The Mantel-Haenszel Procedure Revisited: Models and Generalizations

Several statistical methods have been developed for adjusting the Odds Ratio of the relation between two dichotomous variables X and Y for some confounders Z. With the exception of the Mantel-Haenszel method, commonly used methods, notably binary logistic regression, are not symmetrical in X and Y. The classical Mantel-Haenszel method however only works for confounders with a limited number of discrete strata, which limits its utility, and appears to have no basis in statistical models. Here we revisit the Mantel-Haenszel method and propose an extension to continuous and vector valued Z. The idea is to replace the observed cell entries in strata of the Mantel-Haenszel procedure by subject specific classification probabilities for the four possible values of (X,Y) predicted by a suitable statistical model. For situations where X and Y can be treated symmetrically we propose and explore the multinomial logistic model. Under the homogeneity hypothesis, which states that the odds ratio does not depend on Z, the logarithm of the odds ratio estimator can be expressed as a simple linear combination of three parameters of this model. Methods for testing the homogeneity hypothesis are proposed. The relationship between this method and binary logistic regression is explored. A numerical example using survey data is presented.     

Random-effects models for serial observations with binary response

This paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants. Each subject's serial responses are assumed to arise from a logistic model, but with regression coefficients that vary between subjects. The logistic regression parameters are assumed to be normally distributed in the population. Inference is based upon maximum likelihood estimation of fixed effects and variance components, and empirical Bayes estimation of random effects. Exact solutions are analytically and computationally infeasible, but an approximation based on the mode of the posterior distribution of the random parameters is proposed, and is implemented by means of the EM algorithm. This approximate method is compared with a simpler two-step method proposed by Korn and Whittemore (1979, Biometrics 35, 795-804), using data from a panel study of asthmatics originally described in that paper. One advantage of the estimation strategy described here is the ability to use all of the data, including that from subjects with insufficient data to permit fitting of a separate logistic regression model, as required by the Korn and Whittemore method. However, the new method is computationally intensive.     

Bayesian dose-finding in phase I/II clinical trials using toxicity and efficacy odds ratios

A Bayesian adaptive design is proposed for dose-finding in phase I/II clinical trials to incorporate the bivariate outcomes, toxicity and efficacy, of a new treatment. Without specifying any parametric functional form for the drug dose-response curve, we jointly model the bivariate binary data to account for the correlation between toxicity and efficacy. After observing all the responses of each cohort of patients, the dosage for the next cohort is escalated, deescalated, or unchanged according to the proposed odds ratio criteria constructed from the posterior toxicity and efficacy probabilities. A novel class of prior distributions is proposed through logit transformations which implicitly imposes a monotonic constraint on dose toxicity probabilities and correlates the probabilities of the bivariate outcomes. We conduct simulation studies to evaluate the operating characteristics of the proposed method. Under various scenarios, the new Bayesian design based on the toxicity-efficacy odds ratio trade-offs exhibits good properties and treats most patients at the desirable dose levels. The method is illustrated with a real trial design for a breast medical oncology study.     

Logistic regression for dependent binary observations

The likelihood of a set of binary dependent outcomes, with or without explanatory variables, is expressed as a product of conditional probabilities each of which is assumed to be logistic. The models are called regressive logistic models. They provide a simple but relatively unknown parametrization of the multivariate distribution. They have the theoretical and practical advantage that they can be analyzed and fitted as in logistic regression for independent outcomes, and with the same computer programs. The paper is largely expository and is intended to motivate the development and usage of the regressive logistic models. The discussion includes serially dependent outcomes, equally predictive outcomes, more specialized patterns of dependence, multidimensional tables, and three examples.     

Estimating haplotype effects on dichotomous outcome for unphased genotype data using a weighted penalized log-likelihood approach

OBJECTIVE: To develop a method to estimate haplotype effects on dichotomous outcomes when phase is unknown, that can also estimate reliable effects of rare haplotypes. METHODS: In short, the method uses a logistic regression approach, with weights attached to all possible haplotype combinations of an individual. An EM-algorithm was used: in the E-step the weights are estimated, and the M-step consists of maximizing the joint log-likelihood. When rare haplotypes were present, a penalty function was introduced. We compared four different penalties. To investigate statistical properties of our method, we performed a simulation study for different scenarios. The evaluation criteria are the mean bias of the parameter estimates, the root of the mean squared error, the coverage probability, power, Type I error rate and the false discovery rate. RESULTS: For the unpenalized approach, mean bias was small, coverage probabilities were approximately 95%, power ranged from 15.2 to 44.7% depending on haplotype frequency, and Type I error rate was around 5%. All penalty functions reduced the standard errors of the rare haplotypes, but introduced bias. This trade-off decreased power. CONCLUSION: The unpenalized weighted log-likelihood approach performs well. A penalty function can help to estimate an effect for rare haplotypes.     

