Non-Parametric Procedures in Statistics

Non-parametric procedures

Reasons for non-parametric tests

There are several reasons why non-parametric tests are preferred to parametric tests. First, the non-parametric tests are used when data does not have clear arithmetical understanding. Thus, these tests are most suitable when assessing ordinal data. The second reason is that non-parametric tests are easier to use than the parametric tests. Finally, the non-parametric tests do not follow a probability distribution because they make fewer assumptions than parametric tests.

Statistical power

Even though the use of the non-parametric test has grown, this comes at a price. This can be attributed to the fact that to make a conclusion with the same level of significance. Then a bigger sample size will be needed. This means that the test will be conducted several times before a significant effect is established. Thus, estimating type 1 error will be difficult with non-parametric tests because this error is based on the sampling distribution. Thus, it can be concluded that non-parametric tests are less powerful than parametric tests (Field, 2013).

Appropriate non-parametric counterparts

  1. Independent samples t-test – the appropriate non-parametric equivalent is the Mann-Whitney U test.
  2. Dependent t-test – the suitable non-parametric equivalent is the Wilcoxon signed-rank.
  3. Repeated measures ANOVA (one variable) – the most suitable non-parametric equivalent is the Friedman Test.
  4. One-way ANOVA (Independent) – the fitting non-parametric equivalent is the Kruskal-Wallis H test.
  5. Pearson correlation – the suitable non-parametric equivalent is the Spearman rank correlation coefficient analysis.

The non-parametric version of the dependent t-test

Exploratory data analysis on Creativity Pre-test and Creativity Post-test

In this case, the Wilcoxon signed-rank test will be used.

Null hypothesis

Ho: There is no difference between pre-creativity and post-creativity test scores.

Alternative hypothesis

H1: There is a difference between pre-creativity and post-creativity test scores.

A two-tailed test will be carried out at a 5% level of significance.

Descriptive Statistics
N Mean Std. Deviation Minimum Maximum
Test taken 80 1.50 .503 1 2
Creativitypre&creativitypost 80 41.7500 9.06167 20.00 59.00

Wilcoxon Signed Ranks Test

Ranks
N Mean Rank Sum of Ranks
Creativitypre&creativitypost – Test taken Negative Ranks 0a .00 .00
Positive Ranks 80b 40.50 3240.00
Ties 0c
Total 80
a. Creativitypre&creativitypost < Test taken
b. Creativitypre&creativitypost > Test taken
c. Creativitypre&creativitypost = Test taken
Test Statistics
Creativitypre&creativitypost – Test taken
Z -7.773a
Asymp. Sig. (2-tailed) .000
a. Based on negative ranks.
b. Wilcoxon Signed Ranks Test

Discussion of results

The results of ranks show that the test scores of all participants were higher after the post-creativity training because there were no negative ranks. The test statistics for Z is -7.773 while the p-value (asymp.sig. (2-tailed) is 0.00. Since the p-value is less than the value of alpha, the null hypothesis will be rejected. The conclusion will be that there is a difference between pre-creativity and post-creativity test scores at the 95% confidence interval. Thus, it can be noted that creativity training improved the scores.
Measurement of systolic and diastolic blood pressure
In this case, a comparison will be made between the diastolic and systolic blood pressure. Also, a comparison will be made between the two blood pressures under different conditions.
Between diastolic and systolic blood pressure
Null hypothesis
Ho: There is no difference between the diastolic and systolic blood pressure.
Alternative hypothesis
H1: There is a difference between diastolic and systolic blood pressure.
A two-tailed test will be carried out at a 5% level of significance.
Descriptive Statistics
N Mean Std. Deviation Minimum Maximum
Systolic Blood Pressure 30 124.77 9.031 110 145
Diastolic Blood Pressure 30 82.90 2.833 78 90

Wilcoxon Signed Ranks Test

Ranks
N Mean Rank Sum of Ranks
Diastolic Blood Pressure – Systolic Blood Pressure Negative Ranks 30a 15.50 465.00
Positive Ranks 0b .00 .00
Ties 0c
Total 30
a. Diastolic Blood Pressure < Systolic Blood Pressure
b. Diastolic Blood Pressure > Systolic Blood Pressure
c. Diastolic Blood Pressure = Systolic Blood Pressure
Test Statistics
Diastolic Blood Pressure – Systolic Blood Pressure
Z -4.784a
Asymp. Sig. (2-tailed) .000
a. Based on positive ranks.
b. Wilcoxon Signed Ranks Test

