Repeated chi-square testing

A sequentialized version of the x²; goodness of fit test, called repeated x,²; test, is introduced. The form of the asymptotic distribution of the repeated x² test statistic is given under the null hypothesis as well as under local alternatives. For various numbers of cells Monte Carlo results are given for critical values, power and distribution of stopping time. Finally, the perfor-mance of the repeated and the fixed sample x² test are compared.     

Group sequential x2 statistic for testing the difference of two mean vectors in clinical trials

In this study we discuss the group sequential procedures for comparing two treatments based on multivariate observations in clinical trials. Also we suppose that a response vector on each of two treatments has a multivariate normal distribution with unknown covariance matrix. Then we propose a group sequential x² statistic in order to carry out repeated significance test for hypothesis of no difference between two population mean vectors. In order to realize the group sequential test where average sample number is reduced, we propose another modified group sequential x² statistic by extension of Jennison and Turnbull ( 1991 ). After construction of repeated confidence boundaries for making the repeated significance test, we compare two group sequential procedures based on two statistics regarding the average sample number and the power of the test in the simulations.     

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.     

On limiting distribution laws of statistics analogous to pearson's chi-square

Two test-statistics analogous to Pearson's chi-square test function - given in (1.6) and (1.7) - are investigated. These statistics utilize, apart from the number of sample elements lying in the respective intervals of the partition, their positions within the intervals too. It is shown that the test-statistics are asymptotically distributed - as the sample size N tends to infinity - according to the x ²distribution with parameter r, i.e. the number of intervals chosen. The limiting distribution of the test statistics under the null-hypothesis when N tends to the infinity and r =O(N ^α) (0<α<1), further the consistency of the tests based on these statistics is considered. Some remarks are made concerning the efficiency of the corresponding goodness of fit tests also; the authors intend to return to a more detailed treatment of the efficiency later.     

Partial and inverse autocorrelations in portmanteau-type tests for time series

We present a decomposition of the correlation coefficient between x_t and x_t?k into three terms that include the partial and inverse autocorrelations. The first term accounts for the portion of the autocorrelation that is explained by the inner variables {x_t?1 , x_t?2 , …, x _{t? k+1}}, the second one measures the portion explained by the outer variables {x _t+1, x _t+2, } ∪ {x _t?k?1, x _t?k?2,…} and the third term measures the correlation between x _t and x_t?k given all other variables. These terms, squared and summed, can form the basis of three portmanteau-type tests that are able to detect both deviation from white noise and lack of fit of an entertained model. Quantiles of their asymptotic sample distributions are complicated to derive at an adequate level of accuracy, so they are approximated using the Monte Carlo method. A simulation experiment is carried out to investigate significance levels and power of each test, and compare them to the portmanteau test.     

The exact unconditional z-pooled test for equality of two binomial probabilities: optimal choice of the Berger and Boos confidence coefficient

Exact unconditional tests for comparing two binomial probabilities are generally more powerful than conditional tests like Fisher's exact test. Their power can be further increased by the Berger and Boos confidence interval method, where a p-value is found by restricting the common binomial probability under H ₀ to a 1?γ confidence interval. We studied the average test power for the exact unconditional z-pooled test for a wide range of cases with balanced and unbalanced sample sizes, and significance levels 0.05 and 0.01. The detailed results are available online on the web. Among the values 10^?3, 10^?4, …, 10^?10, the value γ=10^?4 gave the highest power, or close to the highest power, in all the cases we looked at, and can be given as a general recommendation as an optimal γ.     

Comparison op some two sample tests for equality of variances

The F-test, F max-test and Bartlett's test are compared on the basis of power for the purpose of testing the equality of variances in two normal populations. The power of each test is expressed as a linear combination of F-probabilities. Bartlett's test is noted to be unbiased, UMPU, consistent against all alterna¬tives and the test which yields minimum length confidence intervals on the ratio of the variancesλ=σ₁ ² /σ² ₂ The two samples Bartlett critical values, although not recognized as such, are found in the works of other authors. Tables of the powers of each test are given for various values of λ, levels of significance a and the respective sample sizes, n₁ and n₂.     

