An Exact Paired Rank Test

An exact rank test for two dependent samples based on overall mid‐ranks is discussed which can be applied to metric as well as to ordinal data. The exact conditional distribution of the test statistic given the observed vector of rank differences is determined. A recursion formula is given as well as a fast shift algorithm in SAS/IML code. Moreover, it is demonstrated that the paired rank test can be more powerful than other tests for paired samples by means of a simulation study. Finally, the test is applied to a psychiatric trial with longitudinal ordinal data.     

Use of the Gibbs Sampler to Obtain Conditional Tests, with Applications

A random sample is drawn from a distribution which admits aminimal sufficient statistic for the parameters. The Gibbs sampleris proposed to generate samples, called conditionally sufficientor co-sufficient samples, from the conditional distributionof the sample given its value of the sufficient statistic. Theprocedure is illustrated for the gamma distribution. Co-sufficientsamples may be used to give exact tests of fit; for the gammadistribution these are compared for size and power with approximatetests based on the parametric bootstrap.     

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.     

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.     

Nonparametric All‐Pairs Multiple Comparisons

Nonparametric all‐pairs multiple comparisons based on pairwise rankings can be performed in the one‐way design with the Steel‐Dwass procedure. To apply this test, Wilcoxon's rank sum statistic is calculated for all pairs of groups; the maximum of the rank sums is the test statistic. We provide exact calculations of the asymptotic critical values (and P‐values, respectively) even for unbalanced designs. We recommend this asymptotic method whenever large sample sizes are present. For small sample sizes we recommend the use of the new statistic according to Baumgartner , Weiss , and Schindler (1998, Biometrics 54 , 1129–1135) instead of Wilcoxon's rank sum for the multiple comparisons. We show that the resultant procedure can be less conservative and, according to simulation results, more powerful than the original Steel‐Dwass procedure. We illustrate the methods with a practical data set.     

A P-Value for an Exact Two-Sided Test of Hardy-Weinberg Equilibrium

A two-tailed P-value is proposed for testing two-sided departures from Hardy-Weinberg equilibrium at a diallelic locus. The calculation of P uses the exact conditional distribution of the test statistic P, the observed number of heterozygotes in the sample. The proposed P-value is always two-tailed, unlike other P-values proposed in the literature.     

An intrinsic rank test for k independent samples

Considerations on the sensitivity of the KRUSKAL-WALLIS Analysis of Variance by Ranks suggest an alternative method that is less sensitive to spurious interchange of ranks and tied ranks yet offers sufficient power. The Intrinsic Rank Test for k Independent Samples is proposed in this sense as an alternative method to the KRUSKAL- WALLIS test. The Intrinsic Rank Test is based on rank information, but the criterion ranks (intrinsic ranks) are not directly derïved from the observations, but rather indirectly, by means of a further transformation, from the ordinal ranks assigned in the usual manner. The incidence of the intrinsic ranks upon the k samples is then used to calculate a X²-like test statistic, which is readily evaluated with the Gamma distribution.     

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.     

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.     

A New Statistical Goodness-of-Fit Test Based on Graphical Representation

A statistical goodness-of-fit test, based on representing the sample observations by linked vectors, is developed. The direction and the length of the linked vectors are defined as functions of the expected values of the order statistics and sample order statistics, respectively. The underlying method can be used to test distributional assumptions for any location-scale family. A test statistic Q_n is introduced and some of its properties are studied. It is shown that the proposed test can be generalized to test if two or more independent samples come from the same distribution. The test procedure provides a graphical method of identifying the true distribution when the null hypothesis is rejected.     

Exact tests for the analysis of case-control studies of genetic markers

OBJECTIVES: The association of a candidate gene with disease can be evaluated by a case-control study in which the genotype distribution is compared for diseased cases and unaffected controls. Usually, the data are analyzed with Armitage's test using the asymptotic null distribution of the test statistic. Since this test does not generally guarantee a type I error rate less than or equal to the significance level alpha, tests based on exact null distributions have been investigated. METHODS: An algorithm to generate the exact null distribution for both Armitage's test statistic and a recently proposed modification of the Baumgartner-Weiss-Schindler statistic is presented. I have compared the tests in a simulation study. RESULTS: The asymptotic Armitage test is slightly anticonservative whereas the exact tests control the type I error rate. The exact Armitage test is very conservative, but the exact test based on the modification of the Baumgartner-Weiss-Schindler statistic has a type I error rate close to alpha. The exact Armitage test is the least powerful test; the difference in power between the other two tests is often small and the comparison does not show a clear winner. CONCLUSION: Simulation results indicate that an exact test based on the modification of the Baumgartner-Weiss-Schindler statistic is preferable for the analysis of case-control studies of genetic markers.     

A new u-statistic with superior design sensitivity in matched observational studies

Summary In an observational or nonrandomized study of treatment effects, a sensitivity analysis indicates the magnitude of bias from unmeasured covariates that would need to be present to alter the conclusions of a naïve analysis that presumes adjustments for observed covariates suffice to remove all bias. The power of sensitivity analysis is the probability that it will reject a false hypothesis about treatment effects allowing for a departure from random assignment of a specified magnitude; in particular, if this specified magnitude is “no departure” then this is the same as the power of a randomization test in a randomized experiment. A new family of u‐statistics is proposed that includes Wilcoxon's signed rank statistic but also includes other statistics with substantially higher power when a sensitivity analysis is performed in an observational study. Wilcoxon's statistic has high power to detect small effects in large randomized experiments—that is, it often has good Pitman efficiency—but small effects are invariably sensitive to small unobserved biases. Members of this family of u‐statistics that emphasize medium to large effects can have substantially higher power in a sensitivity analysis. For example, in one situation with 250 pair differences that are Normal with expectation 1/2 and variance 1, the power of a sensitivity analysis that uses Wilcoxon's statistic is 0.08 while the power of another member of the family of u‐statistics is 0.66. The topic is examined by performing a sensitivity analysis in three observational studies, using an asymptotic measure called the design sensitivity, and by simulating power in finite samples. The three examples are drawn from epidemiology, clinical medicine, and genetic toxicology.     

