Alternatives to the Chi-Square Test of Homogeneity in 2×2 Tables and to Fisher's Exact Test

In comparing two independent binomial proportions by modified X² tests: Gart's modified likelihood-ratio, Overall and Starbuck's “tailored F”, Pearson's adjusted X²[=X² multiplied by (n - 1)/n] and its skewness-corrected version proposed by Berchtold, it was found that the last two have error probabilities that in the mean are close to the significance level.     

A Two-Tailed P-Value for Fisher's Exact Test

A two-tailed P-value is presented for a significance test in two by two contingency tables. There is no extraneous quasi-observation such as is needed in the exact randomized uniformly most powerful unbiased (UMPU) test of the hypothesis of independence. The proposed P-value can never exceed unity and is always two-tailed, unlike other P-values proposed in the literature     

UMPU and Alternative Tests for Association in 2 × 2 Tables

The use of the uniformly most powerful among the unbiased (UMPU) test was recently suggested for the study of gametic association between two polymorphic loci as an alternative to the Fisher's exact test (Zapata and Alvarez, 1997, Annals of Human Genetics 61, 71-77). However, the proposed test is not UMPU for two-sided alternatives. In this study, we present the UMPU test, discuss criticisms against the use of randomized tests, and compare the power of several tests. We show that, in many practical cases, the use of the UMPU test is less than desirable and propose the alternative adjusted-more extreme tables (A-MET) and the equal-tails (ET) tests. We suggest that some of the general arguments against the use of randomized tests can be alleviated by a newly proposed extended p-value definition.     

Multiple Comparisons of Polynomial Distributions

Asymptotically correct 90 and 95 percentage points are given for multiple comparisons with control and for all pair comparisons of several independent samples of equal size from polynomial distributions. Test statistics are the maxima of the X²-statistics for single comparisons. For only two categories the asymptotic distributions of these test statistics result from DUNNETT'S many-one tests and TUKEY'S range test (cf. MILLER, 1981). The percentage points for comparisons with control are computed from the limit distribution of the test statistic under the overall hypothesis H₀. To some extent the applicability of these bounds is investigated by simulation. The bounds can also be used to improve Holm's sequentially rejective Bonferroni test procedure (cf. HOLM, 1979). The percentage points for all pair comparisons are obtained by large simulations. Especially for 3×3-tables the limit distribution of the test statistic under H₀ is derived also for samples of unequal size. Also these bounds can improve the corresponding Bonferroni-Holm procedure. Finally from SKIDÁK's probability inequality for normal random vectors (cf. SKIDÁK, 1967) a similar inequality is derived for dependent X²-variables applicable to simultaneous X²-tests.     

Comparing Power Behaviours for Some Goodness-of-Fit Test Statistics in Sparse Multinomials

This paper is concerned with the power behaviour of four goodness-of-fit test statistics in sparse multinomials with k cells. Most previous work has been concerned only with both Pearson's X² and the likelihood ratio test statistics. We consider in this study, two additional test statistics, namely, the Cressie-Read test statistic – I(2/3) and the modified Freeman-Tukey test (FT) statistic. Because k ≥ 10 in this study, a Monte Carlo procedure based on 1000 simulated samples is used to estimate the powers for the four test statistics. Alternatives on various line segments are employed. Results suggest that none of the test statistics completely dominate the other and that the choice of which test to use depends on the nature of the alternative hypothesis. These results are consistent with those obtained by West and Kempthorne (1972), although, the Pearson's χ² test statistic may be preferred because of its closer approximation to the χ² distribution in terms of the attained α levels.     

Using Discrete Distributions to Compute Powers for Goodness-of-Fit Test Statistics in a One-Way Multinomial Setting

Both the Bionomial and Poisson distributions are employed in this study to compute approximate powers for goodness-of-fit test statistics. The procedure adopted involves simulating 1000 samples from each of these distributions. These samples are then employed to compute both the randomized nominal critical values and estimated powers. The type I error rates returned from the use of the randomized critical levels Cα fall within the acceptable regions. We illustrate the use of the procedure with the Pearson's X² test statistic and show that this can readily be extended to any of the other well known goodness-of-fit test statistics.     

