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Subject: Nonparametric statistics.

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Now showing 1 - 7 of 7
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    Comparison of Proposed K Sample Tests with Dietz's Test for Nondecreasing Ordered Alternatives for Bivariate Normal Data
    (North Dakota State University, 2011) Zhao, Yanchun
    There are many situations in which researchers want to consider a set of response variables simultaneously rather than just one response variable. For instance, a possible example is when a researcher wishes to determine the effects of an exercise and diet program on both the cholesterol levels and the weights of obese subjects. Dietz (1989) proposed two multivariate generalizations of the Jonckheere test for ordered alternatives. In this study, we propose k-sample tests for nondecreasing ordered alternatives for bivariate normal data and compare their powers with Dietz's sum statistic. The proposed k-sample tests are based on transformations of bivariate data to univariate data. The transformations considered are the sum, maximum and minimum functions. The ideas for these transformations come from the Leconte, Moreau, and Lellouch (1994). After the underlying bivariate normal data are reduced to univariate data, the Jonckheere-Terpstra (JT) test (Terpstra, 1952 and Jonckheere, 1954) and the Modified Jonckheere-Terpstra (MJT) test (Tryon and Hettmansperger, 1973) are applied to the univariate data. A simulation study is conducted to compare the proposed tests with Dietz's test for k bivariate normal populations (k=3, 4, 5). A variety of sample sizes and various location shifts are considered in this study. Two different correlations are used for the bivariate normal distributions. The simulation results show that generally the Dietz test performs the best for the situations considered with the underlying bivariate normal distribution. The estimated powers of MJT sum and JT sum are often close with the MJT sum generally having a little higher power. The sum transformation was the best of the three transformations to use for bivariate normal data.
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    A Comparison of the Ansari-Bradley Test and the Moses Test for the Variances
    (North Dakota State University, 2011) Yuni, Chen
    This paper is aimed to compare the powers and significance levels of two well known nonparametric tests: the Ansari-Bradley test and the Moses test in both situations where the equal-median assumption is satisfied and where the equal-median assumption is violated. R-code is used to generate the random data from several distributions: the normal distribution, the exponential distribution, and the t-distribution with three degrees of freedom. The power and significance level of each test was estimated for a given situation based on 10,000 iterations. Situations with the equal samples of size 10, 20, and 30, and unequal samples of size 10 and 20, 20 and 10, and 20 and 30 were considered for a variety of different location parameter shifts. The study shows that when two location parameters are equal, generally the Ansari-Bradley test is more powerful than the Moses test regardless ofthe underlying distribution; when two location parameters are different, the Moses is generally preferred. The study also shows that when the underlying distribution is symmetric, the Moses test with large subset size k generally has higher power than the test with smaller k; when the underlying distribution is not symmetric, the Moses test with larger k is more powerful for relatively small sample sizes and the Moses test with medium k has higher power for relatively large sample sizes.
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    Comparison of Proposed K Sample Tests with Dietz's Test For Nondecreasing Ordered Alternatives for Bivariate Exponential Data
    (North Dakota State University, 2011) Pothana, Jyothsnadevi
    Comparison of powers is essential to determine the best test that can be used for data under certain specific conditions. Likewise, several nonparametric methods have been developed for testing the ordered alternatives. The Jonckheere-Terpstra (JT) test and the Modified Jonckheere-Terpstra (MJT) test are for testing nondecreasing ordered alternatives for univariate data. The Dietz test is for testing nondecreasing alternatives based on bivariate data. This paper compares various tests when testing for nondecreasing alternatives specifically when the underlying distributions are bivariate exponential. The JT test and the MJT test are applied to univariate data which is derived by reducing bivariate data to univariate data using various transformations. A Monte Carlo simulation study is conducted comparing the estimated powers of JT tests and MJT tests (based on a variety of transformed univariate data) with the estimated powers of Dietz test (based on bivariate data) under a variety of location shifts and sample sizes. The results are compared with Zhao' s (2011) results for bivariate normal data. The overall best test statistic for bivariate data ordered alternatives is discussed in this paper.
