Unconditional Exact Tests for Equivalence or Noninferiority for Paired Binary Endpoints

Problems of establishing equivalence or noninferiority between two medical diagnostic procedures involve comparisons of the response rates between correlated proportions. When the sample size is small, the asymptotic tests may not be reliable. This article proposes an unconditional exact test procedure to assess equivalence or noninferiority. Two statistics, a sample-based test statistic and a restricted maximum likelihood estimation (RMLE)-based test statistic, to define the rejection region of the exact test are considered. We show the p-value of the proposed unconditional exact tests can be attained at the boundary point of the null hypothesis. Assessment of equivalence is often based on a comparison of the confidence limits with the equivalence limits. We also derive the unconditional exact confidence intervals on the difference of the two proportion means for the two test statistics. A typical data set of comparing two diagnostic procedures is analyzed using the proposed unconditional exact and asymptotic methods. The p-value from the unconditional exact tests is generally larger than the p-value from the asymptotic tests. In other words, an exact confidence interval is generally wider than the confidence interval obtained from an asymptotic test.     

A cautionary note on exact unconditional inference for a difference between two independent binomial proportions

Fisher's exact test for comparing response proportions in a randomized experiment can be overly conservative when the group sizes are small or when the response proportions are close to zero or one. This is primarily because the null distribution of the test statistic becomes too discrete, a partial consequence of the inference being conditional on the total number of responders. Accordingly, exact unconditional procedures have gained in popularity, on the premise that power will increase because the null distribution of the test statistic will presumably be less discrete. However, we caution researchers that a poor choice of test statistic for exact unconditional inference can actually result in a substantially less powerful analysis than Fisher's conditional test. To illustrate, we study a real example and provide exact test size and power results for several competing tests, for both balanced and unbalanced designs. Our results reveal that Fisher's test generally outperforms exact unconditional tests based on using as the test statistic either the observed difference in proportions, or the observed difference divided by its estimated standard error under the alternative hypothesis, the latter for unbalanced designs only. On the other hand, the exact unconditional test based on the observed difference divided by its estimated standard error under the null hypothesis (score statistic) outperforms Fisher's test, and is recommended. Boschloo's test, in which the p-value from Fisher's test is used as the test statistic in an exact unconditional test, is uniformly more powerful than Fisher's test, and is also recommended.     

A Comparison of the Three Conditional Exact Tests in Two‐way Contingency Tables Using the Unconditional Exact Power

The conditional exact tests of homogeneity of two binomial proportions are often used in small samples, because the exact tests guarantee to keep the size under the nominal level. The Fisher's exact test, the exact chi‐squared test and the exact likelihood ratio test are popular and can be implemented in software StatXact. In this paper we investigate which test is the best in small samples in terms of the unconditional exact power. In equal sample cases it is proved that the three tests produce the same unconditional exact power. A symmetry of the unconditional exact power is also found. In unequal sample cases the unconditional exact powers of the three tests are computed and compared. In most cases the Fisher's exact test turns out to be best, but we characterize some cases in which the exact likelihood ratio test has the highest unconditional exact power. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interim Treatment Selection in Drug Development

We consider hypothesis testing in a clinical trial with an interim treatment selection. Recently, unconditional and conditional procedures for selecting one treatment as the winner have been proposed when the mean responses are approximately normal. In this paper, we generalize both procedures to multi-winner cases. The distributions of the test statistics are obtained and step-down approaches are proposed. We prove that both unconditional and conditional procedures strongly control the family-wise error rate. We give a brief discussion on power comparisons.     

Testing the ratio of two poisson rates

In this paper we compare the properties of four different general approaches for testing the ratio of two Poisson rates. Asymptotically normal tests, tests based on approximate p -values, exact conditional tests, and a likelihood ratio test are considered. The properties and power performance of these tests are studied by a Monte Carlo simulation experiment. Sample size calculation formulae are given for each of the test procedures and their validities are studied. Some recommendations favoring the likelihood ratio and certain asymptotic tests are based on these simulation results. Finally, all of the test procedures are illustrated with two real life medical examples.     

