, the complexity of the computation increases as the dimension of the random effects in the model increases. We propose a method based on pseudo likelihood that simplifies the computational complexities, allows a wide class of multivariate models, and that can be used for many different data structures including settings where the level of censoring is high. The motivation for this work comes from the need for a joint model to assess the joint effect of pro‐inflammatory and anti‐inflammatory biomarker data on 30‐day mortality status while simultaneously accounting for longitudinal left censoring and correlation between markers in the analysis of Genetic and Inflammatory Markers for Sepsis study conducted at the University of Pittsburgh. Two markers, interleukin‐6 and interleukin‐10, which naturally are correlated because of a shared similar biological pathways and are left‐censored because of the limited sensitivity of the assays, are considered to determine if higher levels of these markers is associated with an increased risk of death after accounting for the left censoring and their assumed correlation. Copyright © 2016 John Wiley & Sons, Ltd.     

Bias-corrected maximum likelihood estimator of the intraclass correlation parameter for binary data

A popular model to analyse over/under-dispersed proportions is to assume the extended beta-binomial model with dispersion (intraclass correlation) parameter phi and then to estimate this parameter by maximum likelihood. However, it is well known that maximum likelihood estimate (MLE) may be biased when the sample size n or the total Fisher information is small. In this paper we obtain a bias-corrected maximum likelihood (BCML) estimator of the intraclass correlation parameter and compare it, by simulation, in terms of bias and efficiency, with the MLE, an estimator Q(2) based on optimal quadratic estimating equations of Crowder and recommended by Paul et al. and a double extended quasi-likelihood (DEQL) estimator proposed by Lee. The BCML estimator has superior bias and efficiency properties in most instances. Analyses of a set of toxicological data from Paul and a set of medical data pertaining to chromosomal abnormalities among survivors of the atomic bomb in Hiroshima from Otake and Prentice show, in general, much improvement in standard errors of the BCML estimates over the other three estimates.     

A model selection approach to analysis of variance and covariance

An alternative to analysis of variance is a model selection approach where every partition of the treatment means into clusters with equal value is treated as a separate model. The null hypothesis that all treatments are equal corresponds to the partition with all means in a single cluster. The alternative hypothesis correspond to the set of all other partitions of treatment means. A model selection approach can also be used for a treatment by covariate interaction, where the null hypothesis and each alternative correspond to a partition of treatments into clusters with equal covariate effects. We extend the partition‐as‐model approach to simultaneous inference for both treatment main effect and treatment interaction with a continuous covariate with separate partitions for the intercepts and treatment‐specific slopes. The model space is the Cartesian product of the intercept partition and the slope partition, and we develop five joint priors for this model space. In four of these priors the intercept and slope partition are dependent. We advise on setting priors over models, and we use the model to analyze an orthodontic data set that compares the frictional resistance created by orthodontic fixtures. Copyright © 2009 John Wiley & Sons, Ltd.     

Multivariate and multilocus variance components method,based on structural relationships to assess quantitative trait linkage via SEGPATH

A general-purpose modeling framework for performing path and segregation analysis jointly, called SEGPATH (Province and Rao [1995] Stat. Med. 7:185-198), has been extended to cover "model-free" robust, variance-components linkage analysis, based on identity-by-descent (IBD) sharing. These extended models can be used to analyze linkage to a single marker or to perform multipoint linkage analysis, with a single phenotype or multivariate vector of phenotypes, in pedigrees. Within a single, consistent approach, SEGPATH models can perform segregation analysis, path analysis, linkage analysis, or combinations thereof. SEGPATH models can incorporate environmental or other measured covariate fixed effects (including measured genotypes), genotype-specific covariate effects, population heterogeneity models, repeated-measures models, longitudinal models, autoregressive models, developmental models, gene-by-environment interaction models, etc., with or without linkage components. The data analyzed can have any missing value structure (assumed missing at random), with entire individuals missing, or missing on one or more measurements. Corrections for ascertainment can be made on a vector of phenotypes and/or other measures. Because of the flexibility of the class of models, the SEGPATH approach can also be used in nongenetic applications where there is a hierarchical structure, such as longitudinal, repeated-measures, time series, or nested models. A variety of specific models are provided, as well as some comparisons with other linkage analysis models. Particular applications demonstrate the importance of correctly accounting for the extraneous sources of familial resemblance, as can be done easily with these SEGPATH models, so as to give added power to detect linkage as well as to protect against spuriously inferring linkage.     

