Statistics: Non-Parametric Tests: When Assumptions Fail
Choosing and using rank-based tests when standard assumptions are not met
Statistics: Non-Parametric Tests: When Assumptions Fail
Choosing and using rank-based tests when standard assumptions are not met
Statistics - Grade 9-12
- 1
A biology class compares the plant heights from two independent groups. Group A used regular water and Group B used fertilizer. The sample sizes are small, and the height data are strongly skewed with one extreme outlier. Should the class use a two-sample t-test or the Mann-Whitney U test? Explain your choice.
Look for the number of groups, whether they are independent, and whether normality is reasonable.
The class should use the Mann-Whitney U test because the groups are independent, the sample sizes are small, and the data are skewed with an outlier. A two-sample t-test depends more strongly on the assumption that the data are approximately normal. - 2
A teacher records student stress ratings before and after a mindfulness activity. The ratings are on a 1 to 5 scale, and the same students are measured twice. Which non-parametric test is more appropriate: Mann-Whitney U or Wilcoxon signed-rank? Explain.
Decide whether the two sets of scores come from the same people or from different groups.
The Wilcoxon signed-rank test is more appropriate because the data are paired. The same students are measured before and after the activity, so each student creates a matched pair of ratings. - 3
Rank the following values from smallest to largest. Use average ranks for ties: 12, 9, 12, 15, 10.
The ranks are 12 = 3.5, 9 = 1, 12 = 3.5, 15 = 5, and 10 = 2. The two 12s are tied for ranks 3 and 4, so each receives the average rank of 3.5. - 4
Two independent groups have the following scores. Group A: 5, 7, 9. Group B: 1, 4, 6. Rank all six values together and calculate the Mann-Whitney U statistic for Group A. Use U_A = R_A - n_A(n_A + 1)/2.
First rank all values together, not within each group separately.
The ranked values are 1(B), 4(B), 5(A), 6(B), 7(A), 9(A). The rank sum for Group A is R_A = 3 + 5 + 6 = 14. Since n_A = 3, U_A = 14 - 3(4)/2 = 14 - 6 = 8. - 5
Using the same data from the previous problem, Group A has U_A = 8. There are 3 scores in each group, so n_A n_B = 9. Find U_B and the smaller U statistic used for the Mann-Whitney U test.
Since U_A + U_B = n_A n_B, U_B = 9 - 8 = 1. The smaller U statistic is 1. - 6
A paired study has the following differences after minus before: 2, -1, 3, 0, 4. For a Wilcoxon signed-rank test, ignore the zero difference, rank the absolute differences, and find W+ and W-.
Zeros are removed before ranking the absolute differences.
Ignoring the zero, the absolute differences are 2, 1, 3, and 4. Their ranks are 2, 1, 3, and 4. The positive differences have ranks 2, 3, and 4, so W+ = 9. The negative difference has rank 1, so W- = 1. - 7
A researcher compares reaction times for three independent groups: no music, classical music, and loud music. The distributions are skewed, and the researcher wants to compare typical performance across all three groups. Which test should be used: Kruskal-Wallis, Wilcoxon signed-rank, or Spearman correlation? Explain.
The Kruskal-Wallis test should be used because there are three independent groups and the data do not meet the normality assumption. Wilcoxon signed-rank is for paired data, and Spearman correlation is for association between two variables. - 8
A scatterplot shows that as hours of practice increase, performance score generally increases, but the pattern is curved rather than linear. Which correlation measure is more appropriate: Pearson correlation or Spearman rank correlation? Explain.
Spearman works well when the relationship generally moves in one direction but is not a straight line.
Spearman rank correlation is more appropriate because it measures a monotonic relationship using ranks. Pearson correlation is designed for linear relationships and can be misleading when the pattern is curved. - 9
Calculate Spearman's rank correlation for these paired ranks: x ranks are 1, 2, 3, 4, 5 and y ranks are 1, 2, 4, 3, 5. Use rho = 1 - [6 sum d^2]/[n(n^2 - 1)].
The rank differences are 0, 0, -1, 1, and 0, so sum d^2 = 2. With n = 5, rho = 1 - [6(2)]/[5(25 - 1)] = 1 - 12/120 = 0.90. The Spearman rank correlation is 0.90. - 10
A survey asks people to rate satisfaction as very dissatisfied, dissatisfied, neutral, satisfied, or very satisfied. Why might a non-parametric test be more appropriate than a test that compares means?
Think about whether the difference between each pair of neighboring response choices is exactly the same.
A non-parametric test may be more appropriate because the responses are ordinal categories. The order matters, but the distances between categories may not be equal, so comparing means can be less meaningful. - 11
A study reports a Mann-Whitney U test result of p = 0.03 using alpha = 0.05. State the decision and interpret it in context.
Because p = 0.03 is less than alpha = 0.05, the study rejects the null hypothesis. There is statistically significant evidence that the two independent groups differ in their distributions or typical values. - 12
A study reports a Wilcoxon signed-rank test result of p = 0.28 using alpha = 0.05. State the decision and interpret it carefully.
Failing to reject the null does not prove that there is no effect.
Because p = 0.28 is greater than alpha = 0.05, the study fails to reject the null hypothesis. The data do not provide strong enough evidence of a median change between the paired measurements. - 13
Observed counts for favorite drink are 18 for water and 12 for juice. If the expected counts are 15 and 15, calculate the chi-square test statistic using sum (O - E)^2/E.
For water, (18 - 15)^2/15 = 9/15 = 0.6. For juice, (12 - 15)^2/15 = 9/15 = 0.6. The chi-square statistic is 0.6 + 0.6 = 1.2. - 14
A Kruskal-Wallis test is used with three groups. The combined ranks give these rank sums: Group A = 3, Group B = 7, Group C = 11. Each group has 2 observations. Which group appears to have the largest typical values, and why?
In rank-based tests, larger data values receive larger ranks.
Group C appears to have the largest typical values because it has the largest rank sum. Since each group has the same sample size, the group with the largest rank sum also has the largest average rank. - 15
For each situation, choose the best non-parametric method from this list: Mann-Whitney U, Wilcoxon signed-rank, Kruskal-Wallis, Spearman rank correlation. Situation 1: compare two independent groups with skewed data. Situation 2: compare before and after scores for the same people. Situation 3: compare four independent groups with ordinal ratings. Situation 4: measure association between two variables with a monotonic but non-linear pattern.
Situation 1 uses the Mann-Whitney U test because there are two independent groups. Situation 2 uses the Wilcoxon signed-rank test because the data are paired. Situation 3 uses the Kruskal-Wallis test because there are more than two independent groups. Situation 4 uses Spearman rank correlation because the association is monotonic but not linear.