Bias In Point Estimates Research Articles

Specification of covariance structure in longitudinal data analysis for randomized clinical trials

Misspecification of the covariance structure for repeated measurements in longitudinal analysis may lead to biased estimates of the regression parameters and under or overestimation of the corresponding standard errors in the presence of missing data. The so-called sandwich estimator can 'correct' the variance, but it does not reduce the bias in point estimates. Removing all assumptions from the covariance structure (i.e. using an unstructured (UN) covariance) will remove such biases. However, an excessive amount of missing data may cause convergence problems for iterative algorithms, such as the default Newton-Raphson algorithm in the popular SAS PROC MIXED. This article examines, both through theory and simulations, the existence and the magnitude of these biases. We recommend the use of UN covariance as the default strategy for analyzing longitudinal data from randomized clinical trials with moderate to large number of subjects and small to moderate number of time points. We also present an algorithm to assist in the convergence when the UN covariance is used.

Influence of control selection in genome-wide association studies: the example of diabetes in the Framingham Heart Study

Epidemiologic study designs represent a major challenge for genome-wide association studies. Most such studies to date have selected controls from the pool of participants without the disease of interest at the end of the study time. These choices can lead to biased estimates of exposure effects. Using data from the Framingham Heart Study (Genetic Analysis Workshop 16 Problem 2), we evaluate the impact on genetic association estimates for designs with control selection based on status at the end of a study (case exclusion (CE) sampling) to control selection based on incidence density (ID) sampling, when controls are selected from the pool of participants who are disease-free at the time a case is diagnosed. Cases are defined as those diagnosed with type 2 diabetes (T2D). We estimated odds ratios for 18 previously confirmed T2D variants using 189 cases selected by ID sampling and using 231 cases selected by CE sampling. We found none of these single-nucleotide polymorphisms to be significantly associated with T2D using either design. Because these empirical analyses were based on a small number of cases and on single-nucleotide polymorphisms with likely small effect sizes, we supplemented this work with simulated data sets of 500 cases from each strategies across a variety of allele frequencies and effect sizes. In our simulated datasets, we show ID sampling to be less biased than CE, although CE shows apparent increased power due to the upward bias of point estimates. We conclude that ID sampling is an appropriate option for genome-wide association studies.

Regression models for public health surveillance data: a simulation study

Objectives:Poisson regression is now widely used in epidemiology, but researchers do not always evaluate the potential for bias in this method when the data are overdispersed. This study used simulated...

Complementary Log–Log Regression for the Estimation of Covariate‐Adjusted Prevalence Ratios in the Analysis of Data from Cross‐Sectional Studies

We assessed complementary log-log (CLL) regression as an alternative statistical model for estimating multivariable-adjusted prevalence ratios (PR) and their confidence intervals. Using the delta method, we derived an expression for approximating the variance of the PR estimated using CLL regression. Then, using simulated data, we examined the performance of CLL regression in terms of the accuracy of the PR estimates, the width of the confidence intervals, and the empirical coverage probability, and compared it with results obtained from log-binomial regression and stratified Mantel-Haenszel analysis. Within the range of values of our simulated data, CLL regression performed well, with only slight bias of point estimates of the PR and good confidence interval coverage. In addition, and importantly, the computational algorithm did not have the convergence problems occasionally exhibited by log-binomial regression. The technique is easy to implement in SAS (SAS Institute, Cary, NC), and it does not have the theoretical and practical issues associated with competing approaches. CLL regression is an alternative method of binomial regression that warrants further assessment.

Author's Response: Response to commentary: Meningioma and mobile phone use--a collaborative case-control study in five North European countries

Dr Milham and Mr Morgan are concerned about the age and gender distribution of controls relative to the cases in our recent paper. In short, the control series in the Interphone study was not selected for meningioma cases only, but the age and sex distribution reflect that of the entire set of cases including also cases of glioma and acoustic neurinoma. All controls were included in the analysis of meningioma in order to increase the statistical power. The age and sex distribution have no influence on the results of the stratified (conditional) analysis. Secondly, Dr Milham points out that dissimilar case and control frequencies are given in different sections of the paper. This refers to numbers of eligible, participating and analysed cases and controls. Indeed, it would be ideal that all eligible subjects would participate and provide complete information, but this rarely occurs in practice. Thirdly, both find the large number of odds ratios <1 disturbing. As outlined by Charles S. Peirce, the scientific method of inference includes the fundamental principle that results cannot be ignored or selected based on prior beliefs. Therefore, whether or not the results support a hypothesis (or confirm prior beliefs) should not justify disregarding the evidence or questioning the value of those results (but such approach would be typical for the ‘method of tenacity’, with fixation of belief regardless of the evidence). This apparent protective effect related to mobile phone use has been discussed in the report and also separate methodological papers. To reiterate, it appears that controls were more likely to participate if they had higher education and/or were mobile phone users. This is likely to result in odds ratios below unity for comparison of evervs never-users of mobile phones. The downward bias in point estimates is, however, likely to be no more than 10%. Nevertheless, its impact on comparisons by amount of use is difficult to assess. A previous simulation study has shown, however, that such selection bias would not substantially distort a dose–effect relation, unless extreme.