An Application of Logistic Models to the Analysis of Ordinal Responses

Logistic probability models—models linear in the log odds of the outcome event—have found extensive application in modelling of unordered categorical responses. This paper illustrates some extensions of logistic models to the modelling of probabilities of ordinal responses. The extensions arise naturally from discrete probability models for the conditional distribution of the ordinal response, as well as from linear modelling of the log odds of response. Methods of estimation and examination of fit developed for the binary logistic model extend in a straightforward manner to the ordinal models. The models and methods are illustrated in an analysis of the dependence of chronic obstructive respiratory disease prevalence on smoking and age.     

LOGISTIC REGRESSION FOR EMPIRICAL STUDIES OF MULTIVARIATE SELECTION

Understanding the mechanics of adaptive evolution requires not only knowing the quantitative genetic bases of the traits of interest but also obtaining accurate measures of the strengths and modes of selection acting on these traits. Most recent empirical studies of multivariate selection have employed multiple linear regression to obtain estimates of the strength of selection. We reconsider the motivation for this approach, paying special attention to the effects of nonnormal traits and fitness measures. We apply an alternative statistical method, logistic regression, to estimate the strength of selection on multiple phenotypic traits. First, we argue that the logistic regression model is more suitable than linear regression for analyzing data from selection studies with dichotomous fitness outcomes. Subsequently, we show that estimates of selection obtained from the logistic regression analyses can be transformed easily to values that directly plug into equations describing adaptive microevolutionary change. Finally, we apply this methodology to two published datasets to demonstrate its utility. Because most statistical packages now provide options to conduct logistic regression analyses, we suggest that this approach should be widely adopted as an analytical tool for empirical studies of multivariate selection.     

Association Models for Clustered Data with Binary and Continuous Responses

Summary .  We consider analysis of clustered data with mixed bivariate responses, i.e., where each member of the cluster has a binary and a continuous outcome. We propose a new bivariate random effects model that induces associations among the binary outcomes within a cluster, among the continuous outcomes within a cluster, between a binary outcome and a continuous outcome from different subjects within a cluster, as well as the direct association between the binary and continuous outcomes within the same subject. For the ease of interpretations of the regression effects, the marginal model of the binary response probability integrated over the random effects preserves the logistic form and the marginal expectation of the continuous response preserves the linear form. We implement maximum likelihood estimation of our model parameters using standard software such as PROC NLMIXED of SAS . Our simulation study demonstrates the robustness of our method with respect to the misspecification of the regression model as well as the random effects model. We illustrate our methodology by analyzing a developmental toxicity study of ethylene glycol in mice.     

Markov models for covariate dependence of binary sequences

Suppose that a heterogeneous group of individuals is followed over time and that each individual can be in state 0 or state 1 at each time point. The sequence of states is assumed to follow a binary Markov chain. In this paper we model the transition probabilities for the 0 to 0 and 1 to 0 transitions by two logistic regressions, thus showing how the covariates relate to changes in state. With p covariates, there are 2(p + 1) parameters including intercepts, which we estimate by maximum likelihood. We show how to use transition probability estimates to test hypotheses about the probability of occupying state 0 at time i (i = 2, ..., T) and the equilibrium probability of state 0. These probabilities depend on the covariates. A recursive algorithm is suggested to estimate regression coefficients when some responses are missing. Extensions of the basic model which allow time-dependent covariates and nonstationary or second-order Markov chains are presented. An example shows the model applied to a study of the psychological impact of breast cancer in which women did or did not manifest distress at four time points in the year following surgery.     

Use of Polytomous Logistic Model in Matched Case-Control Studies

Binary logistic model has been found useful for estimating odds ratio in case of dichotomous exposure variable under matched case-control retrospective design. We describe the use of polytomous logistic model for estimating odds ratios when the exposure of prime interests, relative to disease incidence, has more than two levels. An illustrative example is presented and discussed.     

