Covariate-adjusted generalized pairwise comparisons in small samples

Semiparametric probabilistic index models allow for the comparison of two groups of observations, whilst adjusting for covariates, thereby fitting nicely within the framework of generalized pairwise comparisons (GPC). As with most regression approaches in this setting, the limited amount of data results in invalid inference as the asymptotic normality assumption is not met. In addition, separation issues might arise when considering small samples. In this article, we show that the parameters of the probabilistic index model can be estimated using generalized estimating equations, for which adjustments exist that lead to estimators of the sandwich variance-covariance matrix with improved finite sample properties and that can deal with bias due to separation. In this way, appropriate inference can be performed as is shown through extensive simulation studies. The known relationships between the probabilistic index and other GPC statistics allow to also provide valid inference for example, the net treatment benefit or the success odds.

Goodness-of-fit tests for modified Poisson regression possibly producing fitted values exceeding one in binary outcome analysis

Modified Poisson regression, which estimates the regression parameters in the log-binomial regression model using the Poisson quasi-likelihood estimating equation and robust variance, is a useful tool for estimating the adjusted risk and prevalence ratio in binary outcome analysis. Although several goodness-of-fit tests have been developed for other binary regressions, few goodness-of-fit tests are available for modified Poisson regression. In this study, we proposed several goodness-of-fit tests for modified Poisson regression, including the modified Hosmer-Lemeshow test with empirical variance, Tsiatis test, normalized Pearson chi-square tests with binomial variance and Poisson variance, and normalized residual sum of squares test. The original Hosmer-Lemeshow test and normalized Pearson chi-square test with binomial variance are inappropriate for the modified Poisson regression, which can produce a fitted value exceeding 1 owing to the unconstrained parameter space. A simulation study revealed that the normalized residual sum of squares test performed well regarding the type I error probability and the power for a wrong link function. We applied the proposed goodness-of-fit tests to the analysis of cross-sectional data of patients with cancer. We recommend the normalized residual sum of squares test as a goodness-of-fit test in the modified Poisson regression.

A comparison of bias-adjusted generalized estimating equations for sparse binary data in small-sample longitudinal studies

Using a generalized estimating equation (GEE) can lead to a bias in regression coefficients for a small sample or sparse data. The bias-corrected GEE (BCGEE) and penalized GEE (PGEE) were proposed to resolve the small-sample bias. Moreover, the standard sandwich covariance estimator leads to a bias of standard error for small samples; several modified covariance estimators have been proposed to address this issue. We review the modified GEEs and modified covariance estimators, and evaluate their performance in sparse binary data from small-sample longitudinal studies. The simulation results showed that GEE and BCGEE often failed to achieve convergence, whereas the convergence proportion for PGEE was quite high. The bias for the regression coefficients was generally in the ascending order of PGEE < $$ < $$ BCGEE < $$ < $$ GEE. However, PGEE and BCGEE did not sufficiently remove the bias involving 20-30 subjects with unequal exposure levels with a 5% response rate. The coverage probability (CP) of the confidence interval for BCGEE was relatively poor compared with GEE and PGEE. The CP with the sandwich covariance estimator deteriorated regardless of the GEE methods under the small sample size and low response rate, whereas the CP with the modified covariance estimators-such as Morel's method-was relatively acceptable. PGEE will be the reasonable way for analyzing sparse binary data in small-sample studies. Instead of using the standard sandwich covariance estimator, one should always apply the modified covariance estimators for analyzing these data.

Semiparametric generalized estimating equations for repeated measurements in cross-over designs

A model for cross-over designs with repeated measures within each period was developed. It was obtained using an extension of generalized estimating equations that includes a parametric component to model treatment effects and a non-parametric component to model time and carry-over effects; the estimation approach for the non-parametric component is based on splines. A simulation study was carried out to explore the model properties. Thus, when there is a carry-over effect or a functional temporal effect, the proposed model presents better results than the standard models. Among the theoretical properties, the solution is found to be analogous to weighted least squares. Therefore, model diagnostics can be made by adapting the results from a multiple regression. The proposed methodology was implemented in the data sets of the cross-over experiments that motivated the approach of this work: systolic blood pressure and insulin in rabbits.

To be happy and behave in a healthier way. A longitudinal study about gender differences in the older population

INTRODUCTION

Subjective well-being plays a key role in health. The objectives of this study are to analyse the longitudinal associations between subjective well-being dimensions and healthy behaviours, and to examine gender differences.

METHOD

A representative sample of 1,190 Spanish non-institutionalised adults aged 50+ were interviewed over a 6-year follow-up period. The Cantril scale was used to measure evaluative well-being. The Day Reconstruction Method measured experienced well-being. The Global Physical Activity Questionnaire was used, whereas fruit and vegetables, tobacco and alcohol consumption, and sleep quality were self-reported. The Generalised Estimating Equation was calculated.

