Nonparametric and
Resampling Statistics
Wilcoxon Rank-Sum Test
•
•
•
•
•
•
•
To compare two independent samples
Null is that the two populations are identical
The test statistic is Ws , Table of Critical Vals.
For large samples, there is a normal approx.
Is equivalent to the Mann-Whitney test.
Assumptions are different from those of t.
Rank the scores and sum the ranks in each
group.
• See handout and text for details on computation.
Assumptions
Summary Statement
A Wilcoxon rank-sum test indicated
that babies whose mothers started
prenatal care in the first trimester weighed
significantly more (N = 8, M = 3259 g, Mdn
= 3015 g, s = 692 g) than did those whose
mothers started prenatal care in the third
trimester (N = 10, M = 2576 g, Mdn =
2769 g, s = 757 g), W = 52, p = .034.
Power
• If the assumptions of t are met, power will
be a little greater with t than with W.
• When the assumptions of t are not met,
power may be greater with W.
• Nana & Sawilowsky showed this to be the
case with typical Likert scale data.
Wilcoxon’s Signed-Ranks Test
• Two related groups.
• Rank the absolute values of the difference
scores. Sum of + ranks, sum of – ranks.
• See handout and text for computational
details.
• The test statistic is T.
• Table of Critical Values in text.
• There is a normal approximation for large
samples.
Summary Statement
A Wilcoxon signed-ranks test
indicated that participants who were
injected with glucose had significantly
better recall (M = 7.62, Mdn = 8.5, s =
3.69) than did subjects who were
injected with saccharine (M = 5.81,
Mdn = 6, s = 2.86), T(N = 16) = 14.5,
p = .004.
Kruskal-Wallis ANOVA
• Two or more independent samples.
• Rank the data, sum the ranks in each
group.
• Compute H (see text and handout).
• Evaluate H as a chi-square on k-1 df.
• Pairwise comparisons with Wilcoxon Rank
Sum
• Fisher’s procedure or Bonferroni to cap
FW
Summary Statement
Kruskal-Wallis ANOVA
indicated that type of drug
significantly affected the number
of problems solved, H(2, N = 19)
= 10.36, p = .006. Pairwise
comparisons made with
Wilcoxon’s rank-sum test
revealed that .........
Friedman’s ANOVA
• Two or more related groups.
• Rank the scores within each block.
• Compute F2 on k-1 df (see text and
handout)
• Pairwise comparisons with Wilcoxon
Signed Ranks.
• Fisher’s procedure or Bonferroni to cap
FW
Summary Statement
Friedman’s ANOVA indicated that
judgments of the quality of the
lectures were significantly affected
by the number of visual aids
employed, (2, n = 17) = 10.94, p =
.004. Pairwise comparisons with
Wilcoxon signed-ranks tests
indicated that .......................
SAS
• Run the program Nonpar.sas from my
SAS programs page
Wilcoxon Rank Sum
• SAS reports the sum of ranks in each
group
• It gives both an exact p and a p from a
normal approximation.
• For the normal approximation, it makes a
correction for continuity.
Wilcoxon Matched Pairs
Signed-Ranks Test
• Compute difference scores.
• Point Proc Univariate to the difference
scores.
• SAS reports S, the signed rank statistic
• This can be converted to T, see the
handout.
• A binomial sign test is also provided.
Kruskal-Wallis ANOVA
• The sums and means reported by SAS are
for the ranks, not the original scores.
• I used subsetting IF statements to produce
the three data sets for the pairwise
comparisons.
– data kr12; set kruskal; if group < 3;
– data kr23; set kruskal; if group > 1;
– data kr13; set kruskal; if group NE 2;
Friedman’s Rank Test for
Correlated Samples
• Use Proc Rank to create the ranked data.
• Cochran-Mantel-Haenszel Statistic from
Proc Freq is the test statistic.
• An alternative analysis is a Factorial
ANOVA on the ranks.
– One IV is the blocks, the other is the
Treatment.
– Can easily get pairwise contrasts with this
approach
Resampling Statistics
• The sampling distributions here are
created from the actual data set.
• The approach is simple, but computerintensive.
• We shall use the software provided free by
David Howell.
• See http://core.ecu.edu/psyc/wuenschk/StatHelp/Resampling.htm
Bootstrapping
• Confidence Interval for a Median
• We assume that the population is
distributed exactly as is the sample.
• We randomly draw one score from the
sample of 20. We record it, replace it, and
draw another. We repeat this process 20
times (20 because there are 20 scores in
the sample). We compute the median of
the obtained resample and record it.
• We repeat this process, obtaining a second
sample of 20 scores and computing and
recording a second median.
• We continue until we have obtained a large
number (10,000 or more) of resample medians.
We obtain the probability distribution of these
medians and treat it like a sampling distribution.
• From the obtained resampling distribution, we
find the .025 and the .975 percentiles. These
define the confidence limits.
• Follow My Instructions
Confidence Interval for Pearson r
• Correlation between misanthropy and
attitude about animals (nonidealists).
• Sample, with replacement, from our actual
sample (n = 123), 123 pairs of scores.
• Compute Pearson r on that sample of 123
pairs of scores and then record the value
of r.
• Repeat this process many times, each
time recording the value of r.
• From the resulting resampling distribution
of values of r, we obtain the .025 and .975
percentiles. This is the confidence interval
for our sample r.
• Follow My Instructions
Permutation/Randomization Tests
• Two Independent Samples
• We take all 49 + 18 = 67 scores and throw
them into one pot.
• After mixing them, we select one score
and assign it to the success group.
• Without replacing it, we select a second
score and assign it to the success group.
• We continue this until we have assigned 49
scores to the success group. The remaining
17 scores are assigned to the fail group.
• We compute the medians of the resulting
groups and then the difference between those
medians. We record the difference in medians.
• We put the 67 scores back in the pot and
repeat the process, obtaining and recording a
second difference in medians for the resulting
sample of 49 scores assigned to one group,
17 to the other group.
• We repeat this procedure a large number of
times. The resulting set of differences in
medians is our resampling distribution under
the null of no difference in population medians.
• We map out the nonrejection region (typically
the middle 95% of the resampling distribution.
• If the observed difference in sample medians
falls outside of the nonrejection region, then
the difference is significant.
• Follow My Instructions
Two Correlated Samples
• We conduct a one-sample t test of the null
that the mean difference score is zero.
• For our data, t = 3.121. If we refer this to
the distribution of Student's t, we obtain p
= .006, but we are not going to use
Student's t.
• We are going to construct our own
resampling distribution of t.
• We take the 19 difference scores, strip
them of their signs, and then randomly
assign signs to them.
• We then compute and record a onesample t for the resulting distribution of
difference scores.
• We repeat this process a large number of
times.
• From the resulting sampling distribution
under the null hypothesis of no difference
in population means (that is, a mean
difference score of 0), we compute our p
value by finding what proportion of the
resampled t values differ from 0 by at least
as much as did the t from our actual
samples.
• Follow My Instructions