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.     

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.     

On an Unbiased Variance Estimator for the WILCOXON-MANN-WHITNEY-Statistic Based on Ranks

For some applications of the WILCOXON-MANN-WHITNEY-statistic its variance has to be estimated. So e.g. for the test of POTTHOFF (1963) to detect differences in medians of two symmetric distributions as well as for the computation of approximate, confidence bounds for the probability P(X₁ ≥ X₂), cf. GOVINDARAJULU (1968). In the present paper an easy to compute variance estimator is proposed which as only information uses the ranks of the data with the additional property that it is unbiased for the finite variance. Because of its invariance under any monotone transformation of the data its applicability is not confined to quantitative data. The estimator may be applied to ordinal data just as well. Some properties are discussed and a numerical example is given.     

Generalized Maximum Range Tests for Pairwise Comparisons of Several Populations

A multiple comparison procedure (MCP) is proposed for the comparison of all pairs of several independent samples. This MCP is essentially the closed procedure with union-intersection tests based on given single tests Q_ij for the minimal hypotheses H_ij. In such cases where the α-levels of the nominal tests associated with the MCP can be exhausted, this MCP has a uniformly higher all pair power than any refined Bonferroni test using the same Q_ij. Two different general algorithms are described in section 3. A probability inequality for ranges of i.i.d. random variables which is useful for some algorithms is proved in section 4. Section 5 contains the application to independent normally distributed estimates and section 6 the comparisons of polynomial distributions by multivariate ranges. Further applications are possible. Tables of the 0.05-bounds for the tests of section 5 and 6 are enclosed.     

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.     

A Mixed Model for Binary Data: The Modified X2-test

In this paper, a statistical model for clinical trials is presented for the special situation that a varying and unstructered number of binary responses is obtained from each subject. The assumptions of the model are the following: 1.) For each subject there is a (constant) individual Bernoulli parameter determining the distribution of the binary responses of this subject. 2.) The Bernoulli parameters associated with the subjects are realizations of independent random variables with distributions P_g in treatment group g(g = 1, 2, …, G). 3.) Given the value of the Bernoulli parameter, the observations are stochastically independent within each subject. Under these assumptions, a test statistic is derived to test the hypothesis H₀:E(P₁) = E(P₂) = … = E(P_G). It is proven and demonstrated by simulations, that the test statistic asymptotically (i.e. for a large number of subjects) follows the X²-distribution.     

Rank Tests for Complete Block Designs

In this article a general univariate K-sample rank test for complete block designs with proportional cell frequencies is derived. It is shown that the test statistic has under H₀ and for arbitrary scores asymptotically a X²-distribution with K — 1 degrees of freedom. Special cases of this test are the Kruskal-Wallis test and the Friedman test. The test is compared with the Benard-van-Elteren test, the Mack-Skillings test and a test proposed by Downton. Finally the application of the test is illustrated by two examples.     

Note on a Test for Trend in Binomial Proportions Using Ranks

For a 2xk contingency table when all the k groups fall into a natural order and for which a sequence of k values × may be assigned, ARMITAGE (1955) has suggested a X²₍₁₎ test statistic for the hypothesis H_t of a trend in the proportions; This note considers a statistic for H_t based on midrank values for the binary observations. It is shown that the proposed rank-order test for H_t coincides with that of ARMITAGE (1955).     

Some Tests for k£2 Upper Outliers in an Exponential Sample

The sequential procedure for testing up to k upper outliers proposed by Kimber (1982) for one-parameter exponential distribution is modified to a two-parameter exponential distribution. Further null distributions of some test statistics for an upper outlier-pair in a complete or censored sample from a two-parameter exponential distribution are given. Percentage points of the statistic T₁ are tabulated.     

Extended Rank Analysis of Covariance as a More Efficient Matched Analysis Considering Trend Information

Classical matched analysis, regarded as analysis of covariance (ANOCOVA) in a broad sense, makes no attempt in modeling and may therefore be inefficient. In this paper, we discuss the relative efficiencies of the ERMP (extended rank and matched‐pair) test (Chen and Quade , 2000) to standard matched methods, and extend it to the case of multivariate covariables X . Taking advantage of trend information between the response Y and the covariables X by ranking after matching, ERMP test achieves better efficiency than a proposed class of weighted matched statistics. When Y is dichotomous, the optimal weighted matched statistic is equivalent to the Mantel‐Haenszel statistic. Example and simulation results also suggest the conclusion.     

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 Modified Exact Test for 2 × 2 Contingency Tables

A modified exact test is proposed for 2×2 contingency tables. This test, which is based on a less conservative definition of the concept of significance (STONE, 1969) is compared with a modified form of Pearson's X² test and with Tocher's randomized exact (UMPU) test. The sizes of the new test lie near the nominal 0.05 levels while those of the X² test usually exceed the nominal level, sometimes by a factor of 2 or more. The power of the modified test is usually close to that of the UMPU test.     

A New Approach to Equivalence Assessment in Standard Comparative Bioavailability Trials by Means of the Mann-Whitney Statistic

By a suitable transformation of the pairs of observations obtained in the successive periods of the trial, bioequivalence assessment in a standard comparative bioavailability study reduces to testing for equivalence of two continuous distributions from which unrelated samples are available. Let the two distribution functions be given by F(x) = P[X ≤ x], G(y) = P[Y ≤ y] with (X, Y) denoting an independent pair of real-valued random variables. An intuitively appealing way of putting the notion of equivalence of F and G into nonparametric terms can be based on the distance of the functional P[X > Y] from the value it takes if F and G coincide. This leads to the problem of testing the null hypothesis H_o P[X > Y] ≤ 1/2 - ε₁ or P[X > Y] ≥ 1/2 + ε₂ versus H₁ : 1/2 ? ε₁ < P[X > Y] < 1/2 + ∈₂, with sufficiently small ε₁, ε₂ ∈ (0, 1/2). The testing procedure we derive for (₀, H₁) and propose to term Mann-Whitney test for equivalence, consists of carrying out in terms of the U-statistics estimator of P[X > Y] the uniformly most powerful level a test for an interval hypothesis about the mean of a Gaussian distribution with fixed variance. The test is shown to be asymptotically distribution-free with respect to the significance level. In addition, results of an extensive simulation study are presented which suggest that the new test controls the level even with sample sizes as small as 10. For normally distributed data, the loss in power as against the optimal parametric procedure is found to be almost as small as in comparisons between the Mann-Whitney and the t-statistic in the conventional one or two-sided setting, provided the power of the parametric test does not fall short of 80%.     

