Comparison of Rank Transformation Test Statistics with Its Nonparametric Counterpart Using Real-Life Data

Over the years, non-parametric test statistics have been the only solution to solve data that do not follow a normal distribution. However, giving statistical interpretation used to be a great challenge to some researchers. Hence, to overcome these hurdles, another test statistics was proposed called Rank transformation test statistics so as to close the gap between parametric and non-parametric test statistics. The purpose of this study is to compare the conclusion statement of Rank transformation test statistics with its equivalent non parametric test statistics in both one and two samples problems using real-life data. In this study, (2018/2019) Post Unified Tertiary Matriculation Examinations (UTME) results of prospective students of Ladoke Akintola University of Technology (LAUTECH) Ogbomoso across all faculties of the institution were used for the analysis. The data were subjected to nonparametric test statistics which include; Asymptotic Wilcoxon sign test and Wilcoxon sum Rank (both Asymptotic and Distribution) using Statistical Packages for Social Sciences (SPSS). In the same vein, R-statistical programming codes were written for Rank Transformation test statistics. Their P-values were extracted and compared with each other with respect to the pre-selected alpha level (α) = 0.05. Results in both cases revealed that there is a significant difference in the median of the scores across all faculties since their type I error rate are less than the preselected alpha level 0.05. Therefore, Rank transformation test statistics is recommended as alternative test statistics to non-parametric test in both one sample and two-sample problems.


Introduction
In social, behavioural and physical sciences, studies have shown the usefulness of statistical analysis in arriving at conclusions in problems solving Odukoya, Omonijo, Olowookere, John & Atayero, 2019; Olowookere, Omonijo, Odukoya & Anyaegbunam, 2020). Also, in statistical analysis, non-normality of the data set may be inevitable, in case of the existence of such data set, the parametric test is not fit for the study. Hence, non-parametric tests or transformation of data becomes necessary or better still use a robust estimator to correct the anomalies.
However, Conover and Iman (1981) came up with another test statistic known as Rank transformation test statistics. Here all parametric test statistics were re-written in a rank form to fill the distance between parametric and non-parametric test statistics. Therefore, it is very imperative to apply some of these test statistics to real life data and compare whether the conclusion statement of rank transformation test statistics by Conover and Iman (1981) in one sample and two samples problems is the same with the already existing non-parametric test statistics. Hence, to accomplish this task the scores of post Unified Tertiary Matriculation Examination (Post-UTME) 2018/2019 academic session prospective students of Ladoke Akintola University of Technology (LAUTECH) across all faculties of the institution were collected and used for the analysis.

