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8.1 Introduction to surveys
8.2 Methodological approaches
8.3 Doing survey research
8.4 Statistical Analysis
8.4.1 Descriptive statistics
8.4.2 Exploring relationships
8.4.3 Analysing samples
8.4.3.1 Generalising from samples
8.4.3.2 Dealing with sampling error
8.4.3.3 Confidence limits
8.4.3.4 Statistical significance
8.4.3.5 Hypothesis testing
8.4.3.6 Significance tests
8.4.3.6.1 Parametric tests of significance
8.4.3.6.2 Non-parametric tests of significance
8.4.3.6.2.1 Chi-square test
8.4.3.6.2.2 Mann Witney U test
8.4.3.6.2.3 Kolmogorov-Smirnov Test
8.4.3.6.2.4 H test
8.4.3.6.2.5 Sign test
8.4.3.6.2.6 Wilcoxon test
8.4.3.6.2.7 Friedman test
8.4.3.6.2.8 Q test
8.4.3.7 Summary of significance testing and association: an example
8.4.4 Report writing
8.5 Summary and conclusion
When to use the Friedman test
Hypotheses
What we need to know
Sampling distribution
Assumptions
Testing Statistic
Critical values
Worked examples
Unworked examples
When to use the Friedman two-way analysis of ranks
The Friedman two-way analysis of ranks (Friedman test) is used for three or more samples of ordinal scale data. It is theoretically applicable for any sample size, which in practice means samples of 5 or more.
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Hypotheses
The null hypothesis (H0) is that the samples are not significantly different.
The alternative hypothesis (HA) is that the samples are significantly different.
Only the two-tail test is considered.
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What we need to know
All the values of the related samples, so they can be ranked in order.
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Sampling distribution
The sampling distribution associated with the Friedman test is the chi-square distribution.
The data in the samples being tested are assumed to come from the same population.
The data is ranked in order for each related category (tied scores being given average rank). The ranks are summed for each sample.
If there is no significant difference, the sum of ranks would be approximately the same.
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Assumptions
There are no assumptions associated with the Friedman test other than that the sampling distribution approximates a chi-square distribution.
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Testing statistic
The Friedman testing statistic is denoted χ2r and is distributed approximately as a chi-square for j-1 degrees of freedom when
χ2r = (12∑R2/(n.j)(j-1)) - (3n(j+1))
Where
n = matched sample size
j = number of samples
R = sum of ranks for any sample
∑R2 = total sum of squared ranks for the j samples
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Critical values
When j>3 or when j=3 and n>10, the critical value of χ2r is from tables of critical χ2 for j-1 degrees of freedom.
When j = 3 and n<10 then the critical value of ∑R2 is given in the table below:
| n |
5% significance |
1% significance |
| 5 |
332 |
342 |
| 6 |
474 |
486 |
| 7 |
638 |
650 |
| 8 |
818 |
840 |
| 9 |
1028 |
1058 |
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Worked examples
1. Twenty four men were asked to give a score out of five that indicated their liking (or not) of the current government economic policy. A high score indicated agreement with national policy. The sample was divided into four groups of six according to occupation and each group had one person in each of six income brackets from low to high. Is there a significant difference of view based on occupation?
| Income group |
Professional |
Self-employed |
Non-manual |
Manual |
| 1 Lowest |
4 |
3 |
2 |
1 |
| 2 |
4 |
5 |
2 |
1 |
| 3 |
3 |
4 |
4 |
2 |
| 4 |
3 |
5 |
3 |
4 |
| 5 |
4 |
4 |
3 |
2 |
| 6 Highest |
4 |
5 |
5 |
1 |
Hypotheses:
H0: The four samples are not significantly different.
HA: There is a significant difference between the four occupation samples.
Significance level: 5%
Testing statistic: χ2r = (12∑R2/(n.j)(j-1)) - (3n(j+1))
Critical value: Critical χ2 for j-1 degrees of freedom (i.e 4-1=3 degrees of freedom) from tables of critical values, χ2 equals 7.815.
Decision rule: Reject H0 if calculated χ2r > 7.815
Computation:
n=6, j=4
Rewrite the table above with ranks replacing scores for each related (income) category:
| Income group |
Professional |
Self-employed |
Non-manual |
Manual |
| 1 Lowest |
4 |
3 |
2 |
1 |
| 2 |
3 |
4 |
2 |
1 |
| 3 |
2 |
3.5 |
3.5 |
1 |
| 4 |
1.5 |
4 |
1.5 |
3 |
| 5 |
3.5 |
3.5 |
2 |
1 |
| 6 Highest |
2 |
3.5 |
3.5 |
1 |
| R |
16 |
21.5 |
14.5 |
8 |
| R2 |
256 |
462.25 |
210.25 |
64 |
∑R2 = 992.5
χ2r = (12∑R2/(n.j)(j-1)) - (3n(j+1)) = [(12)(992.5)/6(4)(5)] - 3(6)(4+1)
χ2r = 99.25 - 90 = 9.25
Decision: Reject the null hypotheses (H0) as calculated χ2r is greater than the critical value of 7.815. There is a significant difference in scores between the four samples.