Discussion of results

The descriptive statistics show that the mean and the standard deviation of systolic blood pressure are higher than that of the diastolic blood pressure. Further, the minimum and maximum values for systolic blood pressure are higher than those of diastolic blood pressure. Further, systolic blood pressure has a higher range than diastolic blood pressure. Thus, in general, the systolic blood pressure is higher than diastolic blood pressure. The test will ascertain if this difference is statistically significant. The table of ranks shows that there were only negative ranks. There were only positive ranks, and this implies that the diastolic blood pressure is higher than the systolic blood pressure. Further, the test statistics for Z is -4.784 while the p-value (asymp.sig. (2-tailed) is 0.00. Since the p-value is less than the value of alpha, the null hypothesis will be rejected. This shows that there exists a statistically significant difference between the diastolic and systolic blood pressure at the 95% confidence level.

For diastolic and systolic blood pressure under the three environments

In this case, the diastolic and systolic blood pressure will be analyzed separately to ascertain if they were different under the three environments.

Null hypothesis

Ho: There is no difference in the blood pressure under the three environments.

Alternative hypothesis

H1: There is a difference in the blood pressure under the three environments.

A two-tailed test will be carried out at a 5% level of significance.

Ranks
N Mean Rank Sum of Ranks
Systolic Blood Pressure – Setting Negative Ranks 0a .00 .00
Positive Ranks 30b 15.50 465.00
Ties 0c
Total 30
Diastolic Blood Pressure – Setting Negative Ranks 0d .00 .00
Positive Ranks 30e 15.50 465.00
Ties 0f
Total 30
a. Systolic Blood Pressure < Setting
b. Systolic Blood Pressure > Setting
c. Systolic Blood Pressure = Setting
d. Diastolic Blood Pressure < Setting
e. Diastolic Blood Pressure > Setting
f. Diastolic Blood Pressure = Setting
Test Statistics
Systolic Blood Pressure – Setting Diastolic Blood Pressure – Setting
Z -4.783a -4.792a
Asymp. Sig. (2-tailed) .000 .000
a. Based on negative ranks.
b. Wilcoxon Signed Ranks Test

Discussion of results

The ranks for both the systolic and diastolic blood pressure were positive. This shows that there was a difference in the blood pressures under the three environments. Further, under the systolic blood pressure, the test statistics for Z is -4.782, while under the diastolic blood pressure, the test statistics for Z is -4.792. On the other hand, the p-value (asymp.sig. (2-tailed) for systolic and diastolic blood pressure are 0.00. Since the p-values for both the blood pressures are less than the value of alpha, then the null hypothesis will be rejected. This shows that there exists a statistically significant difference in the results of systolic and diastolic blood pressure taken at home, in the doctor’s office, and in the classroom environment at the 95% confidence level.

The non-parametric version of independent t-test

Exploratory data analysis on Creativity Pre-test and Creativity Post-test

In this case, the Mann-Whitney test will be used. The test will be used to ascertain if there is a difference in the results of two independent groups.

Null hypothesis

Ho: There is no difference between pre-creativity and post-creativity test scores.

Alternative hypothesis

H1: There is a difference between pre-creativity and post-creativity test scores.

A two-tailed test will be carried out at a 5% level of significance.

Ranks
Test taken N Mean Rank Sum of Ranks
Creativitypre & creativitypost pre-creativity 40 36.22 1449.00
Post-creativity 40 44.78 1791.00
Total 80
Test Statistics
Creativitypre&creativitypost
Mann-Whitney U 629.000
Wilcoxon W 1449.000
Z -1.647
Asymp. Sig. (2-tailed) .100
a. Grouping Variable: Test taken

Discussion of results

The mean and the sum of ranks for the post-creativity test scores are higher than those for pre-creativity test scores. The hypothesis will ascertain if the difference is statistically significant. From the results above, the value of Mann-Whitney U is 629.00. Further, the value test statistics for Z is -1.647 while the p-value (asymp.sig. (2-tailed) is 0.100. It can be observed that the p-value is more than the value of alpha. Thus, the null hypothesis will not be rejected. This shows that there is no statistically significant difference between the post-creativity and pre-creativity test scores at the 95% confidence level (Greene, 2011).

