1. Statistical Inference for more than two groups
Statistics for Health Research
Statistical Inference for more than two groups
Peter T. Donnan
Professor of Epidemiology and Biostatistics

2. Tests to be covered
Chi-squared test
One-way ANOVA
Logrank test

3. Significance testing – general overview
Define the null and alternative hypotheses under the study
Acquire data
Calculate the value of the test statistic
Compare the value of the test statistic to values from a known probability distribution
Interpret the p-value and draw conclusion

5. Categorical data > 2 groups
Unordered categories – Nominal
- Chi-squared test for association
Ordered categories - Ordinal
- Chi squared test for
trend

6. Example Does the proportion of mothers developing
pre-eclampsia vary by parity (birth order)?

7. Contingency table (r x c) Pre-eclampsia Birth Order 1st 2nd 3rd 4th No
Yes
1170 (79.4%)
278 (84.8%)
83 (86.5%)
86 (92.4%)
304 (20.6%)
50 (15.2%)
13 (13.5%) 7
(7.5%)

8. Null Hypotheses
Null hypothesis: No association between pre-eclampsia and birth order
Null hypothesis: There is no trend in pre-eclampsia with parity

9. Test of association
Test of linear trend

10. Conclusions
Strong association between pre-eclampsia and birth order (Χ2 = 15.42, p = 0.001)
Significant linear trend in incidence of pre-eclampsia with parity (Χ2 = 15.03, p < 0.001)
Note 3 degrees of freedom for association test and 1 df for test for trend

11. Contingency table (r x c) Pre-eclampsia Birth Order 1st 2nd 3rd 4th No
Yes
1170 (79.4%)
278 (84.8%)
83 (86.5%)
86 (92.4%)
304 (20.6%)
50 (15.2%)
13 (13.5%) 7
(7.5%)

12. Contingency Tables (r x c)
Tables can be any size. For example SIMD deciles by parity would be a 10 x 4 table
But with very large tables difficult to interpret tests of association
Crosstabulations in SPSS can give Odds ratios as an option with row or column with two categories

13. Numerical data > 2 groups
Compare means from several groups
Single global test of difference in means
Also test for linear trend
1-way analysis of variance (ANOVA)

14. Extend t-test to >2 groups i.e Analysis of Variance (ANOVA)
Consider scores for contribution to energy intake from fat groups, milk groups and alcohol groups
Does the mean score differ across the three categories of intake groups?
Koh ET, Owen WL. Introduction to Nutrition and Health Research Kluwer Boston, 2000

15. One-Way ANOVA of scores
Contributor to Energy Intake
Fat
Milk
Alcohol
n=6
Mean=4.22
n=6
Mean=2.01
n=6
Mean=0.167

16. One-Way ANOVA of Scores
The null hypothesis (H0) is ‘there are no differences in mean score across the three groups’
Use SPSS One-Way ANOVA to carry out this test

17. Assumptions of 1-Way ANOVA
1. Standard deviations are similar
2. Test variable (scores) are approx. Normally distributed
If assumptions are not met, use non-parametric equivalent Kruskal-Wallis test

18. Results of ANOVA
ANOVA partitions variation into Within and Between group components
Results in F-statistic – compared with values in F-tables
F = 108.6, with 2 and 15 df, p<0.001

19. Results of ANOVA
The groups differ significantly and it is clear the Fat group contributes most to energy score with a mean = 4.22
Further pair-wise comparisons can be made (3 possible) using multiple comparisons test e.g. Bonferroni

20. Example 2
Does income vary by highest level
of education achieved?

21. Null Hypothesis and alternative
H0: no difference in mean
income by education level
achieved
H1: mean income varies with
education level achieved

22. Assumptions of 1-Way ANOVA
Standard deviations or variances are similar
Test variable (income) are approx. Normally distributed
If assumptions are not met, use non-parametric equivalent Kruskal-Wallis test

25. Table of Mean income for each level of educational achievement

29. Analysis of Variance Table
F-test gives
P < showing significant difference between mean levels of education

30. Table of each pairwise comparison.
Note lower income for ‘did not complete school’ to all other groups.
All p-values adjusted for multiple comparisons

31. Summary of ANOVA
ANOVA useful if number of groups with continuous summary in each
SPSS does all pairwise group comparisons adjusted for multiple testing
Note that ANOVA is just a form of linear regression – see later

32. Extending Kaplan-Meier and logrank test in SPSS
You need to specify:
Survival time – time from surgery (tfsurg)
Status – Dead = 1, censored = 0 (dead)
Factor – Duke’s stage at baseline (A, B, C, D, Unknown)
Select compare factor and logrank
Optionally select plot of survival

33. Implementing Logrank test in SPSS

34. Select options to obtain plot and median survival
Select Compare Factor to obtain logrank test
Select options to obtain plot and median survival
Select linear trend for this test

35. Overall Comparisons
Chi-Square df Sig.
Log Rank (Mantel-Cox)
The vector of trend weights is -2, -1, 0, 1, 2. This is the default.
The test for trend in survival across Duke’s stage is highly significant

36. Interpret SPSS output
Note the logrank statistic, degrees of freedom and statistical significance (p-value).
Note in which direction survival is worst or best and back up visual information from the Kaplan-Meier plot with median survival and 95% confidence intervals from the output.
Finally, interpret the results!

37. Interpret test result in relation to median survival
Duke’s Stage
Median Survival (days)
Mean Survival
(Days) A
2770
1978 B
1749
1866 C
1120
1304 D
375
646
Unknown
581
1297

38. Output form Kaplan-Meier in SPSS
Note that SPSS gives three possible tests:
Logrank, Tarone-Ware and Breslow
In general, logrank gives greater weight to later events compared to the other two tests.
If all are similar quote logrank test.
If different results, quote more than one test result

39. Editing SPSS output
Note that everything in the SPSS output window can be copied and pasted into Word and Powerpoint.
Double-clicking on plots also allows editing of the plot such as changing axes, colours, fonts, etc.

40. Diabetic patients LDL data
Try carrying out extended Crosstabulations and ANOVA where appropriate in the LDL data…
E.g. APOE genotype

41. Colorectal cancer patients: survival following surgery
Try carrying out Kaplan-Meier plots and logrank tests for other factors such as WHO Functional Performance, smoking, etc…

42. Extending test to more than 2 groups Summary
Define H0 and H1
Choosing the appropriate test according to type of variables
Interpret output carefully