The Binary Regression Quantile Plot: Assessing the Importance of Predictors in Binary Regression Visually

We present a graphical measure of assessing the explanatory power of regression models with a binary response. The binary regression quantile plot and an area defined by it are used for the visual comparison and ordering of nested binary response regression models. The plot shows how well various models explain the data. Two data sets are analyzed and the area representing the fit of a model is shown to agree with the usual likelihood ratio test.     

Designs for Discrimination Between Bivariate Binary Response Models

We propose a test statistic for discrimination between alternative bivariate binary response models and the optimal design procedure which is an extension of T-optimality. Under certain conditions we prove that the maximum value of the power can be-obtained when the degrees of freedom of the test statistic is one. The conclusion is the same as that in discrimination between alternative univariate separate models. However the test statistics are different.     

Tests for Order Restrictions in Binary Data

In this article, a general procedure is presented for testing for equality of k independent binary response probabilities against any given ordered alternative. The proposed methodology is based on an estimation procedure developed in Hwang and Peddada (1994, Annals of Statistics 22, 67-93) and can be used for a very broad class of order restrictions. The procedure is illustrated through application to two data sets that correspond to three commonly encountered order restrictions: simple tree order, simple order, and down turn order.     

Tests for Monotonic Trend from Case‐Control Data: Cochran‐Armitage‐Mantel Trend Test,Isotonic Regression and Single and Multiple Contrast Tests

Tests for a monotonic trend between an ordered categorical exposure and disease status are routinely carried out from case‐control data using the Mantel‐extension trend test or the asymptotically equivalent Cochran‐Armitage test. In this study, we considered two alternative tests based on isotonic regression, namely an order‐restricted likelihood ratio test and an isotonic modification of the Mantel‐extension test extending the recent proposal by Mancuso, Ahn and Chen (2001) to case‐control data. Furthermore, we considered three tests based on contrasts, namely a single contrast (SC) test based on Schaafsma's coefficients, the Dosemeci and Benichou (DB) test, a multiple contrast (MC) test based on the Helmert, reverse‐Helmert and linear contrasts and we derived their case‐control versions. Using simulations, we compared the statistical properties of these five alternative tests to those of the Mantel‐extension test under various patterns including no relationship, as well as monotonic and non‐monotonic relationships between exposure and disease status. In the case of no relationship, all tests had close to nominal type I error except in situations combining a very unbalanced exposure distribution and small sample size, where the asymptotic versions of the three tests based on contrasts were highly anticonservative. The use of bootstrap instead of asymptotic versions corrected this anticonservatism. For monotonic patterns, all tests had close powers. For non monotonic patterns, the DB‐test showed the most favourable results as it was the least powerful test. The two tests based on isotonic regression were the most powerful tests and the Mantel‐extension test, the SC‐ and MC‐tests had in‐between powers. The six tests were applied to data from a case‐control study investigating the relationship between alcohol consumption and risk of laryngeal cancer in Turkey. In situations with no evidence of a monotonic relationship between exposure and disease status, the three tests based on contrasts did not conclude in favour of a significant trend whereas all the other tests did. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Modification of Stone's Test for Trend for Binary Outcome

Stone (1988) suggested the “first isotonic regression estimator” as a tool for drawing inferences on possibly increased cancer case counts among several subregions around a putative source. He assumed the case counts to be Poisson distributed and therefore introduced a rare disease assumption into his approach. However, when analyzing cross sectional data one would rather refer to prevalence estimates among these subregions around a point risk source (for example the origin of chemical fallout). Therefore we applied antitonic regression estimation in Binomial distributions to derive a test statistic and a p value to test for a possible trend in the observed prevalence data around the putative source. The computation of this p value will be illustrated as well as severe difficulties concerning its interpretation. Further the Maximum Likelihood Ratio approach will be used to derive an alternative test statistic.     

Measures of Stability (Trait) and Mutability (State) of Binary Responses in Pre-post-treatment Fourfold Tables

For the discussion of binary responses in pre-post-treatment we define new, simple measures for the stability (Trait) and mutability (State) of the responses, which have very nice and simple properties. We show the important fact that under certain conditions these two measures are uncorrelated. Because of the simplicity of the measures we derive the exact and limit distributions for testing the one respectively the two sample case.     