Comparison of exact tests for association in unordered contingency tables using standard,mid-p,and randomized test versions

Pearson’s chi-square (Pe), likelihood ratio (LR), and Fisher (Fi)–Freeman–Halton test statistics are commonly used to test the association of an unordered r×c contingency table. Asymptotically, these test statistics follow a chi-square distribution. For small sample cases, the asymptotic chi-square approximations are unreliable. Therefore, the exact p-value is frequently computed conditional on the row- and column-sums. One drawback of the exact p-value is that it is conservative. Different adjustments have been suggested, such as Lancaster’s mid-p version and randomized tests. In this paper, we have considered 3×2, 2×3, and 3×3 tables and compared the exact power and significance level of these test’s standard, mid-p, and randomized versions. The mid-p and randomized test versions have approximately the same power and higher power than that of the standard test versions. The mid-p type-I error probability seldom exceeds the nominal level. For a given set of parameters, the power of Pe, LR, and Fi differs approximately the same way for standard, mid-p, and randomized test versions. Although there is no general ranking of these tests, in some situations, especially when averaged over the parameter space, Pe and Fi have the same power and slightly higher power than LR. When the sample sizes (i.e., the row sums) are equal, the differences are small, otherwise the observed differences can be 10% or more. In some cases, perhaps characterized by poorly balanced designs, LR has the highest power.     

Testing the equality of correlation matrices

We present results that extend an existing test of equality of correlation matrices. A new test statistic is proposed and is shown to be asymptotically distributed as a linear combination of independent x ² random variables. This new formulation allows us to find the power of the existing test and our extensions by deriving the distribution under the alternative using a linear combination of independent non-central x ² random variables. We also investigate the null and the alternative distribution of two related statistics. The first one is a quadratic form in deviations from a control group with which the remaining k-1 groups are to be compared. The second test is designed for comparing adjacent groups. Several approximations for the null and the alternative distribution are considered and two illustrative examples are provided.     

Application of boundary calculation methodologies to group sequential testing in general parametric models

For clinical trials with interim analyses, there have been methodologies and software to calculate boundaries for comparing binomial, normal, and survival data from two treatment groups. Jermison & Turnbull (1991) extended Pocock (1977) and O' Brien- Fleming (1979) boundaries to t-tests, x²-tests and F-tests for comparing normal data from several treatment groups. This paper demonstrates that the above boundaries can be applied to a wide variety of test statistics based on general parametric settings. We show that asymptotically the x² boundaries as well as the corresponding nominal significance levels calculated by Jennison & Turnbull can be applied to interim analyses based on the score test, the Wald test, and the likelihood ratio test for general parametric models. Based on the results of this paper, currently available software in group sequential testing can be used to calculate. the nominal significance levels (or boundaries) for group sequential testing based on logistic regression, A NOVA, and other parametric methods.     

Small-sample comparison of the exact and asymptotic upper tail probabilities of chi-squared goodness-of-fit statistics: the binomila and the mixture binomial

The exact and asymptotic upper tail probabilities (α = .10, .05, .01, .001) of the three chi-squared goodness-of-fit statistics Pearson's X ², likelihood ratioG ², and powerdivergence statisticD ²(λ), with λ= 2/3 are compared by complete enumeration for the binomial and the mixture binomial. For the two-component mixture binomial, three cases have been distinguished. 1. Both success probabilities and the mixing weights are unknwon. 2. One of the two success probabilities is known. And 3., the mixing weights are known. The binomial was investigated for the number of cellsk, being between 3 and 6 with sample sizes between 5 and 100, for k = 7 with sample sizes between 5 and 45, and for k = 10 with sample sizes ranging from 5 to 20. For the mixture binomial, solely k = 5 cells were considered with sample sizes from 5 to 100 and k = 8 cells with sample sizes between 4 and 20. Rating the relative accuracy of the chi-squared approximation in terms of ±10% and ±20% intervals around α led to the following conclusions for the binomial: 1. Using G2 is not recommendable. 2. At the significance levels α=.10 and α=.05X ² should be preferred over D ²; D ² is the best choice at α = .01. 3. Cochran's (1954; Biometrics, 10, 417-451) rule for the minimum expectation when using X ² seems to generalize to the binomial for G ² and D ² ; as a compromise, it gives a rather strong lower limit for the expected cell frequencies in some circumstances, but a rather liberal in others. To draw similar conclusions concerning the mixture binomial was not possible, because in that case, the accuracy of the chi-squared approximation is not only a function of the chosen test statistic and of the significance level, but also heavily depends on the numerical value of theinvolved unknown parameters and on the hypothesis to be tested. Thereto, the present study may give rise only to warnings against the application of mixture models to small samples.     

Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law from Ecology

Taylor's law, which originated in ecology, states that, in sets of measurements of population density, the sample variance is approximately proportional to a power of the sample mean. Taylor's law has been verified for many species ranging from bacterial to human. Here, we show that the variance V(x) and the mean M(x) of the primes not exceeding a real number x obey Taylor's law asymptotically for large x. Specifically, V(x) ～ (1/3)(M(x))² as x → ∞. This apparently new fact about primes shows that Taylor's law may arise in the absence of biological processes, and that patterns discovered in biological data can suggest novel questions in number theory. If the Hardy-Littlewood twin primes conjecture is true, then the identical Taylor's law holds also for twin primes. Taylor's law holds in both instances because the primes (and the twin primes, given the conjecture) not exceeding x are asymptotically uniformly distributed on the integers in [2, x]. Hence, asymptotically M(x) ～ x/2, V(x) ～ x²/12. Higher-order moments of the primes (twin primes) not exceeding x satisfy a generalized Taylor's law. The 11,078,937 primes and 813,371 twin primes not exceeding 2 × 10⁸ illustrate these results.     

An empirical comparison of several methods for testing the equality of dependent proportions

Three methods for testing the equality of nonindependent proportions were compared with, the use of Monte Carlo techniques. The three methods included Cochran's test, an ANOVA F test, and Hotelling's T² test. With respect to empirical significance levels, the ANOVA F test is recommended as the preferred method of analysis.

Oftentimes an experimenter is interested in testing the equality of several proportions. When the proportions are independent Kemp and Butcher (1972) and Butcher and Kemp (1974) compared several methods for analysing large sample binomial data for the case of a 3 x 3 factorial design without replication. In addition, Levy and Narula (1977) compared many of the same methods for analyzing binomial data; however, Levy and Narula investigated the relative utility of the methods for small sample sizes.     

DETERMINATION OF DOMAINS OF ATTRACTION BASED ON A SEQUENCE OF MAXIMA

Suppose that the maximum of a random sample from a distribution F(x) may be obtained in each of k equally spaced observation periods. This paper proposes a test to determine the domain of attraction of F(x), and investigates the properties when the sample size is very large and perhaps unknown and k is fixed and small. The test statistic is a function of the spacings between the order statistics based on the sequence of maxima and is suggested by reference to one studied previously when inference was based on the largest k observations of a random sample. A Monte Carlo study shows that the proposed test is more powerful than its main competitor. The test is illustrated by two examples.     

A fully nonparametric diagnostic test for homogeneity of variances

The authors propose a new nonparametric diagnostic test for checking the constancy of the conditional variance function σ²(x) in the regression model Y_i = m(x_i) + σ(x_i)?_i, i = 1,…, m. Their test, which does not assume a known parametric form for the conditional mean function m(x), is inspired by a recent asymptotic theory in the analysis of variance when the number of factor levels is large. The authors demonstrate through simulations the good finite‐sample properties of the test and illustrate its use in a study on the effect of drug utilization on health care costs.     

Testing equality of variances after a preliminary test of significance for correlation coefficient

On the basis of the outcome of a preliminary test of significance for the population correlation coefficient it is decided as to whether the variance ratio or the sample correlation coefficient between u=(x+y)/2 and v=x?y)/2 is to be used as a test statistic for testing the equality of variances. A method for determining the critical points for the preliminary test and the main test has been suggested. The power of the test procedure is compared with those of standard tests.     

