A joint modeling approach to data with informative cluster size: robustness to the cluster size model

In many biomedical and epidemiological studies, data are often clustered due to longitudinal follow up or repeated sampling. While in some clustered data the cluster size is pre-determined, in others it may be correlated with the outcome of subunits, resulting in informative cluster size. When the cluster size is informative, standard statistical procedures that ignore cluster size may produce biased estimates. One attractive framework for modeling data with informative cluster size is the joint modeling approach in which a common set of random effects are shared by both the outcome and cluster size models. In addition to making distributional assumptions on the shared random effects, the joint modeling approach needs to specify the cluster size model. Questions arise as to whether the joint modeling approach is robust to misspecification of the cluster size model. In this paper, we studied both asymptotic and finite-sample characteristics of the maximum likelihood estimators in joint models when the cluster size model is misspecified. We found that using an incorrect distribution for the cluster size may induce small to moderate biases, while using a misspecified functional form for the shared random parameter in the cluster size model results in nearly unbiased estimation of outcome model parameters. We also found that there is little efficiency loss under this model misspecification. A developmental toxicity study was used to motivate the research and to demonstrate the findings.     

Performance of analytical methods for overdispersed counts in cluster randomized trials: Sample size,degree of clustering and imbalance

Many different methods have been proposed for the analysis of cluster randomized trials (CRTs) over the last 30 years. However, the evaluation of methods on overdispersed count data has been based mostly on the comparison of results using empiric data; i.e. when the true model parameters are not known. In this study, we assess via simulation the performance of five methods for the analysis of counts in situations similar to real community‐intervention trials. We used the negative binomial distribution to simulate overdispersed counts of CRTs with two study arms, allowing the period of time under observation to vary among individuals. We assessed different sample sizes, degrees of clustering and degrees of cluster‐size imbalance. The compared methods are: (i) the two‐sample t‐test of cluster‐level rates, (ii) generalized estimating equations (GEE) with empirical covariance estimators, (iii) GEE with model‐based covariance estimators, (iv) generalized linear mixed models (GLMM) and (v) Bayesian hierarchical models (Bayes‐HM). Variation in sample size and clustering led to differences between the methods in terms of coverage, significance, power and random‐effects estimation. GLMM and Bayes‐HM performed better in general with Bayes‐HM producing less dispersed results for random‐effects estimates although upward biased when clustering was low. GEE showed higher power but anticonservative coverage and elevated type I error rates. Imbalance affected the overall performance of the cluster‐level t‐test and the GEE's coverage in small samples. Important effects arising from accounting for overdispersion are illustrated through the analysis of a community‐intervention trial on Solar Water Disinfection in rural Bolivia. Copyright © 2009 John Wiley & Sons, Ltd.     

A simple sample size formula for analysis of covariance in cluster randomized trials

For cluster randomized trials with a continuous outcome, the sample size is often calculated as if an analysis of the outcomes at the end of the treatment period (follow‐up scores) would be performed. However, often a baseline measurement of the outcome is available or feasible to obtain. An analysis of covariance (ANCOVA) using both the baseline and follow‐up score of the outcome will then have more power. We calculate the efficiency of an ANCOVA analysis using the baseline scores compared with an analysis on follow‐up scores only. The sample size for such an ANCOVA analysis is a factor r² smaller, where r is the correlation of the cluster means between baseline and follow‐up. This correlation can be expressed in clinically interpretable parameters: the correlation between baseline and follow‐up of subjects (subject autocorrelation) and that of clusters (cluster autocorrelation). Because of this, subject matter knowledge can be used to provide (range of) plausible values for these correlations, when estimates from previous studies are lacking. Depending on how large the subject and cluster autocorrelations are, analysis of covariance can substantially reduce the number of clusters needed. Copyright © 2012 John Wiley & Sons, Ltd.     

