NONPARAMETRIC TESTS FOR THE SIMPLE TREE ALTERNATIVE FOR SCALE AND LOCATION TESTING AND FOR THE MIXED DESIGN FOR PORPORTOINS TESTING ARE PROPOSED

Abstract

To test for changes in location and/or scale, two nonparametric tests are proposed for the simple tree alternative. The Wald-Wolfowitz runs test, and a modified Ansari-Bradley test are used in developing these tests. The proposed tests are compared in a study to see how well they maintain their significance levels. The proposed tests powers are also estimated for three and four populations under a variety of conditions. Three different types of variable parameters vectors are considered with each vector containing a location and a scale parameter. For symmetric distributions, the Proposed First Test is best for location changes, while the Proposed Second Test is best for scale changes and combined changes. However, for non-symmetric distributions, the Proposed Second Test is best for location changes, while the Proposed First Test is best for scale changes and combined changes. In studies involving proportions, researchers often encounter scenarios where data is collected through a combination of paired samples and independent samples, constituting a mixed design. Existing statistical tests like McNemar's test and the two-sample proportion test are limited in their ability to analyze such mixed data simultaneously. This study proposes five new nonparametric test statistics (𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 ,and 𝑇5), that integrate McNemar's test for paired data with the two-sample proportion test, allowing for a unified analysis under the mixed design framework. When paired and independent sample sizes were equal, the 𝑇1 test, which assigned equal weights to the standardized McNemar's and two-sample proportion tests, exhibited the highest estimated powers, particularly when using a test for a mixed design. As sample size imbalances increased between the paired and independent samples, different tests became more powerful. Specifically, when independent samples were at least two times greater than paired samples, the 𝑇2 test (doubling the weight of the standardized two-sample proportion test) was favored. Conversely, when paired samples were substantially larger, the 𝑇3 test (doubling the weight of the standardized McNemar's test) demonstrated superior powers.