Exact Tests for Comparing Two Paired Proportions with Incomplete Data

Various asymptotic test procedures have been developed previously for testing the equality of two binomial proportions with partially incomplete paired data. Test procedures that discard incomplete observations have been shown to be less powerful than those procedures that utilize all available observations. On the other hand, asymptotic test procedures that utilize all available observations may not be reliable in small‐sample problems or sparse data structures. In this article, unconditional exact test procedures are proposed for testing the equality of two paired binomial proportions with partially incomplete paired data under a random mechanism. The proposed unconditional exact test methods are illustrated with real data from a neurological study. Empirical studies are conducted to investigate the performance of these and other test procedures with respect to size and power. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Exact Unconditional test for the Hardy-Weinberg Equilibrium

An exact test based on the unconditional distribution of a test statistic for the Hardy-Weinberg equilibrium is introduced. This test is usually more powerful and requires less extensive tabulations compared with the (ordinary) exact conditional test. Tables of critical values are given for N = 5(1)100 and α =·10, ·05, ·01. Sample sizes required to attain a given power with the unconditional and conditional exact tests are provided.     

Unconditional versions of several tests commonly used in the analysis of contingency tables

By focusing on a confidence interval for a nuisance parameter, Berger and Boos (1994, Journal of the American Statistical Association 89, 1012-1016) proposed new unconditional tests. In particular, they showed that, for a 2 x 2 table, this procedure generally was more powerful than Fisher's exact test. This paper utilizes and extends their approach to obtain unconditional tests for combining several 2 x 2 tables and testing for trend and homogeneity in a 2 x K table. The unconditional procedures are compared to the conditional ones by reanalyzing some published biomedical data.     

Evaluating the effects of rater and subject factors on measures of association

Large‐scale agreement studies are becoming increasingly common in medical settings to gain better insight into discrepancies often observed between experts' classifications. Ordered categorical scales are routinely used to classify subjects' disease and health conditions. Summary measures such as Cohen's weighted kappa are popular approaches for reporting levels of association for pairs of raters' ordinal classifications. However, in large‐scale studies with many raters, assessing levels of association can be challenging due to dependencies between many raters each grading the same sample of subjects' results and the ordinal nature of the ratings. Further complexities arise when the focus of a study is to examine the impact of rater and subject characteristics on levels of association. In this paper, we describe a flexible approach based upon the class of generalized linear mixed models to assess the influence of rater and subject factors on association between many raters' ordinal classifications. We propose novel model‐based measures for large‐scale studies to provide simple summaries of association similar to Cohen's weighted kappa while avoiding prevalence and marginal distribution issues that Cohen's weighted kappa is susceptible to. The proposed summary measures can be used to compare association between subgroups of subjects or raters. We demonstrate the use of hypothesis tests to formally determine if rater and subject factors have a significant influence on association, and describe approaches for evaluating the goodness‐of‐fit of the proposed model. The performance of the proposed approach is explored through extensive simulation studies and is applied to a recent large‐scale cancer breast cancer screening study.     

Tests for Trends in Incidence Rate Ratios

Breslow (1984) described an efficient score test for trend in incidence density rate ratios for cohort studies under a conditional Poisson or binomial model employing maximum likelihood estimation of the rate parameters. In this communication, an alternative derivation of this statistic that is based on an unconditional approach is provided, along with an examination of associated goodness-of-fit tests and methods of confidence interval estimation. The procedures are illustrated by a cohort study of ischemic heart disease mortality following industrial exposure to carbon disulfide.     