Measurement of systolic and diastolic blood pressure

Systolic and diastolic blood pressure observed at home (control) and in the doctor’s office

Null hypothesis

Ho: There is no difference in the result of systolic and diastolic blood pressure when observed at home and in the doctor’s office.

Alternative hypothesis

H1: There is a difference in the result of systolic and diastolic blood pressure when observed at home and in the doctor’s office.

A two-tailed test will be carried out at a 5% level of significance.

Ranks
Setting N Mean Rank Sum of Ranks
Systolic Blood Pressure Home (control) 10 7.50 75.00
Doctor’s office 10 13.50 135.00
Total 20
Diastolic Blood Pressure Home (control) 10 10.30 103.00
Doctor’s office 10 10.70 107.00
Total 20
Test Statistics
Systolic Blood Pressure Diastolic Blood Pressure
Mann-Whitney U 20.000 48.000
Wilcoxon W 75.000 103.000
Z -2.274 -.152
Asymp. Sig. (2-tailed) .023 .879
Exact Sig. [2* (1-tailed Sig.)] .023a .912a
a. Not corrected for ties.
b. Grouping Variable: Setting

Discussion of results

In the table of ranks, it can be observed that the mean and sum of ranks of systolic blood pressure taken at the doctor’s office is higher than that taken at home. A similar trend was observed in the diastolic blood pressure. In the case of systolic blood pressure, the value of Mann-Whitney U is 20.00. Further, the value of test statistics for Z is -2.274 while the p-value (asymp.sig. (2-tailed) is 0.023. It can be observed that the p-value is less than the value of alpha. This implies that the null hypothesis will be rejected. The conclusion is that there is a statistically significant difference between the systolic blood pressure taken in the doctor’s office and at home. On the other hand, the value of Mann-Whitney U for diastolic blood pressure is 48.00. Further, the value of test statistics for Z is -0.152 while the p-value (asymp.sig. (2-tailed) is 0.879. It can be observed that the p-value is more than the value of alpha. This implies that the null hypothesis will not be rejected. The conclusion is that there is no statistically significant difference between the diastolic blood pressure taken in the doctor’s office and at home. These results are based on the 95% confidence level (Greene, 2011).

For diastolic blood pressure and systolic blood pressure under the home (control) and class environment

Ranks
Setting N Mean Rank Sum of Ranks
Systolic Blood Pressure Home (control) 10 12.25 122.50
Classroom 10 8.75 87.50
Total 20
Diastolic Blood Pressure Home (control) 10 10.95 109.50
Classroom 10 10.05 100.50
Total 20
Test Statistics
Systolic Blood Pressure Diastolic Blood Pressure
Mann-Whitney U 32.500 45.500
Wilcoxon W 87.500 100.500
Z -1.327 -.343
Asymp. Sig. (2-tailed) .184 .731
Exact Sig. [2* (1-tailed Sig.)] .190a .739a
a. Not corrected for ties.
b. Grouping Variable: Setting

Discussion of results

In the table of ranks, it can be observed that the mean and sum of ranks of systolic blood pressure taken at home is higher than the results taken in the classroom. A similar trend was observed in the diastolic blood pressure. In the case of systolic blood pressure, the value of Mann-Whitney U is 32.5. Further, the value of the test statistic for Z is -1.327 while, the p-value (asymp.sig. (2-tailed) is 0.184. It can be observed that the p-value is more than the value of alpha. This implies that the null hypothesis will not be rejected. The conclusion is that there is no statistically significant difference between the systolic blood pressure taken at home and in the classroom. On the other hand, the value of Mann-Whitney U for diastolic blood pressure is 45.50. Further, the value of test statistics for Z is -0.731 while the p-value (asymp.sig. (2-tailed) is 0.831. It can be observed that the p-value is more than the alpha. This implies that the null hypothesis will not be rejected. The conclusion is that there is no statistically significant difference between the diastolic blood pressure taken at home and in the classroom. These results are based on the 95% confidence level.

