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.     

Point Estimation on Relative Risk under Inverse Sampling

When the underlying disease is rare, to control the coefficient of variation for the sample proportion of cases, we may wish to apply inverse sampling. In this paper, we derive the uniformly minimum variance unbiased estimator (UMVUE) of relative risk and its variance in closed form under inverse sampling. On the basis of a Monte Carlo simulation, we demonstrate that using the UMVUE of relative risk can substantially reduce the mean-squared-error of using the maximum likelihood estimator, especially when the number of index cases in both comparison samples is small. For a given fixed total cost, we include a program that can be used to find the optimal allocation for the number of index cases to minimize the variance of the UMVUE as well.     

Confidence Intervals for the Risk Ratio in Cohort Studies under Inverse Sampling

Three simple interval estimates for the risk ratio in inverse sampling are considered. The first two interval estimates are derived on the basis of Fieller's Theorem and the delta method with the logarithmic transformation, respectively. The third interval estimate is derived on the basis of an F-test statistic proposed by BENNETT (1981) for testing equal probabilities of a disease between two comparison groups when the disease is rare. To evaluate the performance of these three methods, a Monte Carlo simulation is used to compare the actual coverage probability with the nominal confidence level for each method and to estimate the expected length of the corresponding confidence interval in a variety of situations. On the basis of the results found in the simulation, we have concluded that the method with the logarithmic transformation is either equivalent to or better than the other two methods for all situations considered here.     

Tests for Homogeneity of Variances Using Robust Weighted Likelihood Estimates

The classical normal-theory tests for testing the null hypothesis of common variance and the classical estimates of scale have long been known to be quite nonrobust to even mild deviations from normality assumptions for moderate sample sizes. Levene (1960) suggested a one-way ANOVA type statistic as a robust test. Brown and Forsythe (1974) considered a modified version of Levene's test by replacing the sample means with sample medians as estimates of population locations, and their test is computationally the simplest among the three tests recommended by Conover , Johnson , and Johnson (1981) in terms of robustness and power. In this paper a new robust and powerful test for homogeneity of variances is proposed based on a modification of Levene's test using the weighted likelihood estimates (Markatou , Basu , and Lindsay , 1996) of the population means. For two and three populations the proposed test using the Hellinger distance based weighted likelihood estimates is observed to achieve better empirical level and power than Brown-Forsythe's test in symmetric distributions having a thicker tail than the normal, and higher empirical power in skew distributions under the use of F distribution critical values.     

Hypothesis Testing Procedures for a Series of Independent Fourfold Tables under Inverse Sampling

This paper considers four summary test statistics, including the one recently proposed by Bennett (1986, Biometrical Journal 28, 859–862), for hypothesis testing of association in a series of independent fourfold tables under inverse sampling. This paper provides a systematic and quantitative evaluation of the small-sample performance for these summary test statistics on the basis of a Monte Carlo simulation. This paper notes that the test statistic developed by Bennett (1986) can be conservative and thereby possibly lose the power when the underlying disease is not rare. This paper also finds that for given a fixed total number of cases in each table, the conditional test statistic is the best in controlling type I error among all test statistics considered here.     

Unconditional Confidence Interval for the Difference between Two Proportions

In applied statistics it is customary to have to obtain a one‐ or two‐tail confidence interval for the difference d = p₂ – p₁ between two independent binomial proportions. Traditionally, one is looking for a classic and non‐symmetric interval (with respect to zero) of the type d ∈ [δ_L;δ_U], d ≤ δ₀ or d ≥ δ₀. However, in clinical trials, equivalence studies, vaccination efficacy studies, etc., and even after performing the classic homogeneity test, intervals of the type |d| ≤ Δ₀ or |d| ≥ Δ₀, where Δ₀ > 0, may be necessary. In all these cases it is advisable to obtain the interval by inverting the appropriate test. The advantage of this procedure is that the conclusions obtained using the test are compatible with those obtained using the interval. The article shows how this is done using the new exact and asymptotic unconditional tests published. The programs for performing these tests may be obtained at URL http://www.ugr.es/~bioest/software.htm.     

Notes on Confidence Limits for the Odds Ratio in Case-Control Studies under Inverse Sampling

When the disease of interest is rare or chronic, we often apply the case control study design and the odds ratio to determine the possibly risk factors. In this paper, three simple closed-form interval calculation procedures for the odds ratio under inverse sampling are proposed. Results on the basis of a Monte Carlo simulation to evaluate and compare the performance of these three procedures are presented. A discussion on the limitation of the proposed procedures when the probability of exposure is extremely common in both the case and the control groups and a suggestion to overcome this limitation are included as well.     