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    A Nonparametric Test for the Non-Decreasing Alternative in an Incomplete Block Design
    (North Dakota State University, 2011) Ndungu, Alfred Mungai
    The purpose of this paper is to present a new nonparametric test statistic for testing against ordered alternatives in a Balanced Incomplete Block Design (BIBD). This test will then be compared with the Durbin test which tests for differences between treatments in a BIBD but without regard to order. For the comparison, Monte Carlo simulations were used to generate the BIBD. Random samples were simulated from: Normal Distribution; Exponential Distribution; T distribution with three degrees of freedom. The number of treatments considered was three, four and five with all the possible combinations necessary for a BIBD. Small sample sizes were 20 or less and large sample sizes were 30 or more. The powers and alpha values were then estimated after 10,000 repetitions.The results of the study show that the new test proposed is more powerful than the Durbin test. Regardless of the distribution, sample size or number of treatments, the new test tended to have higher powers than the Durbin test.
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    Comparing Tests for a Mixed Design with Block Effect
    (North Dakota State University, 2009) Zhao, Hui
    Tests Comb and Comb II are used to test the equality of means in a mixed design which is a combination of randomized complete block design and completely randomized design. The powers of Comb and Comb II for a mixed design have already been compared with Page's test (Magel, Terpstra, Wen (2009)) when there was little or no block effect added to the portion that was analyzed as a completely randomized design. In this paper, we wish to compare the tests when the portion of the design analyzed as a completely randomized design actually has a block effect. A Monte Carlo simulation study was conducted to compare the power of the three tests where Page's test was used only on data from the randomized complete block portion. A variety of situations were considered. Three underlying distributions were included in the simulation study. These included the normal distribution, exponential distribution, and t distribution with degree of freedom equal to 3. For every distribution, 16, 32 and 40 blocks were used in the randomized complete block design portion where the equal sample size of completely randomized data portion was 1/8, 1/4 and 1/2 the number of blocks considered. Unequal sample sizes for the completely randomized design portion were also considered. Powers were estimated for different location parameter arrangements for 3, 4 and 5 populations. Two variances, 0.25 and I, for the block effect were used. The block factor added into the completely randomized design portion didn't change the test with highest rejection percentage for the equal sample size cases, although the powers of the two tests for the mixed design decreased. For most of unequal sample size cases, Page's test has the highest rejection percentage. Overall, it was concluded that it was better to use one of the two tests for mixed design instead of Page's test when there were equal sample sizes for portion analyzed as a completely randomized design. When there were not equal size samples, but the first sample size was twice the size of the others, it was generally better to use Comb over Page's unless the number of populations became very large or there was a large block effect variance.
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    A Proposed Nonparametric Test for Simple Tree Alternative in a BIBD Design
    (North Dakota State University, 2011) Wang, Zhuangli
    A nonparametric test is proposed to test for the simple tree alternative in a Balanced Incomplete Block Design (BIBD). The details of the test statistic when the null hypothesis is true are given. The paper also introduces the calculations of the means and variances under a variety of situations. A Monte Carlo simulation study based on SAS is conducted to compare the powers of the new proposed test and the Durbin test. The simulation study is used to generate the BIBD data from three distributions: the normal distribution, the exponential distribution, and the Student's t distribution with three degrees of freedom. The powers of the proposed test and the Durbin test are both estimated based on 10,000 iterations for three, four, and five treatments, and for different location shifts. According to the results of simulation study, the Durbin test is better when at least one treatment mean is close to or equal to the control mean: otherwise, the proposed test is better.
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    Nonparametric Test for the Umbrella Alternative in a Randomized Complete Block and Balanced Incomplete Block Mixed Design
    (North Dakota State University, 2012) Hemmer, Michael Toshiro
    Nonparametric tests have served as robust alternatives to traditional statistical tests with rigid underlying assumptions. If a researcher expects the treatment effects to follow an umbrella alternative, then the test developed in this research will be applicable in the Balanced Incomplete Block Design (Hemmer’s test). It is hypothesized that Hemmer’s test will prove to be more powerful than the Durbin test when the umbrella alternative is true. A mixed design consisting of a Balanced Incomplete Block Design and a Randomized Complete Block Design will also be considered, where two additional test statistics are developed for the umbrella alternative. Monte Carlo simulation studies were conducted using SAS to estimate powers. Various underlying distributions were used with 3, 4, and 5 treatments, and a variety of peaks and mean parameter values. For the mixed design, different ratios of complete to incomplete blocks were considered. Recommendations are given.