Statistics for Health Research
Statistical Inference
for more than two
groups
Peter T. Donnan
Professor of Epidemiology and Biostatistics
Tests to be covered
•Chi-squared test
•One-way ANOVA
•Logrank test
Significance testing – general
overview
1. Define the null and alternative hypotheses
under the study
2. Acquire data
3. Calculate the value of the test statistic
4. Compare the value of the test statistic to
values from a known probability distribution
5. Interpret the p-value and draw conclusion
Categorical data > 2 groups
Unordered categories – Nominal
- Chi-squared test for association
Ordered categories - Ordinal
- Chi squared test for
trend
Example
Does the proportion of
mothers developing
pre-eclampsia vary by
parity (birth order)?
Contingency table
(r x c)
Preeclampsia
1st
No 1170
(79.4%)
Yes 304
(20.6%)
Birth Order
2nd
3rd
4th
278
(84.8%)
83
(86.5%)
86
(92.4%)
50
(15.2%)
13
(13.5%)
7
(7.5%)
Null Hypotheses
1. Null hypothesis: No
association between preeclampsia and birth order
2. Null hypothesis: There is
no trend in pre-eclampsia
with parity
Test of
association
Test of
linear
trend
Conclusions
1. Strong association between preeclampsia and birth order (Χ2 =
15.42, p = 0.001)
2. Significant linear trend in
incidence of pre-eclampsia with
parity (Χ2 = 15.03, p < 0.001)
3. Note 3 degrees of freedom for
association test and 1 df for test
for trend
Contingency table
(r x c)
Preeclampsia
1st
No 1170
(79.4%)
Yes 304
(20.6%)
Birth Order
2nd
3rd
4th
278
(84.8%)
83
(86.5%)
86
(92.4%)
50
(15.2%)
13
(13.5%)
7
(7.5%)
Contingency Tables
(r x c)
1. Tables can be any size. For
example SIMD deciles by parity
would be a 10 x 4 table
2. But with very large tables
difficult to interpret tests of
association
3. Crosstabulations in SPSS can give
Odds ratios as an option with row
or column with two categories
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)
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
One-Way ANOVA of scores
Contributor to Energy Intake
Fat
Milk
Alcohol
n=6
n=6
n=6
Mean=4.22
Mean=2.01
Mean=0.167
One-Way ANOVA of Scores
The null hypothesis (H0) is ‘there are no
differences in mean score across the
three groups’
x1  x2  x3
Use SPSS One-Way ANOVA to
carry out this test
Assumptions of 1-Way
ANOVA
1. Standard deviations are similar
2. Test variable (scores) are approx.
Normally distributed
If assumptions are not met, use nonparametric equivalent Kruskal-Wallis
test
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
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
Example 2
Does income vary by
highest level
of education achieved?
Null Hypothesis and
alternative
H0: no difference in mean
income by education level
achieved
H1: mean income varies with
education level achieved
Assumptions of 1-Way
ANOVA
1. Standard deviations or
variances are similar
2. Test variable (income) are
approx. Normally distributed
If assumptions are not met, use nonparametric equivalent Kruskal-Wallis
test
Table of Mean
income for each
level of educational
achievement
Analysis of
Variance Table
F-test gives
P < 0.001
showing
significant
difference
between mean
levels of
education
Table of
each
pairwise
comparison.
Note lower
income for
‘did not
complete
school’ to
all other
groups.
All p-values
adjusted
for multiple
comparisons
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
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
Implementing Logrank test in
SPSS
Select
Compare
Factor to
obtain
logrank test
Select linear trend for
this test
Select options
to obtain plot
and median
survival
Overall Comparisons
Log Rank (Mantel-Cox)
Chi-Square df
80.534 1
Sig.
.000
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
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!
Interpret test result in
relation to median survival
Duke’s
Stage
Median Mean
Survival Survival
(days)
(Days)
A
2770
1978
B
1749
1866
C
1120
1304
D
375
646
Unknown 581
1297
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
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.
Diabetic patients LDL data
• Try carrying out extended
Crosstabulations and ANOVA
where appropriate in the LDL
data…
• E.g. APOE genotype
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…
Extending test to more than
2 groups
Summary
• Define H0 and H1
• Choosing the appropriate test
according to type of variables
• Interpret output carefully