Literature Review
In order to find out how sensitive and robust some inferential statistics (one sample test statistics) are to outliers, Ayinde et al, (2016) carried out a simulation studies in this respect. Test statistics considered are student t-test, z-test, sign test, wilcoxon signed rank test (both distribution and asymptotic), rank transformation test and trimmed test statistics. In their study, the experiment was replicated five thousand times using montecarlo study and conducted at eight levels of sample sizes n = 10, 15, 20, 25, 30, 35, 40 and 50. The data used in the study were simulated from Gaussian distribution with the aid of R-statistical programming codes. 10 and 20 percent of some randomly selected data were polluted with twenty-one magnitudes of outliers ranges from -10 to +10 and compared the type I error rate of the test statistics with the most commonly used preselected alpha levels 0.1, 0.05 and 0.01. Their results revealed that the type I error rates of student t-test, Rank transformation test, Asymptotic wilcoxon test are good at all the preselected alpha levels. However, parametric tests were spotted to be sensitive to outliers. Also, at 0.1 level of significant; sign test distribution and its asymptotic, at 0.05 level of significant trimmed t-test, wilcoxon signed and at 0.01 level of significant; trimmed t-test and sign test asymptotic are robust to outliers respectively and thereby recommended.
In  (8) levels of  sample sizes namely: 10, 15, 20, 25, 30, 35, 40 and 50 whereby data were simulated from Gaussian distribution with the aid of R-statistical programming codes. Also, to exhibit a different degree of correlations between the paired samples, the levels of correlations reconsidered are; 0, 0.3, 0.6, 0.9, 0.95, and 0.99. At each sample size, ten and twenty percent of the simulated data were invoked with twenty-one (21) various magnitude (k) of outliers ranges from -10 to +10. The three (3) commonly used preselected levels of significance used were 0.1, 0.05, and 0.01. The Type I error rate of the inferential test statistics was determined when there was no outlier in the data sets. While to assess the sensitivity and robustness of the test statistics hence, Power rate was determined. Also, in their study, they affirmed that a test was examined to be robust if its estimated type I error rate approximately equal to the true error rate and it has the highest number of times it approximates the error rate when counted over the preselected alpha level otherwise sensitive. Their results revealed that paired t-test and Asymptotic wilcoxon sign test have better Type I error rate across different categories of correlation and that Asymptotic sign test, Rank transformation test and distribution sign test, and Trimmed t-test statistics are respectively robust to outliers at all alpha levels.
As a means for understanding the nature of rank transform, this made Michael (1990) to make use of rank transform asymptotic version of it for testing problems in some two factor designs. In his investigation, he found out that this version works well and helps to suggest which Analysis of Variance (ANOVA) procedure should be employed. He applied this method on the balanced and unbalanced nested model including their two way layout with and without interaction.
Attention was paid to Aligned Rank Transform (ART) by Mausouri, et al (2007). They considered ART as a testing procedure in linear model where two way layout and multiple comparison techniques were considered. The authenticity of the technique was ascertained through simulation study and results revealed that the technique procedure was found to be quiet robust against violations of the assumption of continuous error distribution. Edgar and Holger (2012) explicitly form a test statistic which was derived with the aid of partial rank transform and its asymptotic distribution was also determined. Their research was centered on the two factor mixed models (Rank procedures) where fixed treatment effects as a problem was considered in two factors, mixed model with interaction and unequal cell frequencies in the absence of normality assumptions. Their further research revealed that when cell frequencies are not equal, they regarded generalization of Friedman's test as a proposed test statistic. In their research, they compared the proposed test with the equivalent already existing test under the assumption of normality by the criterion of asymptotic relative efficiency. Finally, the exact condition distribution was discovered and proposed are the estimators and confidence intervals for the shift effect.
An investigation was carried out by Haiko (2017) on the implication of discrete distribution on the error rates. The analysis was centered on the two-way layouts Analysis of Variance (ANOVA) design, the Aligned Rank Transform (ART) and compared it with the parametric F-test. Also, some other nonparametric tests considered are; Rank Transform, Inverse Normal Transform, combination of ART and INT, combination Puri and Sen's L statistics, Van der Waerden and Akritas and Brunners ATS. In the same vein, two major continuous distributions were used which include; Uniform and exponential distributions. He restricted his scaling impact to ART and the combination of ART and INT methods. In his analysis, two effects were revealed; first one with increasing cell counts where their error rates rise beyond any acceptable limit up to 20 percent and more. Likewise, the error rates of the second one rise as the number of the dependent variables decrease. But for underlying exponential distribution, its behavior is distinctively severe than that of uniform distribution. He finally claimed that if the mean cell frequencies are more than ten (10) ART should not be applied.
A most powerful sequential rank test was derived in (2017) by Jan. the test was meant for the hypothesis of randomness against a general alternative which include as a special case for the alternative hypothesis regression in location or the two samples difference in location. As the classical results of Hajek and Sidak (1967), he also derived for the one sample problem for independence of two samples in an analogous spirit. For a fixed sample size similarly, he derived its test and the results obtained brought about argument in favour of already existing ones.
Rayner and Best (2013) did a thorough work on rank transformation and Analysis of variance by extending the procedures of rank transformation for balanced designs. Their research was centered on the construction of table of counts for the midranks of data where they made use of contingency table constructed by Davy (1998, 1999) so as to partition the test statistic of Pearson.
Furthermore, in the further application of rank transformation, Mitchell et al, (1994) compared and contrasted the results of rank transformation test statistic with parametric test when applied to genotoxicity data with examples from cultured Mammalian cells. In their study, the results of rank transformation test revealed that there was no loss of power and compete favourably with parametric test. Likewise, when there is 2x2x2 fixed effects Analysis of Variance, Sawilowsky et al, (1989) investigated the power and type I error rates of the rank transformation statistic. Their research showcased that in some situation, rank transformation was declared robust and not robust at times under some certain circumstances.
In order to determine the efficacy and efficiency of rank transformation test relative to Ftest, Thompson and Ammann (1989) carried out a research in this direction. They concluded that the efficacy of rank transformation coincides with the Kruskal -Wallis test. Also, in (1999) Thompson embarked on an in-depth study on rank transform statistic proposed by Conover and Iman (1981) which was based on testing for interaction in a balanced twoway classification. He was able to find out that rank transformation is asymptotically chisquared divided by its degrees of freedom even when there is only one main effect and exactly two levels of both main effect. He further corroborated that as the sample size is increasing, the expected value of the rank transformation is tending to infinity.

