SW388R6

1
Nonparametric Correlated Groups Tests

• Wilcoxon Matched-pairs Signed Ranks Test
• Sign Test

2
Alternatives to the Correlated Groups T-test

• As we have seen for other types of t-tests, there
  are alternative statistics available for testing
  differences between correlated groups that do not
  require the assumption of normality.
• The Wilcoxon matched-pairs signed ranks test
  compares difference in both direction and
  magnitude of the difference.
• The Sign Test compares differences only in terms
  of direction (plus or minus).
• The Wilcoxon matched-pairs signed ranks test is
  less powerful at detecting statistical
  differences than is the correlated groups t-test,
  and the sign test is less powerful than both.

3
Computing the Wilcoxon matched-pairs test

• First, a difference score is calculated for each
  case by subtracting the actual score for the
  first variable in the pair from the actual score
  for the second variable in the pair. If the
  difference score is zero because the actual
  scores are the same, the case is dropped from the
  analysis.
• Second, the difference scores are rank ordered
  from low to high by size, ignoring the sign of
  the difference scores (I.e. using absolute
  values)
• Third, the ranks associated with the negative
  difference scores are summed and the ranks
  associated with the positive difference scores
  are summed.
• Fourth, the probability of the Wilcoxon
  match-pairs signed ranks test statistic is
  computed, using the smaller of the total summed
  ranks in the formula.
• Fifth, since the test is always done with the
  smaller of the summed ranks, we must link the
  relationship in the research question to the
  pattern of summed ranks to make certain that our
  finding is consistent with the relationship we
  wanted to test.

4
Problem 1

• 1. Based on the dataset OMAHA.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic? Use 0.05 as the level
  of significance. Base your answer on the output
  for the Wilcoxon matched-pairs signed ranks test.
• For the population represented by this sample,
  victims of domestic violence in Omaha were less
  likely to agree that they were a person of worth
  at one week after the domestic violence incident
  compared to six months after the domestic
  violence incident.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

5
Request the Wilcoxon matched-pairs test
To compute the Wilcoxon match-pairs signed ranks
test in SPSS, select the Nonparametric Tests
Related Samples command from the Analyze menu.
6
Select the specifications for the test
First, select the pair of variables for the
comparison and move them to the Test Pairs List.
Third, click on the OK button to complete the
request and produce the output.
Second, mark the checkbox for the Wilcoxon test.
7
Research question related to difference scores
Our research question states that victims were
less likely to agree at one week compared to six
months. This implies that their scores at one
week would tend to be lower (e.g. 1 or 2) than
their scores at six months (e.g. 3 or 4). To
compute the difference scores, we subtract actual
scores at week one from actual scores at month
six. For example Subject one week score
six months score difference score 1
1 (strongly disagree) 2 (disagree)
1 2 2 (disagree) 1
(strongly disagree) -1 3
2 (disagree) 4 (strongly agree)
2 4 1 (strongly disagree) 4
(strongly agree) 3 If
the actual scores at one week are really lower,
we would expect to find a larger number of
positive difference scores (subjects 1, 3, and
4) than negative difference scores (subject 2).
8
Research question related to sum of ranks
In addition, we would expect the size of the
positive difference scores to be larger than the
negative difference scores. Since larger
difference scores are assigned a higher rank
(irrespective of the plus or minus sign), the sum
of the ranks associated with the positive
difference scores (1.5 2 3 6.5) would tend
to be larger than the sum of the ranks associated
with the negative difference scores (1.5
1.5). Subject one week score six months
score difference score rank 1 1
(strongly disagree) 2 (disagree) 1
1.5 2 2 (disagree) 1
(strongly disagree) -1 1.5 3
2 (disagree) 4 (strongly agree)
2 2 4 1 (strongly
disagree) 4 (strongly agree) 3
3
9
The research question as a Wilcoxon matched-pairs
test hypothesis
The question in the problem translates to a test
of the research hypothesis for the Wilcoxon
match-pairs signed-rank test that the total sum
of ranks associated with positive difference
scores is larger than the total sum of ranks
associated with negative difference scores. The
null hypothesis would state that the sum of the
ranks associated with positive difference scores
is equal to the sum of the ranks for negative
difference scores.
10
The output for Wilcoxon matched-pairs test
In the table of ranks, we see that the sum of
ranks associated with positive difference scores
(7,551) is larger than the sum of ranks
associated with negative difference scores
(6,110). The sum of ranks supports the
direction of the relationship stated in the
research hypothesis.
11
The output for Wilcoxon matched-pairs test
Since the hypothesis states a direction, the
one-tailed probability for the test statistic is
calculated by dividing the two-tailed probability
in half 0.146 / 2 0.073
The probability of the test statistic is greater
than the level of significance of 0.05. We fail
to reject the null hypotheses and cannot support
the research hypothesis that the sum of ranks for
positive differences was larger, and, therefore,
ratings were lower at one week than at six months.
The answer to the question is false.
12
Problem 2

• 5. Based on the dataset OMAHA.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic? Use 0.05 as the level
  of significance. Base your answer on the output
  for correlated groups t-test.
• For the population represented by this sample,
  victims of domestic violence in Omaha were more
  likely to agree that they don't have much to be
  proud of at one week after the domestic violence
  incident compared to six months after the
  domestic violence incident.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