Minimum sample sizes for identifying chromosomal fragile sites from individuals: Monte Carlo estimation

A Monte Carlo simulation procedure was used to estimate the exact level of the standardized X ² test statistic (X _s ²) for randomness in the FSM methodology for the identification of fragile sites from chromosomal breakage data for single individuals. A random-number generator was used to simulate 10 000 chromosomal breakage data sets, each corresponding to the null hypothesis of no fragile sites for numbers of chromosomal breaks (n) from 1 to 2000 and at three levels of chromosomal band resolution (k). The reliability of the test was assessed by comparisons of the empirical and nominal α levels for each of the corresponding values of n and k. These analyses indicate that the sparse and discrete nature of chromosomal breakage data results in large and unpredictable discrepancies between the empirical and nominal α levels when fragile site identifications are based on small numbers of breaks (n < 0.5 k). With n≥ 0.5 k, the distribution of X _s ² appears to be stable and non-significant differences in the empirical and nominal α levels are generally obtained. These results are inherent to the nature of the data and are, therefore, relevant to any statistical model for the identification of fragile sites from chromosomal breakage data. For FSM identification of fragile sites at α = 0.05, we suggest that n≥ 0.5 k is the minimum reliable number of mapped chromosomal breaks per individual. Received: 28 April 1997 / Accepted: 1 July 1997     

Analysis of Proportions in One- or Two-Way Tables with Spoiled Data

This paper discusses the analysis of data on proportions in contingency tables. The X² ‘index of dispersion test’ (e.g. Fisher, 1954) is developed in these situations and compared with the use of the ‘logit’ transformation. An example using Osborn's (1979) data is given, illustrating the estimation of one or more missing observations.     

Small Sample Properties of the Mantel-Haenszel Test Statistic for Density Follow-Up Studies

For the analysis of combinations of 2×2 non-contingency tables as obtained from density follow-up studies (relating a number of events to a number of person-years of follow-up) an analogue of the Mantel-Haenszel test for 2×2 contingency tables is widely used. In this paper the small sample properties of this test, both with and without continuity correction, are evaluated. Also the improvement of the test-statistic by using the first four cumulants via the Edgeworth expansion was studied. Results on continuity correction agree with similar studies on the Mantel-Haenszel statistic for 2×2 contingency tables: Continuity correction gives a p-value which approximates the exact p-value better than the p-value obtained without this correction; both the exact test and its approximations show considerable conservatism in small samples; the uncorrected Mantel-Haenszel test statistic gives a p-value that agrees more with the nominal significance level, but can be anti-conservative. The p-value based on the first four cumulants gives a better approximation of the exact p-value than the continuity corrected test, especially when the distribution has marked skewness.     

Comparisons of Some Chi-squared Tests for the Test of Independence in Sparse Two-Way Contingency Tables

We consider in this paper, the behaviour of a class of the CRESSIE READ (1984) power divergence test statistics indexed by parameter λ - I (λ), with the modified X₂ test statistics (LU) proposed by LAWAL and UPTON (1984), for sparse contingency tables ranging from the 3×3 to the 10×10. We present a sample of our results here. The results indicate that the LU test out-performs either the Cressie-Read suggested test I(2/3) or the Pearson's test - I(1). Our results further show that the modification to the likelihood ratio test [Y₂ = I'(0)] proposed by WILLIAMS (1976) performs like the parent Y₂ test, very poorly compared with either the I(2/3), X₂ or the LU test statistics. Power results also indicate that the powers of the LU test are in all cases considered in this study slightly higher than those of X₂ and I(2/3) tests. The LU test is therefore strongly recommended for use with sparse two-way contingency tables because in all of the cases considered, none of the other test statistics consistently out-performs the LU test with respect to attained α level or power.     

The Exact Distribution of the Anderson-Kannemann Statistic

The Anderson-Kannemann test is a rank test for treatment effects in a randomized block design with K treatments and N blocks. In this paper, an algorithm for computing the exact distribution of the Anderson-Kannemann test statistic under the null hypothesis is deduced. Then, the exact distribution is compared with the asymptotic χ²-distribution, and it is shown that the exact distribution is approximated fairly well by the asymptotic distribution. Tables of the exact distribution are given for K = 3, N = 3(1)15; K = 4, N = 3(1)11; K = 5, N = 3(1)7; and K = 6, N = 3(1)5.     

