Properties of the estimated variance component for subject-by-formulation interaction in studies of individual bioequivalence

Characteristics of the variance component for the subject-by-formulation interaction (sigma(2)(D)), estimated in simulated studies of individual bioequivalence and in three- and four-period cross-over trials reported by the FDA, were compared. sigma(2)(D) was estimated by (i) restricted maximum likelihood (REML) and (ii) the method of moments (MM). Variation of the variance component, estimated by both procedures (s(2)(D)) and for both the simulated and FDA data, increased with rising intra-individual variation. Consequently, a constant level of s(2)(D) (such as 0.0225 suggested by the FDA) may not be regarded as a basis for demonstrating substantial interactions. Features of the FDA and simulated parameters were similar. The results suggested that the FDA data were compatible with assuming sigma(D)=0.05 or perhaps 0.00. Therefore, there is no foundation for concerns about public health. Both simulations and calculations demonstrated that s(2)(D) estimated by MM was unbiased and its variance was proportional to sigma(4)(WF) when sigma(2)(D)=0.     

The application of REML in clinical trials

Residual maximum likelihood (REML) is a technique for estimating variance components in multi-classified data. In contrast to analysis of variance it can be routinely applied to unbalanced data and avoids some of the problems of biased variance estimates found with standard maximum likelihood estimation. The full REML method is of particular value for the analysis of unbalanced clinical trials as it allows recovery of all the available information on treatment effects which can lead to significant improvements in their precision. The use of REML has until recently been limited by heavy computational requirements and lack of readily available software. This is no longer such a restriction, however, as REML procedures are now available in several widely-used statistical packages, including BMDP, Genstat and SAS. This paper describes the REML technique and discusses its application to three common types of clinical trial: crossover, repeated measures and multicentre.     

Simultaneous assessment of occupational exposures from multiple worker groups

The methods developed by Rappaport et al. [Ann. Occup. Hyg. 39 (1995) 469] and Lyles et al. [J. Agri. Bio. Environ. Stat. 2 (1997a) 64; Ann. Occup. Hyg. 41 (1997b) 63]) for assessing workplace exposures on a group-by-group basis are extended to allow for the simultaneous assessment of data from multiple worker groups within the same industry. These extended methods allow models to be fit simultaneously to data on all groups in a study, even when some of the groups might not contribute adequate information to be modeled separately. We assume that the exposures are log-normally distributed, and that they can be adequately modeled by a mixed effects regression model with parameters for exposure levels and for between- and within-worker variance components. Simultaneously analyzing data from multiple groups is only advantageous when at least one of these variance components can be assumed to be homogeneous across the groups. Here, we advocate testing an assumption of homogeneous within-worker variance components, sigma(2)(w,h), using a likelihood ratio test to choose between a full model (distinct sigma(2)(w,h) for each group) and a reduced model (common sigma(2)(w) across groups). We then develop a procedure, which is conditional on the results of the likelihood ratio test, for testing whether or not each group of workers is overexposed to the contaminant of interest. This modeling and testing procedure was applied to 39 different data sets, each containing data for multiple groups, from a wide variety of industries. For these data, the testing procedure generally resulted in the same conclusion regarding overexposure under both models, even in those data sets where the within-worker variance components appeared to be quite heterogeneous. We also conducted a small simulation study to estimate the significance level of the proposed testing procedure, and found that the significance levels tended to be adequately close to the specified nominal level when a likelihood ratio test with significance level of at least 0.01 was used as a preliminary test. Additionally, we make specific recommendations for designing studies and suggest a method for determining whether engineering and administrative controls or individual-level interventions would be of most benefit to an overexposed group of workers.     

Estimating variance parameters from multivariate normal variables subject to limit of detection: MLE,REML, or Bayesian approaches?

Likelihood‐based approaches, which naturally incorporate left censoring due to limit of detection, are commonly utilized to analyze censored multivariate normal data. However, the maximum likelihood estimator (MLE) typically underestimates variance parameters. The restricted maximum likelihood estimator (REML), which corrects the underestimation of variance parameters, cannot be easily extended to analyze censored multivariate normal data. In the light of the connection between the REML and a Bayesian approach discovered in 1974 by Dr Harville, this paper describes a Bayesian approach to censored multivariate normal data. This Bayesian approach is justified through its link to the REML via Laplace's approximation and its performance is evaluated through a simulation study. We consider the Bayesian approach as a valuable alternative because it yields less biased variance parameter estimates than the MLE, and because a solid REML is technically difficult when data are left censored. Copyright © 2009 John Wiley & Sons, Ltd.     