Non-parametric
The alternative test to parametric tests when the distribution from which data are collected are indisposed is the non-parametric test. Non-parametric test has being in existence as early as the 13th century or earlier use in estimation by Edward Wright, 1599. John Arbuthnot, (1710) in his early analysis used non-parametric statistics which include the median and the sign test in analyzing the human sex ratio at birth.
Moreover, nonparametric tests are usually used to test the hypothesis that has nothing to do with population parameters but rather based on the shape of the population frequency distributions. It is generally known that, if the data do not satisfy the properties of parametric test (that is normality assumptions, equal variance, and continuous), then such data may be analyzed with a nonparametric test. But in a real sense, if a nonparametric test is required, additional data may be needed to make the same general conclusion. (Ayinde, et al, 2016).

Characteristics of nonparametric
The non-parametric test has some characteristics which are highlighted as follows: (v) It is usually useful when data is nominal or ordinal even when measured on a weak measurement scale. (vi) The procedure is very easy to understand even for the researchers who are not mathematically or statistically oriented. (vii) Its Computations are easy to perform even without automated instrument such as calculators or computers compared to parametric test. (viii) They are meant for small numbers of data, such as; classifications, counts, and ratings. This enables us to find the difference based on a specific significance level. Then, we can assert whether the null hypothesis should be rejected or not.
(vii) The results are explained and a conclusion is drawn out. (Hesse et al, 2018).

The sign test
One sample parametric test that is used to compare a single sample with some hypothesized value is called the sign test. The sign test is being used in a situation whereby one sample or paired t-test is majorly applied. To each observation it assigns a sign either positive or negative according to whether it is greater or less than some hypothesized median value and considers whether this is substantially different from what we would expect by chance. If any observation is exactly equal to the hypothesized median they are ignored and dropped from the sample size. The test statistics are + or − as the case may be. If values with negative signs are the least, then our test statistic is − , otherwise the test statistic is + . if P (t ≤ + ) < or P (t ≤ − ) < Under null hypothesis, binomial distribution is used and null hypothesis is hereby rejected. When the test is two-tail, 2 ⁄ is used instead of . The sign test has its distribution following binomial (n, 1 2 ⁄ ) asymptotically. (Ayinde, et al, 2016).

The asymptotic test
The asymptotic test statistic for sign test is

Procedures:
To run a sign test, the following assumptions must hold: (i) Observations must be measured on at least ordinal scale. (ii) The sample must be randomly drawn from the population. (iii) The probability of values falling on the hypothesized median is low.

Wilcoxon rank sum test
This is a test that is quick and easy for two independent samples. It is a good alternative test to the student t-test when the normality assumptions about the data could not hold. The numerical alternative test to it is Mann-Whitney U-test. If only ordinal data are available the test can also be performed. With respect to the median, it tests the null hypothesis that the two distributions are identical against the alternative hypothesis that the two distributions differ.
The test is based on the Wilcoxon rank sum test statistic ''W'', which is the sum of the ranks of one of the samples. The procedures:

Wilcoxon rank sum test assumptions
(i) All data are ranked from smallest up to the largest.
(ii) Add the ranks in the smaller group to get the test statistic W, if one group has fewer values than the other such as n 1 <n 2 and if n 1 = n 2 , add the ranks in the group containing the most smallest ranks. (iii) Check the appropriate table for the statistic W based on the sample sizes and determine its probability accordingly. (iv) Reject H 0 or accept H 0 based on the p-value, the statistic W becomes approximately normal as the two sample sizes increase.
By standardizing W the test Z-statistic is given as; Where, When the null hypothesis (no difference in distributions) is true, P-values for the Wilcoxon test are based on the sampling distribution of the rank sum statistic W. Also, from special tables, software or a normal approximation (with continuity correction)P-value can be calculated.

Rank Transformation
According to Conover and Iman, (1981) they defined Rank Transformation as one in which the parametric procedure is applied to the ranks of the data instead of the data themselves. They developed some parametric tests in their rank form which was considered as a viable tool for developing new non-parametric procedures to solve new problems. Non-parametric methods as an integral part of an introductory course in statistics this new approach provides a useful pedagogical technique.
Also in a situation where parametric assumptions could not hold, this approach is viewed as a useful tool for developing new non-parametric methods. (Conover and Iman, 1981 and Ayinde, et al, (2016)).