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2. A sample of 24 manual workers was comprised of 8 in each of three occupation groups, matched by current income. The terminal education of the men is shown in the table below. Is there any difference in terminal education between the three occupation groups?
| Income group |
Skilled manual |
Semi-skilled manual |
Unskilled manual |
| 1 Lowest |
1 |
5 |
1 |
| 2 |
3 |
4 |
1 |
| 3 |
2 |
2 |
3 |
| 4 |
5 |
1 |
1 |
| 5 |
6 |
3 |
2 |
| 6 |
6 |
4 |
1 |
Key to terminal qualifications: 1. no qualifications. 2. Two or fewer GCSEs. 3 Three to five GCSEs. 4 More then five GCSEs. 5 One or more Advanced level qualification 6. Further education diploma. 7. Higher education degree.
Hypotheses:
H0: The three samples do not differ significantly.
HA: The three samples do differ significantly.
Significance level: 5%
Testing statistic: The Friedman test is used but as n<10 and j=3, it is only necessary to calculate ∑R2 and compare it with the critical values for small samples in the table above
Critical value: Critical value, when j=3 and n=8, for ∑R2= 818
Decision rule: reject H0 if calculated ∑R2> 818
Computation:
Rewrite the table above with ranks replacing scores for each related (income) category:
| Income group |
Skilled manual |
Semi-skilled manual |
Unskilled manual |
| 1 Lowest |
1.5 |
3 |
1.5 |
| 2 |
2 |
3 |
1 |
| 3 |
1.5 |
1.5 |
3 |
| 4 |
3 |
1.5 |
1.5 |
| 5 |
3 |
2 |
1 |
| 6 |
3 |
2 |
1 |
| 7 |
3 |
1 |
2 |
| 8 Highest |
1.5 |
1.5 |
3 |
| R |
18.5 |
15.5 |
14 |
| R2 |
342.25 |
240.25 |
196 |
∑R2 = 778.5
Decision: Cannot reject the null hypotheses (H0) as calculated ∑R2 is less than the critical value of 818. There is no significant difference in the three samples.
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Unworked examples
1. The table below shows the results of a sample of 20 students at the same school taking end-of-year examinations in five subjects, graded A to G, with A being the highest grade. Is there any significant difference in results between the subjects?
| Pupil number |
English |
Mathematics |
French |
History |
Chemistry |
| 1 |
A |
E |
E |
A |
A |
| 2 |
C |
D |
F |
B |
B |
| 3 |
A |
A |
C |
C |
C |
| 4 |
C |
A |
B |
A |
A |
| 5 |
B |
A |
E |
A |
A |
| 6 |
F |
C |
D |
C |
B |
| 7 |
D |
A |
A |
A |
A |
| 8 |
E |
C |
A |
C |
C |
| 9 |
D |
B |
A |
A |
B |
| 10 |
D |
F |
C |
C |
F |
| 11 |
F |
D |
A |
B |
D |
| 12 |
A |
C |
C |
F |
E |
| 13 |
B |
B |
B |
D |
D |
| 14 |
C |
A |
F |
A |
A |
| 15 |
F |
B |
A |
B |
B |
| 16 |
A |
C |
B |
C |
C |
| 17 |
A |
A |
C |
A |
A |
| 18 |
C |
A |
A |
A |
A |
| 19 |
D |
B |
A |
C |
D |
| 20 |
A |
B |
E |
G |
A |
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2. Six social workers selected at random from a large city were asked questions about the local authority's housing arrangements. Each of the six respondents were re-interviewed, after a year and then again a further year later, to see if experience changed their views. Part of the interview related to fairness of allocation of local authority owned housing (with a scale from 0–9, with a high score indicating a fair allocation). The results are in the table below. Is there a significant change in attitude over time?
| Social Worker |
First interview |
Second interview |
Third interview |
| 1 |
7 |
5 |
3 |
| 2 |
4 |
5 |
4 |
| 3 |
8 |
7 |
2 |
| 4 |
6 |
5 |
2 |
| 5 |
3 |
0 |
0 |
| 6 |
4 |
1 |
3 |
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