13
Research question related to difference scores
Our research question states that victims were
more likely to agree at one week compared to six
months, implying that they had higher scores at
week one. Higher scores at week one would
results in a larger number of negative difference
scores when we subtract the larger week one
scores from the smaller month six scores. When
we rank the difference scores to take size of the
differences into account, we would expect the sum
of the ranks for negative difference scores to be
higher than the sum of the ranks for positive
difference scores. The research hypothesis for
the Wilcoxon matched-pairs signed ranks test
states that the total sum of ranks associated
with negative difference scores is larger than
the total sum of ranks associated with positive
difference scores. The null hypothesis would
state that the sum of the ranks associated with
negative difference scores is equal to the sum of
the ranks for positive difference scores.
14
The output for Wilcoxon matched-pairs test
In the table of ranks, we see that the sum of
ranks associated with negative difference scores
(12,818) is larger than the sum of ranks
associated with positive difference scores
(6,488). The sum of ranks supports the
direction of the relationship stated in the
research hypothesis.
15
The output for Wilcoxon matched-pairs test
Since the test for a directional relationship is
a one-tailed test, the probability of the test
statistic is calculated by dividing the
two-tailed probability in half 0.000 / 2 0.000
The probability of the test statistic is less
than the level of significance of 0.05. We reject
the null hypotheses and support the research
hypothesis that the sum of ranks for negative
differences was larger, and, therefore, ratings
were higher at one week than at six months.
The answer to the question is true.
16
Solving Wilcoxon matched-pairs signed ranks
problems - 1
The following is a guide to the decision process
for answering Wilcoxon matched-pairs signed
ranks test problems
Is the probability of the test statistic less
than or equal to the level of significance?
Yes
17
Solving Wilcoxon matched-pairs signed ranks
problems - 2
Does the direction of the relationship
represented by the sum of rank agree with the
research question stated in the problem?
18
The sign test

• The sign test compares the number of negative
  differences in scores between matched pairs to
  the number of positive differences in scores.
• To compute the sign test
• First, a difference score is calculated for each
  case by subtracting the actual score for the
  first variable in the pair from the actual score
  for the second variable in the pair. If the
  difference score is zero because the actual
  scores are the same, the case is dropped from the
  analysis.
• Second, the number of positive differences and
  the number of negative differences are counted.
• Third, the probability of the sign statistic is
  computed, using the larger of the number of
  positive or negative differences in the formula.

19
Problem 3

• 1. Based on the dataset OMAHA.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic? Use 0.05 as the level
  of significance. Base your answer on the output
  for the sign signed ranks test.
• For the population represented by this sample,
  victims of domestic violence in Omaha were less
  likely to agree that they were a person of worth
  at one week after the domestic violence incident
  compared to six months after the domestic
  violence incident.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

20
Request the sign test
To compute the sign test in SPSS, select the
Nonparametric Tests Related Samples command
from the Analyze menu.
21
Select the specifications for the test
First, select the pair of variables for the
comparison and move them to the Test Pairs List.
Third, click on the OK button to complete the
request.
Second, mark the checkbox for the Sign test.
22
The research question as a sign test hypothesis
Our research question states that victims were
less likely to agree at one week compared to six
months. This implies that their scores at one
week would be lower (e.g. 1 or 2) than their
scores at six months (e.g. 3 or 4). Subtracting
the lower one week scores from the higher six
months scores would result in positive difference
scores. Our research question translates to a
test of the research hypothesis that there will
be more positive differences than negative
differences. The null hypothesis would state
that the number of positive differences is equal
to the number of negative differences.
23
The output for sign test
In the table of frequencies, we see that the
number of positive differences (90) is greater
than the number of negative differences (76).
The tally of difference scores supports the
direction of the relationship stated in the
research hypothesis.
24
The output for sign test
Since the test for a directional relationship is
a one-tailed test, the probability of the test
statistic is calculated by dividing the
two-tailed probability in half 0.313 / 2 0.157
The probability of the test statistic is greater
than the level of significance of 0.05. We fail
to reject the null hypotheses and cannot support
the research hypothesis that there are more
positive differences and, therefore, ratings were
lower at one week than at six months.
The answer to the question is false.
25
Problem 4

• 5. Based on the dataset OMAHA.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic? Use 0.05 as the level
  of significance. Base your answer on the output
  for correlated groups t-test.
• For the population represented by this sample,
  victims of domestic violence in Omaha were more
  likely to agree that they don't have much to be
  proud of at one week after the domestic violence
  incident compared to six months after the
  domestic violence incident.
• 1. True
• 2. True with caution
• 3. False
• 4. Incorrect application of a statistic

26
The research question as a sign test hypothesis
Our research question states that victims were
more like to agree at one week compared to six
months. This implies that their scores at one
week would be higher than their scores at six
months. Subtracting the lower one week scores
from the higher six months scores would result in
negative difference scores. Our research
question translates to a test of the research
hypothesis that there will be more negative
differences than positive differences. The null
hypothesis would state that the number of
positive differences is equal to the number of
negative differences.
27
The output for sign test
In the table of frequencies, we see that the
number of negative differences (130) is greater
than the number of positive differences (66).
The tally of difference scores supports the
direction of the relationship stated in the
research hypothesis.
28
The output for sign test
Since the test for a directional relationship is
a one-tailed test, the probability of the test
statistic is calculated by dividing the
two-tailed probability in half 0.000 / 2 0.000
The probability of the test statistic is less
than the level of significance of 0.05. We reject
the null hypotheses and support the research
hypothesis that there more negative ratings and,
therefore, ratings were higher at one week than
at six months.
The answer to the question is true.
29
Solving sign test problems - 1
The following is a guide to the decision process
for answering sign test problems
Is the probability of the test statistic less
than or equal to the level of significance?
Yes
30
Solving sign test problems - 2
Does the direction of the relationship agree with
the research question stated in the problem?