A Shift Algorithm for Exact Computation in Sequential Tests with Binary Outcome

A simple shift algorithm is described enabling the exact determination of power functions and sample size distributions for a large variety of closed sequential two‐sample designs with a binary outcome variable. The test statistics are assumed to be based on relative frequencies of successes or failures, but the number of interim analyses, the monitoring times, and the continuation regions may be specified as desired. To give examples, exact properties of designs proposed by the program package EaSt (Cytel , 1992) are determined, and plans with interim analyses are considered where decisions are based on the conditional power given the observations obtained so far.     

More Powerful Likelihood Ratio Tests for Isotonic Binomial Proportions

Binomial group testing involves pooling individuals into groups and observing a binary response on each group. Results from the group tests can then be used to draw inference about population proportions. Its use as an experimental design has received much attention in recent years, especially in public‐health screening experiments and vector‐transfer designs in plant pathology. We investigate the benefits of group testing in situations wherein one desires to test whether or not probabilities are increasingly ordered across the levels of an observed qualitative covariate, i.e., across strata of a population or among treatment levels. We use a known likelihood ratio test for individual testing, but extend its use to group‐testing situations to show the increases in power conferred by using group testing when operating in this constrained parameter space. We apply our methods to data from an HIV study involving male subjects classified as intraveneous drug users.     

Binary regression: Total gain in positive and negative predictive values

Models that predict disease incidence or disease recurrence are attractive for clinicians as well as for patients. The usefulness of a risk prediction model is linked to the two questions whether the observed outcome is confirmed by the prediction and whether the risk prediction is accurate in predicting the future outcome, respectively. The first phrasing of the question is linked to considering sensitivity and specificity and the latter to the positive and negative predictive values. We present the measures of standardized total gain in positive and negative predictive values dealing with the performance or accuracy of the prediction model for a binary outcome. Both measures provide a useful tool for assessing the performance or accuracy of a set of predictor variables for the prediction of a binary outcome. This concept is a tool for evaluating the optimal prediction model in future research.     

Treatment Response Predicted from Polarity Patterns in Depressives by Complementary Prediction CFA

A sample of N = 32 depressive patients was observed as to their polarity sequences defined as 4 binary independent variables. After Lithium treatment the dependent variable was a yes-no improvement rating. Nonparametric evaluation of the multivariate regression was made by complementary version of predictive configural frequency analysis. It was confirmed that the ocurrence of manic-depressive episodes prior to treatment was favor of improvement, as hypothesized by the clinician. A construction of exact tests is suggested.     

A Generalized Logistic Model for Quantal Response Bioassay

In bioassay, where different levels of the stimulus may represent different doses of a drug, the binary response is the death or survival of an individual receiving a specified dose. In such applications, it is common to model the probability of a positive response P at the stimulus level x by P = F(x′β), where F is a cumulative distribution function and β is a vector of unknown parameters which characterize the response function. The two most popular models used for modelling binary response bioassay involve the probit model [BLISS (1935), FINNEY (1978)], and the logistic model [BERKSON (1944), BROWN (1982)]. However, these models have some limitations. The use of the probit model involves the inverse of the standard normal distribution function, making it rather intractable. The logistic model has a simple form and a closed expression for the inverse distribution function, however, neither the logistic nor the probit can provide a good fit to response functions which are not symmetric or are symmetric but have a steeper or gentler incline in the central probability region. In this paper we introduce a more realistic model for the analysis of quantal response bioassay. The proposed model, which we refer to it as the generalized logistic model, is a family of response curves indexed by shape parameters m₁ and m₂. This family is rich enough to include the probit and logistic models as well as many others as special cases or limiting distributions. In particular, we consider the generalized logistic three parameter model where we assume that m₁ = m, m is a positive real number, and m₂ = 1. We apply this model to various sets of data, comparing the fit results to those obtained previously by other dose-response curves such as the logistic and probit, and showing that the fit can be improved by using the generalized logistic.     

Logistic and Nonlogistic Density Functions in Binary Regression with Nonstochastic Covariates

A binary random variable depends on nonstochastic covariates through a density function. The equations that determine the maximum likelihood estimators of the parameters are intractable and difficult to solve iteratively. We develop modified maximum likelihood estimators for both logistic and nonlo-gistic densities. These estimators are explicit functions of sample observations and are, therefore, easy to compute. They are asymptotically fully efficient and, for small samples, are almost fully efficient. The appropriateness of the logistic density function is also discussed.     

