Psychology 318 Quiz Section3/10/09
Ch 18: Wilcoxon Rank Sum Test
Notes:
The Wilcoxon Rank Sum test (equivalent to the MWU-test) isthe distribution-freer alternative to the independent-groups t-test.
1. It does not assume the population to be normally distributed.
2. It does not assume H.O.V.
The Rank Sum-test is also a non-parametric test because it does not test a null hypothesis about a population parameter and doesn’t rely on estimating any population parameters.
The null hypothesis forthe Rank Sum-test: most commonly stated as
H0: the populations are identically distributed
and rejecting the null hypothesis is most commonly interpreted as indicating that
scores in one population are “shifted above” scores in the other population.
When should we use Wilcoxon Rank Sum-test instead of the independent-group t-test?
- When data are not at least intervally scaled (i.e., data are in ranks or provide only ordinal information and should be converted to ranks)
2. When sample size is small and you’re concerned about violating distributional assumptions.
3. When you want to test a null hypothesis about medians or stochastic equality.
Steps: 1. rank the observations across both groups
2. sum the ranks for each group
3. the smaller of these groups is Ws (if the ns are equal in both groups, use the smaller of the sums)
4. compute Wʹs
= n1(n1+n2+1) - Ws
the smaller of Ws and Wʹs is your test statistic
- if your test statistic is smaller than the critical value tabled in Appendix Ws, reject Ho; if not, fail to reject Ho
Example
Eighteen autistic 6-year-olds were randomly assigned to two conditions. In the experimental condition, a token economy was established to reinforce eye contact. In the control condition, such reinforcement was not applied. The dependent variable was the numbers of seconds of eye contact in a 5-minute interview with a social worker. Although there were initially nine children in each of the conditions, six children dropped out from the experimental condition and four children dropped out form the control condition.
Data:Ranks:
Experimental / Control121 / 52
36 / 0
119 / 16
41
6
Mean / 92 / 23
Variance / 2353 / 508
Experimental / Control
Sum
Mix PatternCompute W′s:
ConditionRank / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8
*Entire data set was ranked all togetherWhich is smaller, Ws or W′s?
1. Which test is most appropriate? Why? Why might you not want to use the independent groups t test?
2. State the null hypothesis.
3. What is the numerical value of the test statistic you should use? ______
What is the appropriate critical value for a test with α = .05, two tailed? ______
4. Make a statistical decision:
5. Draw a conclusion:
The Wilcoxon Signed Ranks T Test
Notes:
The Wilcoxon T test is the non-parametric alternative to (analog to) the matched pairs t-test.
Dependent samples can be obtained in designs employing either
matched pairs or
repeated measures on the same sample of subjects
The null hypothesis for the Wilcoxon T test: (most commonly stated as)
H0: the population distributions are identical
and rejecting the null hypothesis is most commonly interpreted as indicating that
scores in one population are “shifted above” scores in the other population.
When should we use the Wilcoxon T test instead of the dependent-group t-test?
- When data are at best ordinally scaled
2. When sample sizes are very small and you have concerns about the normality of the
population you’ve sampled from
3. When you wish to test a hypothesis about medians or stochastic equality.
Steps: 1. compute difference scores for each pair; remove difference scores of 0
2. remove the signs from the difference scores and rank them; assign
average rank to ties
3. move the signs from the difference scores to the ranks
4. sum the negative signed ranks and the positive signed ranks separately
5. T is the smaller in absolute value of these two sums
6. compare T to T crit in Appendix T
Example
A sample of 9 geriatric patients at a health clinic is measured on the Beck Depression Inventory before and after 6-week program of daily 30-minute exercise. The following table shows the pre and post ratings for each subject. Higher scores indicate greater depression.
Subj / Pre / Post / Difference / Difference w/o signs / Rank w/o signs / R+ / R-1 / 46 / 44
2 / 35 / 30
3 / 22 / 24
4 / 28 / 29
5 / 18 / 12
6 / 16 / 12
7 / 30 / 18
8 / 22 / 20
9 / 10 / 18
ΣR + = / ΣR - =
1. State the null hypothesis.
2. Compute the test statistic.
3. Make a decision. Let α = .05, two tailed.
4. Draw a conclusion