RESULTS

Women show significantly worse subjective well-being than men longitudinally. Higher scores in life satisfaction and positive affect were significantly related to a higher level of physical activity and better-quality sleep for both women and men. Associations between a higher life satisfaction and an adequate intake of fruits and vegetables and being a non-smoker was only found in women (OR = 1.05; 95% IC = 1.00, 1.10 and OR = 1.16; 95% IC = 1.09, 1.23, respectively).

CONCLUSION

Subjective well-being levels and frequencies in healthy behaviours are different in women and men. Subjective well-being interventions should take into account these differences in the frequency of healthy-unhealthy behaviours.

Analysis of hospital readmissions with competing risks

The 30-day hospital readmission rate has been used in provider profiling for evaluating inter-provider care coordination, medical cost effectiveness, and patient quality of life. Current profiling analyzes use logistic regression to model 30-day readmission as a binary outcome, but one disadvantage of this approach is that this outcome is strongly affected by competing risks (e.g., death). Thus, one, perhaps unintended, consequence is that if two facilities have the same rates of readmission, the one with the higher rate of competing risks will have the lower 30-day readmission rate. We propose a discrete time competing risk model wherein the cause-specific readmission hazard is used to assess provider-level effects. This approach takes account of the timing of events and focuses on the readmission rates which are of primary interest. The quality measure, then is a standardized readmission ratio, akin to a standardized mortality ratio. This measure is not systematically affected by the rate of competing risks. To facilitate the estimation and inference of a large number of provider effects, we develop an efficient Blockwise Inversion Newton algorithm, and a stabilized robust score test that overcomes the conservative nature of the classical robust score test. An application to dialysis patients demonstrates improved profiling, model fitting, and outlier detection over existing methods.

Performance of the Beta-Binomial Model for Clustered Binary Responses: Comparison with Generalized Estimating Equations

This study examined performance of the beta-binomial model in comparison with GEE using clustered binary responses resulting in non-normal outcomes. Monte Carlo simulations were performed under varying intracluster correlations and sample sizes. The results showed that the beta-binomial model performed better for small sample, while GEE performed well under large sample.

Assessment of a Modified Sandwich Estimator for Generalized Estimating Equations with Application to Opioid Poisoning in MIMIC-IV ICU Patients

Longitudinal data is encountered frequently in many healthcare research areas to include the critical care environment. Repeated measures from the same subject are expected to correlate with each other. Models with binary outcomes are commonly used in this setting. Regression models for correlated binary outcomes are frequently fit using generalized estimating equations (GEE). The Liang and Zeger sandwich estimator is often used in GEE to produce unbiased standard error estimation for regression coefficients in large sample settings, even when the covariance structure is misspecified. The sandwich estimator performs optimally in balanced designs when the number of participants is large with few repeated measurements. The sandwich estimator’s asymptotic properties do not hold in small sample and rare-event settings. Under these conditions, the sandwich estimator underestimates the variances and is biased downwards. Here, the performance of a modified sandwich estimator is compared to the traditional Liang-Zeger estimator and alternative forms proposed by authors Morel, Pan, and Mancl-DeRouen. Each estimator’s performance was assessed with 95% coverage probabilities for the regression coefficients using simulated data under various combinations of sample sizes and outcome prevalence values with independence and autoregressive correlation structures. This research was motivated by investigations involving rare-event outcomes in intensive care unit settings.

Sample size estimation for modified Poisson analysis of cluster randomized trials with a binary outcome

The modified Poisson regression coupled with a robust sandwich variance has become a viable alternative to log-binomial regression for estimating the marginal relative risk in cluster randomized trials. However, a corresponding sample size formula for relative risk regression via the modified Poisson model is currently not available for cluster randomized trials. Through analytical derivations, we show that there is no loss of asymptotic efficiency for estimating the marginal relative risk via the modified Poisson regression relative to the log-binomial regression. This finding holds both under the independence working correlation and under the exchangeable working correlation provided a simple modification is used to obtain the consistent intraclass correlation coefficient estimate. Therefore, the sample size formulas developed for log-binomial regression naturally apply to the modified Poisson regression in cluster randomized trials. We further extend the sample size formulas to accommodate variable cluster sizes. An extensive Monte Carlo simulation study is carried out to validate the proposed formulas. We find that the proposed formulas have satisfactory performance across a range of cluster size variability, as long as suitable finite-sample corrections are applied to the sandwich variance estimator and the number of clusters is at least 10. Our findings also suggest that the sample size estimate under the exchangeable working correlation is more robust to cluster size variability, and recommend the use of an exchangeable working correlation over an independence working correlation for both design and analysis. The proposed sample size formulas are illustrated using the Stop Colorectal Cancer (STOP CRC) trial.

