A simple test of association for contingency tables with multiple column responses

Loughin and Scherer (1998, Biometrics 54, 630-637) investigated tests of association in two-way tables when one of the categorical variables allows for multiple-category responses from individual respondents. Standard chi-squared tests are invalid in this case, and they developed a bootstrap test procedure that provides good control of test levels under the null hypothesis. This procedure and some others that have been proposed are computationally involved and are based on techniques that are relatively unfamiliar to many practitioners. In this paper, the methods introduced by Rao and Scott (1981, Journal of the American Statistical Association 76, 221-230) for analyzing complex survey data are used to develop a simple test based on a corrected chi-squared statistic.     

Regression analysis of doubly censored failure time data using the additive hazards model

Doubly censored failure time data arise when the survival time of interest is the elapsed time between two related events and observations on occurrences of both events could be censored. Regression analysis of doubly censored data has recently attracted considerable attention and for this a few methods have been proposed (Kim et al., 1993, Biometrics 49, 13-22; Sun et al., 1999, Biometrics 55, 909-914; Pan, 2001, Biometrics 57, 1245-1250). However, all of the methods are based on the proportional hazards model and it is well known that the proportional hazards model may not fit failure time data well sometimes. This article investigates regression analysis of such data using the additive hazards model and an estimating equation approach is proposed for inference about regression parameters of interest. The proposed method can be easily implemented and the properties of the proposed estimates of regression parameters are established. The method is applied to a set of doubly censored data from an AIDS cohort study.     

A note on one-sided tests with multiple endpoints

Testing problems with multivariate one-sided alternative hypotheses are common in clinical trials with multiple endpoints. In the case of comparing two treatments, treatment 1 is often preferred if it is superior for at least one of the endpoints and not biologically inferior for the remaining endpoints. Bloch et al. (2001, Biometrics57, 1039-1047) propose an intersection-union test (IUT) for this testing problem, but their test does not utilize the appropriate multivariate one-sided test. In this note we modify their test by an alternative IUT that does utilize the appropriate one-sided test. Empirical and graphical evidence show that the proposed test is more appropriate for this testing problem.     

Analysis of ordered categorical data: two score-independent approaches

SUMMARY: A trend test is often employed to analyze ordered categorical data, in which a set of increasing scores is assigned a priori. There is a drawback in this approach, because how to choose a set of scores is not clear. There have been debates on which scores should be used (e.g., Graubard and Korn, 1987, Biometrics 43, 471-476; Ivanova and Berger, 2001, Biometrics 57, 567-570; Senn, 2007, Biometrics 63, 296-298). Conflicting conclusions are often obtained with different sets of scores. Two approaches, which have been applied to genetic case-control studies, are appealing for ordered categorical data, because they take into account the natural order in the data, are score independent, and not contingent on asymptotic theory. These two approaches are applied to a prospective study for detecting association between maternal drinking and congenital malformations.     

Estimators and tests in the analysis of multiple nonindependent 2 x 2 tables with partially missing observations

We present methods for the analysis of a K-variate binary measure for two independent groups where some observations may be incomplete, as in the case of K repeated measures in a comparative trial. For the K 2 X 2 tables, let theta = (theta 1,..., theta K) be a vector of association parameters where theta k is a measure of association that is a continuous function of the probabilities pi ik in each group (i = 1, 2; k = 1,..., K), such as the log odds ratio or log relative risk. The asymptotic distribution of the estimates theta = (theta 1,..., theta K) is derived. Under the assumption that theta k = theta for all k, we describe the maximally efficient linear estimator theta of the common parameter theta. Tests of contrasts on the theta are presented which provide a test of homogeneity Ha: theta k = theta l for all k not equal to l. We then present maximally efficient tests of aggregate association Hb: theta = theta 0, where theta 0 is a given value. It is shown that the test of aggregate association Hb is asymptotically independent of the preliminary test of homogeneity Ha. These methods generalize the efficient estimators of Gart (1962, Biometrics 18, 601-610), and the Cochran (1954, Biometrics 10, 417-451), Mantel-Haenszel (1959, Journal of the National Cancer Institute 22, 719-748), and Radhakrishna (1965, Biometrics 21, 86-98) tests to nonindependent tables. The methods are illustrated with an analysis of repeated morphologic evaluations of liver biopsies obtained in the National Cooperative Gallstone Study.     

On the use of historical control data for trend test in carcinogenicity studies

Historical control data are often available in carcinogenicity studies and are included for testing dose effects in current studies. A new method is developed for incorporating the historical control information into a dose effect test. The method generalizes the test procedures proposed by Tarone (1982, Biometrics 38, 215-220) and Ibrahim and Ryan (1996, Biometrics 52, 1478-1485) by taking into account the variation resulting from parameter estimation based on historical data. Two examples are discussed for illustrating the proposed method.     