T-test for Rank Transformation in one sample
Let D 1 , D 2 ,…, D n be independent random variables with the same mean but in the case of matched pairs (X i , Y i ); D i = X i , -Y i . For the case of one sample D i = X i -0 where 0 is the hypothesized mean value. For the Wilcoxon signed rank test the D i 's are replaced by the signed ranks R i , where R i = (sign D i ) x (rank of / D i /). When the test statistic T is too large or too small as measured by the normal approximation, the hypothesis is hereby rejected.
Alternatively t-test statistic is computed on the signed ranks as follows; This is compared with the t-distribution (n-1) df (degree of freedom).

T-test for Rank Transformation in two independent samples
Assuming that X 1 ,------X n and Y 1, -------Y n are two independent random samples. To test the hypothesis that 1 = 2 the parametric procedure employs the two sample t-statistic.
The non-parametric Wilcoxon Mann-Whitney twosamples test requires by replacing the data by the ranks R i from 1 to N, and uses the statistic in its standard form with the adjustment for ties incorporated.
Where = 1 + 2 − 2 and S = ∑ =1 is denoted to be the sum of ranks of the independent variables (X's). The statistic is compared with the standard normal distribution and exact tables may be used for S if there are no ties and the sample sizes are less than 20. (Conover, 1980). Its procedure is based by computing t on the rank of R i 688 Comparison of Rank Transformation Test Statistics with Its Nonparametric Counterpart Using Real-Life Data to get the statistic given in equation (7).
With 1 + 2 − 2 degree of freedom. According to Conover and Iman (1981), algebraically an important relationship between tR and T is revealed as: by mathematical expression, tR is a monotonically increasing function of T. Therefore, with all of Wilcoxon -Mann -Whitney test good properties, it may be performed using tR as a test statistic instead of T. (Conover and Iman, 1981)

Descriptive Analysis
The descriptive analysis of results of (2018/2019) Post UTME of Ladoke Akintola University of Technology (LAUTECH) prospective students across all faculties of the institution was carried out using SPSS. The following results were obtained as seen in table 2:  figure 1) and table 2 respectively, indicating the faculty of Basic Medical science as the highest faculty applied for by the applicant and Management science as the least faculty applied for by the applicant.

Test of Normality across all faculties
Hypothesis test H 0 : The applicant score are normally distributed H 1 : The applicant score are not normally distributed

One Sample Problem
H 0 : The median score is not significantly different H 1 : The median score is significantly different  Table 4 shows that there is no significant difference in the median scores across all faculties in both test statistics since their P-values are greater than the preselected alpha level 0.05. Therefore, rank transformation can also be used as an alternative test statistic to a non-parametric test statistics in one sample problem.

Two Samples Problem
H 0 : There is no significant difference between the medians of the two groups H 1 : There is a significant difference between the medians of the two groups.  From table 5, it can be seen that there is a significant difference between the medians of the two groups (faculties) in both test statistics since their type I error rate is less than the preselected alpha level 0.05. Thus, rank transformation can also be used as an alternative test statistic to non-parametric test statistics in two samples problem.

Conclusions
The data used in the analysis for each faculty are described using descriptive analysis which shows that faculty of Basic Medical sciences has the highest number of applicants with 3772 number of applicants, faculty of Engineering and Technology was the second with 1522 number of applicants, faculty of pure and applied science was the third with 631 number of applicants, faculty of Agricultural Science was the fourth with 347 number of applicants, faculty of Environmental science was the fifth with 204 number of applicants and faculty of Management science has the least applicants with a total number of 72 applicants.
The test of normality was carried out on applicant post UTME score for each faculty using Shapiro-wilk test for the data and it was discovered that the score is not normally distributed across all faculties and thereby shows that parametric test is not suitable for the analysis.
The data were subjected to Non-parametric tests using Wilcoxon asymptotic sign test, Wilcoxon sum rank (Asymptotic and distribution), and rank transformation test statistics in one and two sample problems.
The results of the analysis show that all scores for each faculty is not normally distributed.
Also, sequel to the outcome of the results in both nonparametric and Rank transformation test statistics in one sample problem, there is no significant difference in the median score since their type I error rates is greater than the preselected alpha level 0.05. Also, in two sample problems, there is a significant difference between the medians of the two groups (faculties) in both test statistics since their type I error rate is less than the preselected alpha level 0.05.
Consequently, Rank transformation test statistics can be used as an alternative hypothesis testingto the nonparametric test when normality assumptions are violated in the data set.