Relative efficiencies of two-stage sampling schemes for mean estimation in multilevel populations when cluster size is informative

In multilevel populations, there are two types of population means of an outcome variable ie, the average of all individual outcomes ignoring cluster membership and the average of cluster-specific means. To estimate the first mean, individuals can be sampled directly with simple random sampling or with two-stage sampling (TSS), that is, sampling clusters first, and then individuals within the sampled clusters. When cluster size varies in the population, three TSS schemes can be considered, ie, sampling clusters with probability proportional to cluster size and then sampling the same number of individuals per cluster; sampling clusters with equal probability and then sampling the same percentage of individuals per cluster; and sampling clusters with equal probability and then sampling the same number of individuals per cluster. Unbiased estimation of the average of all individual outcomes is discussed under each sampling scheme assuming cluster size to be informative. Furthermore, the three TSS schemes are compared in terms of efficiency with each other and with simple random sampling under the constraint of a fixed total sample size. The relative efficiency of the sampling schemes is shown to vary across different cluster size distributions. However, sampling clusters with probability proportional to size is the most efficient TSS scheme for many cluster size distributions. Model-based and design-based inference are compared and are shown to give similar results. The results are applied to the distribution of high school size in Italy and the distribution of patient list size for general practices in England.     

Missing data in the 2 x 2 table: patterns and likelihood-based analysis for cross-sectional studies with supplemental sampling

Standard measures of crude association in the context of a cross-sectional study are the risk difference, relative risk and odds ratio as derived from a 2x 2 table. Most such studies are subject to missing data on disease, exposure, or both, introducing bias into the usual complete-case analysis. We describe several scenarios distinguished by the manner in which missing data arise, and for each we adjust the natural multinomial likelihood to properly account for missing data. The situations presented allow for increasing levels of generality with regard to the missing data mechanism. The final case, quite conceivable in epidemiologic studies, assumes that the probability of missing exposure depends on true exposure and disease status, as well as upon whether disease status is missing (and conversely for the probability of missing disease information). When parameters relating to the missing data process are inestimable without strong assumptions, we propose maximum likelihood analysis subsequent to collecting supplemental data in the spirit of a validation study. Analytical results give insight into the bias inherent in complete-case analysis for each scenario, and numerical results illustrate the performance of likelihood-based point and interval estimates in the most general case. Adjustment for potential confounders via stratified analysis is also discussed.     

Retrospective analysis of case-control studies when the population is in Hardy-Weinberg equilibrium

Association analysis of genetic polymorphisms has been mostly performed in a case-control setting in connection with the traditional logistic regression analysis. However, in a case-control study, subjects are recruited according to their disease status and their past exposures are determined. Thus the natural model for making inference is the retrospective model. In this paper, we discuss some retrospective models and give maximum likelihood estimators of exposure effects and estimators of asymptotic variances, when the frequency distribution of exposures in controls contains information about the parameters of interest. Two situations about the control population are considered in this paper: (a) the control population or its subpopulations are in Hardy-Weinberg equilibrium; and (b) genetic and environmental factors are independent in the control population. Using the concept of asymptotic relative efficiency, we shall show the precision advantages of such retrospective analysis over the traditional prospective analysis. Maximum likelihood estimates and variance estimates under retrospective models are simple in computation and thus can be applied in many practical applications. We present one real example to illustrate our methods.     

Sample size requirement to detect an intervention effect at the end of follow‐up in a longitudinal cluster randomized trial

It is often anticipated in a longitudinal cluster randomized clinical trial (cluster‐RCT) that the course of outcome over time will diverge between intervention arms. In these situations, testing the significance of a local intervention effect at the end of the trial may be more clinically relevant than evaluating overall mean differences between treatment groups. In this paper, we present a closed‐form power function for detecting this local intervention effect based on maximum likelihood estimates from a mixed‐effects linear regression model for three‐level continuous data. Sample size requirements for the number of units at each data level are derived from the power function. The power function and the corresponding sample size requirements are verified by a simulation study. Importantly, it is shown that sample size requirements computed with the proposed power function are smaller than that required when testing group mean difference using data only at the end of trial and ignoring the course of outcome over the entire study period. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimation of the effect size in meta‐analysis with few studies

Meta‐analysis is often conducted with only a small number of studies. Adjustments of the (restricted) maximum likelihood estimator of the effect size are derived and their gains in efficiency are explored. The proposed estimators are applied to three sets of studies. Copyright © 2009 John Wiley & Sons, Ltd.     

