Practicability Of Parametric Test Based On Rank Transform Statistic

Chapter 1 Practicability of Parametric Test Based on Rank Transform Statistic for Common Quantitative Data.BackgroundStatistical test was divided into parametric test and nonparametric test. Parametric test usually need to meet certain conditions for quantitative data, such as normality and homoscedasticity. When data violated these conditions, we have two methods to chance. One is data transformation, the other is nonparametric test. Actually, if data transformations don’t meet the conditions of parametric test and there is no nonparametric test for it, like factorial design, orthogonal design and so on. At this time, parametric test is inappropriate. As a result, parametric test based on rank transform statistic (semi-parametric test) may be considered as an alternative method for common quantitative data.The development of rank transformation has nearly 8 decades. Rank transformations for data which was named of method of ranks, was first proposed by Friedman in 1937. Wilcoxon (1945) proposed nonparametric test of paired-sample (one-sample) design and two independent sample designs firstly, which was extended to unequal sample size by Mann and Whitney in 1947. Nonparametric test of two independent sample designs was spread to K independent sample designs by Kruskal and Wallis in 1952, Kruskal-Wallis test. Conover (1981) concluded the relationship between nonparametric test and semi-parametric test, and presented four coding methods of ranks using widely at present.① The entire set of observations is ranked from smallest to largest, with the smallest observation having rank 1, the second smallest rank 2, and so on. Average ranks are assigned in case of ties. ② The observations are partitioned into subsets and each subset is ranked within itself independently of the other subsets. ③ This rank transformation is ① applied after some appropriate reexpression of data.④ the ② type is applied to some appropriate reexpression of data. In fact, we found that these four series rank method can be summarized into two categories, namely overall rank or different combinations rank. In practice, overall rank was used to paired-sample (one-sample) design, two independent sample designs as well as three or more independent sample designs generally. But randomized block design is usually adopted by different combinations rank. Iman (1974,1984) compared randomized block design ANOVA, nonparametric tests and semi-parametric test. Conover (1982) discussed robustness and power of the covariance analysis of variance based on the rank transformations for quantitative data of covariance design. Scheirer (1976), Blair (1987), Sawilowsky (1989), Thompson (1991,1993) and so on, compared all kinds of methods of quantitative of factorial designs studied. Zimmerman (2012) made a more comprehensive simulation of rank transformations for one-sample sample, two independent sample designs as well as three or more independent sample and revealed the relationship between nonparametric test and semi-parametric test.As we all know, parametric test is the best one for quantitative data, which meet normality and homoscedasticity. However, if data transformations don’t meet the conditions of parametric test, both correction parametric method (e.g. Welch) and nonparametric test all are considered as well as semi-parametric test. Although Conver revealed the relationship between nonparametric test and semi-parametric test, the application of semi-parametric test is still a worth question to exploring. In this paper, for the common data types, paired-sample (one-sample) design, two independent sample designs as well as three or more independent sample designs, we comprehensively introduces semi-parametric test and compares the three methods (parametric test, nonparametric and semi-parametric test)by mean of Monte-Carlo simulation, in order to practicability of semi-parametric test in a variety of circumstances.ObjectiveTo explore the practicability of parametric test, nonparametric test and semi-parametric test for quantitative data of paired-sample (one-sample) design, two independent sample designs as well as three or more independent sample designs when data violate normality and homoscedasticity.MethodsIntroducing the theory of parametric test based on rank transformation and comparing type I error and power of the three kind methods by means of Monte Carlo Simulation considering that data are normality or negative skewness and homoscedasticity or heteroscedasticity.We considered four factors, distribution, mean, variance and sample size and four conditions,① negative distribution(paired-sample or one-sample);② homoscedasticity and negative distribution;③ heteroscedasticity and normal distributionl;④ heteroscedasticity and negative distribution.Sample size:5,6,8,10,12,14,16,20,25,30,35,40,50 and 100; there are two kinds of design, balanced design and unbalanced design. There are three levels for three independent sample designs. When data have same ties, we get average instead. We use welch test when data are heteroscedasticity. Each case is replicated 10000 times to insure the stability of the simulated sampling distributions