Marginal analysis of bivariate mixed responses with measurement error and misclassification

Bivariate responses with mixed continuous and binary variables arise commonly in applications such as clinical trials and genetic studies. Statistical methods based on jointly modeling continuous and binary variables have been available. However, such methods ignore the effects of response mismeasurement, a ubiquitous feature in applications. It has been well studied that in many settings, ignorance of mismeasurement in variables usually results in biased estimation. In this paper, we consider the setting with a bivariate outcome vector which contains a continuous component and a binary component both subject to mismeasurement. We propose estimating equation approaches to handle measurement error in the continuous response and misclassification in the binary response simultaneously. The proposed estimators are consistent and robust to certain model misspecification, provided regularity conditions. Extensive simulation studies confirm that the proposed methods successfully correct the biases resulting from the error-in-variables under various settings. The proposed methods are applied to analyze the outbred Carworth Farms White mice data arising from a genome-wide association study.

Comparison of small-sample standard-error corrections for generalised estimating equations in stepped wedge cluster randomised trials with a binary outcome: A simulation study

Generalised estimating equations with the sandwich standard-error estimator provide a promising method of analysis for stepped wedge cluster randomised trials. However, they have inflated type-one error when used with a small number of clusters, which is common for stepped wedge cluster randomised trials. We present a large simulation study of binary outcomes comparing bias-corrected standard errors from Fay and Graubard; Mancl and DeRouen; Kauermann and Carroll; Morel, Bokossa, and Neerchal; and Mackinnon and White with an independent and exchangeable working correlation matrix. We constructed 95% confidence intervals using a t-distribution with degrees of freedom including clusters minus parameters (DFC-P), cluster periods minus parameters, and estimators from Fay and Graubard (DFFG), and Pan and Wall. Fay and Graubard and an approximation to Kauermann and Carroll (with simpler matrix inversion) were unbiased in a wide range of scenarios with an independent working correlation matrix and more than 12 clusters. They gave confidence intervals with close to 95% coverage with DFFG with 12 or more clusters, and DFC-P with 18 or more clusters. Both standard errors were conservative with fewer clusters. With an exchangeable working correlation matrix, approximated Kauermann and Carroll and Fay and Graubard had a small degree of under-coverage.

A comparison study on modeling of clustered and overdispersed count data for multiple comparisons

Data collected in various scientific fields are count data. One way to analyze such data is to compare the individual levels of the factor treatment using multiple comparisons. However, the measured individuals are often clustered - e.g. according to litter or rearing. This must be considered when estimating the parameters by a repeated measurement model. In addition, ignoring the overdispersion to which count data is prone leads to an increase of the type one error rate. We carry out simulation studies using several different data settings and compare different multiple contrast tests with parameter estimates from generalized estimation equations and generalized linear mixed models in order to observe coverage and rejection probabilities. We generate overdispersed, clustered count data in small samples as can be observed in many biological settings. We have found that the generalized estimation equations outperform generalized linear mixed models if the variance-sandwich estimator is correctly specified. Furthermore, generalized linear mixed models show problems with the convergence rate under certain data settings, but there are model implementations with lower implications exists. Finally, we use an example of genetic data to demonstrate the application of the multiple contrast test and the problems of ignoring strong overdispersion.

Relationship between subjective well-being and healthy lifestyle behaviours in older adults: a longitudinal study

Objectives: People who report better subjective well-being tend to be healthier in their daily behaviours. The objective of this study is to assess whether different components of subjective well-being are prospectively associated with different healthy lifestyle behaviours and to assess whether these associations differ by age.Method: A total of 1,892 participants aged 50+ living in Spain were interviewed in 2011-12 and 2014-15. Life satisfaction was measured with the Cantril Self-Anchoring Striving Scale. Positive and negative affect were assessed using the Day Reconstruction Method. Physical activity was assessed with the Global Physical Activity Questionnaire version 2. The remaining healthy lifestyle behaviours were self-reported. Generalised Estimating Equations (GEE) models were run.Results: Not having a heavy episodic alcohol drinking was the healthy lifestyle behaviour most fulfilled (97.97%), whereas the intake of five or more fruits and vegetables was the least followed (33.12%). GEE models conducted over the 50-64 and the 65+ age groups showed that a higher life satisfaction was significantly related to a higher physical activity in both groups. Relationships between a higher negative affect and presenting a lower level of physical activity, and a higher positive affect and following the right consumption of fruits and vegetables and being a non-daily smoker, were only found in the older group.Conclusion: The relationship between subjective well-being and healthy lifestyle behaviours was found fundamentally in those aged 65+ years. Interventions focused on incrementing subjective well-being would have an impact on keeping a healthy lifestyle and, therefore, on reducing morbidity and mortality.

Youth Depression Alleviation with Anti-inflammatory Agents (YoDA-A): a randomised clinical trial of rosuvastatin and aspirin

BACKGROUND

Inflammation contributes to the pathophysiology of major depressive disorder (MDD), and anti-inflammatory strategies might therefore have therapeutic potential. This trial aimed to determine whether adjunctive aspirin or rosuvastatin, compared with placebo, reduced depressive symptoms in young people (15-25 years).

