Nonparametric Tests with Abnormal Data Distribution

Questions

Reasons for Selecting a Non-parametric Test

The nature of data determines the preference for non-parametric tests over parametric tests. The non-normal distribution of data is a major reason for choosing a non-parametric test. In essence, non-parametric tests apply when the distribution of data does not meet the assumption of normality required by parametric tests (Rana, Singhal, & Dua, 2016). The presence of outliers requiring consideration during analysis is another reason that allows the use of a non-parametric test. Small sample sizes, which do not permit the application of the central limit theorem, are robust for non-parametric tests (Rana et al., 2016).

When the median is an appropriate measure of central tendency in distribution, a non-parametric test is suitable. Unlike parametric tests that require data to be on a continuous scale, non-parametric tests work well with data on nominal and ordinal scales.

Statistical Power

Statistical power is an integral aspect of statistical analysis because it determines the ability of a test to discriminate between significant and insignificant results. A comparison of the statistical power shows that parametric tests are more powerful than non-parametric tests when the assumption of normality holds (Rana et al., 2016). The normality of data and the use of continuous scales increase the statistical power of parametric tests because they provide accurate data about a given population. Non-parametric tests are prone to type II errors because they fail to reject null hypotheses when false. However, the statistical power of parametric tests varies according to the degree of violation of key assumptions relating to a specific test.

The Appropriate Non-parametric Counterparts

Parametric Test Non-Parametric Counterpart
Dependent t-test Wilcoxon signed-rank test
Independent samples t-test Mann‑Whitney U‑test
Repeated measures ANOVA (one-variable) Friedman’s Test
One-way ANOVA (independent) Kruskal‑Wallis
Pearson Correlation Spearman’s rho

SPSS Assignment

Activity 8A: The Wilcoxon Signed-Rank Test

Descriptive statistics (Table 1) depict that the creativity of participants (N = 40) increased from the pre-test score (M = 40.15, SD = 8.304) to the post-test score (M = 43.35, SD = 9.598).

Table 1. Descriptive Statistics.
N Mean Std. Deviation Minimum Maximum Percentiles
25th 50th (Median) 75th
Creativity pre-test 40 40.15 8.304 26 56 34.00 38.00 47.75
Creativity post-test 40 43.35 9.598 20 59 36.00 44.00 51.00

Table 2 indicates that 9 participants had higher creativity scores before the test than after the test, while 28 participants had greater creativity scores after the test when compared to before the test. However, 3 participants did not show variation in creativity scores before and after tests.

Table 2. Ranks
N Mean Rank Sum of Ranks
Creativity post-test – Creativity pre-test Negative Ranks 9a 15.67 141.00
Positive Ranks 28b 20.07 562.00
Ties 3c
Total 40
a. Creativity post-test < Creativity pre-test
b. Creativity post-test > Creativity pre-test
c. Creativity post-test = Creativity pre-test

The Wilcoxon signed-rank test (Table 3) indicates that creativity scores improved statistically significantly among participants from the pre-test to the post-test (Z = -3.179, p = 0.001).

Table 3. Test Statistics.
Creativity post-test – Creativity pre-test
Z -3.179b
Asymp. Sig. (2-tailed) .001
a. Wilcoxon Signed Ranks Test
b. Based on negative ranks.

Activity 8B: The Mann-Whitney U test

Descriptive statistics (Table 4) show that the mean of creativity scores is 41.75 (SD = 9.062) with maximum and minimum values of 59 and 20, respectively.

Table 4. Descriptive Statistics.
N Mean Std. Deviation Minimum Maximum Percentiles
25th 50th (Median) 75th
Creativity 80 41.7500 9.06167 20.00 59.00 34.2500 41.0000 49.7500
Test 80 1.50 .503 1 2 1.00 1.50 2.00

According to Table 5, the pretest score had a lower mean rank of 36.23 than that of the post-test of 44.78. Hence, the results of the ranks table depict the existence of an apparent increase in creativity scores among participants.

Table 5. Ranks.
Test N Mean Rank Sum of Ranks
Creativity Pretest 40 36.23 1449.00
Posttest 40 44.78 1791.00
Total 80

The Mann-Whitney U test (Table 6) points out that the mean rank of creativity scores in the post-test was not statistically significantly higher than in the pre-test (U = 629, p = 0.100).

Table 6. Test Statistics.
Creativity
Mann-Whitney U 629.000
Wilcoxon W 1449.000
Z -1.647
Asymp. Sig. (2-tailed) .100

Activity 8C: The Kruskal-Wallis H Test

Table 7 displays that systolic blood pressure has a mean of 124.77 (SD = 9.031), while diastolic blood has a mean of 82.90 (SD = 2.833).

Table 7. Descriptive Statistics.
N Mean Std. Deviation Minimum Maximum Percentiles
25th 50th (Median) 75th
Systolic Blood Pressure 30 124.77 9.031 110 145 118.00 123.50 130.00
Diastolic Blood Pressure 30 82.90 2.833 78 90 81.00 82.00 85.00
Setting 30 2.00 .830 1 3 1.00 2.00 3.00

In the ranks table (Table 8), the doctor’s office has the highest mean rank followed by home and classroom settings in both systolic and diastolic blood pressure. In systolic blood pressure, the doctor’s office, home, and classroom settings have the mean ranks of 22.80, 14.25, and 9.45, correspondingly. Comparatively, doctor’s office, home, and classroom settings have the mean ranks of 16.30, 15.75, and 14.45 in diastolic blood pressure, respectively.

Table 8. Ranks.
Setting N Mean Rank
Systolic Blood Pressure Home (control) 10 14.25
Doctor’s office 10 22.80
Classroom 10 9.45
Total 30
Diastolic Blood Pressure Home (control) 10 15.75
Doctor’s office 10 16.30
Classroom 10 14.45
Total 30

The Kruskal-Wallis H Test (Table 9) settings have a statistically significant influence on systolic blood pressure, χ(30) = 10.40, p = 0.006. Nevertheless, settings have no statistically significant effect on diastolic blood pressure, χ(30) = 0.268, p = 0.875.

Table 9. Test Statistics.
Systolic Blood Pressure Diastolic Blood Pressure
N 30 30
Median 123.50 82.00
Chi-Square 10.400 .268c
df 2 2
Asymp. Sig. .006 .875

Reference

Rana, R. K., Singhal, R., & Dua, P. (2016). Deciphering the dilemma of parametric and nonparametric tests. Journal of Practical of Cardiovascular Sciences, 2(2), 95-98. Web.