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.     

The Method of Unweighted Means in Unbalanced Analysis of Variance

For factorial experiments with unbalanced data the method of unweighted means is an alternate method of analysis which is computationally simpler than likelihood ratio based F-tests. The quality of the null approximation to the F-distribution of the statistic for the method of unweighted means and the power of this test relative to the likelihood ratio test are discussed. Recommendations are made of when in the course of data analysis the superior operating characteristics of the likelihood ratio test may not outweigh the computational simplicity of the method of unweighted means test.     

Tables for Testing the Equality of Two Proportions when Prior Information on their Common Value May be Available

In testing the equality of two proportions, one may define a rejection point T? such that if the test statistic T (in this paper, the ordinary Pearsonian chi-squared) exceeds T? then the hypothesis may safely be rejected, whatever the common value p of the two proportions may be; and similarly define an acceptance point A? such that if T ≦ T? then one may safely accept. These points may be refined if prior information is available, for example that p must lie in the central interval (c, 1 – c) or one of the extremal intervals (v, w) and (1 – w, 1 – v). Smallsample tables are provided both for the unrestricted case and for situations where one has such prior information.     

Estimation of the Number of Classes in a Population

The problem of estimating M, the number of classes in a population, is formulated as an occupancy problem in which N items are drawn from M urns. Under the assumption of a uniform distribution for the number of classes in the population, the sufficient statistic for M, which is the number of distinct classes observed, does not depend upon the number of repetitions in the sample. Point and interval estimates of M are developed using maximum likelihood and the method of moments. Both techniques give rise to the same basic equation which requires a simple iterative solution. These same techniques are used in the more general situation in which the classes can be further subdivided according to type.     

Bootstrap and Second‐Order Tests of Risk Difference

Summary Clinical trials data often come in the form of low‐dimensional tables of small counts. Standard approximate tests such as score and likelihood ratio tests are imperfect in several respects. First, they can give quite different answers from the same data. Second, the actual type‐1 error can differ significantly from nominal, even for quite large sample sizes. Third, exact inferences based on these can be strongly nonmonotonic functions of the null parameter and lead to confidence sets that are discontiguous. There are two modern approaches to small sample inference. One is to use so‐called higher order asymptotics ( Reid, 2003 , Annal of Statistics 31 , 1695–1731) to provide an explicit adjustment to the likelihood ratio statistic. The theory for this is complex but the statistic is quick to compute. The second approach is to perform an exact calculation of significance assuming the nuisance parameters equal their null estimate ( Lee and Young, 2005 , Statistic and Probability Letters 71 , 143–153), which is a kind of parametric bootstrap. The purpose of this article is to explain and evaluate these two methods, for testing whether a difference in probabilities p₂? p₁ exceeds a prechosen noninferiority margin δ₀ . On the basis of an extensive numerical study, we recommend bootstrap P‐values as superior to all other alternatives. First, they produce practically identical answers regardless of the basic test statistic chosen. Second, they have excellent size accuracy and higher power. Third, they vary much less erratically with the null parameter value δ₀ .     

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.     

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.     

Use of the Likelihood Ratio Test on the Uniform Distribution

In this article we give a simple procedure to determine the exact distribution of the likelihood ratio test of a statistical hypothesis regarding the parameter of the uniform distribution. The resulting distribution will be shown to serve as an approximation to the distribution of the likelihood ratio statistic for testing the equality of scale parameters of k independent Exponential populations.     

Use of the tetrachoric correlation model in assessing clinical tests

Categorical data on n patients are classified according to the results on an initial clinical test T, and also with respect to a subsequent and definitive diagnosis D. In this paper the sensitivity (=), specificity (=) and predictive values (=_v) are discussed with reference to the tetrachoric correlation model based on the bivariate normal density (e.g. PEARSON, 1901; KENDALL and STUART, 1972, II, p. 317). The result of HAMDAN (1970) concerning the equivalence of the tetrachoric r_t and the maximum likelihood estimate of the correlation coefficient is utilized in this paper to obtain a test of significance concerning r_t, and also the relation between the tetrachoric function w and the relative risk ψ for the 2X2 table.     

Yolk protein in the pharaoh's ant: Influence of larvae and workers on vitellogenin and vitellin content in queens

Summary Vitellin from the pharaoh's ant,Monomorium pharaonis (L.), was found to contain a single apoprotein with aM _r of 198.3 kDa. The protein is a glycoprotein exposing mannose containing carbohydrate groups.Antibodies to pharaoh's ant vitellin (v _t), raised in rabbits, were used to determine the head-, thorax-, and abdominal contents of vitellogenin and vitellin (v _g+v_t) in queens in relation to the presence or absence of larvae and workers using an ELISA test. Thev _g+v_t contents were also compared to the egg-laying rate and the weight of the queens.The results confirm the existence of a positive correlation betweenv _g+v_t contents in the queens and their access to larvae, probably related to the queens' preferential feeding on larval secretions. In queens without larvae the abdominalv _g+v_t contents declined in concordance with a low oviposition rate of 6–8 eggs/day. In spite of cessation of egg-laying within 24 hours after removal of both larvae and workers, the queens maintained basal contents ofv _g+vt. This may indicate that the presence of larvae is not only essential for the nutrition of the queens, but also for the uptake of vitellogenin in the growing oocytes. This additional stimultive factor may be based on the queens' response to primer pheromones liberated by the larvae. v _g+v_t could not be demonstrated in workers or larvae with the ELISA test. If anyv _g+vt is present in larvae and workers the amount is lower than the detection limit (1–2 ng/individual). This seems to rule out the possibility of transfer of proteins of this kind to the queens.     