METHODS

YoDA-A, Youth Depression Alleviation with Anti-inflammatory Agents, was a 12-week triple-blind, randomised, controlled trial. Participants were young people (aged 15-25 years) with moderate to severe MDD (MADRS mean at baseline 32.5 ± 6.0; N = 130; age 20.2 ± 2.6; 60% female), recruited between June 2013 and June 2017 across six sites in Victoria, Australia. In addition to treatment as usual, participants were randomised to receive aspirin (n = 40), rosuvastatin (n = 48), or placebo (n = 42), with assessments at baseline and weeks 4, 8, 12, and 26. The primary outcome was change in the Montgomery-Åsberg Depression Rating Scale (MADRS) from baseline to week 12.

RESULTS

At the a priori primary endpoint of MADRS differential change from baseline at week 12, there was no significant difference between aspirin and placebo (1.9, 95% CI (- 2.8, 6.6), p = 0.433), or rosuvastatin and placebo (- 4.2, 95% CI (- 9.1, 0.6), p = 0.089). For rosuvastatin, secondary outcomes on self-rated depression and global impression, quality of life, functioning, and mania were not significantly different from placebo. Aspirin was inferior to placebo on the Quality of Life Enjoyment and Satisfaction Questionnaire (Q-LES-Q-SF) at week 12. Statins were superior to aspirin on the MADRS, the Clinical Global Impressions Severity Scale (CGI-S), and the Negative Problem Orientation Questionnaire scale (NPOQ) at week 12.

CONCLUSIONS

The addition of either aspirin or rosuvastatin did not to confer any beneficial effect over and above routine treatment for depression in young people. Exploratory comparisons of secondary outcomes provide limited support for a potential therapeutic role for adjunctive rosuvastatin, but not for aspirin, in youth depression.

TRIAL REGISTRATION

Australian New Zealand Clinical Trials Registry, ACTRN12613000112763. Registered on 30/01/2013.

Effect Partitioning in Cross-Sectionally Clustered Data Without Multilevel Models

Effect partitioning is almost exclusively performed with multilevel models (MLMs) - so much so that some have considered the two to be synonymous. MLMs are able to provide estimates with desirable statistical properties when data come from a hierarchical structure; but the random effects included in MLMs are not always integral to the analysis. As a result, other methods with relaxed assumptions are viable options in many cases. Through empirical examples and simulations, we show how generalized estimating equations (GEEs) can be used to effectively partition effects without random effects. We show that more onerous steps of MLMs such as determining the number of random effects and the structure for their covariance can be bypassed with GEEs while still obtaining identical or near-identical results. Additionally, violations of distributional assumptions adversely affect estimates with MLMs but have no effect on GEEs because no such assumptions are made. This makes GEEs a flexible alternative to MLMs with minimal assumptions that may warrant consideration. Limitations of GEEs for partitioning effects are also discussed.

A transition care coordinator model reduces hospital readmissions and costs

BACKGROUND

The optimal structure and intensity of interventions to reduce hospital readmission remains uncertain, due in part to lack of head-to-head comparison. To address this gap, we evaluated two forms of an evidence-based, multi-component transitional care intervention.

METHODS

A quasi-experimental evaluation design compared outcomes of Transition Care Coordinator (TCC) Care to Usual Care, while controlling for sociodemographic characteristics, comorbidities, readmission risk, and administrative factors. The study was conducted between January 1, 2013 and April 30, 2015 as a quality improvement initiative. Eligible adults (N = 7038) hospitalized with pneumonia, congestive heart failure, or chronic obstructive pulmonary disease were identified for program evaluation via an electronic health record algorithm. Nurse TCCs provided either a full intervention (delivered in-hospital and by post-discharge phone call) or a partial intervention (phone call only).

RESULTS

A total of 762 hospitalizations with TCC Care (460 full intervention and 302 partial intervention) and 6276 with Usual Care was examined. In multivariable models, hospitalizations with TCC Care had significantly lower odds of readmission at 30 days (OR = 0.512, 95% CI 0.392 to 0.668) and 90 days (OR = 0.591, 95% CI 0.483 to 0.723). Adjusted costs were significantly lower at 30 days (difference = $3969, 95% CI $5099 to $2691) and 90 days (difference = $5684, 95% CI $7602 to $3627). The effect was similar whether patients received the full or partial intervention.

CONCLUSION

An evidence-based multi-component intervention delivered by nurse TCCs reduced 30- and 90-day readmissions and associated health care costs. Lower intensity interventions delivered by telephone after discharge may have similar effectiveness to in-hospital programs.