多水平模型在生物等效性评价中的应用(Ⅰ)

探讨多水平模型在生物等效性评价中的应用价值.以2×4试验设计的抗高血压药生物等效性评价为研究实例,研究多水平模型对效应指标值的变异即方差的分解方式,并与FDA推荐的矩法所获得的方差分量进行比较.对比传统FDA推荐的生物等效性评价标准,研究利用多水平模型直接进行平均等效性、群体等效性和个体等效性评价的可行性.对于2×4试验设计的单变量两水平模型获得ln(AUC)指标的方差分量如试验药T的总方差σ_(TT)~2、个体间方差σ_(BT)~2和个体内方差σ_(WT)~2以及参比药R的总方差σ_(TR)~2、个体间方差σ_(BR)~2和个体内方差σ_(WR)~2,与FDA推荐的矩法所获得的结果非常接近.实际应用中,根据FDA提出的生物等效性评价的标准和程序进行评价,直接用多水平模型的估计值进行平均、群体和个体等效性评价,两者结果一致.多水平模型适合于交叉设计的生物等效性评价,相对于FDA推荐的方法,多水平模型对于复杂的有影响因素的交叉试验设计更容易估计方差分量,进而可以评价平均、群体和个体等效性,实际应用上更具有灵活性,为生物等效性评价提供了新的思路和方法.     

Meta‐analysis of binary data: which within study variance estimate to use?

We applied a mixed effects model to investigate between- and within-study variation in improvement rates of 180 schizophrenia outcome studies. The between-study variation was explained by the fixed study characteristics and an additional random study effect. Both rate difference and logit models were used. For a binary proportion outcome p(i) with sample size n(i) in the ith study, (circumflexp(i)(1-circumflexp(i))n)(-1) is the usual estimate of the within-study variance sigma(i)(2) in the logit model, where circumflexpi) is the sample mean of the binary outcome for subjects in study i. This estimate can be highly correlated with logit(circumflexp(i)). We used (macronp(i)(1-macronp)n(i))(-1) as an alternative estimate of sigma(i)(2), where macronp is the weighted mean of circumflexp(i)'s. We estimated regression coefficients (beta) of the fixed effects and the variance (tau(2)) of the random study effect using a quasi-likelihood estimating equations approach. Using the schizophrenia meta-analysis data, we demonstrated how the choice of the estimate of sigma(2)(i) affects the resulting estimates of beta and tau(2). We also conducted a simulation study to evaluate the performance of the two estimates of sigma(2)(i) in different conditions, where the conditions vary by number of studies and study size. Using the schizophrenia meta-analysis data, the estimates of beta and tau(2) were quite different when different estimates of sigma(2)(i) were used in the logit model. The simulation study showed that the estimates of beta and tau(2) were less biased, and the 95 per cent CI coverage was closer to 95 per cent when the estimate of sigma(2)(i) was (macronp(1-macronp)n(i))(-1) rather than (circumflexp(i)(1-circumflexp)n(i))(-1). Finally, we showed that a simple regression analysis is not appropriate unless tau(2) is much larger than sigma(2)(i), or a robust variance is used.     

交叉设计资料的混合效应模型分析

目的 放宽交叉设计方差分析对残留效应假定和探讨交叉设计混合效应分析模型。方法 建立三个模型以适应残留效应的不同假定，即残留效应为零、相等或不等，通过构造对数似然函数以及利用Fisher记分迭代算法可求得卢和口的极大似然估计与限制极大似然估计。结果 用SAS程序实现了交叉设计混合效应模型分析，得到了有关参数的估计值和直接处理效应的比较结果，实例分析表明，混合效应模型能够提供更多的有效信息。结论 混合效应模型较一般方差分析有更强的适应性，可完善和丰富交叉设计资料的分析方法。     