For diastolic blood pressure and systolic blood pressure under the doctor’s office and class environment

Ranks
Setting N Mean Rank Sum of Ranks
Systolic Blood Pressure Doctor’s office 10 14.80 148.00
Classroom 10 6.20 62.00
Total 20
Diastolic Blood Pressure Doctor’s office 10 11.10 111.00
Classroom 10 9.90 99.00
Total 20
Test Statistics
Systolic Blood Pressure Diastolic Blood Pressure
Mann-Whitney U 7.000 44.000
Wilcoxon W 62.000 99.000
Z -3.254 -.458
Asymp. Sig. (2-tailed) .001 .647
Exact Sig. [2* (1-tailed Sig.)] .000a .684a
a. Not corrected for ties.
b. Grouping Variable: Setting

Discussion of results

In the table of ranks, it can be observed that the mean and sum of ranks of systolic blood pressure taken at the doctor’s office are higher than the results taken in the classroom. A similar trend was observed in the diastolic blood pressure. In the case of systolic blood pressure, the value of Mann-Whitney U is 7.0. Further, the value of test statistics for Z is -3.254 while the p-value (asymp.sig. (2-tailed) is 0.001. It can be observed that the p-value is less than the value of alpha. This implies that the null hypothesis will be rejected. The conclusion is that there is a statistically significant difference between the systolic blood pressure taken in the doctor’s office and in the classroom. On the other hand, the value of Mann-Whitney U for diastolic blood pressure is 44.00. Further, the value of test statistics for Z is -0.458 while the p-value (asymp.sig. (2-tailed) is 0.647. It can be observed that the p-value is more than the value of alpha. This implies that the null hypothesis will not be rejected. The conclusion is that there is no statistically significant difference between the diastolic blood pressure taken in the doctor’s office and in the classroom. These results are based on the 95% confidence level.

The non-parametric version of single-factor ANOVA

The Kruskal-Wallis H test will be used as a non-parametric version of the single factor ANOVA.

Exploratory data analysis on Creativity Pre-test and Creativity Post-test

In this case, the test will be used to ascertain if there is a difference between pre-creativity and post-creativity test scores.

Null hypothesis

Ho: There is no difference between pre-creativity and post-creativity test scores.

Alternative hypothesis

H1: There is a difference between pre-creativity and post-creativity test scores.

A two-tailed test will be carried out at a 5% level of significance.

Ranks
Test taken N Mean Rank
Creativitypre&creativitypost pre-creativity 40 36.22
Post-creativity 40 44.78
Total 80
Test Statistics,b
Creativitypre&creativitypost
Chi-Square 2.713
Df 1
Asymp. Sig. .100
a. Kruskal Wallis Test
b. Grouping Variable: Test taken

Discussion of results

In the above tables, it can be observed that the mean rank for post-activity test scores is higher than those for pre-creativity. Further, the degree of freedom is 1. The test statistics for chi-square are 2.713, while the p-value is 0.100. It can be observed that the p-value is more than the value of alpha. Thus, the null hypothesis will not be rejected. This shows that there is no statistically significant difference between the post-creativity and pre-creativity test scores at the 95% confidence level.

Measurement of systolic and diastolic blood pressure

Null hypothesis

Ho: There is no difference in the blood pressure taken under the three environments. Alternative hypothesis

H1: There is a difference in the blood pressure taken under the three environments.

A two-tailed test will be carried out at a 5% level of significance.

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
Test Statistics,b
Systolic Blood Pressure Diastolic Blood Pressure
Chi-Square 11.851 .237
Df 2 2
Asymp. Sig. .003 .888
a. Kruskal Wallis Test
b. Grouping Variable: Setting

Discussion of results

In both cases, the mean rank for blood pressure taken in the doctor’s office is higher than those taken in other environments. Further, the degree of freedom in both cases is 2. Under systolic blood pressure, the test statistics for chi-square are 11.851, while the p-value is 0.003. It can be observed that the p-value is less than the value of alpha. Therefore, the null hypothesis will be rejected. This shows that there is a statistically significant difference in the blood pressure taken under the three environments at the 95% confidence level. In the case of diastolic blood pressure, the test statistics for chi-square are 0.237, while the p-value is 0.888. It can be observed that the p-value is more than the value of alpha; the null hypothesis will be rejected (Greene, 2011). This shows that there is no statistically significant difference in the blood pressure taken under the three environments at the 95% confidence level.

References

Field, A. (2013). Discovering statistics using IBM SPSS statistics. London: SAGE Publication.

Greene, W. (2011). Econometric analysis. NJ, USA: Pearson Education.