Stable and invariant adjusted directed likelihoods

For the purpose of making inferences for a one-dimensional interestparameter, or constructing approximate complementary ancillariesor residuals, the directed likelihood or signed square rootof the likelihood ratio statistic can be adjusted so that theresulting modified directed likelihood is under ordinary repeatedsampling approximately standard normal with error of O(n^–3/2),conditional on a suitable ancillary statistic and hence unconditionally.In general, suitable specification of the ancillary statisticmay be difficult. We introduce two adjusted directed likelihoodswhich are similar to the modified directed likelihood but donot require the specification of the ancillary statistic. Theerror of the standard normal approximation to the distributionof these new adjusted directed likelihoods is O(n^–1),conditional on any reasonable ancillary statistic, which isstill an improvement over the unadjusted directed likelihoods.     

Photosynthetic performance of cold-sensitive mutants of maize at low temperature

The virescent character is a genetic variant in pigmentation characterized by a delay in greening. Seedlings of the virescent mutants v1, v2, v3, v4, v13, v16, v18, v19 and v26 of maize exhibit chlorosis when grown at low temperature. Chlorotic leaves contain plastids that appear to have been arrested at an early stage of development. The results indicated that V16, V2, V3 and V4 loci control early stages of chloroplast development while V1, V13 and V19 may play a role at the end of development. The mutations in the V18 and V26 loci may control an intermediate step. At the pigment level, the virescent mutants of maize differ widely from analogous mutations existing in other plants. The mutations were characterized by a reduced amount of chlorophyll a and b (up to 100 times in v16) and chlorophyll a/b ratio above normal (up to 13.7 in v16). Lutein content was reduced in all mutants (less than 3% in v16 compared to wild type) but v13, while pigments of the xanthophyll cycle were found at higher levels in v1 and v13 (more than 10 and 90%, respectively). The v2, v3, v4, v16 and v18 mutants that are most depleted in β-carotene (36 times less in average than wild type) are also deprived in D1 and D2 polypeptides. Moreover, the v2, v3, v4, v16 and v18 mutants characterized by a lower accumulation in lutein are most depleted of light-harvesting complex II. All mutants possess a functioning, fully reversible, non-photochemical quenching mechanism. This is most developed in the v13 and v19 mutants (φ_n = 0.48 and 0.44, respectively). These two mutants also have a relatively high primary photochemical yield for photosystem II and a functioning photosystem I (φ_p = 0.23 and 0.39, respectively). The most interesting mutant is v13 that shows severe chlorosis and possesses the most effective non-photochemical quenching mechanism(s), which is thought to provide protection against excess photon absorption by photosystem II.     

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.     

Testing the Intraclass Version of Kappa Coeffcient of Agreement with Binary Scale and Sample Size Determination

The intraclass version of kappa coefficient has been commonly applied as a measure of agreement for two ratings per subject with binary outcome in reliability studies. We present an efficient statistic for testing the strength of kappa agreement using likelihood scores, and derive asymptotic power and sample size formula. Exact evaluation shows that the score test is generally conservative and more powerful than a method based on a chi‐square goodness‐of‐fit statistic (Donner and Eliasziw , 1992, Statistics in Medicine 11 , 1511–1519). In particular, when the research question is one directional, the one‐sided score test is substantially more powerful and the reduction in sample size is appreciable.     

Using a Normal Approximation to Test for the Binomial Distribution

The commonly used method to test for the binomial distribution is the x²-test. In this paper, we introduce an alternative method to test for the binomial distribution. Suppose N is the number of sample groups with n individuals each, x_ij is the jth sample in ith group, a Bernoulli variable with parameter and VVI=s²/[m(1 - m)/n]. Then it is well know that the asymptotic distribution of the statistic (N - 1) VVI is x²(N - 1) under the hypothesis p₁ = p₂ = … = p_N. Here we find that VVI has an asymptotic normal distribution N(1, 2(1 - 1/n)/(N - 1)). Unlike the x²-statistic, the variance of the normal test statistic is a function of n. This method is convenient in detecting spatial patterns and dispersion in the study of diseased organisms (e.g., plants) in field samples.     

Estimation of Linkage and Association from Allele Transmission Data

The TDT provides a hypothesis test for the presence of linkage or association (linkage disequilibrium). However, since the TDT is a single test statistic, it cannot be used to separate association and linkage. The importance of this difficulty, following a significant TDT result, has been recently emphasized by Whittaker , Denham and Morris (2000), who alert the community to the possibility that a significant TDT may result from loose linkage and strong association, or from tight linkage and weak association. To attack this problem we start with the parametric model for family‐based allele transmission data of Sham and Curtis (1995) (or Sham (1998)) and find that the parameters in the model are not always identifiable. So we introduce a reparameterization that resolves the identifiability issues and leads to a valid likelihood ratio (LR) test for linkage. Since the linkage and association parameters are both of interest, we next introduce and apply an integrated likelihood (IL) approach to provide separate point estimates and confidence intervals for these parameters. The estimates are shown to have generally small bias and mean square error, while the confidence intervals have good average length and coverage probabilities. We compare the power of the IL approach for testing linkage and, separately association, with the TDT and LR.     

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.     

The estimation of Wright's fixation index from genotypic frequencies

For populations in which the genotypic frequencies are estimated, it is customary to test the deviation of the estimates from those predicted by the Hardy-Weinberg Law using the x² goodness-of-fit statistic. Maximum likelihood estimation ofWright's F statistic and its variance furnishes a similar test. Following the derivation of the variance of , this note reviews the relation between the two tests. The F statistic possesses the advantages of full sufficiency and an easier formulation of the power function. Its bias is negligible when sample size exceeds 20.     

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.