Logistic regression in capture-recapture models

The effect of population heterogeneity in capture-recapture, or dual registration, models is discussed. An estimator of the unknown population size based on a logistic regression model is introduced. The model allows different capture probabilities across individuals and across capture times. The probabilities are estimated from the observed data using conditional maximum likelihood. The resulting population estimator is shown to be consistent and asymptotically normal. A variance estimator under population heterogeneity is derived. The finite-sample properties of the estimators are studied via simulation. An application to Finnish occupational disease registration data is presented.     

Misclassification in Logistic Regression with Discrete Covariates

We study the effect of misclassification of a binary covariate on the parameters of a logistic regression model. In particular we consider 2 × 2 × 2 tables. We assume that a binary covariate is subject to misclassification that may depend on the observed outcome. This type of misclassification is known as (outcome dependent) differential misclassification. We examine the resulting asymptotic bias on the parameters of the model and derive formulas for the biases and their approximations as a function of the odds and misclassification probabilities. Conditions for unbiased estimation are also discussed. The implications are illustrated numerically using a case control study. For completeness we briefly examine the effect of covariate dependent misclassification of exposures and of outcomes.     

Measurement of interrater agreement with adjustment for covariates

The kappa coefficient measures chance-corrected agreement between two observers in the dichotomous classification of subjects. The marginal probability of classification by each rater may depend on one or more confounding variables, however. Failure to account for these confounders may lead to inflated estimates of agreement. A multinomial model is used that assumes both raters have the same marginal probability of classification, but this probability may depend on one or more covariates. The model may be fit using software for conditional logistic regression. Additionally, likelihood-based confidence intervals for the parameter representing agreement may be computed. A simple example is discussed to illustrate model-fitting and application of the technique.     

GOLDmineR: improving models for classifying patients with chest pain

The laboratory is dealing with reporting tests as information needed to make clinical decisions. The traditional statistical quality control measures which assigns reference ranges based on 95 percent confidence intervals is insufficient for diagnostic tests that assign risk. We construct a basis for risk assignment by a method that builds on the 2 x 2 contingency table used to calculate the C2 goodness-of-fit and Bayesian estimates. The widely used logistic regression is a subset of the regression method, as it only considers dichotomous outcome choices. We use examples of multivalued predictor(s) and a multivalued as well as dichotomous outcome. Outcomes analyses are quite easy using the ordinal logit regression model.     

Climate Change: Believing and Seeing Implies Adapting

Knowledge of factors that trigger human response to climate change is crucial for effective climate change policy communication. Climate change has been claimed to have low salience as a risk issue because it cannot be directly experienced. Still, personal factors such as strength of belief in local effects of climate change have been shown to correlate strongly with responses to climate change and there is a growing literature on the hypothesis that personal experience of climate change (and/or its effects) explains responses to climate change. Here we provide, using survey data from 845 private forest owners operating in a wide range of bio-climatic as well as economic-social-political structures in a latitudinal gradient across Europe, the first evidence that the personal strength of belief and perception of local effects of climate change, highly significantly explain human responses to climate change. A logistic regression model was fitted to the two variables, estimating expected probabilities ranging from 0.07 (SD ±0.01) to 0.81 (SD ±0.03) for self-reported adaptive measures taken. Adding socio-demographic variables improved the fit, estimating expected probabilities ranging from 0.022 (SD ±0.008) to 0.91 (SD ±0.02). We conclude that to explain and predict adaptation to climate change, the combination of personal experience and belief must be considered.     

Efficient adaptively weighted analysis of secondary phenotypes in case-control genome-wide association studies

We propose and compare methods of analysis for detecting associations between genotypes of a single nucleotide polymorphism (SNP) and a dichotomous secondary phenotype (X), when the data arise from a case-control study of a primary dichotomous phenotype (D), which is not rare. We considered both a dichotomous genotype (G) as in recessive or dominant models and an additive genetic model based on the number of minor alleles present. To estimate the log odds ratio β(1) relating X to G in the general population, one needs to understand the conditional distribution [D ∣ X, G] in the general population. For the most general model, [D ∣ X, G], one needs external data on P(D = 1) to estimate β(1). We show that for this 'full model', the maximum likelihood (FM) corresponds to a previously proposed weighted logistic regression (WL) approach if G is dichotomous. For the additive model, WL yields results numerically close, but not identical, to those of the maximum likelihood FM. Efficiency can be gained by assuming that [D ∣ X, G] is a logistic model with no interaction between X and G (the 'reduced model'). However, the resulting maximum likelihood (RM) can be misleading in the presence of interactions. We therefore propose an adaptively weighted approach (AW) that captures the efficiency of RM but is robust to the occasional SNP that might interact with the secondary phenotype to affect the risk of the primary disease. We study the robustness of FM, WL, RM and AW to misspecification of P(D = 1). In principle, one should be able to estimate β(1) without external information on P(D = 1) under the reduced model. However, our simulations show that the resulting inference is unreliable. Therefore, in practice one needs to introduce external information on P(D = 1), even in the absence of interactions between X and G.     