Fragmentation by weirs in a riverine system: A study of genetic variation in time and space among populations of European grayling (Thymallus thymallus) in a Danish river system

Human induced habitat alterations affect the genetic structure of many fish populations. Weirs in particular have caused fragmented populations previously connected by gene-flow. We studied the effects of weirs on the distribution of genetic variation within European grayling (Thymallus thymallus) populations from the Skjern River, Denmark. We compared microsatellite data from DNA extracted from historic scale samples collected 60 years ago with data from contemporary samples. Pairwise multilocus F _ST estimates between all contemporary population samples were significant as well as exact tests for population differentiation. Assignment tests of individuals to a set of baseline samples showed correct assignment to the population of origin of between 54 and 79%. Assignment of individuals from recent samples to the historic population samples showed highly variable results (3–83%) of correct assignment suggesting different population histories. Pairwise multilocus F _ST estimates were significantly correlated with the number of intervening weirs but not with waterway distance. A simulation procedure was used to estimate differences in relative population sizes, which indicated that the main river population was approximately three times larger than those of the tributaries. There were no signs of any loss of genetic variation for the river system as a whole during these 60 years. The results show that weirs can be an important factor for creating the current distribution patterns of genetic variation among grayling populations, most likely by only allowing passive downstream drift of fry and obstructing active upstream migration. This revised version was published online in July 2006 with corrections to the Cover Date.     

Signed-Rank Tests in the Presence of Ties (with extended tables)

We begin with a review of the areas of application of the signed-rank tests (SRTs) and we conclude that the results are exact only if no ties of non-null differences exist. In order to apply the SRTs according to WILCOXON and according to PRATT also in the presence of ties, by assigning midranks, we derive their null distributions. As special cases the null distributions for the problem without ties are obtained. In order to save the practising statistician the time-consuming calculations of the distribution functions, we compute tables of critical values (for reasons of volume they will be published as part of the reprints only). For N₀ = 0 (1) 5 null differences and M = = 1(1) 10 non-null differences the critical values of all distributions with all possible tie vectors are calculated. Instructions are provided and an example serves to illustrate the use of the table. The extension of the tables are obtained by means of counting formulas given in the text. Approximations are provided in order to make the application of tests possible for larger samples as well. It is shown that the approximation of the null distribution in the presence of ties by the null distributions under the assumption of no ties in some cases overstates and sometimes understates the exact rejection probability. For N₀ = 0 (1) 10 and M = 1 (1) 10 all distributions with all possible tie vectors for the SRTs with WILCOXON and PRATT ranking are examined with respect to the lattice type of the test statistic. The result is given in table 6. It is evident that the portion of PRATT -distributions with lattice character decreases as the number of null differences increases. Continuity corrections are obtained for the asymptotic normal distribution which take into account the lattice character of the distribution of the test statistic.     

A remark on a paper by misra

MISRA (1978) sets confidence intervals for a double linear compound of multivariate normal regression coefficients by using ROY'S maximum root test criterion. The exact test statistic to be used is STUDENT'S t. The t statistic gives narrower confidence bounds than those given by ROY's maximum root statistic. A result given by MORRISON (1975, p. 18, equation 10) for profile analysis is also obtained by using the STUDENT'S t test.     

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.     

Exact Simultaneous Comparisons with Control in an r × c Contingency Table

Let categorical data coming from a control group and (r - 1) treated groups be given in an r × c contingency table. A simultaneous test procedure of the (r - 1) hypotheses that the probabilities of all c categories do not differ between the i-th treated group and the control is derived. For small tables and small cell frequencies it is exactly performed by generation of all tables having the given marginal sums. If 2 categories or 2 groups only are given the asymptotic distribution of the test statistic is known; otherwise its distribution may be simulated if the computational expenditure of performing an exact test is too large. By means of a Monte Carlo study it is shown that this method meets its level more reliably and that it has a better power than others.     

Approximating the Non-Null Distribution of the Likelihood Ratio Test Statistic

This paper is concerned with investigation into the behavior of the likelihood ratio test statistic G² when the alternative hypothesis M (QQ) is the true model. Exact moments of G² are computed empirically and three approximations are considered for approximating the non-null distribution of G². Our results show that the two parameter gamma distribution provides a closer approximation to the exact powers of G². A randomized procedure was employed to obtain critical values based on 1000 simulated samples.     

A Maximum Rank-Sum Test for One-Pulse Variation in Monthly Data

A nonparametric test to detect a pulse in monthly data is presented. This test is a maximum rank-sum test. The test statistic can be computed from frequencies or rates. The exact null distribution of the test statistic is tabulated for pulses that last 3, 4, 5, or 6 months. Estimates from a simulation study of the test's type I error rate and power are presented. The statistical modeling of the data is discussed. Several examples are given to illustrate the application of the test and the modeling procedures. Practical matters such as the treatment of tied observations, the effect of unequal lengths in the months, sample-size calculation, and post-test power analysis are discussed and illustrated with examples.