Estimation and Verification of Hypotheses in Some Zyskind-Martin Models with Missing Values

Several theorems on estimation and verification of linear hypotheses in some Zyskind-Martin (ZM) models are given. The assumptions are as follows. Let y = Xβ + e or (y, Xβ, σ²V) be a fixed model where y is a vector of n observations, X is a known matrix nXp with rank r(X) = r ≦ p < n, where p is a number of coordinates of the unknown parameter vector β, e is a random vector of errors with covariance matrix σ²V, where σ² is unknown scalar parameter, V is a known non-negative definite matrix such that R(X) ? R(V). Symbol R(A) denotes a vector space generated by columns of matrix A. The expected value of y is Xβ. In this paper four following Zyskind-Martin (ZM) models are considered: ZMd, ZMa, ZMc and ZMqd (definitions in sec. 1) when vector y y₁ y₂ involves a vector y₁ of m missing values and a vector y₂ with (n — m) observed values. A special transformation of ZM model gives again ZM model (cf. theorem 2.1). Ten properties of actual (ZMa) and complete (ZMc) Zyskind-Martin models with missing values (cf. theorem 2.2) test functions F are given in (2.11)) are presented. The third propriety constitutes a generalization of R. A. Fisher's rule from standard model (y, Xβ, σ²I) to ZM model. Estimation of vector y₁ (cf. 3.3) of vector β (cf. th. 3.2) and of scalar σ² (cf. th. 3.4) in actual ZMa model and in diagonal quasi-ZM model (ZMqd) are presented. Relation between y? ₁ and β is given in theorem 3.1. The results of section 2 are illustrated by numerical example in section 4.     

A Confidence‐Limit‐Based Approach to the Assessment of Hardy–Weinberg Equilibrium

The classical χ²‐procedure for the assessment of Hardy–Weinberg equilibrium (HWE) is tailored for detecting violations of HWE. However, many applications in genetic epidemiology require approximate compatibility with HWE. In a previous contribution to the field (Wellek, S. (2004). Biometrics, 60 , 694–703), the methodology of statistical equivalence testing was exploited for the construction of tests for problems in which the assumption of approximate compatibility of a given genotype distribution with HWE plays the role of the alternative hypothesis one aims to establish. In this article, we propose a procedure serving the same purpose but relying on confidence limits rather than critical bounds of a significance test. Interval estimation relates to essentially the same parametric function that was previously chosen as the target parameter for constructing an exact conditional UMPU test for equivalence with a HWE conforming genotype distribution. This population parameter is shown to have a direct genetic interpretation as a measure of relative excess heterozygosity. Confidence limits are constructed using both asymptotic and exact methods. The new approach is illustrated by reanalyzing genotype distributions obtained from published genetic association studies, and detailed guidance for choosing the equivalence margin is provided. The methods have been implemented in freely available SAS macros.     

Bivariate Sign Tests Sensitive to Homeo- and Heteropoetic Treatment Effects

Starting from N paired differences of a continuous variable, BENNETT 's (1962) bivariate sign test is identified as a test against so called homeopoetic treatment effects (or location shifts in one or both variables X and Y). As a complement, a new sign test is identified to be sensitive to so called heteropoetic treatment effects (shifts in dispersion). Both tests are shown to be important in biomedical and psychosocial research. A numerical example is given from psychopharmacology using the new bivariate sign test.     

Modified GEE and Goodness of the Marginal Fit (GOMF) Test with Correlated Binary Responses for Contingency Tables

This paper addresses testing the goodness of fit of models for marginal probabilities estimated by generalized estimating equations. We develop a modified version of generalized estimating equation and a goodness‐of‐fit test based on the fitted marginal means. The test statistic is easy to compute and has a simple reference distribution. Its performance is evaluated asymptotically and in small samples. It is also compared to the deviance and Pearson X² statistics. Example applications are given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Robust Test for Multi-Sample Location Problems with Unequal Group Variances and Nonnormality

A robust test (to be referred to as M^* test) is proposed for testing equality of several group means without assuming normality and equality of variances. This test statistic is obtained by combining Tiku's MML robust procedure with the James statistic. Monte Carlo simulation studies indicate that the M^* test is more powerful than the Welch test, the James test, and the tests based on Huber's M-estimators over a wide range of nonnormal universes. It is also more powerful than the Brown and Forsythe test under most of nonnormal distributions and has substantially the same power as the Brown and Forsythe test under normal distribution. Comparing with Tan-Tabatabai test, M^* is almost as powerful as Tan-Tabatabai test.     