A comparison of Bayesian and likelihood-based methods for fitting multilevel models

We use simulation studies, whose design is realistic for educational andmedicalresearch(aswellasotherfleldsofinquiry),tocompareBayesianand likelihood-basedmethodsforflttingvariance-components(VC)andrandom-efiects logistic regression (RELR) models. The likelihood (and approximate likelihood) approachesweexaminearebasedonthemethodsmostwidelyusedincurrentap- plied multilevel (hierarchical) analyses: maximum likelihood (ML) and restricted ML(REML)forGaussianoutcomes,andmarginalandpenalizedquasi-likelihood (MQL and PQL) for Bernoulli outcomes. Our Bayesian methods use Markov chain Monte Carlo (MCMC) estimation, with adaptive hybrid Metropolis-Gibbs sampling for RELR models, and several difiuse prior distributions (i i1 (†;†) and U(0; 1 ) priors for variance components). For evaluation criteria we consider bias of point estimates and nominal versus actual coverage of interval estimates in re- peated sampling. In two-level VC models we flnd that (a) both likelihood-based and Bayesian approaches can be made to produce approximately unbiased esti- mates, although the automatic manner in which REML accomplishes this is an advantage, but (b) both approaches had di-culty achieving nominal coverage in smallsamplesandwithsmallvaluesoftheintraclasscorrelation. Withthethree- levelRELRmodelsweexamineweflndthat(c)quasi-likelihoodmethodsforesti- mating random-efiects variances perform badly with respect to bias and coverage intheexamplewesimulated,and(d)Bayesiandifiuse-priormethodsleadtowell- calibratedpointandintervalRELRestimates. Whileitistruethatthelikelihood- based methods we study are considerably faster computationally than MCMC, (i) steady improvements in recent years in both hardware speed and e-ciency of MonteCarloalgorithmsand(ii)thelackofcalibrationoflikelihood-basedmethods insomecommonhierarchicalsettingscombinetomakeMCMC-basedBayesianflt- tingofmultilevelmodelsanattractiveapproach,evenwithratherlargedatasets. Other analytic strategies based on less approximate likelihood methods are also possible butwouldbeneflt fromfurtherstudy ofthe type summarized here.

Comparison of different estimation procedures for proportional hazards model with random effects

Proportional hazards models with multivariate random effects (frailties) acting multiplicatively on the baseline hazard are a topic of intensive research. Several estimation procedures have been proposed to deal with this type of models. Four procedures used to fit these models are compared in two real-life datasets and in a simulation study. The performance of the four methods is investigated in terms of the bias of point estimates, their empirical variability and the bias of the estimation of the variability.

Using Data Augmentation to Correct for Non-Ignorable Non-Response When Surrogate Data are Available: An Application to the Distribution of Hourly Pay

SummaryThe paper develops a data augmentation method to estimate the distribution function of a variable, which is partially observed, under a non-ignorable missing data mechanism, and where surrogate data are available. An application to the estimation of hourly pay distributions using UK Labour Force Survey data provides the main motivation. In addition to considering a standard parametric data augmentation method, we consider the use of hot deck imputation methods as part of the data augmentation procedure to improve the robustness of the method. The method proposed is compared with standard methods that are based on an ignorable missing data mechanism, both in a simulation study and in the Labour Force Survey application. The focus is on reducing bias in point estimation, but variance estimation using multiple imputation is also considered briefly.

On Confidence Intervals for Genotype Relative Risks and Attributable Risks from Case Parent Trio Designs for Candidate-Gene Studies

Scherag et al. [Hum Hered 2002;54:210–217] recently proposed point estimates and asymptotic as well as exact confidence intervals for genotype relative risks (GRRs) and the attributable risk (AR) in case parent trio designs using single nucleotide polymorphism (SNP) data. The aim of this study was the investigation of coverage probabilities and bias in estimates if the marker locus is not identical to the disease locus. Using a variety of parameter constellations, including marker allele frequencies identical to and different from the SNP at the disease locus, we performed an analytical study to quantify the bias and a Monte-Carlo simulation study for quantifying both bias and coverage probabilities. No bias was observed if marker and trait locus coincided. Two parameters had a strong impact on coverage probabilities of confidence intervals and bias in point estimates if they did not coincide: the linkage disequilibrium (LD) parameter δ and the allele frequency at the marker SNP. If marker allele frequencies were different from the allele frequencies at the functional SNP, substantial biases occurred. Further, if δ between the marker and the disease locus was lower than the maximum possible δ, estimates were also biased. In general, biases were towards the null hypothesis for both GRRs and AR. If one GRR was not increased, as e.g. in a recessive genetic model, biases away from the null could be observed. If both GRRs were in identical directions and if both were substantially larger than 1, the bias always was towards the null. When applying point estimates and confidence intervals for GRRs and AR in candidate gene studies, great care is needed. Effect estimates are substantially biased towards the null if either the allele frequencies at the marker SNP and the true disease locus are different or if the LD between the marker SNP and the disease locus is not at its maximum. A bias away from the null occurs only in uncommon study situations; it is small and can therefore be ignored for applications.