Exact unconditional inference for risk ratio in a correlated 2 x 2 table with structural zero

In this article, we consider small-sample statistical inference for rate ratio (RR) in a correlated 2 x 2 table with a structural zero in one of the off-diagonal cells. Existing Wald's test statistic and logarithmic transformation test statistic will be adopted for this purpose. Hypothesis testing and confidence interval construction based on large-sample theory will be reviewed first. We then propose reliable small-sample exact unconditional procedures for hypothesis testing and confidence interval construction. We present empirical results to evince the better confidence interval performance of our proposed exact unconditional procedures over the traditional large-sample procedures in small-sample designs. Unlike the findings given in Lui (1998, Biometrics 54, 706-711), our empirical studies show that the existing asymptotic procedures may not attain a prespecified confidence level even in moderate sample-size designs (e.g., n = 50). Our exact unconditional procedures on the other hand do not suffer from this problem. Hence, the asymptotic procedures should be applied with caution. We propose two approximate unconditional confidence interval construction methods that outperform the existing asymptotic ones in terms of coverage probability and expected interval width. Also, we empirically demonstrate that the approximate unconditional tests are more powerful than their associated exact unconditional tests. A real data set from a two-step tuberculosis testing study is used to illustrate the methodologies.     

The 2 x 2 matched-pairs trial: exact unconditional design and analysis

An exact unconditional method for the design and analysis of the 2 x 2 matched-pairs trial is presented. Unlike the exact conditional method, which is based on only the number of discordant pairs, the exact unconditional method uses the total number N of sampled pairs. This unconditional test, based on a simple Z statistic, yields sample sizes that are generally smaller than those produced by the exact conditional test for the cases tabulated herein, namely one-sided alpha = .01, .025, and .05 along with 80% and 90% power. Moreover, it is found to be uniformly more powerful than the latter for all the combinations of parameters considered in this paper, namely alpha = .01, .025, and .05, and N = 10(1)200. The method is illustrated by assessing various design options of an in vitro study of the effectiveness of antimicrobial agents. Some numerical examples of matching efficiency are also given.     

Stratified Fisher's exact test and its sample size calculation

Chi‐squared test has been a popular approach to the analysis of a 2 × 2 table when the sample sizes for the four cells are large. When the large sample assumption does not hold, however, we need an exact testing method such as Fisher's test. When the study population is heterogeneous, we often partition the subjects into multiple strata, so that each stratum consists of homogeneous subjects and hence the stratified analysis has an improved testing power. While Mantel–Haenszel test has been widely used as an extension of the chi‐squared test to test on stratified 2 × 2 tables with a large‐sample approximation, we have been lacking an extension of Fisher's test for stratified exact testing. In this paper, we discuss an exact testing method for stratified 2 × 2 tables that is simplified to the standard Fisher's test in single 2 × 2 table cases, and propose its sample size calculation method that can be useful for designing a study with rare cell frequencies.     

Bias-reduced estimators of conditional odds ratios in matched case-control studies with unmatched confounding

We study bias-reduced estimators of exponentially transformed parameters in general linear models (GLMs) and show how they can be used to obtain bias-reduced conditional (or unconditional) odds ratios in matched case-control studies. Two options are considered and compared: the explicit approach and the implicit approach. The implicit approach is based on the modified score function where bias-reduced estimates are obtained by using iterative procedures to solve the modified score equations. The explicit approach is shown to be a one-step approximation of this iterative procedure. To apply these approaches for the conditional analysis of matched case-control studies, with potentially unmatched confounding and with several exposures, we utilize the relation between the conditional likelihood and the likelihood of the unconditional logit binomial GLM for matched pairs and Cox partial likelihood for matched sets with appropriately setup data. The properties of the estimators are evaluated by using a large Monte Carlo simulation study and an illustration of a real dataset is shown. Researchers reporting the results on the exponentiated scale should use bias-reduced estimators since otherwise the effects can be under or overestimated, where the magnitude of the bias is especially large in studies with smaller sample sizes.     

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.     