Predictive Configural Frequency Analysis Evaluated by Simultaneous Berchtold-Corrected Fourfold X2-Tests

For testing against outlying cells in r × c contingency tables in predictive configural frequency analysis, it is suggested not to replace the usual PEARSON fourfold X²-test by YATES ' discontinuity corrected test but by BERCHTOLD'S test correcting for skewed hypergeometric distributions. In general, the BERCHTOLD correction is most efficient for tables involving three small and one large fourfold frequency. The correction is illustrated by configural frequency analysis of data from psychotherapy research.     

Comparison of stool antigen immunoassay and serology for screening for Helicobacter pylori infection in intellectually disabled children

Diagnosis of active Helicobacter pylori infection in intellectually disabled (ID) children is problematic because they are unable to cooperate with performance of invasive tests. In this study, the non‐invasive methods of measuring serum IgG antibody concentrations and performing stool antigen tests were used to screen for H. pylori infection in ID children. Eighty‐seven children with intellectual disabilities were studied. The amount of serum IgG antibody against H. pylori was measured by the ELISA method. Stool samples were examined using an amplified IDEIA HpStAR kit. To assess categorical variables, X², Fisher's exact and Kappa tests were used. The stool antigen tests showed that 93.1% of the children had H. pylori antigen and the serology test that 85.1% of children were positive for H. pylori IgG antibodies. Agreement between results of H. pylori stool antigen (HpSA) testing and IgG antibody serology was 82.8%; however, according to the kappa measure of agreement this agreement is not statistically significant (value, 0.128; P = 0.19). Discordant results were observed for 15 children (17.2%): 11 (12.6%) who were positive on HpSA test but negative by serology and 4 (4.6%) who were IgG seropositive but had negative HpSA tests. This study showed a notably higher rate of H. pylori infection in ID children than has been reported by others for non‐ID children from the same geographical area. The HpSA test is a valid method for primary screening for H. pylori infection in ID children; it detects the specific antigens shed during active infections and has less cross‐reactivity than serological tests that detect antibodies. HpSA is a sensitive non‐invasive method for detecting infection in ID children and may serve as an accurate alternative to serology.     

Chop-Lump Tests for Vaccine Trials

Summary This article proposes new tests to compare the vaccine and placebo groups in randomized vaccine trials when a small fraction of volunteers become infected. A simple approach that is consistent with the intent‐to‐treat principle is to assign a score, say W, equal to 0 for the uninfecteds and some postinfection outcome X > 0 for the infecteds. One can then test the equality of this skewed distribution of W between the two groups. This burden of illness (BOI) test was introduced by Chang, Guess, and Heyse (1994, Statistics in Medicine 13 , 1807–1814). If infections are rare, the massive number of 0s in each group tends to dilute the vaccine effect and this test can have poor power, particularly if the X's are not close to zero. Comparing X in just the infecteds is no longer a comparison of randomized groups and can produce misleading conclusions. Gilbert, Bosch, and Hudgens (2003, Biometrics 59 , 531–541) and Hudgens, Hoering, and Self (2003, Statistics in Medicine 22 , 2281–2298) introduced tests of the equality of X in a subgroup—the principal stratum of those “doomed” to be infected under either randomization assignment. This can be more powerful than the BOI approach, but requires unexaminable assumptions. We suggest new “chop‐lump” Wilcoxon and t‐tests (CLW and CLT) that can be more powerful than the BOI tests in certain situations. When the number of volunteers in each group are equal, the chop‐lump tests remove an equal number of zeros from both groups and then perform a test on the remaining W's, which are mostly >0. A permutation approach provides a null distribution. We show that under local alternatives, the CLW test is always more powerful than the usual Wilcoxon test provided the true vaccine and placebo infection rates are the same. We also identify the crucial role of the “gap” between 0 and the X's on power for the t‐tests. The chop‐lump tests are compared to established tests via simulation for planned HIV and malaria vaccine trials. A reanalysis of the first phase III HIV vaccine trial is used to illustrate the method.     

The Exact Evaluation of 2-way Cross-classifications: An Algorithmic Solution

An algorithm is presented which permits the exact evaluation of sparse cross-classifications and contingency tables. It may be used to compute the exact point and cumulative probabilities of observed values of the X²-like test statistic, or even of the observed tables themselves. For sparser cases, it may also be used to generate part or all of the exact distributions. As a corollary, an operational solution is presented to the enumeration problem of rectangular integer matrices with arbitrarily fixed row and column sums. Exact results obtained with the algorithm accompany this presentation, giving, further support to the recommendation that the Gamma distribution be preferred to the Chi-square in sparse cases.