On the Asymptotic Relative Efficiency of the Mantel-Haenszel Test

The Mantel-Haenszel test is optimal when the odds ratio is constant. This paper investigates the effects of departures from the assumption of a constant odds ratio on the behavior of the Mantel-Haenzel test. A simple approximation is proposed for the non-null distribution of the test statistic. Based on this approximation, the asymptotic relative efficiency of the Mantel-Haenszel test, compared to the overall χ² test for no partial association, is calculated. For the case of 2 strata, it is shown that the Mantel-Haenszel test is efficient as long as the logarithms of the odds ratios are of the same sign and their absolute values exceed 1.     

On Estimation of Relative Risk in Case Control Studies

An estimator of relative risk in a case control study has been proposed in terms of observed cell frequencies and the probability of disease. The bias of the usual estimator i.e odds ratio as compared to the new estimator has been workedout. The expression of Mean Square Error of proposed estimator has been derived in situations where probability of disease is exactly known and when it is estimated through an independent survey. It has been observed that there is a serious error using odds ratio as an estimate of relative risk when probability of disease is not negligible. In such situations the proposed estimator can be used with advantage.     

On cross-odds ratio for multivariate competing risks data

The cross-odds ratio is defined as the ratio of the conditional odds of the occurrence of one cause-specific event for one subject given the occurrence of the same or a different cause-specific event for another subject in the same cluster over the unconditional odds of occurrence of the cause-specific event. It is a measure of the association between the correlated cause-specific failure times within a cluster. The joint cumulative incidence function can be expressed as a function of the marginal cumulative incidence functions and the cross-odds ratio. Assuming that the marginal cumulative incidence functions follow a generalized semiparametric model, this paper studies the parametric regression modeling of the cross-odds ratio. A set of estimating equations are proposed for the unknown parameters and the asymptotic properties of the estimators are explored. Non-parametric estimation of the cross-odds ratio is also discussed. The proposed procedures are applied to the Danish twin data to model the associations between twins in their times to natural menopause and to investigate whether the association differs among monozygotic and dizygotic twins and how these associations have changed over time.     

Aspects of the design and analysis of high-dimensional SNP studies for disease risk estimation

The state of readiness for high-dimensional single nucleotide polymorphism (SNP) epidemiologic association studies is described, as background for a discussion of statistical aspects of case-control study design and analysis. Specifically, the important role that multistage designs can play in the elimination of false-positive associations and in the control of study costs will be noted. Also, the trade-offs associated with using pooled DNA at early design stages for additional important cost reductions will be discussed in some detail. An odds ratio approach to relating SNP alleles to disease risk using pooled DNA will be proposed, in conjunction with a simple empirical variance estimator, based on comparisons among log-odds ratio estimators from distinct pairs of case and control pools. Simulation studies will be presented to evaluate the moderate sample size properties of such multistage designs and estimation procedures. The design of an ongoing three-stage study in the Women's Health Initiative to relate 250,000 SNPs to the risk of coronary heart disease, stroke, and breast cancer will provide illustration, and will be used to motivate the choice of simulation configurations.     

Effects of cluster sampling on epidemiologic analysis in population-based case-control studies

We consider population-based case-control designs in which controls are selected by one of three cluster sampling plans from the entire population at risk. The effects of cluster sampling on classical epidemiologic procedures are investigated, and appropriately modified procedures are developed. In particular, modified procedures for testing the homogeneity of odds ratios across strata, and for estimating and testing a common odds ratio are presented. Simulations that use the data from the 1970 Health Interview Survey as a population suggest that classical procedures may be fairly robust in the presence of cluster sampling. A more extreme example based on a mixed multinomial model clearly demonstrates that the classical Mantel-Haenszel (1959, Journal of the National Cancer Institute 22, 719-748) and Woolf-Haldane tests of no exposure effect may have sizes exceeding nominal levels and confidence intervals with less than nominal coverage under an alternative hypothesis. Classical estimates of odds ratios may also be biased with non-self-weighting cluster samples. The modified procedures we propose remedy these defects.     

Statistical methods for the analysis of two-arm non-inferiority trials with binary outcomes