A comparison of recent methods for the analysis of small-sample cross-over studies

The standard analysis of variance (ANOVA) method is usually applied to analyse continuous data from cross-over studies. The method, however, has been known to be not robust for general variance-covariance structure. The simple empirical generalized least squares (EGLS) method, proposed in an attempt to improve the precision of the standard ANOVA method for general variance-covariance structure, is usually insufficient for small-sample cross-over trials. In this paper we compare the following commonly used or recent approaches: standard ANOVA; simple EGLS; modified ANOVA method derived from a modified approximate F-distribution; and a modified EGLS method adjusted by the Kenward and Roger procedure in terms of robustness and power while applying to small-sample cross-over studies (say, the sample size is less than 40) over a variety of variance-covariance structures by simulation. We find that the unconditional modified ANOVA method has robust performance for all of the simulated small-sample cross-over studies over the various variance-covariance structures, and has comparable power with the standard ANOVA method whenever they are comparable in type I error rate. The EGLS method (simple or modified) is not reliable when the sample size of a cross-over study is too small, say, less than 24 in the simulation, unless a simple covariance structure is correctly assumed. Given a relatively larger sample size, the modified EGLS method, assuming an unstructured covariance matrix, demonstrates robust performance over the various variance-covariance structures in the simulation and provides more powerful tests than those of the modified (or standard) ANOVA method.     

Bayesian methods for analysis of binary outcome data in cluster randomized trials on the absolute risk scale

A Bayesian hierarchical modelling approach to the analysis of cluster randomized trials has advantages in terms of allowing for full parameter uncertainty, flexible modelling of covariates and variance structure, and use of prior information. Previously, such modelling of binary outcome data required use of a log-odds ratio scale for the treatment effect estimate and an approximation linking the intracluster correlation (ICC) to the between-cluster variance on a log-odds scale. In this paper we develop this method to allow estimation on the absolute risk scale, which facilitates clinical interpretation of both the treatment effect and the between-cluster variance. We describe a range of models and apply them to data from a trial of different interventions to promote secondary prevention of coronary heart disease in primary care. We demonstrate how these models can be used to incorporate prior data about typical ICCs, to derive a posterior distribution for the number needed to treat, and to consider both cluster and individual level covariates. Using these methods, we can benefit from the advantages of Bayesian modelling of binary outcome data at the same time as providing results on a clinically interpretable scale.     

A comparison of confidence interval methods for the intraclass correlation coefficient in cluster randomized trials

This study compared different methods for assigning confidence intervals to the analysis of variance estimator of the intraclass correlation coefficient (rho). The context of the comparison was the use of rho to estimate the variance inflation factor when planning cluster randomized trials. The methods were compared using Monte Carlo simulations of unbalanced clustered data and data from a cluster randomized trial of an intervention to improve the management of asthma in a general practice setting. The coverage and precision of the intervals were compared for data with different numbers of clusters, mean numbers of subjects per cluster and underlying values of rho. The performance of the methods was also compared for data with Normal and non-Normally distributed cluster specific effects. Results of the simulations showed that methods based upon the variance ratio statistic provided greater coverage levels than those based upon large sample approximations to the standard error of rho. Searle's method provided close to nominal coverage for data with Normally distributed random effects. Adjusted versions of Searle's method to allow for lack of balance in the data generally did not improve upon it either in terms of coverage or precision. Analyses of the trial data, however, showed that limits provided by Thomas and Hultquist's method may differ from those of the other variance ratio statistic methods when the arithmetic mean differs markedly from the harmonic mean cluster size. The simulation results demonstrated that marked non-Normality in the cluster level random effects compromised the performance of all methods. Confidence intervals for the methods were generally wide relative to the underlying size of rho suggesting that there may be great uncertainty associated with sample size calculations for cluster trials where large clusters are randomized. Data from cluster based studies with sample sizes much larger than those typical of cluster randomized trials are required to estimate rho with a reasonable degree of precision.     

A comparison of the statistical power of different methods for the analysis of cluster randomization trials with binary outcomes

