6 - NONPARAMETRIC TESTS FOR COMPARING TWO POPULATIONS

In situations where the normality of the population(s) is suspect or the sample sizes are so small that checking normality is not really feasible, it is sometimes preferable to use nonparametric tests to make inferences about “average” value.

Wilcoxon Rank Sum Test (a.k.a. Mann-Whitney U Test)
This test is an alternative to the two-sample t-test for comparing the “average” value of two populations where the samples from each population are taken independently.

The hypotheses tested can be stated as follows:

[pic] The distribution of population 1 and population 2 are identical. If the populations are symmetric (but not necessarily normal) the null hypothesis can be expressed in terms of the population medians as: [pic]

[pic] The distribution of population 1 and population 2 are different. (two-tailed) [pic]

or

[pic] The distribution of population 1 is shifted to the right of the distribution for population 2, i.e. the population 1 values are generally larger than the population 2 values. (right-tailed) [pic]

or

[pic] The distribution of population 1 is shifted to the left of the distribution for population 2, i.e. the population 1 values are generally smaller than the population 2 values. (left-tailed) [pic]

The tests statistic is based on the sum of the ranks assigned to the observed data from each population when the combined sample is ranked from smallest to largest. We will always assume that the sample size (m) for population 1 is less than or equal to the sample size (n) from population 2.

Example: Anticipated Length of Office Visit and Weight Status of Patients
Researchers wanted to compare the anticipated office visit time for patients whose BMI indicates normal weight vs. those whose BMI indicates the patient is overweight. It is hypothesized that doctors will report a shorter anticipated office visit time for patients who are classified as overweight. Stating this hypothesis in terms of medians the research hypothesis would be that median office visit time for normal weight patients is greater than that for overweight patients.

[pic] vs. [pic]

The data below are the anticipated office visit time (min) for these two groups of patients.

Normal: 20 25 30 35 40 45 50

Overweight: 5 10 15 15 20 30

The sum of the ranked appointment lengths for normal weight patients is: _________.

The sum of the ranked appointment length for overweight patients is: ____________.

The sum of the ranks for the overweight patients is smaller than the rank sum for the normal weight patients but this would be expected even if the null hypothesis were true. Why?

The test statistic,[pic], is the sum of the ranks for population O, the overweight patients. We can use the table on the following page to determine whether to reject the null or not. Intuitively we will reject the null hypothesis if the sum of the ranked appointment lengths for the overweight patients is “small”. The table tells what “small” is for a given significance level ([pic]).

For m = 6 and n = 7 we find the following from the table:

From Wilcoxon Rank Sum Table: 1-tail α ’ .025 α ’ .050 2-tail α ’ .050 α ’ .100 m n W d P W d P
6 7 27 57 7 .0175 29 55 9 .0367

The table says we will reject the null at the [pic]level if:
[pic] for [pic] (
[pic] > 55 for [pic]
[pic]< 27 or [pic] for [pic]

We have evidence to conclude that the anticipated office visit times are generally smaller than the anticipated office times for patients with BMI’s considered normal (p < .05).

WILCOXON RANK SUM TEST IN JMP
[pic]

Wilcoxon Signed Rank Test
This test is an alternative to the paired t-test which is used when we do not wish to assume that the population of paired differences is normally distributed. As with the Mann-Whitney U test, the Wilcoxon Signed-Rank Test use ranks based on the paired differences rather than the actual values.

Example: Effect of Togetherness on the Heart Rate of Rats

|Rat |Alone Rate |Together Rate |di = Ti – Ai |Sign || di | |Rank |di| |Signed Rank |
|1 |463 |523 |60 | | | | |
|2 |462 |494 |32 | | | | |
|3 |462 |461 |-1 | | | | |
|4 |456 |535 |79 | | | | |
|5 |450 |476 |26 | | | | |
|6 |426 |454 |28 | | | | |
|7 |418 |448 |30 | | | | |
|8 |415 |408 |-7 | | | | |
|9 |409 |470 |61 | | | | |
|10 |402 |437 |35 | | | | |

We then calculate [pic] = the sum of the positive signed ranks = _______________

and [pic] = the sum of the negative signed ranks = _______________

Are hypotheses can be stated in terms of the median of the paired differences. Listed below are the hypotheses along with the test statistic based on the signed rank sums used to test it.

Intuitively we will conclude there has been a heart rate increase if….

Details of the Wilcoxon Signed Rank Test

Let [pic]

[pic] vs. [pic] (two-tailed) Test statistic [pic] In practical terms the null says there is no change in the rats heart rate after the change in environment, the alternative says there is a shift up or down in their heart rate.

[pic] vs. [pic] (right-tailed) Test statistic [pic]
In practical terms the alternative says that there is an increase or shift up in their heart rate as the difference is defined to be Together – Alone.

[pic] vs. [pic] (left-tailed) Test statistic [pic]
In practical terms the alternative says that there is a decrease or shift down in their heart rate as the difference is defined to be Together – Alone.

For this example, if had originally hypothesized that the heart rate of a rat will increase when it is placed in a social environment then we have the right-tailed alternative and our test statistic W = _______.

The Wilcoxon Signed Rank Test Table (handed out) give p-values associated with an observed test statistic value w for a given sample size, i.e. number of pairs, n.
[pic]

Here our p-value = ____________, thus we reject the null hypothesis and conclude that the heart rate of a rat will generally increase when it is taken from a solitary confinement and placed in a social environment with other rats.

WILCOXON SIGNED RANK TEST IN JMP

We first use JMP to form the paired differences as we did for the paired t-test.
[pic]

Select Distribution > Test Mean > Enter 0 for the hypothesized value and check the nonparametric test box. [pic]

The results of the test are shown below.
[pic]
Conclusion:
TABLE FOR WILCOXON RANK SUM TEST (Page 1)
[pic]

TABLE FOR WILCOXON RANK SUM TEST (Page 2)
[pic]

TABLE FOR WILCOXON SIGNED RANK TEST (Page 1)
[pic]

TABLE FOR WILCOXON SIGNED RANK TEST (Page 2)
[pic]

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Data Table
[pic]

Select Nonparametric > Wilcoxon Test
[pic]

The p-values for the upper-tailed t-Test and the Wilcoxon signed-rank test have been highlighted. The test statistic reported by JMP for Wilcoxon test =[pic]
Why? I don’t know, but we only need the p-value anyway.