The aim of this contribution is to give an overview of approaches to testing for non-inferiority of one out of two binomial distributions as compared to the other in settings involving independent samples (the paired samples case is not considered here but the major conclusions and recommendations can be shown to hold for both sampling schemes). In principle, there is an infinite number of different ways of defining (one-sided) equivalence in any multiparameter setting. In the binomial two-sample problem, the following three choices of a measure of dissimilarity between the underlying distributions are of major importance for real applications: the odds ratio (OR), the relative risk (RR), and the difference (DEL) of both binomial parameters. It is shown that for all three possibilities of formulating the hypotheses of a non-inferiority problem concerning two binomial proportions, reasonable testing procedures providing exact control over the type-I error risk are available. As a particularly useful and versatile way of handling mathematically nonnatural parametrizations like RR and DELTA, the approach through Bayesian posterior probabilities of hypotheses with respect to some non-informative reference prior has much to recommend it. In order to ensure that the corresponding testing procedure be valid in the classical, i.e. frequentist sense, it suffices to use straightforward computational techniques yielding suitably corrected nominal significance levels. In view of the availability of testing procedures with satisfactory properties for all parametrizations of main practical interest, the discussion of the pros and cons of these methods has to focus on the question of which of the underlying measures of dissimilarity should be preferred on grounds of logic and intuition. It is argued that the OR clearly merits to be given preference also with regard to this latter kind of criteria since the non-inferiority hypotheses defined in terms of the other parametric functions are bounded by lines which cross the boundaries of the parameter space. From this fact, we conclude that the exact Fisher type test for one-sided equivalence provides the most reasonable approach to the confirmatory analysis of non-inferiority trials involving two independent samples of binary data. The marked conservatism of the nonrandomized version of this test can largely be removed by using a suitably increased nominal significance level (depending, in addition to the target level, on the sample sizes and the equivalence margin), or by replacing it with a Bayesian test for non-inferiority with respect to the odds ratio.     

Power calculations for matched case-control studies

Power calculations are derived for matched case-control studies in terms of the probability po of exposure among the control patients, the correlation coefficient phi for exposure between matched case and control patients, and the odds ratio psi for exposure in case and control patients. For given Type I and Type II error probabilities alpha and beta, the odds ratio that can be detected with a given sample size is derived as well as the sample size needed to detect a specified value of the odds ratio. Graphs are presented for paired designs that show the relationship between sample size and power for alpha = .05, beta = .2, and different values of po, phi, and psi. The sample size needed for designs involving M matched control patients can be derived from these graphs by means of a simple equation. These results quantify the loss of power associated with increasing correlation between the exposure status of matched case and control patients. Sample size requirements are also greatly increased for values of po near 0 or 1. The relationship between sample size, psi, phi, and po is discussed and illustrated by examples.     

Residual plots for log odds ratio regression models.

This article describes graphical diagnostic methods for log odds ratio regression models. To study the effects of an additional covariate on log odds ratio regression analysis, three types of residual plots based on weighted least squares (WLS) are discussed: (i) added variable plot (partial regression plot), (ii) partial residual plot, and (iii) augmented partial residual plot. These plots provide diagnostic procedures for identifying heterogeneity of error variances, outliers, or nonlinearity of the model. They are especially useful for clarifying whether including a covariate as a linear term is appropriate, or whether quadratic or other nonlinear transformations are preferable. A well-known data set for case-control studies is analyzed to illustrate the residual plots.     

Effect Sizes for 2×2 Contingency Tables

Sample size calculations are an important part of research to balance the use of resources and to avoid undue harm to participants. Effect sizes are an integral part of these calculations and meaningful values are often unknown to the researcher. General recommendations for effect sizes have been proposed for several commonly used statistical procedures. For the analysis of tables, recommendations have been given for the correlation coefficient for binary data; however, it is well known that suffers from poor statistical properties. The odds ratio is not problematic, although recommendations based on objective reasoning do not exist. This paper proposes odds ratio recommendations that are anchored to for fixed marginal probabilities. It will further be demonstrated that the marginal assumptions can be relaxed resulting in more general results.     

Estimation of Relative Risk Under Variable Multiple Matching

When disease incidence is low, the odds ratio, which closely approximates relative risk, is estimated optimally in a closed form using a variable matching ratio in a retrospective design. The model also enables one to assess the homogeneity of the odds ratio from the same one matched sample. Analogous procedures are shown to hold for the prospective design.     

Notes on Estimation of the General Odds Ratio and the General Risk Difference for Paired‐Sample Data

Under the matched‐pair design, this paper discusses estimation of the general odds ratio OR_G for ordinal exposure in case‐control studies and the general risk difference RD_G for ordinal outcomes in cross‐sectional or cohort studies. To illustrate the practical usefulness of interval estimators of OR_G and RD_G developed here, this paper uses the data from a case‐control study investigating the effect of the number of beverages drunk at “burning hot” temperature on the risk of possessing esophageal cancer, and the data from a cross‐sectional study comparing the grade distributions of unaided distance vision between two eyes. Finally, this paper notes that using the commonly‐used statistics related to odds ratio for dichotomous data by collapsing the ordinal exposure into two categories: the exposure versus the non‐exposure, tends to be less efficient than using the statistics related to OR_G proposed herein.     