核心家系中数量性状遗传方差分量模型的研究

目的 研究核心家系中带协变量的数量性状的遗传方差分量模型，探索数量性状的遗传因素和环境因素作用大小的分析方法。方法 将线性涨合模型应用于核心家系资料，根据核心家系成员的遗传关系建立遗传方差分量模型，运用SAS软件中的Mixed模块和WinBUGS软件进行参数估计，分析各影响因素作用大小。结果 模拟研究表明REML法和MCMC法都可得到近似无偏的参数估计。REML法计算耗时较少，但该法对各参数的区间估计具有一定的局限性，而MCMC法可方便地得到各参数的可信区间。结论 该模型可应用于实际家系资料，分析遗传因素和环境因素对数量表型的影响大小。     

A small sample confidence interval approach to assess individual bioequivalence

The concept of interchangeable pharmaceutical products has been examined in great detail in the literature. Anderson and Hauck proposed a statistical random coefficient model to study 'switchability', and coined the phrase 'individual bioequivalence' which they defined with a probability-based inequality. Since that paper there has been considerable work and discussion. The Food and Drug Administration has recommended the introduction of individual bioequivalence (IBE) and population bioequivalence (PBE) methods in a draft guidance document. The proposal in the draft guidance includes criteria for IBE and PBE and recommends the use of non-parametric bootstrap 95 per cent upper confidence intervals for the conclusion of either IBE or PBE. However, this method requires intensive computations. We have developed an alternative confidence interval procedure to assess IBE by the FDA recommended criteria. This method utilizes Howe's approximation I to a Cornish-Fisher expansion. Our proposed method is applicable to balanced or unbalanced data in a broad class of extended cross-over designs, and can be easily programmed using readily available software.     

Clinical heterogeneity in random-effect meta-analysis: Between-study boundary estimate problem

Random-effect meta-analysis is commonly applied to estimate overall effects with unexplained heterogeneity across studies. However, standard methods, including (restricted) maximum likelihood (ML or REML), frequently produce (near) zero estimates for between-study variance parameters. Consequently, these methods are reduced to simple and unrealistic fixed-effect models, resulting in an ignorance of the substantial clinical heterogeneity and sometimes leading to incorrect conclusions. To solve the boundary estimate problem, we propose (1) an adjusted maximum likelihood method for the between-study variance that maximizes a likelihood defined as a product of a standard likelihood and a Gaussian class of adjustment factor and (2) a framework using sensitivity analysis by developing a new criterion to check for the occurrence of the boundary estimate. Although the adjustment introduces bias to the overall effects to ensure strictly positive estimates of the between-study variance when the number of studies K is small, the bias asymptotically approaches zero, resulting in the same estimates derived from the REML method. Moreover, the adjusted maximum likelihood estimator of the between-study variance is consistent for large K, and interestingly, the REML method and our method are equivalent in terms of mean squared error criterion, up to O(K⁻¹). We illustrate our approach with a motivating example to examine the controversial result of a meta-analysis for 24 randomized controlled trials of human albumin. Numerical evaluations show that our approach produces no boundary estimates but similar synthesized results with the standard maximum likelihood methods as those produced by conventional methods, especially with a small number of studies.     

Parametric bootstrap and penalized quasi-likelihood inference in conditional autoregressive models

This paper discusses a variety of conditional autoregressive (CAR) models for mapping disease rates, beyond the usual first-order intrinsic CAR model. We illustrate the utility and scope of such models for handling different types of data structures. To encourage their routine use for map production at statistical and health agencies, a simple algorithm for fitting such models is presented. This is derived from penalized quasi-likelihood (PQL) inference which uses an analogue of best-linear unbiased estimation for the regional risk ratios and restricted maximum likelihood for the variance components. We offer the practitioner here the use of the parametric bootstrap for inference. It is more reliable than standard maximum likelihood asymptotics for inference purposes since relevant hypotheses for the mapping of rates lie on the boundary of the parameter space. We illustrate the parametric bootstrap test of the practically relevant and important simplifying hypothesis that there is no spatial autocorrelation. Although the parametric bootstrap requires computational effort, it is straightforward to implement and offers a wealth of information relating to the estimators and their properties. The proposed methodology is illustrated by analysing infant mortality in the province of British Columbia in Canada.     