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.     

Characterization of the Negative Binomial and Gamma Distributions

Characterization of the negative binomial and gamma distributions by a conditional distribution and a linear regression, and the gamma distribution by the negative binomial distribution are given. An application to a random shock model is discussed.     

Exact Power of Conditional Tests for Several Binomial Populations

An efficient recursive polynomial multiplication method is proposed for exact unconditional power calculation for unordered 2 × K contingency table with up to moderate sample size. Our method can be applied to the family of cell-additive statistics which includes the Freeman-Halton statistic, the Pearson χ² statistic and the likelihood ratio statistic. We illustrate our proposed method by several numerical examples.     

On the exact null distribution of the generalised likelihood ratio test for analysing unreplicated factorial designs

Al-Shiha and Yang (1999) proposed a multistage procedure for analysing unreplicated factorial experiments, which is based on the statistic that is derived from the generalised likelihood ratio test statistic under the assumption of normality. It was shown by their simulation study that the method is quite competitive with Lenth's (1989) method. In their paper, because of the difficulty of determining the null distribution analytically, the quantiles of the null distribution were empirically simulated. In this paper, we give the exact null distribution of their test statistic, which makes it possible to calculate the critical values of the test.     

Precision Intervals for Estimates of the Difference in Success Rates for Binary Random Variables Based on the Permutation Principle

Fisher's exact test is a very commonly applied test in clinical trials with a binary outcome variable (e.g. success/failure). However confidence statements about the difference of success rates are usually based on the normal approximation. This may sometimes lead to the confusing statement that the test is statistically significant at a prespecified level while the corresponding confidence interval includes the zero difference and vice versa. Here, we construct precision intervals for the difference of success rates from two independent samples based on the permutation principle which are in perfect agreement with the discrete (permutation) test and compare it to examples from the literature. APL programs are provided.     

On small-sample confidence intervals for parameters in discrete distributions

The traditional definition of a confidence interval requires the coverage probability at any value of the parameter to be at least the nominal confidence level. In constructing such intervals for parameters in discrete distributions, less conservative behavior results from inverting a single two-sided test than inverting two separate one-sided tests of half the nominal level each. We illustrate for a variety of discrete problems, including interval estimation of a binomial parameter, the difference and the ratio of two binomial parameters for independent samples, and the odds ratio.     

Monte Carlo Investigations for Relative Efficiency of Stratified Random Sampling Scheme for Estimation of Relative Risk in Case-Control Studies

Monte Carlo Investigations have been widely used in Sample Surveys in Comparing the efficiency of various methods when exact mathematical comparisons are not possible. In this paper the same has been used for comparing the efficiency of Stratified Random Sampling with respect to Simple Random Sampling for estimation of Relative Risk in Case-Control Studies. The data used relate to a Case Control study on peptic ulcer. On the basis of Monte Carlo Investigations on 50 samples of size 10–20 (Cases and Controls), it has been observed that there is considerable gain in efficiency in using Stratified Random Sampling over Simple Random Sampling. The sensitivity of the results with the change in Sample Size has also been investigated.     

A Test for Homogeneity of Odds Ratios in Ordered 2 × 2 Tables

Consider K ordered 2 × 2 contingency tables. A new test of the null hypothesis that the odds ratios of these tables are equal vs the alternative hypothesis that the odds ratios are nondecreasing, is recommended. The test is exact (non‐asymptotic), is easily carried out (software is available), and has other favorable properties. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

亲代传递的独立性与传递不平衡的边缘齐性检验

传递不平衡检验其实质是边缘齐性检验,需同时考察家系中两个亲代向受累子代的传递情形．但是,其零假设成立时,同一家系中父亲的传递与母亲的传递并不一定相互独立．本文探讨了父母传递的独立性条件对检验统计量的影响．结果表明,即使父母传递不独立时,边缘齐性检验仍然适合．     

An exact test for all-way interaction in a 2M contingency table: application to interval capture-recapture estimation of population size

An exact conditional test for an M-way log-linear interaction in a fully observed 2M contingency table is formulated. From this is derived a procedure for interval estimation of the total count N in a 2M contingency table, one of whose entries is unobserved. This procedure has an immediate application to interval estimation of the size of a closed population from incomplete, overlapping lists of records, as in capture-recapture analysis of epidemiological data. Data on the prevalence of spina bifida in live births in upstate New York in 1969-1974 illustrate this application.