The logistic transform for bounded outcome scores

The logistic transformation, originally suggested by Johnson (1949), is applied to analyze responses that are restricted to a finite interval (e.g. (0,1)), so-called bounded outcome scores. Bounded outcome scores often have a non-standard distribution, e.g. J- or U-shaped, precluding classical parametric statistical approaches for analysis. Applying the logistic transformation on a normally distributed random variable, gives rise to a logit-normal (LN) distribution. This distribution can take a variety of shapes on (0,1). Further, the model can be extended to correct for (baseline) covariates. Therefore, the method could be useful for comparative clinical trials. Bounded outcomes can be found in many research areas, e.g. drug compliance research, quality-of-life studies, and pain (and pain relief) studies using visual analog scores, but all these scores can attain the boundary values 0 or 1. A natural extension of the above approach is therefore to assume a latent score on 0,1) having a LN distribution. Two cases are considered: (a) the bounded outcome score is a proportion where the true probabilities have a LN distribution on (0,1) and (b) the bounded outcome score on [0,1] is a coarsened version of a latent score with a LN distribution on (0,1). We also allow the variance (on the transformed scale) to depend on treatment. The usefulness of our approach for comparative clinical trials will be assessed in this paper. It turns out to be important to distinguish the case of equal and unequal variances. For a bounded outcome score of the second type and with equal variances, our approach comes close to ordinal probit (OP) regression. However, ignoring the inequality of variances can lead to highly biased parameter estimates. A simulation study compares the performance of our approach with the two-sample Wilcoxon test and with OP regression. Finally, the different methods are illustrated on two data sets.     

Diagnostic testing revisited: pathways through uncertainty

To aid physicians who may be having difficulty applying the principles of decision analysis to diagnostic data according to the methods published in the past several years, the authors of this paper set out a few principles and schemes for using and interpreting diagnostic data obtained from dichotomous tests. They also present a simple BASIC program for calculating post-test probabilities from likelihood ratios and pretest probabilities that a particular disease is present in a particular patient; the program can be adapted for use on microcomputers.     

Generalized Walker-Duncan Procedure for Obtaining Maximum Likelihood Estimates of Parameters of Logistic Discriminant Function

A recursive method of obtaining the maximum likelihood estimates of the parameters of the quadratic logistic discriminant function is presented. This method is an extension of the Walker and Duncan procedure (1967) proposed for the linear logistic discriminant function in a dichotomous case. A generalization of the method to the problem of discrimination between several populations is also given in the paper. It works for both linear and quadratic logistic discriminant function. After an estimation of the parameters of the logistic function a classification can be performed. An example of application of the method to automatic diagnosis of some respiratory diseases is presented. Comparison with the standard procedures used for the estimation is done by a short simulation study.     

Point and interval estimation of baseline risk and treatment effect based on logistic model for observational studies

In observational studies with dichotomous outcome of a population, researchers usually report treatment effect alone, although both baseline risk and treatment effect are needed to evaluate the significance of the treatment effect to the population. In this article, we study point and interval estimates including confidence region of baseline risk and treatment effect based on logistic model, where baseline risk is the risk of outcome of the population under control treatment while treatment effect is measured by the risk difference between outcomes of the population under active versus control treatments. Using approximate normal distribution of the maximum‐likelihood (ML) estimate of the model parameters, we obtain an approximate joint distribution of the ML estimate of the baseline risk and the treatment effect. Using the approximate joint distribution, we obtain point estimate and confidence region of the baseline risk and the treatment effect as well as point estimate and confidence interval of the treatment effect when the ML estimate of the baseline risk falls into specified range. These interval estimates reflect nonnormality of the joint distribution of the ML estimate of the baseline risk and the treatment effect. The method can be easily implemented by using any software that generates normal distribution. The method can also be used to obtain point and interval estimates of baseline risk and any other measure of treatment effect such as risk ratio and the number needed to treat. The method can also be extended from logistic model to other models such as log‐linear model.