Nonparametric Tests for Ordering in Completely Randomized and Randomized Block Mixed Designs

Two nonparametric tests are proposed in testing for k nondecreasing treatment effects under a mixed design consisting of a completely randomized portion and a randomized block portion. The randomized block portion may contain missing observations for some of the treatments within a block. A simulation study is conducted to estimate the powers of the two tests for 3, 4, and 5 treatments under a variety of different treatment effects and underlying distributions. Powers are estimated when the randomized block portion is 1/4, 1/3, 1/2, 1.0, 1.5, 2.0, 3.0 and 4.0 times the completely randomized portion. A recommendation as to which test has higher powers is given. Citation: Magel R, Ndungu A (2013) Nonparametric Tests for Ordering in Completely Randomized and Randomized Block Mixed Designs. J Biomet Biostat 4: 170. doi:10.4172/2155-6180.1000170 J Biomet Biostat ISSN: 2155-6180 JBMBS, an open access journal Page 2 of 4 Volume 4 • Issue 4 • 1000170 which the treatment effects were different, but were not nondecreasing. Terpstra, Terpstra et al. [9] developed another test for the nondecreasing alternative in a CRD based on Spearman’s coefficient. The Wilcoxon test for paired samples [10] or the Friedman test [11,12] tests for a difference in k treatment effects in a randomized complete block design (RCBD). Page’s test [13] was designed to test the nondecreasing alternative in (1) under an RCBD design. The Durbin test [14] was designed to test the differences in k treatment effects under a balanced incomplete block design (BICB). Cao et al. [15] proposed a test for the nondecreasing alternative in a balanced incomplete block design. Alvo and Cabilio [16] introduced a test for nondecreasing treatment effects in a randomized block design with missing observations. Dubnicka et al. [1] developed a nonparametric test for a mixed two-sample design consisting of two independent samples and a paired sample testing the differences in two treatment effects. Their test statistic is formed by adding the unstandardized versions of the MannWhitney test [4], and the Wilcoxon paired sample test [10], together and then standardizing this sum by subtracting the combined mean and dividing by the square root of the combined variance. Under the null hypothesis of no difference in the two treatment effects, the test statistic will have an asymptotic standard normal distribution. Magel and Fu [17] introduced a test for this design which consists of adding the standardized versions of the Mann-Whitney test and the paired Wilcoxon test together and then dividing by the square root of two. Their test statistic also has an asymptotic standard normal distribution under the null hypothesis. When the sample size for the completely randomized portion is equal to or higher than the number of blocks, the Magel and Fu test has higher powers. When the number of blocks is higher than the equal sample sizes for the completely randomized design portions, the Dubnicka et al. [1] test has higher powers. Magel et al. [18] introduced nonparametric tests for the mixed design consisting of an RCBD and a CRB for the general alternative and for the umbrella alternative. Magel et al. [19] introduced nonparametric tests for the same design when testing for nondecreasing treatment effects as in (1). The tests proposed by Magel et al. [18] were for a mixed design consisting of a completely randomized portion and, a randomized complete block portion or a repeated measures complete portion. If the block portion or repeated measures portion had missing observations, then the tests could not be used unless the blocks that were not complete were discarded. In this research, we will propose a test for a mixed design consisting of a completely randomized portion, and a randomized block portion when there may be missing observations within a block. Introduction of Two Tests Alvo and Cabilio [16] introduced an extension of Page’s [13] nonparametric test to include both complete and incomplete blocks. Observations are ranked within a block from smallest to largest with missing observations assigned the average rank. The Alvo and Cabilio test statistic is given by i AC i*R =∑ summed over i=1,2,...,k (time periods) (2) with Ri equal to the sum of ranks assigned to treatment i (or time period i). The variance of the AC test for block i is given by 2 i i il i k (k+1)/12*(k +1)* (o average(o )) ∗ − ∑ (3) where ki is the number of treatments appearing in the block, k is the number of treatments, average (oi) is the average of the treatment numbers, not ranks, appearing in each block, and oil is the treatment number. The values of oil range from 1 to k. The asymptotic distributions of both the AC test and Page’s test when the null hypothesis is true, are normal. Two test statistics are proposed for the mixed design consisting of a completely randomized portion and a randomized block portion. The proposed test statistics are combinations of the Jonckheere-Terpstra test and the Alvo and Cabilio test. Let ZJT and ZAC represent the standardized versions of the JT and AC tests, respectively. The JT test will be applied on the completely randomized portion and the AC test will be applied on the randomized block portion. The first proposed test is given by JT AC Z = (Z + Z )/sqrt(2) first (4) Zfirst has an asymptotic standard normal distribution when the null hypothesis is true. Let E(JT) and the E(AC) denote the means of the JT and AC tests, respectively. Also, let Var(JT) and Var(AC) denote the variances of the JT and the AC tests, respectively. The second proposed test is given by Z = ((JT+AC)-(E(JT)+E(AC))/sqrt(Var(JT)+Var(AC)) last JT AC = (sqrt(Var(JT))*Z +sqrt(Var(AC))*Z )/sqrt((VarJT)+Var(AC)) The asymptotic distribution of Zlast when the null hypothesis is true is also a standard normal distribution. The question becomes, which test statistic has higher powers? Is Zfirst better is some cases and Zlast better in other cases? It is noted that both Zfirst and Zlast are sums of weighted values of ZJT and ZAC. Zfirst gives equal weights to both ZJT and ZAC. The weights that Zlast give to both ZJT and ZAC vary depending upon the variances of each of these statistics. When the variance of the JT statistic is larger, more weight is given to ZJT. When the variance of the AC statistic is larger, more weight is given to ZAC.