Bias-reduced and separation-proof GEE with small or sparse longitudinal binary data

Generalized estimating equation (GEE) is a popular approach for analyzing correlated binary data. However, the problems of separation in GEE are still unknown. The separation created by a covariate often occurs in small correlated binary data and even in large data with rare outcome and/or high intra-cluster correlation and a number of influential covariates. This paper investigated the consequences of separation in GEE and addressed them by introducing a penalized GEE, termed as PGEE. The PGEE is obtained by adding Firth-type penalty term, which was originally proposed for generalized linear model score equation, to standard GEE and shown to achieve convergence and provide finite estimate of the regression coefficient in the presence of separation, which are not often possible in GEE. Further, a small-sample bias correction to the sandwich covariance estimator of the PGEE estimator is suggested. Simulations also showed that the GEE failed to achieve convergence and/or provided infinitely large estimate of the regression coefficient in the presence of complete or quasi-complete separation, whereas the PGEE showed significant improvement by achieving convergence and providing finite estimate. Even in the presence of near-to-separation, the PGEE also showed superior properties over the GEE. Furthermore, the bias-corrected sandwich estimator for the PGEE estimator showed substantial improvement over the standard sandwich estimator by reducing bias in estimating type I error rate. An illustration using real data also supported the findings of simulation. The PGEE with bias-corrected sandwich covariance estimator is recommended to use for small-to-moderate size sample (N ≤ 50) and even can be used for large sample if there is any evidence of separation or near-to-separation.

On the analysis of very small samples of Gaussian repeated measurements: an alternative approach

The analysis of very small samples of Gaussian repeated measurements can be challenging. First, due to a very small number of independent subjects contributing outcomes over time, statistical power can be quite small. Second, nuisance covariance parameters must be appropriately accounted for in the analysis in order to maintain the nominal test size. However, available statistical strategies that ensure valid statistical inference may lack power, whereas more powerful methods may have the potential for inflated test sizes. Therefore, we explore an alternative approach to the analysis of very small samples of Gaussian repeated measurements, with the goal of maintaining valid inference while also improving statistical power relative to other valid methods. This approach uses generalized estimating equations with a bias-corrected empirical covariance matrix that accounts for all small-sample aspects of nuisance correlation parameter estimation in order to maintain valid inference. Furthermore, the approach utilizes correlation selection strategies with the goal of choosing the working structure that will result in the greatest power. In our study, we show that when accurate modeling of the nuisance correlation structure impacts the efficiency of regression parameter estimation, this method can improve power relative to existing methods that yield valid inference. Copyright © 2017 John Wiley & Sons, Ltd.

Generalized estimating equations in cluster randomized trials with a small number of clusters: Review of practice and simulation study

Background/aims:

Generalized estimating equations are a common modeling approach used in cluster randomized trials to account for within-cluster correlation. It is well known that the sandwich variance estimator is biased when the number of clusters is small (≤40), resulting in an inflated type I error rate. Various bias correction methods have been proposed in the statistical literature, but how adequately they are utilized in current practice for cluster randomized trials is not clear. The aim of this study is to evaluate the use of generalized estimating equation bias correction methods in recently published cluster randomized trials and demonstrate the necessity of such methods when the number of clusters is small.

Methods:

Review of cluster randomized trials published between August 2013 and July 2014 and using generalized estimating equations for their primary analyses. Two independent reviewers collected data from each study using a standardized, pre-piloted data extraction template. A two-arm cluster randomized trial was simulated under various scenarios to show the potential effect of a small number of clusters on type I error rate when estimating the treatment effect. The nominal level was set at 0.05 for the simulation study.

Results:

Of the 51 included trials, 28 (54.9%) analyzed 40 or fewer clusters with a minimum of four total clusters. Of these 28 trials, only one trial used a bias correction method for generalized estimating equations. The simulation study showed that with four clusters, the type I error rate ranged between 0.43 and 0.47. Even though type I error rate moved closer to the nominal level as the number of clusters increases, it still ranged between 0.06 and 0.07 with 40 clusters.

Conclusions:

Our results showed that statistical issues arising from small number of clusters in generalized estimating equations is currently inadequately handled in cluster randomized trials. Potential for type I error inflation could be very high when the sandwich estimator is used without bias correction.

Improving power in small-sample longitudinal studies when using generalized estimating equations

Generalized estimating equations (GEE) are often used for the marginal analysis of longitudinal data. Although much work has been performed to improve the validity of GEE for the analysis of data arising from small-sample studies, little attention has been given to power in such settings. Therefore, we propose a valid GEE approach to improve power in small-sample longitudinal study settings in which the temporal spacing of outcomes is the same for each subject. Specifically, we use a modified empirical sandwich covariance matrix estimator within correlation structure selection criteria and test statistics. Use of this estimator can improve the accuracy of selection criteria and increase the degrees of freedom to be used for inference. The resulting impacts on power are demonstrated via a simulation study and application example. Copyright © 2016 John Wiley & Sons, Ltd.

Killer Cell Immunoglobulin-Like Receptor Alleles Alter HIV Disease in Children

BACKGROUND

HLA class I molecules are ligands for killer cell immunoglobin like receptors (KIR) that control the antiviral response of natural killer (NK) cells. However, the effects of KIR and HLA (KIR/HLA) alleles on HIV disease of children have not been studied.

METHODS

993 antiretroviral naïve children with symptomatic HIV infection from PACTG protocols P152 and P300 were genotyped for KIR and HLA alleles using the Luminex platform. Linear regression was used to test the association between genotypes and baseline pre-ART HIV RNA, CD4+ lymphocyte count, and cognitive score, adjusting for age, race/ethnicity and study. The interaction between genetic markers and age was investigated. To account for multiple testing the false discovery rate (FDR) was controlled at 0.05.

