C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Kruskall-Wallis and...
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C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
1
Kruskall-Wallis and Friedman Tests
• Non-parametric statistical tests exist for studies were there are more than two samples• Kruskall Wallis - Between Groups Designs• Friedman - Within Subjects & Matched Designs
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
2
Rationale of the Kruskall Wallis
• When the null hypothesis is true we expect a random distribution of ranks across the groups
• When the null hypothesis is false we expect a systematic distribution of ranks across the groups
Rank
Group A B C B A C C B A
1 2 3 4 5 6 7 8 9
A
B
C
Group Rank Sum
15
14
16
Rank
Group A A B A B C B C C
1 2 3 4 5 6 7 8 9
A
B
C
Group Rank Sum
7
15
23
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
3
The Rationale of the Kruskall Wallis
• The Null Hypothesis is True• The average rank of each cell in the design should be
equal• The Null Hypothesis is False
• The average rank of each cell in the design should be different
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
4
An Example of the Kruskall Wallis
Type of ReinforcementControl Praise CriticismScore Rank Score Rank Score Rank10 4 12 6.5 11 513 8.5 14 11.5 9 2.514 11.5 15 14.5 12 6.515 14.5 14 11.5 9 2.514 11.5 16 16.5 8 116 16.5 17 18 13 8.5
Total 66.5 Total 78.5 Total 26.0Mean 11.08 Mean 13.08 Mean 4.33
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
5
Formula for the Kruskall Wallis
• The critical value of the Kruskall-Wallis is calculated using the following formula:
• Where is the total number of scores is the mean rank for each level of the variable is the number of scores in each level of the variable
KW 12
N(N 1)nR 2
3(N 1)
N
R
n
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
6
Calculating the Value of the Kruskall Wallis
• Given the formula
• We substitute the values
KW 1218 181
(6(11.082)6(13.082)6(4.332)
3181
Total Number of Subjects
Number of Subjects in Each Group
Group Rank Means
KW 12
N(N 1)nR 2
3(N 1)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
7
The Significance of the Kruskall Wallis
• The observed value equals 8.81.• Find the critical value of the test statistic in tables• In this case the critical value is 5.99.• If the observed value is greater than the critical value then
reject the null hypothesis.• We reject the null hypotheses and state that the groups are
different
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
8
Interpreting a Kruskall Wallis
• Plotting the Mean Rank we find:
• We know that the three levels of the independent variable produce different outcomes.
• We don't know exactly what or where the differences are.
Mean Rank0
2
4
6
8
10
12
14
Control
Praise
Criticism
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
9
Interpreting a Kruskall Wallis
• We can ask a variety of questions: • Is praise is better than no
reinforcement?• Is criticism worse than no
reinforcement?• Is praise better than criticism?
• The Kruskall Wallis statistic doesn't give us the answers
Mean Rank0
2
4
6
8
10
12
14
Control
Praise
Criticism
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
10
Post Hoc Comparisons
• We can test the significance of individual pairs of conditions • These tests are known as post hoc pairwise comparisons• A critical difference value is calculated
• If the difference between two rank means is greater than the critical difference then it is significant
• If the difference between two rank means is less than the critical difference then it is not significant
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
11
Post Hoc Comparisons
• The critical difference value is given by the following formula:
• Where is the total number of subjects in the experiment is the number of subjects in one group is the number of subjects in the other group
Critical Difference 2.394N(N +1)
12
1
n1
1
n2
N
n1
n2
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
12
Post Hoc Comparisons
• If the difference between a pair of Rank Means is greater than or equal to the right hand side of this equation then they are significantly different at p<0.05.