Longitudinal Studies of Binary Response Data Following Case–Control and Stratified Case–Control Sampling: Design and Analysis

Summary We discuss design and analysis of longitudinal studies after case–control sampling, wherein interest is in the relationship between a longitudinal binary response that is related to the sampling (case–control) variable, and a set of covariates. We propose a semiparametric modeling framework based on a marginal longitudinal binary response model and an ancillary model for subjects' case–control status. In this approach, the analyst must posit the population prevalence of being a case, which is then used to compute an offset term in the ancillary model. Parameter estimates from this model are used to compute offsets for the longitudinal response model. Examining the impact of population prevalence and ancillary model misspecification, we show that time‐invariant covariate parameter estimates, other than the intercept, are reasonably robust, but intercept and time‐varying covariate parameter estimates can be sensitive to such misspecification. We study design and analysis issues impacting study efficiency, namely: choice of sampling variable and the strength of its relationship to the response, sample stratification, choice of working covariance weighting, and degree of flexibility of the ancillary model. The research is motivated by a longitudinal study following case–control sampling of the time course of attention deficit hyperactivity disorder (ADHD) symptoms.     

Binary Responses and the Three-period Cross-over

Problems with carry-over effects in the simple two-period cross-over have lead to interest in more complex cross-over designs. A method for analysing the optimum two-treatment three-period design with binary response variables is given by making a simple extension to Gart's logistic model. The method gives independent tests for, and estimates of the difference in treatment and first-order carry-over effects. An example of the analysis is given, using the loglinear models facility in GLIM.     

Nonparametric Tests for Trend: Jonckheere's Test,a Modification and a Maximum Test

Jonckheere's test is a frequently used nonparametric trend test for the evaluation of preclinical studies and clinical dose-finding trials. In this paper, a modification of Jonckheere's test is proposed. If the exact permutation distribution is used for inference, the modified test can fill out the level of the type I error in a much more complete way and is substantially more powerful than the common Jonckheere test. If the asymptotic normality is used for inference, the modified test is slightly more powerful. In addition, a maximum test is investigated which is more robust concerning an a priori unknown dose-response shape. The robustness is advantageous, especially in a closed testing procedure. The different tests are applied to two example data sets.     

Choice of Scores in Trend Tests for Case‐Control Studies of Candidate‐Gene Associations

When applying the Cochran‐Armitage (CA) trend test for an association between a candidate allele and a disease in a case‐control study, a set of scores must be assigned to the genotypes. Sasieni (1997, Biometrics 53 , 1253–1261) suggested scores for the recessive, additive, and dominant models but did not examine their statistical properties. Using the criteria of minimizing the required sample size of the CA trend test to achieve prespecified type I and type II errors, we show that the scores given by Sasieni (1997) are optimal for the recessive and dominant models and locally optimal for the additive one. Moreover, the additive scores are shown to be locally optimal for the multiplicative model. The tests are applied to a real dataset.     

On Exact Unconditional Test for Linear Trend in Dose‐Response Studies

The one‐degree‐of‐freedom Cochran‐Armitage (C‐A) test statistic for linear trend has been widely applied in various dose‐response studies (e.g., anti‐ulcer medications and short‐term antibiotics, animal carcinogenicity bioassays and occupational toxicant studies). This approximate statistic relies, however, on asymptotic theory that is reliable only when the sample sizes are reasonably large and well balanced across dose levels. For small, sparse, or skewed data, the asymptotic theory is suspect and exact conditional method (based on the C‐A statistic) seems to provide a dependable alternative. Unfortunately, the exact conditional method is only practical for the linear logistic model from which the sufficient statistics for the regression coefficients can be obtained explicitly. In this article, a simple and efficient recursive polynomial multiplication algorithm for exact unconditional test (based on the C‐A statistic) for detecting a linear trend in proportions is derived. The method is applicable for all choices of the model with monotone trend including logistic, probit, arcsine, extreme value and one hit. We also show that this algorithm can be easily extended to exact unconditional power calculation for studies with up to a moderately large sample size. A real example is given to illustrate the applicability of the proposed method.