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On a test for multivariate normality and on certain statistical procedures valid for complex surveys
Abstract
A complex survey design (e.g., two-stage cluster sampling design) induces complex correlation structure in the sample observations and so the Shapiro-Wilk test which is based on the assumption of independence is not valid for such a complex design. Through a Monte Carlo simulation study, it has been demonstrated that the consequence of using the standard Shapiro-Wilk test for checking normality of complex survey data could be severe in that the test fails to control the type I error. As a solution to this problem, first a transformation is suggested on the original sample so that the transformed observations are uncorrelated and then the standard Shapiro-Wilk test is applied to the transformed sample observations. The new test performs well in controlling the type I error and its power performance is much better than the original Shapiro-Wilk test. Many commonly used multivariate methods rely on the assumption of multivariate normality and so it is crucial to assess the multivariate normality of a given random sample before using such methods. The Malkovich-Afifi test for multivariate normality proposed twenty five years back is revisited. The test was constructed by exploiting the well-known fact that a random vector follows multivariate normal distribution if and only if all possible linear combinations of the component random variables follow univariate normal distributions. However, this test has received very little attention possibly because of the computational difficulties associated with the test. A Monte Carlo simulation study shows that the power of the Malkovich-Afifi test could be much higher than that of a rival test proposed ten years later. In order to implement the test, a computational algorithm and a SAS program is developed. A goodness of fit test is proposed along the lines of the Pearson's famous $\chi\sp2$ test but in which both the cell probabilities and cell frequencies are estimated. The asymptotic null distribution of the test statistic can be obtained under certain regularity conditions. The asymptotic null distribution of the proposed test statistic is not $\chi\sp2.$ Based on the result, a method is developed to check the arbitrary distributional assumptions about the random effects and pure errors in a mixed linear model. The method is further examined by a simulation study. The well-known Wilks $\Lambda$ test for testing the regression coefficients in a multivariate linear regression is valid when the observations are obtained by simple random sampling. The dissertation addresses the problem when sample observations are collected using a two-stage cluster sampling design. Using a suitable multivariate transformation, the original multivariate observation vectors are first transformed into new multivariate observation vectors which have simpler correlation structure. Using these new transformed multivariate observations, a test similar to the Wilks $\Lambda$ is proposed.
Subject Area
Statistics
Recommended Citation
Wu, Chien-Hua, "On a test for multivariate normality and on certain statistical procedures valid for complex surveys" (1998). ETD collection for University of Nebraska-Lincoln. AAI9903788.
https://digitalcommons.unl.edu/dissertations/AAI9903788