Nonrandom sampling in genetic epidemiology: maximum likelihood methods for multifactorial analysis of quantitative data ascertained through truncation

Three types of nonrandom sampling of family data are described, and appropriate maximum likelihood methods are proposed for each. The three types arise depending on whether the selection of probands, based on truncation, is applied directly to the phenotypic distribution, to the distribution of a correlated trait, or to the liability distribution of an associated disease. Family data ascertained through random and nonrandom sampling can be analyzed together in a unified approach. Results of a Monte Carlo study are presented that demonstrate the utility of the proposed methods. In particular, likelihood ratio tests of null hypotheses are shown to be distributed as chi-square, even in samples as small as 50 families (with variable sibship size).     

Hierarchic analysis of variance in the classification of industrial environmental hazards

Summary The paper presents the method of statistical analysis used for assessment of environmental data. The method is very useful and rather simple design for exploring in more detailed way the main sources of variation in the dust survey analysis. It is especially recommended in the course of epidemiological prospective observations.This reasearch was carried out under the patronage of the Scientific Program Council for Studies on Chronic Respiratory Diseases, Kraków (Chairman: Prof. Dr. J. Kostrzewski)The work was partly supported by a grant under contract NCHS PL.1 with the National Center for Health Statistics, Washington (Director of the Study: Prof. Dr. F. Sawicki)The idea of the paper was given to me by Prof. F. Sawicki to whom I owe many thanks. The dust measurements were made by the Air Pollution Laboratory of the Sanitary Epidemiologic Station in Kraków     

基于靶向最大似然估计模型在高血压药物疗效评价中的应用

目的  依托山东省胶南市“全人群高血压、糖尿病综合防治项目”建立队列,借助靶向最大似然估计(targeted maximum likelihood estimation, TMLE)模型评价高血压患者服用卡托普利或尼群地平对血压控制的平均因果效应和个体化因果效应,在大数据背景下辅助精准医疗以实现高血压控制。 方法  筛选只服用卡托普利或尼群地平的患者,将其第一次随访血压控制情况作为结局,将年龄、性别、职业、BMI、吸烟、饮酒及运动情况纳入分析,采用嵌入Super Learner组合预测算法的靶向最大似然估计模型拟合条件均值结果的初始估计并进行波动,更新初始拟合,对目标参数做出最优偏差-方差权衡优化模型,从而得到平均因果效应,并进一步分析个体化因果效应。 结果  共纳入13 676名高血压患者。总体上相比服用卡托普利,服用尼群地平更有利于血压控制(OR=1.24, 95% CI: 1.13~1.35, P=0.004)。从个体净效应来看,98.65%的患者使用尼群地平的血压控制效果更好。 结论  靶向最大似然估计模型能够分析平均因果效应和个性化因果效应,为现实世界的因果推断研究提供方法借鉴。     

Functional data analysis with application to periodically stimulated foetal heart rate data. II: functional logistic regression

We present a basis solution for the modelling of a binary response with a functional covariate plus any number of scalar covariates. This can be thought of as singular longitudinal data analysis as there are more measurements on the functional covariate than subjects in the study. The maximum likelihood parameter estimates are found using a basis expansion and a modified Fisher scoring algorithm. This technique has been extended to model a functional covariate with a repeated stimulus. We used periodically stimulated foetal heart rate tracings to predict the probability of a high risk birth outcome. It was found that these tracings could predict 94.1 per cent of the high risk pregnancies and without the stimulus, the heart rates were no more predictive than chance.     