RESULTS

Children with the KIR2DS4*ALL FULL LENGTH (KIR2DS4*AFL) allele had higher CD4+ lymphocyte counts. Among children ≤2 years of age, the KIR2DS4*AFL was associated with lower plasma HIV RNA and higher cognitive index scores. KIR Cent2DS3/5_1 had lower CD4+ lymphocyte counts in children ≤2 years of age, while the presence of Tel1, Tel2DS4_2, Tel2DS4_4, Tel8, Tel2DS4_6 had higher CD4+ lymphocyte counts in all children. Presence of Cent2, Cent4 and Cent8 was associated with increased HIV RNA load in children ≤2 years. Presence of KIR3DL1+Bw4 was associated with higher CD4+ lymphocyte counts in all children. Among children >2 years old, KIR3DS1+Bw4-80I was associated with higher plasma HIV RNA, and Bw6/Bw6 was associated with lower plasma HIV RNA compared to children with KIR3DS1+Bw4-80I.

CONCLUSIONS

Presented data show for the first time that specific KIR alleles independently or combined with HLA ligands are associated with HIV RNA and CD4+ lymphocyte counts in infected, antiretroviral naive children; and many of these effect estimates appear to be age dependent. These data support a role for specific KIR alleles in HIV pathogenesis in children.

Genome-wide gene-environment interactions on quantitative traits using family data

Gene-environment interactions may provide a mechanism for targeting interventions to those individuals who would gain the most benefit from them. Searching for interactions agnostically on a genome-wide scale requires large sample sizes, often achieved through collaboration among multiple studies in a consortium. Family studies can contribute to consortia, but to do so they must account for correlation within families by using specialized analytic methods. In this paper, we investigate the performance of methods that account for within-family correlation, in the context of gene-environment interactions with binary exposures and quantitative outcomes. We simulate both cross-sectional and longitudinal measurements, and analyze the simulated data taking family structure into account, via generalized estimating equations (GEE) and linear mixed-effects models. With sufficient exposure prevalence and correct model specification, all methods perform well. However, when models are misspecified, mixed modeling approaches have seriously inflated type I error rates. GEE methods with robust variance estimates are less sensitive to model misspecification; however, when exposures are infrequent, GEE methods require modifications to preserve type I error rate. We illustrate the practical use of these methods by evaluating gene-drug interactions on fasting glucose levels in data from the Framingham Heart Study, a cohort that includes related individuals.

Covariance estimators for generalized estimating equations (GEE) in longitudinal analysis with small samples

Generalized estimating equations (GEE) is a general statistical method to fit marginal models for longitudinal data in biomedical studies. The variance-covariance matrix of the regression parameter coefficients is usually estimated by a robust "sandwich" variance estimator, which does not perform satisfactorily when the sample size is small. To reduce the downward bias and improve the efficiency, several modified variance estimators have been proposed for bias-correction or efficiency improvement. In this paper, we provide a comprehensive review on recent developments of modified variance estimators and compare their small-sample performance theoretically and numerically through simulation and real data examples. In particular, Wald tests and t-tests based on different variance estimators are used for hypothesis testing, and the guideline on appropriate sample sizes for each estimator is provided for preserving type I error in general cases based on numerical results. Moreover, we develop a user-friendly R package "geesmv" incorporating all of these variance estimators for public usage in practice.

Generalized estimating equations for genome-wide association studies using longitudinal phenotype data

Many longitudinal cohort studies have both genome-wide measures of genetic variation and repeated measures of phenotypes and environmental exposures. Genome-wide association study analyses have typically used only cross-sectional data to evaluate quantitative phenotypes and binary traits. Incorporation of repeated measures may increase power to detect associations, but also requires specialized analysis methods. Here, we discuss one such method-generalized estimating equations (GEE)-in the contexts of analysis of main effects of rare genetic variants and analysis of gene-environment interactions. We illustrate the potential for increased power using GEE analyses instead of cross-sectional analyses. We also address challenges that arise, such as the need for small-sample corrections when the minor allele frequency of a genetic variant and/or the prevalence of an environmental exposure is low. To illustrate methods for detection of gene-drug interactions on a genome-wide scale, using repeated measures data, we conduct single-study analyses and meta-analyses across studies in three large cohort studies participating in the Cohorts for Heart and Aging Research in Genomic Epidemiology consortium-the Atherosclerosis Risk in Communities study, the Cardiovascular Health Study, and the Rotterdam Study.

Generalized Estimating Equations in Longitudinal Data Analysis: A Review and Recent Developments

Generalized Estimating Equation (GEE) is a marginal model popularly applied for longitudinal/clustered data analysis in clinical trials or biomedical studies. We provide a systematic review on GEE including basic concepts as well as several recent developments due to practical challenges in real applications. The topics including the selection of “working” correlation structure, sample size and power calculation, and the issue of informative cluster size are covered because these aspects play important roles in GEE utilization and its statistical inference. A brief summary and discussion of potential research interests regarding GEE are provided in the end.

