Psychology 320: Psychological Statistics

Professor: Howard B. Lee

Lecture Notes

Week 16 : Chapter 12

Nonparametric Statistical Methods:

Kruskal-Wallis H Test

The Kruskal-Wallis test is considered a nonparametric analogue to the paramateric one-way ANOVA presented in Chapter 10. Just as the one-way ANOVA is an extension of the two independent groups t-test, the Kruskal-Wallis test is an extension of the Mann-W hitney U test. The Kruskal-Wallis test handles k-independent groups of samples. Like the Mann-Whiteny U test, this test uses ranks.
The null and alternative hypotheses are stated verbally. For example:
ho: The diet plans A, B and C are equally effective.
h1: At least one of the following is true: A is different from B, A is different from C or B is different from C.

The test statistic for the Kruskal-Wallis H test is obtained through the following steps:

• Rank all the data disregarding group membership, where a rank of "1" is given to the highest score. For observations that are tied (having the same value), compute the average rank and assign this average to each of the tied observations.
• Add up the ranks for each group, designated as r(i), where i is the group number..
• Compute H through the following formula:

  
  

• If k = 3 and n(i)'s are less than or equal to 5, use Table L to determine the critical value for a specified level of significance.
• Decision rule: Reject ho if H < H(alpha), otherwise do not reject ho.
• The conclusion will relate the decision with the actual hypothesis being tested.

If k, the number of groups is greater than 3 or any of the n(i)'s are greater than 5, the critical value is found by entering the chi-square distribution (Table D) with k-1 degrees of fredom. If the test statistic H is greater than the chi-square value, then ho is rejected, otherwise it is not rejected.

Example Three different diets are tested to determine their effectiveness in helping people lose weight. Fifteen overweight men are chosen for the study. Five men each are assigned to a different diet plan. After a 5 month period each are measured on the perce ntage of body weight. The data are given below.

Plan A	23	41	42	36	30
Plan B	20	24	25	26	*
Plan C	40	42	37	*	*

ho: Diet Plans A, B, and C are equally effective
h1: At least one of the following is true: A is different from B, A is different from C or B is different from C.

The ranks assigned to the data are given in the table below:

Plan A	11	3	1.5	5	7
Plan B	12	10	9	8	*
Plan C	4	1.5	6	*	*

n(1) = 5, n(2) = 4, n(3) = 3
r(1) = 11 + 3 + 1.5 + 5 + 7 = 27.5
r(2) = 12 + 10 + 9 + 8 = 39.0
r(3) = 4 + 1.5 + 6 = 11.5
N = 5 + 4 + 3 = 12
NOTE: one subject dropped out of Plan B and 2 subjects dropped out of Plan C.

From Table L, the critical value obtained is 5.6308. This value is found using n(1) =5, n(2) = 4, and n(3) = 3 and alpha = .05.
Decision Rule: Reject ho if H or KW > 5.603, otherwise, do not reject ho.
Decision: Since H = 5.276 is not larger than 5.603, the null hypothesis is not rejected.
Conclusion: The is insufficient evidence that the Diet plans are different in effectiveness.