Avoiding zero between‐study variance estimates in random‐effects meta‐analysis

Fixed‐effects meta‐analysis has been criticized because the assumption of homogeneity is often unrealistic and can result in underestimation of parameter uncertainty. Random‐effects meta‐analysis and meta‐regression are therefore typically used to accommodate explained and unexplained between‐study variability. However, it is not unusual to obtain a boundary estimate of zero for the (residual) between‐study standard deviation, resulting in fixed‐effects estimates of the other parameters and their standard errors. To avoid such boundary estimates, we suggest using Bayes modal (BM) estimation with a gamma prior on the between‐study standard deviation. When no prior information is available regarding the magnitude of the between‐study standard deviation, a weakly informative default prior can be used (with shape parameter 2 and rate parameter close to 0) that produces positive estimates but does not overrule the data, leading to only a small decrease in the log likelihood from its maximum. We review the most commonly used estimation methods for meta‐analysis and meta‐regression including classical and Bayesian methods and apply these methods, as well as our BM estimator, to real datasets. We then perform simulations to compare BM estimation with the other methods and find that BM estimation performs well by (i) avoiding boundary estimates; (ii) having smaller root mean squared error for the between‐study standard deviation; and (iii) better coverage for the overall effects than the other methods when the true model has at least a small or moderate amount of unexplained heterogeneity. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Comparison of methods for analyzing binary data arising from two-sample twin studies

The study of twins is widely used for research into genetic and environmental influences on human outcome measurements. For the study design in which independent samples of monozygotic and dizygotic twins are compared with respect to their similarity on a binary trait, several statistical methods have been proposed. Using a Monte Carlo simulation, we compare the five following procedures: 1) goodness-of-fit method based on the common correlation model, 2) normal approximation of the maximum likelihood estimators of the common correlation coefficients, 3) Ramakrishnan et al. [(1992) Genet Epidemiol 9:273–282] method of odds ratio comparison, 4) generalized estimating equations method of odds ratio estimation, and 5) tetrachoric correlation method. The results show that the goodness-of-fit approach has similar or better performance in both type-one error rates and power than the other methods in all parameter settings. Its advantage with respect to type-one error rates is particularly clear under conditions of small sample sizes, extreme prevalences, or high values of the intraclass correlation coefficients. Therefore, the goodness-of-fit method is recommended for the two-sample twin study design. Genet. Epidemiol. 14:349–363,1997. © 1997 Wiley-Liss, Inc.     

A likelihood approach to estimating sensitivity and specificity for binocular data: application in ophthalmology

Binocular data typically arise in ophthalmology where pairs of eyes are evaluated, through some diagnostic procedure, for the presence of certain diseases or pathologies. Treating eyes as independent and adopting the usual approach in estimating the sensitivity and specificity of a diagnostic test ignores the correlation between eyes. This may consequently yield incorrect estimates, especially of the standard errors. The paper proposes a likelihood-based method of accounting for the correlations between eyes and estimating sensitivity and specificity using a model for binocular or paired binary outcomes. Estimation of model parameters via maximum likelihood is outlined and approximate tests are provided. The efficiency of the estimates is assessed in a simulation study. An extension of the methodology to the case of several diagnostic tests, or the same test measured on several occasions, which arises in multi-reader studies, is given. A further extension to the case of multiple diseases is outlined as well. Data from a study on diabetic retinopathy are analysed to illustrate the methodology.     

A genetic and environmental analysis of a twin family study of alcohol use,anxiety, and depression

Alcohol consumption, anxiety, and depression were measured by questionnaire in 572 twin families ascertained from the Institute of Psychiatry (London) normal twin register, each family consisting of an adult twin pair, their parents, and siblings-a total of 1,742 individuals. A multivariate normal model for pedigree analysis was applied to each variable, with power transformations fitted to maximise the fit with distributional assumptions. The effect of shared twin environment was estimated by considering the measured cohabitation history of twin pairs. For log-transformed alcohol consumption, amongst current drinkers this effect was the same for MZ and DZ pairs but depended on the cohabitation status of pairs. For both anxiety and depression the effect was clearly not the same for MZ and DZ pairs. Therefore the basic assumption of the classical twin method appears to be invalid for all three traits. Estimates of heritability derived from these analyses were compared with those obtained (1) by applying the classical twin method to twin data only, and (2) by a pedigree analysis ignoring the effect of shared twin environment. For all variables there were considerable differences between estimates based on the three models. This study illustrates that data from twins and their relatives which includes information on cohabitation history might distinguish shared genes and shared environment as causes of familial aggregation. In these behavioral traits the effect of shared twin environment may depend on zygosity and play a major role in explaining familial aggregation in twin family data.     

