Psychology 320: Psychological Statistics

Professor: Howard B. Lee

Lecture Notes

Week 15 : Chapter 12

Nonparametric Statistical Methods:

Nonparametric Statistics

Some experiments yield response measurements that defy quantification. That is they generate responses that can be ordered (ranked) but the location of the response on a scale of measurement is arbitrary. The data may only admit pairwise directional comparisons. This will only tell us whether one measurement is bigger than the other. Data may be ranked but we do not know the exact values of measurement. In market research, a consumer could rank order five soap products. The consumer will not be able to clear give precise measurements as to the distance in preference between products. That is, the consumer will state that product F is preferred over product K, but the person cannot give an exact measurement of distance between them. The data collected here are very different from the score-type data that were collected and analyzed in earlier topics in this course. This type of data is most prevalent in the social and behavioral sciences. This type of data is analyzed by nonparametric statistical methods.

Nonparametric statistical methods are also useful in making inferences in studies where there is doubt about the assumptions underlying standard or parametric methods. The Student t-test compares means of two groups. The t-test is performed under the assumption that the populations from which the two means are taken are normally distributed with equal variances. It is this equal variance assumption that allows us to pool the variance terms across the two groups. The researcher will never really know whether the assumptions are met in a practical situation. However, one usually assumes that the difference is not significant. If the researcher has some serious doubts concerning meeting the assumptions of a conventional analysis, this problem can be dealt with through the use of nonparametric methods.

Studies conducted have shown nonparametric statistical tests to be nearly as powerful in detecting differences among populations as parametric methods when the data are normal. They are more powerful in situations where the data does not meet the underlying assumptions of parametric methods. A number of statisticians advocate the use of nonparametric methods over parametric methods because of these findings.

For each parametric test, there is a corresponding nonparametric test. Consider the table below

Parametric	Nonparametric
2 independent sample t-test	Mann-Whitney U Test
2 dependent sample t-test	Wilcoxon Rank Sum Test
One Way Anova	Kruskal-Wallis H Test
Pearson correlation	Spearman Rank Correlation

Mann-Whitney U Test

The Mann-Whitney U test is the nonparametric counterpart to the two independent samples t-test.
The null and alternative hypotheses for this test are stated verbally since there are no parameters per se to estimate.

For example, ho: The distribution of Males coincide with the distribution of Females
h1: The distribution of Males do not coincide with the distribution of Females.

The test statistic for the Mann-Whitney U 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 the both groups. Use only the one with the smaller total.
• Compute U through the following formula: U = s - n(n+1)/2 where s is the sum of the ranks found in step 2 above, n is the sample size of the group with the smallest total.
• Determine the critical value using Table H from pages 413-415 of the textbook. For a two-side test, alpha is divided by 2. So if alpha = .05, alpha/2 = .025. In the Table, n is for the sample size of the first group and m is for the sample size of the second group. Using these two numbers, look in the table and find that part of the table where the two cross. Here you will find the critical value for U(alpha/2). Once U(alpha/2) is found, U(1-alpha/2) is computed by the following formula: U (1-alpha/2) = nm - U(alpha/2). For a one-sided test, one would still find both critical values but using alpha and 1-alpha instead.
• Decision rule: Reject ho if U < U(alpha/2) or U > U(1-alpha/2), otherwise do not reject ho.
• The conclusion will relate the decision with the actual hypothesis being tested.
• Remember in a one-sided test, both critical values are still necessary. Hence the decision rule would be Reject ho if U < U(alpha) or U > U(1-alpha), otherwise do not reject ho.

Example A educational psychologist feels that men are better at abstract reasoning than women. To test this hypothesis, he collected the following percentile scores for 6 women and 7 men. The data are given below:

Women	81	80	50	95	93	85
Men	70	86	60	92	82	69	94

ho: men are no better than women
h1: men are better than women

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

Women	8	9	13	1	3	6
Men	10	5	12	4	7	11	2

The sum of the ranks in the first group (women) is 8 + 9 + 13 + 1 + 3 + 6 = 50.
So s = 50

With n = 6, the value for U, the test statistic is
U = s - n(n+1)/2 = 50 - (6)(7)/2 = 50 - 21 = 29.

If alpha = .05, then we look up in Table H for U(alpha) and compute U(1-alpha)
because the alternative hypothesis indicates it is a one-tailed (directional) test.

U(alpha) = 9
U(1-alpha) = (6)(7) - 9 = 42 - 9 = 33

Decision Rule: Reject ho if U < 9 or U > 33, otherwise do not reject ho.
Since, U = 29 and U > 9 and U < 33, do not reject ho.
Conclusion: There is insufficient evidence that men are better than women at abstract reasoning.

Wilcoxon Rank Sum Test

The Wilcoxon Rank Sum Test is the nonparametric counterpart to the two dependent samples t-test. Remember dependent samples involves matched pairs and repeated measures.

As with the Mann-Whitney U Test, the null and alternative hypotheses are stated verbally. In fact, one cannot tell whether one has independent samples or dependent samples by inspecting the null and alternative hypotheses.

The null hypotheses states that the relative frequency distributions for the two populations are the same. The test statistic is found by executing the following steps:

• For each pair, compute the difference between the pairs.
• Ignoring the sign, find the ranks for each difference score. For tied scores, assign the average of the tied ranks. Drop those pairs with a difference of zero.
• Add up the ranks that correspond to all the positive difference scores found in step 1.
• Add up the ranks that correspond to all the negative difference scores found in step 1.
• Determine W, which equals the smaller of the sums found in Steps 3 and 4.
• If n, the number of pairs (non-zero difference) is less than or equal to 20, use Table I to find the critical value. Enter Table I with n and alpha.
• For a two sided test, reject ho if W is less than W(alpha/2) or greater than W(1-alpha/2), otherwise do not reject ho.
• For a one-sided test, reject ho if W is less than W(alpha) or greater than W(1-alpha), otherwise do not reject ho.

Example
In a product recognition study, the amount of time (in seconds) is recorded for each of two differently colored advertising layouts. Individually, 8 people are subjected to both layouts in random order. The data are given below.
ho: there is no difference between the two layouts
h1: there is a difference between the two layouts.

Person	1	2	3	4	5	6	7	8
Layout A	4	3	1	5	2	6	4	1
Layout B	3	1	2	1	5	1	5	3
d	1	2	-1	4	-3	5	-1	-2
|d|	1	2	1	4	3	5	1	2
Rank of |d|	7	4.5	7	2	3	1	7	4.5

r+ = 7 + 4.5 + 2 + 1 = 14.5
r- = 7 + 3 + 7 + 4.5 = 21.5
W = min(21.5, 14.5) = 14.5

Go to Table I with n = 8, alpha = .05. h1 tells us it is a 2 sided test, so alpha/2 = .025

The critical value for W(.025) = 4 and W(1-alpha/2) = (n)(n+1)/2 - W(.025)
= (8)(9)/2 - 4 = 36 - 4 = 32.

Decision rule: Reject ho if W < 4 or W > 32, otherwise do not reject ho.
Decision: Since W = 14.5 and 14.5 > 4 and < 32 do not reject ho.
Conclusion: There is insuffiicent evidence that there is a difference between the two layouts.