Small sample performance of bias-corrected sandwich estimators for cluster-randomized trials with binary outcomes

The sandwich estimator in generalized estimating equations (GEE) approach underestimates the true variance in small samples and consequently results in inflated type I error rates in hypothesis testing. This fact limits the application of the GEE in cluster-randomized trials (CRTs) with few clusters. Under various CRT scenarios with correlated binary outcomes, we evaluate the small sample properties of the GEE Wald tests using bias-corrected sandwich estimators. Our results suggest that the GEE Wald z-test should be avoided in the analyses of CRTs with few clusters even when bias-corrected sandwich estimators are used. With t-distribution approximation, the Kauermann and Carroll (KC)-correction can keep the test size to nominal levels even when the number of clusters is as low as 10 and is robust to the moderate variation of the cluster sizes. However, in cases with large variations in cluster sizes, the Fay and Graubard (FG)-correction should be used instead. Furthermore, we derive a formula to calculate the power and minimum total number of clusters one needs using the t-test and KC-correction for the CRTs with binary outcomes. The power levels as predicted by the proposed formula agree well with the empirical powers from the simulations. The proposed methods are illustrated using real CRT data. We conclude that with appropriate control of type I error rates under small sample sizes, we recommend the use of GEE approach in CRTs with binary outcomes because of fewer assumptions and robustness to the misspecification of the covariance structure.

A class of distribution-free models for longitudinal mediation analysis

Mediation analysis constitutes an important part of treatment study to identify the mechanisms by which an intervention achieves its effect. Structural equation model (SEM) is a popular framework for modeling such causal relationship. However, current methods impose various restrictions on the study designs and data distributions, limiting the utility of the information they provide in real study applications. In particular, in longitudinal studies missing data is commonly addressed under the assumption of missing at random (MAR), where current methods are unable to handle such missing data if parametric assumptions are violated.In this paper, we propose a new, robust approach to address the limitations of current SEM within the context of longitudinal mediation analysis by utilizing a class of functional response models (FRM). Being distribution-free, the FRM-based approach does not impose any parametric assumption on data distributions. In addition, by extending the inverse probability weighted (IPW) estimates to the current context, the FRM-based SEM provides valid inference for longitudinal mediation analysis under the two most popular missing data mechanisms; missing completely at random (MCAR) and missing at random (MAR). We illustrate the approach with both real and simulated data.

Finite-sample corrected generalized estimating equation of population average treatment effects in stepped wedge cluster randomized trials

Stepped wedge designs are increasingly commonplace and advantageous for cluster randomized trials when it is both unethical to assign placebo, and it is logistically difficult to allocate an intervention simultaneously to many clusters. We study marginal mean models fit with generalized estimating equations for assessing treatment effectiveness in stepped wedge cluster randomized trials. This approach has advantages over the more commonly used mixed models that (1) the population-average parameters have an important interpretation for public health applications and (2) they avoid untestable assumptions on latent variable distributions and avoid parametric assumptions about error distributions, therefore, providing more robust evidence on treatment effects. However, cluster randomized trials typically have a small number of clusters, rendering the standard generalized estimating equation sandwich variance estimator biased and highly variable and hence yielding incorrect inferences. We study the usual asymptotic generalized estimating equation inferences (i.e., using sandwich variance estimators and asymptotic normality) and four small-sample corrections to generalized estimating equation for stepped wedge cluster randomized trials and for parallel cluster randomized trials as a comparison. We show by simulation that the small-sample corrections provide improvement, with one correction appearing to provide at least nominal coverage even with only 10 clusters per group. These results demonstrate the viability of the marginal mean approach for both stepped wedge and parallel cluster randomized trials. We also study the comparative performance of the corrected methods for stepped wedge and parallel designs, and describe how the methods can accommodate interval censoring of individual failure times and incorporate semiparametric efficient estimators.

Associations of host genetic variants on CD4⁺ lymphocyte count and plasma HIV-1 RNA in antiretroviral naïve children

BACKGROUND

CD4 T-lymphocyte (CD4) counts and HIV plasma RNA concentration (RNA) are 2 key HIV disease markers. The complex interplay between virus and host genetics may contribute to differences in viral set point and CD4 status. Determining the effects of host genetic variation on HIV disease markers is often complicated by the use of antiretroviral therapy. In this study, the association between genetic variants and baseline HIV RNA and CD4 counts was examined in a large cohort of antiretroviral naïve children.

METHODS

Specimens from 1053 HIV-infected children were screened for single nucleotide polymorphisms in 78 regions from 17 genes. Linear regression with a robust variance estimator was used to test the association between genetic markers with HIV RNA and CD4 count, controlling for age, race/ethnicity and study. False discovery rate (FDR) controlling was used to adjust for multiple testing.