A maximum likelihood method for studying gene-environment interactions under conditional independence of genotype and exposure

Given the biomedical interest in gene-environment interactions along with the difficulties inherent in gathering genetic data from controls, epidemiologists need methodologies that can increase precision of estimating interactions while minimizing the genotyping of controls. To achieve this purpose, many epidemiologists suggested that one can use case-only design. In this paper, we present a maximum likelihood method for making inference about gene-environment interactions using case-only data. The probability of disease development is described by a logistic risk model. Thus the interactions are model parameters measuring the departure of joint effects of exposure and genotype from multiplicative odds ratios. We extend the typical inference method derived under the assumption of independence between genotype and exposure to that under a more general assumption of conditional independence. Our maximum likelihood method can be applied to analyse both categorical and continuous environmental factors, and generalized to make inference about gene-gene-environment interactions. Moreover, the application of this method can be reduced to simply fitting a multinomial logistic model when we have case-only data. As a consequence, the maximum likelihood estimates of interactions and likelihood ratio tests for hypotheses concerning interactions can be easily computed. The methodology is illustrated through an example based on a study about the joint effects of XRCC1 polymorphisms and smoking on bladder cancer. We also give two simulation studies to show that the proposed method is reliable in finite sample situation.     

A variance components factor model for genetic association studies: A Bayesian analysis

Studies of gene‐trait associations for complex diseases often involve multiple traits that may vary by genotype groups or patterns. Such traits are usually manifestations of lower‐dimensional latent factors or disease syndromes. We illustrate the use of a variance components factor (VCF) model to model the association between multiple traits and genotype groups as well as any other existing patient‐level covariates. This model characterizes the correlations between traits as underlying latent factors that can be used in clinical decision‐making. We apply it within the Bayesian framework and provide a straightforward implementation using the WinBUGS software. The VCF model is illustrated with simulated data and an example that comprises changes in plasma lipid measurements of patients who were treated with statins to lower low‐density lipoprotein cholesterol, and polymorphisms from the apolipoprotein‐E gene. The simulation shows that this model clearly characterizes existing multiple trait manifestations across genotype groups where individuals' group assignments are fully observed or can be deduced from the observed data. It also allows one to investigate covariate by genotype group interactions that may explain the variability in the traits. The flexibility to characterize such multiple trait manifestations makes the VCF model more desirable than the univariate variance components model, which is applied to each trait separately. The Bayesian framework offers a flexible approach that allows one to incorporate prior information. Genet. Epidemiol. 34: 529–536, 2010. © 2010 Wiley‐Liss, Inc.     

Estimation of individual genetic and environmental factor scores

Implicit in the application of the common-factor model as a method for decomposing trait covariance into a genetic and environmental part is the use of factor scores. In multivariate analyses, it is possible to estimate these factor scores for the communal part of the model. Estimation of scores on latent factors in terms of individual observations within the context of a twin/family study amounts to estimation of individual genetic and environmental scores. Such estimates may be of both theoretical and practical interest and may be provided with confidence intervals around the individual estimates. The method is first illustrated with stimulated twin data and next is applied to blood pressure data obtained in a Dutch sample of 59 male adolescent twin pairs. Subjects with high blood pressure can be distinguished into groups with high genetic or high environmental scores.     

Investigating trial and treatment heterogeneity in an individual patient data meta-analysis of survival data by means of the penalized maximum likelihood approach

In a meta-analysis combining survival data from different clinical trials, an important issue is the possible heterogeneity between trials. Such intertrial variation can not only be explained by heterogeneity of treatment effects across trials but also by heterogeneity of their baseline risk. In addition, one might examine the relationship between magnitude of the treatment effect and the underlying risk of the patients in the different trials. Such a scenario can be accounted for by using additive random effects in the Cox model, with a random trial effect and a random treatment-by-trial interaction. We propose to use this kind of model with a general correlation structure for the random effects and to estimate parameters and hazard function using a semi-parametric penalized marginal likelihood method (maximum penalized likelihood estimators). This approach gives smoothed estimates of the hazard function, which represents incidence in epidemiology. The idea for the approach in this paper comes from the study of heterogeneity in a large meta-analysis of randomized trials in patients with head and neck cancers (meta-analysis of chemotherapy in head and neck cancers) and the effect of adding chemotherapy to locoregional treatment. The simulation study and the application demonstrate that the proposed approach yields satisfactory results and they illustrate the need to use a flexible variance-covariance structure for the random effects.     