A test of normality using nonparametrlic residuals

In this paper, we develop a test of the normality assumption of the errors using the residuals from a nonparametric kernel regression. Contrary to the existing tests based on the residuals from a parametric regression, our test is thus robust to misspecification of the regression function. The test statistic proposed here is a Bera-Jarque type test of skewness and kurtosis. We show that the test statistic has the usual x ²(2) limit distribution under the null hypothesis. In contrast to the results of Rilstone (1992), we provide a set of primitive assumptions that allow weakly dependent observations and data dependent bandwidth parameters. We also establish consistency property of the test. Monte Carlo experiments show that our test has reasonably good size and power performance in small samples and perfornu better than some of the alternative tests in various situations.     

Estimation of the Burr type III distribution with application in unified hybrid censored sample of fracture toughness

In this paper, the statistical inference of the unknown parameters of a Burr Type III (BIII) distribution based on the unified hybrid censored sample is studied. The maximum likelihood estimators of the unknown parameters are obtained using the Expectation–Maximization algorithm. It is observed that the Bayes estimators cannot be obtained in explicit forms, hence Lindley's approximation and the Markov Chain Monte Carlo (MCMC) technique are used to compute the Bayes estimators. Further the highest posterior density credible intervals of the unknown parameters based on the MCMC samples are provided. The new model selection test is developed in discriminating between two competing models under unified hybrid censoring scheme. Finally, the potentiality of the BIII distribution to analyze the real data is illustrated by using the fracture toughness data of the three different materials namely silicon nitride (Si₃N₄), Zirconium dioxide (ZrO₂) and sialon (Si_6?xAl_xO_xN_8?x). It is observed that for the present data sets, the BIII distribution has the better fit than the Weibull distribution which is frequently used in the fracture toughness data analysis.     

Data depth‐based nonparametric scale tests

Liu and Singh (1993, 2006) introduced a depth‐based d‐variate extension of the nonparametric two sample scale test of Siegel and Tukey (1960). Liu and Singh (2006) generalized this depth‐based test for scale homogeneity of k ≥ 2 multivariate populations. Motivated by the work of Gastwirth (1965), we propose k sample percentile modifications of Liu and Singh's proposals. The test statistic is shown to be asymptotically normal when k = 2, and compares favorably with Liu and Singh (2006) if the underlying distributions are either symmetric with light tails or asymmetric. In the case of skewed distributions considered in this paper the power of the proposed tests can attain twice the power of the Liu‐Singh test for d ≥ 1. Finally, in the k‐sample case, it is shown that the asymptotic distribution of the proposed percentile modified Kruskal‐Wallis type test is χ² with k ? 1 degrees of freedom. Power properties of this k‐sample test are similar to those for the proposed two sample one. The Canadian Journal of Statistics 39: 356–369; 2011 © 2011 Statistical Society of Canada     

A Bayesian analysis for the Wilcoxon signed-rank statistic

A Bayesian analysis is provided for the Wilcoxon signed-rank statistic (T⁺). The Bayesian analysis is based on a sign-bias parameter φ on the (0, 1) interval. For the case of a uniform prior probability distribution for φ and for small sample sizes (i.e., 6 ? n ? 25), values for the statistic T⁺ are computed that enable probabilistic statements about φ. For larger sample sizes, approximations are provided for the asymptotic likelihood function P(T⁺|φ) as well as for the posterior distribution P(φ|T⁺). Power analyses are examined both for properly specified Gaussian sampling and for misspecified non Gaussian models. The new Bayesian metric has high power efficiency in the range of 0.9–1 relative to a standard t test when there is Gaussian sampling. But if the sampling is from an unknown and misspecified distribution, then the new statistic still has high power; in some cases, the power can be higher than the t test (especially for probability mixtures and heavy-tailed distributions). The new Bayesian analysis is thus a useful and robust method for applications where the usual parametric assumptions are questionable. These properties further enable a way to do a generic Bayesian analysis for many non Gaussian distributions that currently lack a formal Bayesian model.