Null Hypothesis Of Independence Research Articles

Goodness-of-Fit Tests for Large Sparse Multinomial Distributions

Abstract A goodness-of-fit statistic D 2 is introduced for use in multinomial distributions. Pearson's X2 and D 2 are both approximately normally distributed when the sample size N is not large relative to the number of multinomial categories k. Under sequences of local alternative hypotheses the test based on D 2 exhibits moderate power when the X2 test is biased. Application is made to the analysis of large sparse contingency tables. A theorem is presented that describes the likelihood ratio Λ for testing a simple multinomial distribution against a mixture of multinomial distributions. A wide variety of mixing distributions is considered, and the D 2 statistic is a special case of log Λ when testing for Dirichlet mixtures of multinomial distributions. In the case where N is large, relative to k, X2 and D 2 + k behave approximately as chi-squared random variables and differ by a very small amount under the null hypothesis. In this situation X2 and D 2 + k will generally yield the same inference. With a large, sparse multinomial distribution where N and k are both large, X2 and D 2 will usually behave as normal random variables with means and variances that are unrelated to the chi-squared distribution. In the sparse distribution, X2 and D 2 are not equivalent and X2 will accept the null hypothesis too often under certain alternative hypotheses. When testing for independence of rows and columns in a large, sparse two-dimensional contingency table, estimate the cell means conditional on the marginal totals in the usual manner. The [Dcirc]2 statistic resulting from substituting these estimated means for the cell means is highly correlated with D 2. The normalizing mean and variance of [Dcirc]2 under the null hypothesis of independence are given at the end of the article.

Goodness-of-Fit Tests for Large Sparse Multinomial Distributions

Abstract A goodness-of-fit statistic D 2 is introduced for use in multinomial distributions. Pearson's X2 and D 2 are both approximately normally distributed when the sample size N is not large relative to the number of multinomial categories k. Under sequences of local alternative hypotheses the test based on D 2 exhibits moderate power when the X2 test is biased. Application is made to the analysis of large sparse contingency tables. A theorem is presented that describes the likelihood ratio Λ for testing a simple multinomial distribution against a mixture of multinomial distributions. A wide variety of mixing distributions is considered, and the D 2 statistic is a special case of log Λ when testing for Dirichlet mixtures of multinomial distributions. In the case where N is large, relative to k, X2 and D 2 + k behave approximately as chi-squared random variables and differ by a very small amount under the null hypothesis. In this situation X2 and D 2 + k will generally yield the same inference. With a large, sparse multinomial distribution where N and k are both large, X2 and D 2 will usually behave as normal random variables with means and variances that are unrelated to the chi-squared distribution. In the sparse distribution, X2 and D 2 are not equivalent and X2 will accept the null hypothesis too often under certain alternative hypotheses. When testing for independence of rows and columns in a large, sparse two-dimensional contingency table, estimate the cell means conditional on the marginal totals in the usual manner. The [Dcirc]2 statistic resulting from substituting these estimated means for the cell means is highly correlated with D 2. The normalizing mean and variance of [Dcirc]2 under the null hypothesis of independence are given at the end of the article.

Testing For Independence of Observations in Animal Movements

Many analyses of animals movements assume that an animals's position at time t + 1 is independent of its position at time t, but no statistical procedure exists to test this assumption with bivariate data. Using empirically derived critical values for the ratio of mean squared distance between successive observations to mean squared distance from the center of activity, we demonstrate a bivariate test of the independence assumption first proposed by Schoener. For cases in which the null hypothesis of independence is rejected, we present a procedure for determining the time interval at which autocorrelation becomes negligible. To illustrate implementation of the test, locational data obtained from a ratio—tagged adult female cotton rat (Sigmodon hispidus) were used. The test can be used to design an efficient sampling schedule for movement studies, and it is also useful in revealing behavioral phenomena such as home range shifting and any tendency of animals to follow prescribed routes in their daily activities. Further, the test may provide a means of examining how an animal's use of space is affected by its internal clock.

Intraclass rank tests for independence

SUMMARY An intraclass rank test of Spearman type is proposed to test for independence in a bivariate population. The test is useful when there is prior information that the population has the same marginal distributions. It is shown that the proposed test is asymptotically equivalent to Spearman's test under the null hypothesis of independence and its contiguous alternatives and is empirically better than that in the case of small sample sizes.