Incorporating Individual Error Rate into Association Test of Unmatched Case-Control Design

Objectives: Genotyping error commonly occurs and could reduce the power and bias statistical inference in genetics studies. In addition to genotypes, some automated biotechnologies also provide quality measurement of each individual genotype. We studied the relationship between the quality measurement and genotyping error rate. Furthermore, we propose two association tests incorporating the genotyping quality information with the goal to improve statistical power and inference. Methods: 50 pairs of DNA sample duplicates were typed for 232 SNPs by BeadArray technology. We used scatter plot, smoothing function and generalized additive models to investigate the relationship between genotype quality score (q) and inconsistency rate (ĩ) among duplicates. We constructed two association tests: (1) weighted contingency table test (WCT) and (2) likelihood ratio test (LRT) to incorporate individual genotype error rate (Ε_i), in unmatched case-control setting. Results: In the 50 duplicates, we found q and ĩ were in strong negative association, suggesting the genotypes with low quality score were more likely to be mistyped. The WCT improved the statistical power and partially corrects the bias in point estimation. The LRT offered moderate power gain, but was able to correct the bias in odds ratio estimation. The two new methods also performed favorably in some scenarios when Ε_i was mis-specified. Conclusions: With increasing number of genetic studies and application of automated genotyping technology, there is a growing need to adequately account for individual genotype error rate in statistical analysis. Our study represents an initial step to address this need and points out a promising direction for further research.

On combining dose-response data from epidemiological studies by meta-analysis.

Using data from a meta-analysis of the effects of oestrogen replacement therapy on the development of breast cancer, we compared alternative methods for combining dose-response slopes from epidemiological studies. We evaluated issues related both to summarizing data from single studies and to combining results from multiple studies. Findings related to the analysis of individual dose-response studies include: (1) a method of weighing studies that gives greater influence to dose-response slopes that conform to the linear relation of relative risk to duration can lead to large differences in calculated weights as a function of non-linearity; (2) a regression model using a variable-intercept resulted in a mean dose-response slope that increased as much as threefold when compared with the values obtained with a zero-intercept model. When combining results from multiple studies, we found: (1) calculating standard errors of mean dose-response slopes by methods that allow for both among-study and within-study variability (a random-effects type model) gave values different from a method that assumes homogeneity and equal within-study precision (a fixed-effects model); (2) the random-effects model gives mean and standard error results most similar to a bootstrap resampling method as increasing heterogeneity is observed (however, this model could give biased mean estimates compared with the bootstrap method); (3) a components-of-variance model compares favourably with the bootstrap and is easier to apply than the random-effects model. Based on these findings, we recommend the use of methods which incorporate heterogeneity to guard against underestimating the standard error. However, caution is urged because bias in point estimates can occur if extreme heterogeneity is present. Two other observations affect the interpretation of data combined from multiple studies. First, inclusion into a model of quality scores assigned by blinded reviewers had little effect on the mean dose-response slope and its standard error. Second, the number of studies required to achieve desired statistical power, varies with effect size.

Stopping rules and estimation problems in clinical trials

Stopping rules in clinical trials can lead to bias in point estimation of the magnitude of treatment difference. A simulation exercise, based on estimation of the risk ratio in a typical post-myocardial infarction trial, examines the nature of this exaggeration of treatment effect under various group sequential plans and also under continuous naive monitoring for statistical significance. For a fixed treatment effect the median bias in group sequential design is small, but it is greatest for effects that the trial has reasonable power to detect. Bias is evidently greater in trials that stop early and is dramatic under naive monitoring for significance. Group sequential plans lead to a multimodal sampling distribution of treatment effect, which poses problems for incorporating their estimates into meta-analyses. By simulating a population of trials with treatment effects modelled by an underlying distribution of true risk ratios, a Bayesian method is proposed for assessing the plausible range of true treatment effect for any trial based on interim results. This approach is particularly useful for producing shrinkage of the unexpectedly large and imprecise observed treatment effects that arise in clinical trials that stop early. Its implications for trial design are discussed.

Small-Sample Bias of Point Estimators of the Odds Ratio from Matched Sets

The bias of several point estimators of the odds ratio arising from matched-pair data is investigated for small samples. Simple alternatives to the traditional maximum likelihood estimator are suggested, on both the original scale and the logarithm scale. In each case the suggested estimators possess a superior performance in terms of mean square error. Generalizations are given for 1:R matched data sets.

On Bias Reduction in Estimation

Abstract A general procedure for reducing the bias of point estimators is introduced. The technique includes the “jackknife” as a special case. The existing notion of reapplication is shown to lack a desirable bias removal property for which it was originally designed. Proper reapplication is proposed to conform to the general notion of higher order bias elimination and an interesting algorithm for the correct method is defined. Illustrative examples are drawn from ratio estimation, reliability and truncated distributions. The reduced mean square error which attracted some attention to the jackknife is present in the generalization for some applications.