Graft Rejection Rate and Graft Failure Rate of Penetrating Keratoplasty (PKP) vs Lamellar Procedures: A Systematic Review

Purpose

The aim of our investigation was to conduct a quantitative meta-analysis of the present world literature comparing the major surgical outcomes of penetrating keratoplasty (PKP) to lamellar procedures. Our goal is that clinicians, eye bank administrators, and health policy makers will be able to utilize this study in implementing decisions in regards to corneal transplantation.

Methods

Pooled measures of association were with odds ratios and because of study heterogeneity, the pooled effects were assumed to follow a random effects model (DerSimonian-Laird). The comparisons were between 1) PKP’s and all lamellar procedures (anterior AND posterior) and then 2) between PKP’s and all anterior lamellar procedures and 3) PKP and all posterior lamellar procedures.

Results

For PKP vs anterior lamellar procedures, the pooled odds ratio for rejection of PKP over lamellar keratoplasty (LK) was 3.56 (95% CI: 1.76-7.20) and for outright failure, the pooled odds ratio of PKP failure vs LK was 2.85 (95% CI: 0.84-9.66). For posterior lamellar procedures, the pooled odds ratio for rejection of PKP over LK was 1.52 (95% CI: 1.00-2.32). The pooled odds ratio for outright failure of PKP over posterior lamellar procedures was 2.09 (95% CI: 0.57-7.59). The follow up time was significantly longer for full transplants than for lamellar procedures.

Conclusions

For both anterior and posterior lamellar procedures, the odds ratios comparing rejection of full transplants to lamellar procedures (both anterior and posterior individually) were significantly higher in the PKP group. For outright failure, the PKP group also had a higher risk of failure than the lamellar groups but this was not statistically significant in either instance (anterior or posterior). Some of the clinical differences benefitting lamellar procedures may at least be partly explained by follow up time differences between groups and this needs to be accounted for more rigorously in future studies.     

Covariate measurement error adjustment for matched case-control studies

We propose a conditional scores procedure for obtaining bias-corrected estimates of log odds ratios from matched case-control data in which one or more covariates are subject to measurement error. The approach involves conditioning on sufficient statistics for the unobservable true covariates that are treated as fixed unknown parameters. For the case of Gaussian nondifferential measurement error, we derive a set of unbiased score equations that can then be solved to estimate the log odds ratio parameters of interest. The procedure successfully removes the bias in naive estimates, and standard error estimates are obtained by resampling methods. We present an example of the procedure applied to data from a matched case-control study of prostate cancer and serum hormone levels, and we compare its performance to that of regression calibration procedures.     

Logistic regression with exposure biomarkers and flexible measurement error

Regression calibration, refined regression calibration, and conditional scores estimation procedures are extended to a measurement model that is motivated by nutritional and physical activity epidemiology. Biomarker data, available on a small subset of a study cohort for reasons of cost, are assumed to adhere to a classical measurement error model, while corresponding self-report nutrient consumption or activity-related energy expenditure data are available for the entire cohort. The self-report assessment measurement model includes a person-specific random effect, the mean and variance of which may depend on individual characteristics such as body mass index or ethnicity. Logistic regression is used to relate the disease odds ratio to the actual, but unmeasured, dietary or physical activity exposure. Simulation studies are presented to evaluate and contrast the three estimation procedures, and to provide insight into preferred biomarker subsample size under selected cohort study configurations.     

A Note on Estimating Crude Odds Ratios in Case–Control Studies with Differentially Misclassified Exposure

Summary. Morrissey and Spiegelman (1999, Biometrics 55 , 338–344) provided a comparative study of adjustment methods for exposure misclassification in case‐control studies equipped with an internal validation sample. In addition to the maximum likelihood (ML) approach, they considered two intuitive procedures based on proposals in the literature. Despite appealing ease of computation associated with the latter two methods, efficiency calculations suggested that ML was often to be recommended for the analyst with access to a numerical routine to facilitate it. Here, a reparameterization of the likelihood reveals that one of the intuitive approaches, the inverse matrix method, is in fact ML under differential misclassification. This correction is intended to alert readers to the existence of a simple closed‐form ML estimator for the odds ratio in this setting so that they may avoid assuming that a commercially inaccessible optimization routine must be sought to implement ML.     

A General Class of Semiparametric Transformation Frailty Models for Nonproportional Hazards Survival Data

Summary We propose a semiparametrically efficient estimation of a broad class of transformation regression models for nonproportional hazards data. Classical transformation models are to be viewed from a frailty model paradigm, and the proposed method provides a unified approach that is valid for both continuous and discrete frailty models. The proposed models are shown to be flexible enough to model long‐term follow‐up survival data when the treatment effect diminishes over time, a case for which the PH or proportional odds assumption is violated, or a situation in which a substantial proportion of patients remains cured after treatment. Estimation of the link parameter in frailty distribution, considered to be unknown and possibly dependent on a time‐independent covariates, is automatically included in the proposed methods. The observed information matrix is computed to evaluate the variances of all the parameter estimates. Our likelihood‐based approach provides a natural way to construct simple statistics for testing the PH and proportional odds assumptions for usual survival data or testing the short‐ and long‐term effects for survival data with a cure fraction. Simulation studies demonstrate that the proposed inference procedures perform well in realistic settings. Applications to two medical studies are provided.     