Sample size for testing differences in proportions for the paired-sample design

Miettinen (1968, Biometrics 24, 339-352) presented an approximation for power and sample size for testing the differences between proportions in the matched-pair case. Duffy (1984, Biometrics 40, 1005-1015) gave the exact power for this case and showed that Miettinen's approximation tends to slightly overestimate the power or underestimate the sample size necessary for the design power. A simple alternative approximation that is more conservative is presented here. In many cases, the sample size for the independent-sample case provides a conservative approximation for the matched-pair design.     

Exact analysis of dose response for multiple correlated binary outcomes

The neurotoxicity of a substance is often tested using animal bioassays. In the functional observational battery, animals are exposed to a test agent and multiple outcomes are recorded to assess toxicity, using approximately 40 animals measured on up to 30 different items. This design gives rise to a challenging statistical problem: a large number of outcomes for a small sample of subjects. We propose an exact test for multiple binary outcomes, under the assumption that the correlation among these items is equal. This test is based upon an exponential model described by Molenberghs and Ryan (1999, Environmetrics 10, 279-300) and extends the methods developed by Corcoran et al. (2001, Biometrics 57, 941-948) who developed an exact test for exchangeably correlated binary data for groups (clusters) of correlated observations. We present a method that computes an exact p-value testing for a joint dose-response relationship. An estimate of the parameter for dose response is also determined along with its 95% confidence bound. The method is illustrated using data from a neurotoxicity bioassay for the chemical perchlorethylene.     

Sample size determination for case-control studies and the comparison of stratified and unstratified analyses.

Woolson, Bean, and Rojas (1986, Biometrics 42, 927-932) present a simple approximation of sample size for Cochran's (1954, Biometrics 10, 417-451) test for detecting association between exposure and disease. It is useful in the design of case-control studies. We derive a sample size formula for Cochran's statistic with continuity correction which guarantees that the actual Type I error rate of the test does not exceed the nominal level. The corrected sample size is necessarily larger than the uncorrected one given by Woolson et al. and the relative difference between the two sample sizes is considerable. Allocation of equal number of cases and controls within each stratum is asymptotically optimal when the costs per case and control are the same. When any effect of stratification is absent, Cochran's stratified test, although valid, is less efficient than the unstratified one except for the important case of a balanced design.     

Analysis of two-period crossover design in a multicenter clinical trial

A technique is discussed for analyzing a two-period crossover design for a multicenter trial using identical study protocols. The technique is a modification of the analysis originally proposed by Grizzle (1965, Biometrics 21, 467-480; 1974, Biometrics 30, 727) for analyzing a two-period crossover design when study is not a factor. A mixed model using the first baseline as a covariate is analyzed to increase the power of the test of significance of the treatment-by-period interaction. The baseline values are also used in a preliminary test.     

Sample size determination for testing whether an identified treatment is best

Laska and Meisner (1989, Biometrics 45, 1139-1151) dealt with the problem of testing whether an identified treatment belonging to a set of k + 1 treatments is better than each of the other k treatments. They calculated sample size tables for k = 2 when using multiple t-tests or Wilcoxon-Mann-Whitney tests, both under normality assumptions. In this paper, we provide sample size formulas as well as tables for sample size determination for k > or = 2 when t-tests under normality or Wilcoxon-Mann-Whitney tests under general distribution assumptions are used.     

Approximate interval estimation of the difference in binomial parameters: correction for skewness and extension to multiple tables

Recently, Beal (1987, Biometrics 43, 941-950) found Mee's modification of Anbar's approximate interval estimation for the difference in binomial parameters to be a good choice in small sample sizes. As this method can be derived from the score theory of Bartlett, it is easily corrected for skewness. Exact numerical evaluation shows that this correction is not as important for this case as for the ratio of binomial parameters (Gart and Nam, 1988, Biometrics 44, 323-338). The score theory is also used to extend this method to the stratified or multiple-table case. Thus, good approximate interval estimates for differences, ratios, and odds ratios of binomial parameters can all be derived from the same general theory.     

Some goodness-of-fit methods for the Poisson plus added zeros distribution

Methods for making inferences about the Poisson plus added zeros distribution and the truncated Poisson distribution are presented and illustrated with bacteriological data. Some of the methods are designed for testing the compatibility of the zero frequency with the Poisson distribution, whereas others are given for testing the goodness of fit for the truncated Poisson. In particular, a modified form of the Fisher index of dispersion is presented which is suitable for the truncated case. It is shown that the use of the usual expression of the index of dispersion for testing the adequacy of the truncated Poisson is not correct and leads to accepting inadequate fits more frequently than expected on the basis of test of significance. Furthermore, three test statistics are presented for testing the compatability of the zero frequency with the Poisson distribution. The results of the simulation show that two test statistics, one due to Cochran (W. G. Cochran, Biometrics 10:417-451, 1954) and the other to Rao and Chakravarti (C. R. Rao and I. M. Chakravarti, Biometrics 12:264-282, 1956), are preferable to those from the likelihood ratio test.     