Cluster randomization trials are randomized controlled trials (RCTs) in which intact clusters of subjects are randomized to either the intervention or to the control. Cluster randomization trials require different statistical methods of analysis than do conventional randomized controlled trials due to the potential presence of within-cluster homogeneity in responses. A variety of statistical methods have been proposed in the literature for the analysis of cluster randomization trials with binary outcomes. However, little is known about the relative statistical power of these methods to detect a statistically significant intervention effect. We conducted a series of Monte Carlo simulations to examine the statistical power of three methods that compare cluster-specific response rates between arms of the trial: the t-test, the Wilcoxon rank sum test, and the permutation test; and three methods that compare subject-level response rates: an adjusted chi-square test, a logistic-normal random effects model, and a generalized estimating equations (GEE) method. In our simulations we allowed the number of clusters, the number of subjects per cluster, the intraclass correlation coefficient and the magnitude of the intervention effect to vary. We demonstrated that the GEE approach tended to have the highest power for detecting a statistically significant intervention effect. However, in most of the 240 scenarios examined, the differences between the competing statistical methods were negligible. The largest mean difference in power between any two different statistical methods across the 240 scenarios was 0.02. The largest observed difference in power between two different statistical methods across the 240 scenarios and 15 pair-wise comparisons of methods was 0.14.     

Comparison of methods for estimating the effect of salvage therapy in prostate cancer when treatment is given by indication

For patients who were previously treated for prostate cancer, salvage hormone therapy is frequently given when the longitudinal marker prostate‐specific antigen begins to rise during follow‐up. Because the treatment is given by indication, estimating the effect of the hormone therapy is challenging. In a previous paper we described two methods for estimating the treatment effect, called two‐stage and sequential stratification. The two‐stage method involved modeling the longitudinal and survival data. The sequential stratification method involves contrasts within matched sets of people, where each matched set includes people who did and did not receive hormone therapy. In this paper, we evaluate the properties of these two methods and compare and contrast them with the marginal structural model methodology. The marginal structural model methodology involves a weighted survival analysis, where the weights are derived from models for the time of hormone therapy. We highlight the different conditional and marginal interpretations of the quantities being estimated by the three methods. Using simulations that mimic the prostate cancer setting, we evaluate bias, efficiency, and accuracy of estimated standard errors and robustness to modeling assumptions. The results show differences between the methods in terms of the quantities being estimated and in efficiency. We also demonstrate how the results of a randomized trial of salvage hormone therapy are strongly influenced by the design of the study and discuss how the findings from using the three methodologies can be used to infer the results of a trial. Copyright © 2013 John Wiley & Sons, Ltd.     

Hierarchical cluster analysis for exposure assessment of workers in the semiconductor health study

The fabrication of integrated circuits in the semiconductor industry involves worker exposures to multiple chemical and physical agents. The potential for a high degree of correlation among exposure variables was of concern in the Semiconductor Health Study. Hierarchical cluster analysis was used to identify groups or “clusters” of correlated variables. Several variations of hierarchical cluster analysis were performed on 14 chemical and physical agents, using exposure data on 882 subjects from the historical cohort of the epidemiological studies. Similarity between agent pairs was determined by calculating two metrics of dissimilarity, and hierarchical trees were constructed using three clustering methods. Among subjects exposed to ethylene-based glycol ethers (EGE), xylene, or n-butyl acetate (nBA), 83% were exposed to EGE and xylene, 86% to EGE and nBA, and 94% to xylene and nBA, suggesting that exposures to EGE, xylene, and nBA were highly correlated. A high correlation was also found for subjects exposed to boron and phosphorus (80%). The trees also revealed cluster groups containing agents associated with work-group exposure categories developed for the epidemiologic analyses.     

Multiple imputation for the comparison of two screening tests in two-phase Alzheimer studies

Two-phase designs are common in epidemiological studies of dementia, and especially in Alzheimer research. In the first phase, all subjects are screened using a common screening test(s), while in the second phase, only a subset of these subjects is tested using a more definitive verification assessment, i.e. golden standard test. When comparing the accuracy of two screening tests in a two-phase study of dementia, inferences are commonly made using only the verified sample. It is well documented that in that case, there is a risk for bias, called verification bias. When the two screening tests have only two values (e.g. positive and negative) and we are trying to estimate the differences in sensitivities and specificities of the tests, one is actually estimating a confidence interval for differences of binomial proportions. Estimating this difference is not trivial even with complete data. It is well documented that it is a tricky task. In this paper, we suggest ways to apply imputation procedures in order to correct the verification bias. This procedure allows us to use well-established complete-data methods to deal with the difficulty of the estimation of the difference of two binomial proportions in addition to dealing with incomplete data. We compare different methods of estimation and evaluate the use of multiple imputation in this case. Our simulation results show that the use of multiple imputation is superior to other commonly used methods. We demonstrate our finding using Alzheimer data. Copyright (c) 2006 John Wiley & Sons, Ltd.     