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.     

Test Statistics for Simple MARKOV Chains. A Monte Carlo Study

The statistical method for the analysis of frequency tables which has been published by GRIZZLE, STARMER, and KOCH (1969) is outlined for the case of linear functions of relative frequencies. Its use for the analysis of aggregate time series data under the model of simple MARKOV chains is suggested. Weighty objections against the resulting test statistics were grounds for the planning and performance of a Monte Carlo study. The results of the study demonstrated that the test statistic SS ( Cb =0) which is used for the comparison of different MARKOV chains is in fact X²-distributed. The test statistic SS ( F (p) = Xb ) for evaluation of the lack of fit of the model also follows a X²-distribution when a simple MARKOV chain ～ is used instead of the observed time series ～ can be determined by simply iterating the original GSK algorithm.     

A Truncated Proportional Hazards Test

We present two truncated proportional hazards tests. The first, which is applicable to those cases in which the time of action of an agent or treatment is known, was studied in two forms: One uses expected Fisher information and the other observed Fisher information. In this case the expected information statistic, C₁(a₀), had properties superior to the observed information statistic and is recommended. The use of the X² distribution with one degree of freedom for percentiles appears to be satisfactory. When the time of action is unknown, a statistic based on the maximum of the C₁(a) statistics is used. A simulation study gives empirical percentiles for the Max C₁(a) statistic which agree with those given in a study by Muenz, Green and Byar (1977).     

Various X2-Tests of Homogeneity: Some Comments

Methodological issues in the analysis of incidence rates or prevalence proportions for count data, presented in a form of a sequence of 2×2 tables, corresponding to levels (strata) of a specified variable (risk factor) X, are discussed. Suppose λ_1i and λ_2i are the incidence rates of an event D in the ith stratum for populations 1 and 2, respectively. The homogeneity (null) hypothesis is formulated in the form: H_0:λ1i=λ_2i for all i (i = 1, 2, …, I). Three X²-tests for H₀ and their theoretical bases are discussed: X_Total² which is sensitive to alternatives HA :λ1i± λ2i for at least some i; X_Comb² which is sensitive to alternatives H A : λ1iλ2i² or < λ_2i but not both for all i; and X_Diff² which is sensitive to alternatives H_A:λ1i>λ2i³ for some i and λ_1i′ < λ_2i′ for some i′ (i≠i′). These statistics satisfy the relation X_Total² = X_Comb² + X_Diff². Also, X′²-statistic for pooled data is calculated, which in conjunction with X_Comb² can serve for detecting confounding. Although most of these techniques are known, they are rather scattered in the literature, and not always considered jointly, as it is emphasized in the present paper. It is hoped that these comments will be helpful to biostatisticians as well as to epidemiologists and medical researchers in the analysis of mortality and morbidity data. For illustration, two examples with large sets of epidemiological data are given.     

The Point Evaluation Method for Approximating Function Means,Variances and Covariances

A perennial problem in statistics is the determination of biases, variances and covariances for functions of random variables X₁, X₂, …, X_n which themselves have a known distribution. A common approach is through equations based upon Taylor series approximations but a “point evaluation” method may sometimes be a useful alternative. This involves approximating the multivariate distribution of the X variables by the 2ⁿ points given by X₁=μ₁±₁, X₂ = μ₂ ±₂, …, X_n = = μ_n μ_n, where μ_i is the mean and σ_i the standard deviation of Xi, with appropriate point weights. An advantage over the Taylor series approach is that function derivatives do not have to be explicitely calculated. The point evaluation method is particularly useful in cases where the X variables are uncorrelated. Then the evaluation of the 2ⁿ points can be replaced by the evaluation of 2n points. The point evaluation method is illustrated with powers of a normally distributed variable, and with estimation of gene frequencies from ABO blood group frequencies.     

Characterization of the Mixtures of Gompertz Distributions by Conditional Expectation of Order Statistics

The mixtures of Gompertz random variables (Gompertz , 1825) X₁ and X₂ are identified in terms of relations between the conditional expectation of [exp (αX_2:2) — exp (αX_1:2)]^k given X_1:2 and the hazard rate function of the distribution, k is a positive integer and α < 0. Here X_1:2 and X_2:2 denote the corresponding order statistics. In addition, we also mention some related theorems to characterize the mixtures of Gompertz distributions. Finally, when the sample size is n, the above results are also valid and we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems).     

A Modified Chi-squared Approach for Fitting WEIBULL Models to Synthetic Life Tables1

WEIBULL models are fitted to synthetic life table data by applying weighted least squares analysis to log log functions which are constructed from appropriate underlying contingency tables. As such, the resulting estimates and test statistics are based on the linearized minimum modified X²₁-criterion and thus have satisfactory properties in moderately large samples. The basic methodology is illustrated in terms of an example which is bivariate in the sense of involving two simultaneous, but non-competing, vital events. For this situation, the estimation of WEIBULL model parameters is described for both marginal as well as certain conditional distributions either individually or jointly.