Confidence interval estimation of the intraclass correlation coefficient for binary outcome data

We obtain closed-form asymptotic variance formulae for three point estimators of the intraclass correlation coefficient that may be applied to binary outcome data arising in clusters of variable size. Our results include as special cases those that have previously appeared in the literature (Fleiss and Cuzick, 1979, Applied Psychological Measurement 3, 537-542; Bloch and Kraemer, 1989, Biometrics 45, 269-287; Altaye, Donner, and Klar, 2001, Biometrics 57, 584-588). Simulation results indicate that confidence intervals based on the estimator proposed by Fleiss and Cuzick provide coverage levels close to nominal over a wide range of parameter combinations. Two examples are presented.     

An inferential procedure for the Poisson intervention parameter.

To describe a health chance mechanism whose incidence rate is altered in the middle of a data collection period due to preventive treatments taken by health service agencies, a model called Intervened Poisson Distribution (cf. Shanmugam, 1985, Biometrics 41, 1025-1029) was introduced. In this article, we derive a statistic to test the hypothesis concerning the effectiveness of such preventive treatments, and compare its power with another suggested test (cf. Streit, 1987, Biometrics 43, 999-1000).     

Testing for a treatment effect in the presence of nonresponders

Good (1979, Biometrics 35, 483-489) introduced a new randomization test for the two-sample problem where a proportion 1 - p of the treatment group does not respond to the treatment, and suggested that the Wilcoxon test is not effective for this situation. We show to the contrary that the Wilcoxon test is quite useful when p greater than or equal to .6 and point out an error in his definition of a one-tailed randomization test.     

Some goodness-of-fit methods for the Poisson plus added zeros distribution.

Methods for making inferences about the Poisson plus added zeros distribution and the truncated Poisson distribution are presented and illustrated with bacteriological data. Some of the methods are designed for testing the compatibility of the zero frequency with the Poisson distribution, whereas others are given for testing the goodness of fit for the truncated Poisson. In particular, a modified form of the Fisher index of dispersion is presented which is suitable for the truncated case. It is shown that the use of the usual expression of the index of dispersion for testing the adequacy of the truncated Poisson is not correct and leads to accepting inadequate fits more frequently than expected on the basis of test of significance. Furthermore, three test statistics are presented for testing the compatability of the zero frequency with the Poisson distribution. The results of the simulation show that two test statistics, one due to Cochran (W. G. Cochran, Biometrics 10:417-451, 1954) and the other to Rao and Chakravarti (C. R. Rao and I. M. Chakravarti, Biometrics 12:264-282, 1956), are preferable to those from the likelihood ratio test.     

The ML estimation and testing of generalized ABO-like data with no observed double recessives

The general method of the discrepancy or heterogeneity chi-square is applied to ABO-like data in which there are no observed double blanks in either the disease or the control group. When the recessive gene frequency is assumed zero, this method leads to an approximate chi-square test identical to that suggested by Smouse and Williams (1982, Biometrics 38, 757-768). When this assumption is relaxed, there arise two cases which are determined by whether the maximum likelihood estimate of this frequency is zero or not. It is shown that the value of the simple score statistic of Gart and Nam (1984, Biometrics 40, 887-894) discriminates between the two cases. The various omnibus test statistics for comparing groups are shown to differ little in several practical examples. However, under the more general assumption the appropriate degrees of freedom is one more than the number previously suggested.     

Partial Knowledge and Answer-Until-Correct Tasks in Birds and Humans

Jolliffe and Jolliffe (1997, Biometrics 53, 1136-1142) and Ridout (1999, Biometrics 55, 660-662) have proposed models applicable to memory in coal tits. It is now argued that the birds' task is analogous to answer-until-correct tests used in educational settings in humans, and models from the latter context may be used with the birds' data.     

The equivalence of two tests and models for HLA data with no observed double blanks

Gart and Nam (1984, Biometrics 40, 887-894) consider the analysis of an ABO-like model based on the Hardy-Weinberg law in the commonly occurring case of human leukocyte antigen (HLA) data where there are no double blanks, that is, no recessive homozygotes. They derive a score test, based on the truncated likelihood, of the hypothesis that the true recessive gene (or allele) frequency is zero. Yasuda (1968, Biometrics 24, 915-935) considers a similar codominant system wherein the true recessive gene frequency is assumed zero, but the Hardy-Weinberg law does not hold. In particular, he considers the possibility of a nonzero inbreeding coefficient. We show that the two models are equivalent; each likelihood can be shown to be a reparameterization of the other. Furthermore, the score test of the zero gene frequency in Gart and Nam is identical to the score test for a zero inbreeding coefficient given by Yasuda. The results are applied to an example wherein it appears that the Hardy-Weinberg model is appropriate. Thus, it is not possible in this population to identify homozygous individuals without error from phenotypic data alone.