Notes on Testing Equality in Dichotomous Data with Matched Pairs

In attempting to improve the efficiency of McNemar's test statistic, we develop two test procedures that account for the information on both the discordant and concordant pairs for testing equality between two comparison groups in dichotomous data with matched pairs. Furthermore, we derive a test procedure derived from one of the most commonly‐used interval estimators for odds ratio. We compare these procedures with those using McNemar's test, McNemar's test with the continuity correction, and the exact test with respect to type I error and power in a variety of situations. We note that the test procedures using McNemar's test with the continuity correction and the exact test can be quite conservative and hence lose much efficiency, while the test procedure using McNemar's test can actually perform well even when the expected number of discordant pairs is small. We also find that the two test procedures, which incorporate the information on all matched pairs into hypothesis testing, may slightly improve the power of using McNemar's test without essentially losing the precision of type I error. On the other hand, the test procedure derived from an interval estimator of adds ratio with use of the logarithmic transformation may have type I error much larger than the nominal α‐level when the expected number of discordant pairs is not large and therefore, is not recommended for general use.     

Improved exact confidence intervals for the odds ratio in two independent binomial samples

For two independent binomial samples, the usual exact confidence interval for the odds ratio based on the conditional approach can be very conservative. Recently, Agresti and Min (2002) showed that the unconditional intervals are preferable to conditional intervals with small sample sizes. We use the unconditional approach to obtain a modified interval, which has shorter length, and its coverage probability is closer to and at least the nominal confidence coefficient.     

Conditional Maximum Likelihood Estimate and Exact Test of the Common Relative Difference in Combination of 2 × 2 Tables under Inverse Sampling

On the basis of the conditional distribution, given the marginal totals of non-cases fixed for each of independent 2 × 2 tables under inverse sampling, this paper develops the conditional maximum likelihood (CMLE) estimator of the underlying common relative difference (RD) and its asymptotic conditional variance. This paper further provides for the RD an exact interval calculation procedure, of which the coverage probability is always larger than or equal to the desired confidence level and for investigating whether the underlying common RD equals any specified value an exact test procedure, of which Type I error is always less than or equal to the nominal α-level. These exact interval estimation and exact hypothesis testing procedures are especially useful for the situation in which the number of index subjects in a study is small and the asymptotically approximate methods may not be appropriate for use. This paper also notes the condition under which the CMLE of RD uniquely exists and includes a simple example to illustrate use of these techniques.     

A Score Regression Approach to Assess Calibration of Continuous Probabilistic Predictions

Summary Calibration, the statistical consistency of forecast distributions and the observations, is a central requirement for probabilistic predictions. Calibration of continuous forecasts is typically assessed using the probability integral transform histogram. In this article, we propose significance tests based on scoring rules to assess calibration of continuous predictive distributions. For an ideal normal forecast we derive the first two moments of two commonly used scoring rules: the logarithmic and the continuous ranked probability score. This naturally leads to the construction of two unconditional tests for normal predictions. More generally, we propose a novel score regression approach, where the individual scores are regressed on suitable functions of the predictive variance. This conditional approach is applicable even for certain nonnormal predictions based on the Dawid–Sebastiani score. Two case studies illustrate that the score regression approach has typically more power in detecting miscalibrated forecasts than the other approaches considered, including a recently proposed technique based on conditional exceedance probability curves.     

Extension of Peto's Test by Attribution of Tumor Lethality in the Absence of Cause‐of‐Death Information

A new statistical testing approach is developed for rodent tumorigenicity assays that have a single terminal sacrifice or occasionally interim sacrifices but not cause‐of‐death data. For experiments that lack cause‐of‐death data, statistically imputed numbers of fatal tumors and incidental tumors are used to modify Peto's cause‐of‐death test which is usually implemented using pathologist‐assigned cause‐of‐death information. The numbers of fatal tumors are estimated using a constrained nonparametric maximum likelihood estimation method. A new Newton‐based approach under inequality constraints is proposed for finding the global maximum likelihood estimates. In this study, the proposed method is concentrated on data with a single sacrifice experiment without implementing further assumptions. The new testing approach may be more reliable than Peto's test because of the potential for a misclassification of cause‐of‐death by pathologists. A Monte Carlo simulation study for the proposed test is conducted to assess size and power of the test. Asymptotic normality for the statistic of the proposed test is also investigated. The proposed testing approach is illustrated using a real data set.