of the statistics involved. Simulations were carried out on R software.We use adjusted methods of parametric test when data are heteroscedasticity. The probability density function of negative distribution is as follows: Where mean: skewness= we can use rsn (n=,xi=, omega=, alpha=, tau=, dp=) function of R software to generate negative distribution data.ResultsThe results indicate that he three methods contributed to type Ⅰ error well no matter whether homoscedasticity or not, but the power of parametric test is clearly lower than other methods. Parametric test is superior to two others when data are normality but heteroscedasticity, parametric test based on rank transformation is worse, nonparametric methods is the worst. Nonparametric test has a good consistency to parametric test based on rank transformation in different designs.ConclusionType Ⅰ error and power of parametric test based on rank transformation is nearly equal to that of the nonparametric test when data are applicable to nonparametric test.Chapter 2 Practicability of Parametric Test Based on Rank Transform Statistic for Factorial Design.BackgroundThere is no suitable non-parametric method for quantitative data of factorial design currently. Actually, we generally use analysis of variance, but data should meet certain assumption, such as normality and homogeneity of variance. Some scholars (1976) mentioned that, When data does not meet the parametric test, we can use Kruskal-Wallis test, but this method cannot get the results of the main effects and interactions. Blair(1987) found that the parametric test based on rank transform statistic has serious biases along with the increasing of the sample size and effect size for factorial design studies. It is found in Shlomo S(1989) that the size and type Ⅱ error of parametric test based on rank transform statistic are influenced by the counts of non-zero effective sizes, sample size and the value of effect sizes with consideration of normality, mix normality, t distribution, uniform distribution and exponential distribution. Thompson(1991,1993) studied the interaction effect in factorial design. Hence applicability of parametric test based on rank transform statistic in heterogenicity and/or unbalance design study calls for further research.ObjectiveIntroduction the theory of based on rank transformation statistic for factorial design. At the same time, use simulation methods to compare parametric method with based on rank transformation method, in order to provide a new way to analysis for data of factorial design.MethodsIntroducing the theory of parametric test based on rank transformation and comparing type Ⅰ error and power of the two kind methods by means of Monte Carlo Simulation considering that data are normality or negative skewness and homoscedasticity or heteroscedasticity.We considered four factors, distribution, mean, variance and sample size and three conditions, ①heteroscedasticity and normal distributionl ② homoscedasticity and negative distribution; ③ heteroscedasticity and negative distribution.Sample size:2,4,6,8,10,15,20,30 and 50, there are two kinds of design, balanced design and unbalanced design. There are two levels for each factors and there are two factors. When data have same ties, we get average instead. Each case is replicated 10000 times to insure the stability of the simulated sampling distributions of the statistics involved. Simulations were carried out on R software.Type I error rates and power of the two tests were assessed under the following conditions:a. All effects null; that is to say, c=0 for all effects. b. The A main effect nonnull; all effects null. c. The A and B main effect nonnull; all other effects null. d. The A main effects and AxB interaction effect nonnull; all other effects null. e. All effects nonnull.The model used for data generation is as follows:yij=μ+a1+bj+abij+εij. a. all effects were modeled by setting 0. b. The A main effects were modeled by setting a1=C,a2=-c, where c took on the values 0.25,0.5,0.75,1,all other effects 0. c. a1=c,a2=-c,b1=c,b2=-c, all other effects 0. d. a1=c,a2=-c, a1b1=c, a1b2=-c,a2b1=-c, a2b2=c, all other effects 0. e. a1=c,a2=-c, b1=c,b2=-c,a1b1=c,a1b2=-c,a2b1=-c,a2b2=cResultsThe result of this study indicated that parametric test based on rank transformation performs well at detecting interactions and main effects when data is heteroscedasticity no matter normal distribution or negative distribution. Parametric test and parametric test based on rank transformation are not robust in all cases. When data is negative distribution but heteroscedasticity, type I errors of parametric test will inflate, while parametric test based on rank transformation will more suitable in this condition. Parametric test is better than parametric test based on rank transformation only in one effect is nonnull, in other cases, parametric test based on rank transformation is better when data is normality but heteroscedasticity. When data is negative distribution but homoscedasticity, two methods have a good consistency.ConclusionParametric test based on rank transform statistic performs well at detecting interactions and main effects when data violates homogeneity of variance.