Testing independence of several groups of variables
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The null distribution of Wilks' likelihood-ratio criterion for testing independence of several groups of variables in a multivariate normal population is derived. Percentage points are tabulated for various values of the sample sizeN and partitions of p, the number of variables. This paper extends Mathai and Katiya's (1979) “sphericity” results and tables.Keywords:
Sphericity
Independence
Conditional independence
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Measuring conditional dependence is an important topic in statistics with broad applications including graphical models. Under a factor model setting, a new conditional dependence measure is proposed. The measure is derived by using distance covariance after adjusting the common observable factors or covariates. The corresponding conditional independence test is given with the asymptotic null distribution unveiled. The latter gives a somewhat surprising result: the estimating errors in factor loading matrices, while of root$-n$ order, do not have material impact on the asymptotic null distribution of the test statistic, which is also in the root$-n$ domain. It is also shown that the new test has strict control over the asymptotic significance level and can be calculated efficiently. A generic method for building dependency graphs using the new test is elaborated. Numerical results and real data analysis show the superiority of the new method.
Conditional independence
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Graphical model
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Independence
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Conditional probability distribution
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Objective: In study designs, the statistical power to detect a desired effect size with a specified Type-1 error is computed with the assumption that the p-value distribution under the null hypothesis follows Uniform[0.1]. However, even small departures from this assumption may inflate or deflate the statistical power beyond expectations. In this study, we illustrated the departure of the p-value distribution from Uniform[0,1] for common tests and we proposed an empirical correction to the null p-value distribution. Material and Methods: Using statistical simulation techniques, we illustrated the p-value distributions of numerous commonly used hypothesis tests under the null hypothesis, quantified their departures from Uniform[0,1], and proposed a p-value correction algorithm called 'Uniformitization'. We then graphically illustrated and discussed the level of correction with this Uniformitization approach in the corresponding p-value distribution. Results: Other than Z-test as expected and the Student t-test to most degree, all other tests we used showed non-ignorable departures from Uniform[0.1]. Our Uniformitization approach corrects the p-value distribution and brings them much closer to Uniform[0,1] especially for continuous response. Although still substantial, the correction level is limited with binary and survival response variables due to the discrete nature of these outcome variables. Conclusions: The requirement that the null-distribution of p-values be Uniform[0,1] is an indispensable one to make sure that the obtained statistical power is really where it should be, and our Uniformitization approach provides such corrections in the null distribution of p-values when they deviate from what is theoretically assumed.
p-value
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This chapter addresses the statistical analyses used in single-sample research projects. The single-sample methodology for testing a hypothesis about a population mean has its place in the social and behavioral sciences. The chapter highlights the facts about the sampling distribution of means. Specifically, the z test and z table are used to arrive at a decision about whether or not to reject the null hypothesis. The null hypothesis is rejected when it is the most reasonable conclusion given the relationship between the observed sample mean and the null hypothesized population mean. In hypothesis testing, a Type I error is committed if a true null hypothesis is rejected and a Type II error is committed when a false null hypothesis is not rejected. A t distribution is theoretically established by transforming every mean of a sampling distribution into a t statistic. The chapter presents the assumptions for the single-sample t test and the z test.
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As increasingly complex hypothesis-testing scenarios are considered in many scientific fields, analytic derivation of null distributions is often out of reach. To the rescue comes Monte Carlo testing, which may appear deceptively simple: as long as you can sample test statistics under the null hypothesis, the $p$-value is just the proportion of sampled test statistics that exceed the observed test statistic. Sampling test statistics is often simple once you have a Monte Carlo null model for your data, and defining some form of randomization procedure is also, in many cases, relatively straightforward. However, there may be several possible choices of a randomization null model for the data and no clear-cut criteria for choosing among them. Obviously, different null models may lead to very different $p$-values, and a very low $p$-value may thus occur due to the inadequacy of the chosen null model. It is preferable to use assumptions about the underlying random data generation process to guide selection of a null model. In many cases, we may order the null models by increasing preservation of the data characteristics, and we argue in this paper that this ordering in most cases gives increasing $p$-values, that is, lower significance. We denote this as the null complexity principle. The principle gives a better understanding of the different null models and may guide in the choice between the different models.
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In this paper new asymptotic expansions of the distributions of the sphericity test criterion are obtained in the null and the non-null case when the alternatives are close to the hypothesis. These expansions are obtained for the first time in terms of beta distributions. These appear to be better than the ones available in the literature.
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Asymptotic Analysis
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Abstract A P value is the result of a significance test. It measures the strength of the evidence concerning some proposition about the populations. The null hypothesis is stated, that there is no relationship or difference in the population from which the sample is drawn. A test statistic is found which would follow a known distribution if the null hypothesis were true. P is the probability of a test statistic as far from what would be expected as that observed, if the null hypothesis were true. If P is small, usually taken to be less than 0.05, we have evidence against the null hypothesis. The smaller P is, the stronger is the evidence. If the probability of a value as extreme as that observed is high we have weak evidence against the null hypothesis, the data are consistent with the null hypothesis and the relationship or difference is not significant. If the difference is not significant we have failed to show that there is a difference in the population, but one may still exist. The sample may not be large enough to detect it. We have not proved that there is no difference.
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In a two-way contingency table, the analyst is most interested in the hypotheses of either homogeneity or independence. For testing this as a null hypothesis, Pearson's statistic is most commonly used in practice. Once the null Hypothesis is rejected, he will further search forcells which caused the rejection of the null hypothesis. For this purpose, so called cell components are used. In this paper, we derive the influence function of an obsevation to the statistic, with which cells with high influence can be identified.
Contingency table
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