A comparison of generalized linear mixed model procedures with estimating equations for variance and covariance parameter estimation in longitudinal studies and group randomized trials.

Response data in longitudinal studies and group randomized trials are gathered on units that belong to clusters, within which data are usually positively correlated. Therefore, estimates and confidence intervals for intraclass correlation or variance components are helpful when designing a longitudinal study or group randomized trial. Data simulated from both study designs are used to investigate the estimation of variance and covariance parameters from the following procedures: for continuous outcomes, restricted maximum likelihood (REML) and estimating equations (EE); for binary outcomes, restricted pseudo-likelihood (REPL) and estimating equations (EE). We evaluate these procedures to see which provide valid and precise estimates as well as correct standard errors for the intraclass correlation coefficient or variance components. REML seems the better choice for estimating terms related to correlation for models with normal outcomes, especially in group randomized trial situations. Results for REML and EE are mixed when outcomes are continuous and non-normal. With binary outcomes neither REPL nor EE provides satisfactory estimation or inference in longitudinal study situations, while REPL is preferable for group randomized trials.     

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.     

Variance estimation for clustered recurrent event data with a small number of clusters

Often in biomedical studies, the event of interest is recurrent and within-subject events cannot usually be assumed independent. In semi-parametric estimation of the proportional rates model, a working independence assumption leads to an estimating equation for the regression parameter vector, with within-subject correlation accounted for through a robust (sandwich) variance estimator; these methods have been extended to the case of clustered subjects. We consider variance estimation in the setting where subjects are clustered and the study consists of a small number of moderate-to-large-sized clusters. We demonstrate through simulation that the robust estimator is quite inaccurate in this setting. We propose a corrected version of the robust variance estimator, as well as jackknife and bootstrap estimators. Simulation studies reveal that the corrected variance is considerably more accurate than the robust estimator, and slightly more accurate than the jackknife and bootstrap variance. The proposed methods are used to compare hospitalization rates between Canada and the U.S. in a multi-centre dialysis study. Copyright (c) 2005 John Wiley & Sons, Ltd.     

Bootstrap方法在Cox模型参数估计中的应用

目的探讨自变量分布极偏时Cox模型的参数以及区间估计是否准确,引入Bootstrap法对其加以验证。方法拟合Cox模型研究痰检结果与肺癌发病风险的关系,使用非参数Bootstrap法验证Cox模型的参数估计值、可信区间是否有效。结果两者的参数估计值和大部分可信区间基本一致,个别变量的可信区间不一致,Bootstrap可信区间的下界更低,甚至变为负值。结论当自变量分布极偏时,Cox模型所估计的参数可信区间可能有偏,Bootstrap可信区间更准确。     

Bootstrap和Permutation方法在样本率多重比较中的应用

目的介绍Bootstrap和Permutation方法在样本率多重比较中的应用。方法调用SAS中的MULTFEST过程，编写程序实现样本率的两两比较和与控制组比较，并通过实例说明效果。结果运用Bootstrap和Permutation方法能较好解决样本率的多重比较问题。结论使用Bootstrap和Permutation方法的SAS程序，简单明了，结果准确，使用方便。     

One-stage individual participant data meta-analysis models for continuous and binary outcomes: Comparison of treatment coding options and estimation methods

A one-stage individual participant data (IPD) meta-analysis synthesizes IPD from multiple studies using a general or generalized linear mixed model. This produces summary results (eg, about treatment effect) in a single step, whilst accounting for clustering of participants within studies (via a stratified study intercept, or random study intercepts) and between-study heterogeneity (via random treatment effects). We use simulation to evaluate the performance of restricted maximum likelihood (REML) and maximum likelihood (ML) estimation of one-stage IPD meta-analysis models for synthesizing randomized trials with continuous or binary outcomes. Three key findings are identified. First, for ML or REML estimation of stratified intercept or random intercepts models, a t-distribution based approach generally improves coverage of confidence intervals for the summary treatment effect, compared with a z-based approach. Second, when using ML estimation of a one-stage model with a stratified intercept, the treatment variable should be coded using “study-specific centering” (ie, 1/0 minus the study-specific proportion of participants in the treatment group), as this reduces the bias in the between-study variance estimate (compared with 1/0 and other coding options). Third, REML estimation reduces downward bias in between-study variance estimates compared with ML estimation, and does not depend on the treatment variable coding; for binary outcomes, this requires REML estimation of the pseudo-likelihood, although this may not be stable in some situations (eg, when data are sparse). Two applied examples are used to illustrate the findings.     