Elderly patients' experiences using adaptive conjoint analysis software as a decision aid for osteoarthritis of the knee

Background

Decision making in knee osteoarthritis, with many treatment options, challenges patients and physicians alike. Unfortunately, physicians cannot describe in detail each treatment's benefits and risks. One promising adjunct to decision making in osteoarthritis is adaptive conjoint analysis (ACA).

Objective

To obtain insight into the experiences of elderly patients who use adaptive conjoint analysis to explore treatment options for their osteoarthritis.

Design

Participants, all 65 and older, completed an ACA decision aid exploring their preferences with regard to the underlying attributes of osteoarthritis interventions. We used focus groups to obtain insight into their experiences using this software.

Results

Content analysis distributed our participants' concerns into five areas. The predicted preferred treatment usually agreed with the individual's preference, but our participants experienced difficulty in four other domains: the choices presented by the software were sometimes confusing, the treatments presented were not the treatments of most interest, the researchers' claims about treatment characteristics were unpersuasive and cumulative overload sometimes developed.

Conclusion

Adaptive conjoint analysis presented special challenges to our elderly participants; we believe that their relatively low level of computer comfort was a significant contributor to these problems. We suggest that other researchers choose the software's treatments and present the treatment attributes with care. The next and equally vital step is to educate participants about what to expect, including the limitations in choice and apparent arbitrariness of the trade‐offs presented by the software. Providing participants with a sample ACA task before undertaking the study task may further improve participant understanding and engagement.     

Sample size determination for jointly testing a cause‐specific hazard and the all‐cause hazard in the presence of competing risks

This article considers sample size determination for jointly testing a cause‐specific hazard and the all‐cause hazard for competing risks data. The cause‐specific hazard and the all‐cause hazard jointly characterize important study end points such as the disease‐specific survival and overall survival, which are commonly used as coprimary end points in clinical trials. Specifically, we derive sample size calculation methods for 2‐group comparisons based on an asymptotic chi‐square joint test and a maximum joint test of the aforementioned quantities, taking into account censoring due to lost to follow‐up as well as staggered entry and administrative censoring. We illustrate the application of the proposed methods using the Die Deutsche Diabetes Dialyse Studies clinical trial. An R package “powerCompRisk” has been developed and made available at the CRAN R library.     

A comparison of permutation and mixed-model regression methods for the analysis of simulated data in the context of a group-randomized trial

Our first purpose was to determine whether, in the context of a group-randomized trial (GRT) with Gaussian errors, permutation or mixed-model regression methods fare better in the presence of measurable confounding in terms of their Monte Carlo type I error rates and power. Our results indicate that given a proper randomization, the type I error rate is similar for both methods, whether unadjusted or adjusted, even in small studies. However, our results also show that should the investigator face the unfortunate circumstance in which modest confounding exists in the only realization available, the unadjusted analysis risks a type I error; in this regard, there was little to distinguish the two methods. Finally, our results show that power is similar for the two methods and, not surprisingly, better for the adjusted tests.Our second purpose was to examine the relative performance of permutation and mixed-model regression methods in the context of a GRT when the normality assumptions underlying the mixed model are violated. Published studies have examined the impact of violation of this assumption at the member level only. Our findings indicate that both methods perform well when the assumption is violated so long as the ICC is very small and the design is balanced at the group level. However, at ICC>or=0.01, the permutation test carries the nominal type I error rate while the model-based test is conservative and so less powerful. Binomial group- and member-level errors did not otherwise change the relative performance of the two methods with regard to confounding.     

A comparison of tests for restricted orderings in the three‐class case

A variety of methods for comparing three distributions have been proposed in the literature. These methods assess the same null hypothesis of equal distributions but differ in the alternative hypothesis they consider. The alternative hypothesis can be that measurements from the three classes are distributed according to unequal distributions or that measurements between the three classes follow a specific monotone ordering, an inverse‐U‐shaped (umbrella) ordering, or a U‐shaped (tree) ordering. This paper compares these tests with respect to power and test size under different simulation scenarios. In addition, the methods are illustrated in two applications generated by different research questions with data from three classes suggesting monotone and umbrella orders. Additionally, proposals for the appropriate application of these tests are provided. Copyright © 2009 John Wiley & Sons, Ltd.