RESULTS

The study population was 60% black, 26% Hispanic and 13% white; median age 2.35 years; 55% female. Baseline median CD4 count was 780/mm; median log10 HIV RNA was 5.17 copies/mL. For analyses of the associations of genetic makers with baseline CD4 count, 6 HLA and 4 additional markers exhibited P < 0.05, but none met the criteria for statistical significance with FDR controlled at 0.05. For baseline HIV RNA, HLA DRB1*15, DRB1*10, B-27/57, B-14, Cw-8, B-57 were statistically significant with FDR controlled at 0.05.

CONCLUSIONS

These results provide strong evidence that HLA DRB1*15, DRB1*10, B-27/57, B-14, Cw-8, B-57 are associated with HIV RNA and play a role in HIV pathogenesis in infected children.

Testing for association with multiple traits in generalized estimation equations, with application to neuroimaging data

There is an increasing need to develop and apply powerful statistical tests to detect multiple traits-single locus associations, as arising from neuroimaging genetics and other studies. For example, in the Alzheimer's Disease Neuroimaging Initiative (ADNI), in addition to genome-wide single nucleotide polymorphisms (SNPs), thousands of neuroimaging and neuropsychological phenotypes as intermediate phenotypes for Alzheimer's disease, have been collected. Although some classic methods like MANOVA and newly proposed methods may be applied, they have their own limitations. For example, MANOVA cannot be applied to binary and other discrete traits. In addition, the relationships among these methods are not well understood. Importantly, since these tests are not data adaptive, depending on the unknown association patterns among multiple traits and between multiple traits and a locus, these tests may or may not be powerful. In this paper we propose a class of data-adaptive weights and the corresponding weighted tests in the general framework of generalized estimation equations (GEE). A highly adaptive test is proposed to select the most powerful one from this class of the weighted tests so that it can maintain high power across a wide range of situations. Our proposed tests are applicable to various types of traits with or without covariates. Importantly, we also analytically show relationships among some existing and our proposed tests, indicating that many existing tests are special cases of our proposed tests. Extensive simulation studies were conducted to compare and contrast the power properties of various existing and our new methods. Finally, we applied the methods to an ADNI dataset to illustrate the performance of the methods. We conclude with the recommendation for the use of the GEE-based Score test and our proposed adaptive test for their high and complementary performance.

Fast and accurate modelling of longitudinal and repeated measures neuroimaging data

Despite the growing importance of longitudinal data in neuroimaging, the standard analysis methods make restrictive or unrealistic assumptions (e.g., assumption of Compound Symmetry--the state of all equal variances and equal correlations--or spatially homogeneous longitudinal correlations). While some new methods have been proposed to more accurately account for such data, these methods are based on iterative algorithms that are slow and failure-prone. In this article, we propose the use of the Sandwich Estimator method which first estimates the parameters of interest with a simple Ordinary Least Square model and second estimates variances/covariances with the "so-called" Sandwich Estimator (SwE) which accounts for the within-subject correlation existing in longitudinal data. Here, we introduce the SwE method in its classic form, and we review and propose several adjustments to improve its behaviour, specifically in small samples. We use intensive Monte Carlo simulations to compare all considered adjustments and isolate the best combination for neuroimaging data. We also compare the SwE method to other popular methods and demonstrate its strengths and weaknesses. Finally, we analyse a highly unbalanced longitudinal dataset from the Alzheimer's Disease Neuroimaging Initiative and demonstrate the flexibility of the SwE method to fit within- and between-subject effects in a single model. Software implementing this SwE method has been made freely available at http://warwick.ac.uk/tenichols/SwE.

On assessing model fit for distribution-free longitudinal models under missing data

The generalized estimating equation (GEE), a distribution-free, or semi-parametric, approach for modeling longitudinal data, is used in a wide range of behavioral, psychotherapy, pharmaceutical drug safety, and healthcare-related research studies. Most popular methods for assessing model fit are based on the likelihood function for parametric models, rendering them inappropriate for distribution-free GEE. One rare exception is a score statistic initially proposed by Tsiatis for logistic regression (1980) and later extended by Barnhart and Willamson to GEE (1998). Because GEE only provides valid inference under the missing completely at random assumption and missing values arising in most longitudinal studies do not follow such a restricted mechanism, this GEE-based score test has very limited applications in practice. We propose extensions of this goodness-of-fit test to address missing data under the missing at random assumption, a more realistic model that applies to most studies in practice. We examine the performance of the proposed tests using simulated data and demonstrate the utilities of such tests with data from a real study on geriatric depression and associated medical comorbidities.

Distribution-free models for longitudinal count responses with overdispersion and structural zeros

Overdispersion and structural zeros are two major manifestations of departure from the Poisson assumption when modeling count responses using Poisson log-linear regression. As noted in a large body of literature, ignoring such departures could yield bias and lead to wrong conclusions. Different approaches have been developed to tackle these two major problems. In this paper, we review available methods for dealing with overdispersion and structural zeros within a longitudinal data setting and propose a distribution-free modeling approach to address the limitations of these methods by utilizing a new class of functional response models. We illustrate our approach with both simulated and real study data.