A comparison of heterogeneity variance estimators in combining results of studies

For random effects meta-analysis, seven different estimators of the heterogeneity variance are compared and assessed using a simulation study. The seven estimators are the variance component type estimator (VC), the method of moments estimator (MM), the maximum likelihood estimator (ML), the restricted maximum likelihood estimator (REML), the empirical Bayes estimator (EB), the model error variance type estimator (MV), and a variation of the MV estimator (MVvc). The performance of the estimators is compared in terms of both bias and mean squared error, using Monte Carlo simulation. The results show that the REML and especially the ML and MM estimators are not accurate, having large biases unless the true heterogeneity variance is small. The VC estimator tends to overestimate the heterogeneity variance in general, but is quite accurate when the number of studies is large. The MV estimator is not a good estimator when the heterogeneity variance is small to moderate, but it is reasonably accurate when the heterogeneity variance is large. The MVvc estimator is an improved estimator compared to the MV estimator, especially for small to moderate values of the heterogeneity variance. The two estimators MVvc and EB are found to be the most accurate in general, particularly when the heterogeneity variance is moderate to large.     

Definition of individual bioequivalence: occasion-to-occasion versus mean switchability

Two moment-based scaled definitions of individual bioequivalence are discussed. Based on a mixed effects linear model, their evaluations respectively lead to an unweighted (theta(11)) and a parametric (theta(15)) metric. The two metrics are estimated with respect to study design and two estimation methods. Results show that the two IBE metrics perform equivalently in the fully replicated design. In the semi-replicated design, the definition of theta(11) may not be valid while the evaluation of theta(15) results in a reduction of the weights in the mean difference and switchability components of the metric. Percentage rejection rates in the latter design indicate that theta(11) is more conservative than theta(15). This is because there is an increase of about 15 per cent in the producer risk in theta(11) relative to theta(15) compared to a 7 per cent increase in the consumer risk in theta(15) relative to theta(11). A further disadvantage of the design is that there is a 33 per cent loss in the subject-by-treatment variance efficiency which is reflected in a similar amount of decreased sensitivity to departures from perfect bioequivalence even when more subjects are used to equalize the number of exposure occasions in the two designs. It is concluded that a mean switchability criterion may be more appropriate from an interpretability perspective, the bootstrap resampling method used to evaluate individual bioequivalence based on theta(11) may need to be bias-corrected and that the semi-replicated design should be used cautiously.     

Analysis of Health Utility Data When Some Subjects Attain the Upper Bound of 1: Are Tobit and CLAD Models Appropriate?

BackgroundHealth utility data often show an apparent truncation effect, where a proportion of individuals achieve the upper bound of 1. The Tobit model and censored least absolute deviations (CLAD) have both been used as analytic solutions to this apparent truncation effect. These models assume that the observed utilities are censored at 1, and hence that the true utility can be greater than 1. We aimed to examine whether the Tobit and CLAD models yielded acceptable results when this censoring assumption was not appropriate.MethodsUsing health utility (captured through EQ5D) data from a diabetes study, we conducted a simulation to compare the performance of the Tobit, CLAD, ordinary least squares (OLS), two-part and latent class estimators in terms of their bias and estimated confidence intervals. We also illustrate the performance of semiparametric and nonparametric bootstrap methods.ResultsWhen the true utility was conceptually bounded above at 1, the Tobit and CLAD estimators were both biased. The OLS estimator was asymptotically unbiased and, while the model-based and semiparametric bootstrap confidence intervals were too narrow, confidence intervals based on the robust standard errors or the nonparametric bootstrap were acceptable for sample sizes of 100 and larger. Two-part and latent class models also yielded unbiased estimates.ConclusionsWhen the intention of the analysis is to inform an economic evaluation, and the utilities should be bounded above at 1, CLAD, and Tobit methods were biased. OLS coupled with robust standard errors or the nonparametric bootstrap is recommended as a simple and valid approach.