Simultaneous confidence intervals for comparing binomial parameters

SUMMARY: To compare proportions with several independent binomial samples, we recommend a method of constructing simultaneous confidence intervals that uses the studentized range distribution with a score statistic. It applies to a variety of measures, including the difference of proportions, odds ratio, and relative risk. For the odds ratio, a simulation study suggests that the method has coverage probability closer to the nominal value than ad hoc approaches such as the Bonferroni implementation of Wald or "exact" small-sample pairwise intervals. It performs well even for the problematic but practically common case in which the binomial parameters are relatively small. For the difference of proportions, the proposed method has performance comparable to a method proposed by Piegorsch (1991, Biometrics 47, 45-52).     

Segregostat: a novel concept to control phenotypic diversification dynamics on the example of Gram-negative bacteria

Controlling and managing the degree of phenotypic diversification of microbial populations is a challenging task. This task not only requires detailed knowledge regarding diversification mechanisms but also advanced technical set-ups for the real-time analyses and control of population behaviour on single-cell level. In this work, set-up, design and operation of the so called segregostat are described which, in contrast to a traditional chemostat, allows the control of phenotypic diversification of microbial populations over time. Two exemplary case studies will be discussed, i.e. phenotypic diversification dynamics of Eschericia coli and Pseudomonas putida based on outer membrane permeabilization, emphasizing the applicability and versatility of the proposed approach. Upon nutrient limitation, cell population tends to diversify into several subpopulations exhibiting distinct phenotypic features (non-permeabilized and permeabilized cells). Online analysis leads to the determination of the ratio between cells in these two states, which in turn triggers the addition of glucose pulses in order to maintain a predefined diversification ratio. These results prove that phenotypic diversification can be controlled by means of defined pulse-frequency modulation within continuously running bioreactor set-ups. This lays the foundation for systematic studies, not only of phenotypic diversification but also for all processes where dynamics single-cell approaches are required, such as synthetic co-culture processes.     

Fungal endophytes associated with three South American Myrtae (Myrtaceae) exhibit preferences in the colonization at leaf level

Fungal endophytes associated with Myrtaceae from Brazil and Argentina were isolated at three levels of nesting: leaf, individual host trees, and site collection. The alternating logistic regression (ALR) was used to model the data because it offers a computationally convenient method for fitting regression structures involving large clusters. The objectives of this study were to determine: (i) whether the colonization pattern is influenced by environmental variables, (ii) if there is some leaf part they prefer to colonize; (iii) if there is some fungal endophyte aggregation between hierarchical levels; (iv) what the distance effect is on the fungal association. The environmental variables were statistically significant only for Xylaria, i.e., when the elevation and water precipitation increase and the temperature decreases, the odds ratio of finding another fungal endophyte of that genus previously found increases. Sordariomycetes, Xylariales, and Xylaria exhibited leaf fragment preference to petiole and tip. Fungal endophytes showed association within leaf. The horizontal transmission mode and the dispersal limitation may explain this association at the leaf level. Moreover, our results suggest that when a fungal endophyte infects a leaf or host tree individual, the odds ratio of dispersal inside them is greater.     

Missing Exposure Data in Stereotype Regression Model: Application to Matched Case–Control Study with Disease Subclassification

Summary With advances in modern medicine and clinical diagnosis, case–control data with characterization of finer subtypes of cases are often available. In matched case–control studies, missingness in exposure values often leads to deletion of entire stratum, and thus entails a significant loss in information. When subtypes of cases are treated as categorical outcomes, the data are further stratified and deletion of observations becomes even more expensive in terms of precision of the category‐specific odds‐ratio parameters, especially using the multinomial logit model. The stereotype regression model for categorical responses lies intermediate between the proportional odds and the multinomial or baseline category logit model. The use of this class of models has been limited as the structure of the model implies certain inferential challenges with nonidentifiability and nonlinearity in the parameters. We illustrate how to handle missing data in matched case–control studies with finer disease subclassification within the cases under a stereotype regression model. We present both Monte Carlo based full Bayesian approach and expectation/conditional maximization algorithm for the estimation of model parameters in the presence of a completely general missingness mechanism. We illustrate our methods by using data from an ongoing matched case–control study of colorectal cancer. Simulation results are presented under various missing data mechanisms and departures from modeling assumptions.