• Substituting the appropriate values:
• The critical difference we need to exceed is 7.38
Critical Difference2.394 18(181)12
161
6
Critical Difference 2.394N(N +1)
12
1
n1
1
n2
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
13
Interpreting a Kruskall Wallis
• The differences between the mean ranks are:• Control vs Praise = 2• Control vs Criticism = 6.75• Praise vs Criticism = 8.75
• The difference between Praise vs Criticism exceeds the critical difference
• Therefore there is only one significant difference at p<0.05
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
14
The Rationale of the Friedman
• In the Friedman there are two possible sampling strategies:• Each subject contributes several scores• A matched group of subjects provide one score each
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
15
S1
S2
S3
Rank Total
Subject Level A Level B Level C
1
2
3
6
2
1
2
5
3
3
1
7
S1
S2
S3
Rank Total
Subject Level A Level B Level C
1
2
1
4
2
1
2
5
3
3
3
9
• When the null hypothesis is true we expect a random distribution of the ranks across the subjects
• When the null hypothesis is false we expect a systematic distribution of the ranks across the subjects
Rationale of the Friedman• Each individual subjects scores are ranked
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
16
The Rationale of the Friedman
• When the Null hypothesis is true• The rank totals of the different levels of the IV will be
about equal• When the Null hypothesis is false
• The rank totals of the different levels of the IV will be different
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
17
An Example of the Friedman
Type of ReinforcementControl Praise Criticism
Group Score Rank Score Rank Score Rankg1 10 1 12 3 11 2g2 13 2 14 3 9 1g3 14 2 15 3 12 1g4 15 3 14 2 9 1g5 14 2 16 3 8 1g6 16 2 17 3 13 1
Total 12 Total 17 Total 7
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
18
Formula for the Friedman
• The critical value of the Friedman is calculated using the following formula:
• Where N is the total number of subjects or groups R is the rank total for each level of the independent
variables k is the number levels of the independent variables
Fr 12Nk(k1)
R2
3N(k1)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
19
Calculating the Value of the Friedman
• Give the formula
• The Friedman is calculated as follows
Fr 12Nk(k1)
R2
3N(k1)
Subjects/Groups Levels of the IV
Group RankTotals
Fr 12(6)(3)(31)
(12217272)
(3)(6)(31)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
20
The Significance of the Friedman
• The observed value is 8.33• Find the critical value of the test statistic in tables• The critical value is 7.00• If the observed value is greater than the critical value then
reject the null hypothesis.• We reject the null hypothesis and conclude that the three
levels of the treatment variable are different
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
21
Interpreting a Friedman
• Plotting the Rank Totals we find:• We know that the three levels
of the independent variable produce different outcomes.
• We don't know exactly what or where the differences are.
12
17
7
Total Rank0
2
4
6
8
10
12
14
16
18
Control
Praise
Criticism
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
22
Interpreting a Friedman
• We can ask a variety of questions: • Is praise is better than no reinforcement?• Is criticism worse than no reinforcement?• Is praise better than criticism?
• The Friedman statistic doesn't give us the answers
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
23
Post Hoc Comparisons
• Post Hoc Comparisons • We can test the significance of individual pairs of
conditions • A critical difference value is calculated
• If the difference between two rank totals is greater than the critical difference then it is significant
• If the difference between two rank totals is less than the critical difference then it is not significant
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
24
Post Hoc Comparisons
• The critical difference value is given by the following formula:
• Where• N is the total number of subjects or matched groups
in the experiment• k is the number of levels of the independent variable
Critical Difference2.394 Nk(k1)6
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
25
Post Hoc Comparisons
• If the difference between a pair of Rank Totals is greater than or equal to the right hand side of this equation then they are significantly different at p<0.05.
• Substituting the appropriate values:
• The critical difference we need to exceed is 8.29
Critical Difference2.394 (6)(3)(4)6
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
26
Interpreting a Friedman
• The differences between the rank totals are:• Control vs Praise = 5• Control vs Criticism = 5• Praise vs Criticism = 10
• The difference between Praise vs Criticism exceeds the critical difference
• Therefore there is only one significant difference at p<0.05
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
27
Summary
• There are two non-parametric statistical tests that apply experiments with more than two levels of the independent variable• Kruskall-Wallis - K levels of a between group (independent
samples) design• Friedman - K levels of a within group (or matched
samples) design• For both these tests further exploration is required to
establish the differences between pairs of conditions.