Statistical validity for testing associations between genetic markers and quantitative traits in family data

In genetic analysis it is often of interest to analyze associations between traits of unknown genetic etiology and genetic markers from pedigree data. Statistical methods that assume independence of pedigree members cannot be used because they disregard the statistical dependencies of members in a pedigree. For quantitative traits, a regression model proposed by George and Elston [Genet Epidemiol 4:193–201, 1987] uses an asymptotic likelihood ratio test and incorporates a correlation structure that allows for statistical dependence among the pedigree members. The statistical validity of this test is assessed for finite samples by measuring the discrepancy between the empirical and theoretical chi-square distributions. The variance of the mean of the dependent variable is determined to be related to this discrepancy and can be used to determine whether a pedigree structure is large enough for making valid statistical inferences on the basis of the asymptotic test. A multi-generational pedigree of 200 or so individuals should in many cases be sufficient for valid results when using the asymptotic likelihood ratio test for the association between markers and continuous traits. © 1995 Wiley-Liss, Inc.     

The analysis of binary longitudinal data with time-dependent covariates

We consider longitudinal studies with binary outcomes that are measured repeatedly on subjects over time. The goal of our analysis was to fit a logistic model that relates the expected value of the outcomes with explanatory variables that are measured on each subject. However, additional care must be taken to adjust for the association between the repeated measurements on each subject. We propose a new maximum likelihood method for covariates that may be fixed or time varying. We also implement and make comparisons with two other approaches: generalized estimating equations, which may be more robust to misspecification of the true correlation structure, and alternating logistic regression, which models association via odds ratios that are subject to less restrictive constraints than are correlations. The proposed estimation procedure will yield consistent and asymptotically normal estimates of the regression and correlation parameters if the correlation on consecutive measurements on a subject is correctly specified. Simulations demonstrate that our approach can yield improved efficiency in estimation of the regression parameter; for equally spaced and complete data, the gains in efficiency were greatest for the parameter associated with a time-by-group interaction term and for stronger values of the correlation. For unequally spaced data and with dropout according to a missing-at-random mechanism, MARK1ML with correctly specified consecutive correlations yielded substantial improvements in terms of both bias and efficiency. We present an analysis to demonstrate application of the methods we consider. We also offer an R function for easy implementation of our approach.     

Age-period-cohort analysis of cervical cancer incidence in Hong Kong from 1972 to 2001 using maximum likelihood and Bayesian methods

OBJECTIVE: To examine the secular effects of opportunistic screening for cervical cancer in a rich, developed community where most other such populations have long adopted organised screening. Design, setting, and PARTICIPANTS: The analysis was based on 15 140 cases of invasive cervical cancer from 1972 to 2001. The effects of chronological age, time period, and birth cohort were decomposed using both maximum likelihood and Bayesian methods. RESULTS: The overall age adjusted incidence decreased from 24.9 in 1972-74 to 9.5 per 100,000 in 1999-2001, in a log-linear fashion, yielding an average annual reduction of 4.0% (p<0.001) during the 30 year period. There were two second order and thus identifiable changes: (1) around the mid-1920s cohort curve representing an age-period interaction masquerading as a cohort change that denotes the first availability of Pap testing during the 1960s concentrated among women in their 40s; (2) a hook around the calendar years 1982-83 when cervical cytology became a standard screening test for pregnant women. CONCLUSIONS: Hong Kong's cervical cancer rates have declined since Pap tests first became available in the 1960s, most probably because of increasing population coverage over time and in successive generations in a haphazard fashion and punctuated by the systematic introduction of routine cytology as part of antenatal care in the 1980s.