ON THE METHODOLOGY OF TESTING FOR INDEPENDENCE IN FUTURE PRICES: COMMENT

The Journal of FinanceVolume 31, Issue 3 p. 977-979 Comment ON THE METHODOLOGY OF TESTING FOR INDEPENDENCE IN FUTURE PRICES: COMMENT Peter D. Praetz, Peter D. Praetz Faculty of Economics and Politics, Monash University, Australia.Search for more papers by this author Peter D. Praetz, Peter D. Praetz Faculty of Economics and Politics, Monash University, Australia.Search for more papers by this author First published: June 1976 https://doi.org/10.1111/j.1540-6261.1976.tb01942.xCitations: 12 Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article.Citing Literature Volume31, Issue3June 1976Pages 977-979 RelatedInformation

On the Asymptotic Normality of Kendall's Rank Correlation Statistic

Kendall's rank correlation statistic $T_n = \sum i > j \operatorname{sgn} (X_i - X_j)\cdot$ sgn $(Y_i - Y_j)$ is well known to be asymptotically normally distributed under the null hypothesis of independence as the sample size $n\rightarrow\infty$. In the note it is shown that this assertion can be obtained easily from the recurrence formula $p_n(t) = (1/n) \sum^n_{k =1}p_{n - 1}(t - 2k + n + 1)$ for the probability distribution $p_n$ of $T_n$ (see Kendall (1970), e.g.). This recurrence formula implies that $T_n$ has the same distribution as a sum of $(n - 1)$ well defined independent random variables to which the Lyapunov criterion applies.

Approximating the Distribution of the Sample R 2 in Best Subset Regressions

This note presents research on the problem of determining the distribution of the usual sample R2 statistic in multiple regression studies where the variables to be included in the regression equation are the subset of k variables, from a set of m variables, which maximize the sample R2 value or satisfy some similar criterion. A Monte-Carlo approach was used to estimate certain percentile points of the distribution of R2 under the null hypothesis of independence between the dependent variable and the m independent variables. A function has been developed which appears to provide a good approximation to percentile points of the R2 distribution.

Approximating the Distribution of the Sample R2in Best Subset Regressions

This note presents research on the problem of determining the distribution of the usual sample R 2 statistic in multiple regression studies where the variables to be included in the regression equation are the subset of k variables, from a set of m variables, which maximize the sample R 2 value or satisfy some similar criterion. A Monte—Carlo approach was used to estimate certain percentile points of the distribution of R 2 under the null hypothesis of independence between the dependent variable and the m independent variables. A function has been developed which appears to provide a good approximation to percentile points of the R 2 distribution.

An Analysis of Variance for Categorical Data

Abstract A measure of variation for categorical data is discussed. We develop an analysis of variance for a one-way table, where the response variable is categorical. The data can be viewed alternatively as falling in a two-dimensional contingency table with one margin fixed. Components of variation are derived, and their properties are investigated under a common multinomial model. Using these components, we propose a measure of the variation in the response variable explained by the grouping variable. A test statistic is constructed on the basis of these properties, and its asymptotic behavior under the null hypothesis of independence is studied. Empirical sampling results confirming the asymptotic behavior and investigating power are included.

Canonical Association Between Nominal Measurements

Abstract The problem of the measurement of association between two attributes is of importance in social sciences [5]. Following Hotelling [6, 7], Lancaster [9], and Kendall [8], the canonical structure of a two-way classification is obtained. Based on the canonical correlations, three measures of association are suggested. Their metric properties are summarized. One of the measures is shown to reduce to the measure suggested by Cramér [1]. Their large sample distributions are approximated on the null hypothesis of independence between the two attributes.

Canonical Association between Nominal Measurements

Abstract The problem of the measurement of association between two attributes is of importance in social sciences [5]. Following Hotelling [6, 7], Lancaster [9], and Kendall [8], the canonical structure of a two-way classification is obtained. Based on the canonical correlations, three measures of association are suggested. Their metric properties are summarized. One of the measures is shown to reduce to the measure suggested by Cramer [1]. Their large sample distributions are approximated on the null hypothesis of independence between the two attributes.