1. Case-control studies: statistics
Kath Bennett
16 April 2008

2. Outline Statistical methods
Odds ratios in unmatched and matched studies
Adjusted Odd ratio
Mantel-Haenszel method
Logistic regression
Conditional logistic regression

3. Examples of case-control studies
Doll and Hill. Lung cancer and smoking.
INTERHEART study. Large international standardised case-control study on aetiology of CHD.

4. Data from a case-control study are expressed as:
Past exposure to suspected factor
Cases (with disease under study)
Controls (without disease under study)
Total
Yes
(a) 200
(c) 50
250 No
(b)300
(d)450
750
500
1000

5. Data from unmatched case-control studies are analysed in terms of:
Comparison of the frequency of exposure between the cases and controls
Odds ratios (OR), i.e, the ratio of  the odds in favour of exposure among the cases compared to the odds in favour of exposure among the controls: 
a / b  =  a d
c / d b c
For a rare disease, or short observation time (which makes the disease effectively rare), the odds-ratio (OR) approximates RR.

6. Odds ratio Using the example above:
40% (200/500) of the cases were exposed to the suspected risk factor vs. 10% (50/500) of the controls. 
The odds ratio is calculated as:  200/300 = 200 x 450 = 6  50/450     300 x 50
This is interpreted as: the cases are 6 times more likely than the controls to have been exposed to the risk factor under study. 

7. Example: OR in unmatched case
Age at First Full-Term Pregnancy and Breast Cancer
Age(years)
Cases
Control
<25
131 (35%)
600 (48%)
>=25
243 (65%)
662 (52%)
Total
374 (100%)
1262 (100%)
More cases than controls had their first child after the age of 25 years: 65% vs. 52%.
Odds ratio (OR):
243 / 131
=1.7
662 / 600

8. Odds Ratio in a Matched Case-Control Study (with matched pairs)
History of Cigarette Smoking and Chronic Obstructive Pulmonary Disease (COPD)
Cases with COPD
Smoker Nonsmoker Total
Controls (No COPD) Smoker
Nonsmoker
Total
Pairs: a. Case smoker, control smoker = 14
b. Case smoker, control nonsmoker =17
c. Case nonsmoker, control smoker =5
d. Case nonsmoker, control nonsmoker = 16

9. Odds Ratio in a Matched Case-Control Study (with matched pairs)
Pairs: a. Case smoker, control smoker = 14
b. Case smoker, control nonsmoker = 17
c. Case nonsmoker, control smoker = 5
d. Case nonsmoker, control nonsmoker = 16
Only the discordant pairs contribute relevant information
i.e. those where the case is a smoker and the control is a non-smoker (b) or visa-versa (c).
The OR is the ratio of pairs in which the case is a smoker and the control is a non-smoker (b) to pairs in which the case is a non-smoker and the control is a smoker (c). OR=b/c.
Therefore, the odds ratio based on the discordant pairs is 17/5 = 3.4

10. Odds ratio and 95% confidence limits
An odds ratio >1 indicates a positive association between a factor and the disease.
An odds ratio <1 indicates that the factor is protective.
An odds ratio of 1.0 indicates that there is no association between the factor and the disease.
Confidence limits are used to interpret the significance of relative risks and odds ratios. Relative risks and odds ratios are interpreted as statistically significant if confidence intervals do not include 1.
OR = 7.0 ( )
OR = 7.0 ( )
The second odds ratio is statistically significant, the first one is not.

11. Adjusted ORs
Mantel-Haenszel (stratified) method can be applied to deal with confounding. Assumption that the OR for association does not differ by level of confounding variable.
Logistic regression for unmatched case-control studies, deal with confounding and interaction
Conditional logistic regression for matched case-control studies.

12. Example: Mantel-Haenszel method
Is smoking a risk factor for pulmonary carcinoma after adjustment for occupation and age?
Cases (diseased)
Controls (no disease)
Occupation
Age
Non-smoker
>=1 pack/day
Housewives
<45
45-54
55-64
65+ 2 5 6 11 3 7 24 49 42 1
White collar 4 18 23
Other occupation 10 12 19 15
M-H Adjusted OR=10.68 (95% CI 4.47, 39.26)

13. Logistic regression Binary outcome e.g. Case/control
Logistic regression used to predict the likelihood of being a case vs control in relation to explanatory variables (risk factors).
The resulting beta or regression coefficients are log(OR), therefore adjusted OR and 95% CI can be presented.
Similar in many respects to ordinary regression methods.

14. Logistic Regression
Logistic regression allows us to do 2x2 table analysis, and much more:
Let y = 1 if MI, 0 if not
Let x = 1 if new risk factor, 0 if no risk factor
What is difference between an “exposed” and “unexposed” pair of individuals?

15. Logistic Regression What is the interpretation of β1?
That was simplest case
Logistic regression allows us much more freedom:
x’s can be anything (continuous, binary, etc.)
And we need to adjust for x2 = BMI, x3 = smoking, x4 = alcohol, x5 = years before diagnosis etc..
What is the interpretation of β1?

16. Interpretation of coefficients
β1 is the log odds ratio comparing risk of MI for those with new risk factor vs no risk factor, adjusted for BMI, smoking, alcohol, and years since diagnosis.
e β = 2.4
Individuals with new risk factor are at 2.4 times the risk of MI compared to those without risk factor, adjusting for BMI, smoking, alcohol, and years since diagnosis.

17. Conditional logistic regression
Conditional logistic regression is similar to the usual logistic regression except it keeps track of which case was matched to which control
The computer output and presentation of conditional logistic regression is similar to logistic regression

18. Example: Study of exposure to medication and risk of falls
N=1,486 cases in nursing home with injurious fall matched to 1,486 matched controls
Controls drawn from similar population matched on:
Sex, Age, level of dependency, year and Duration of residence, chronic disorder

19. Example: results from conditional logistic regressionOR
95% CI
Inotrophic agents
0.69**
BBs
1.04
Anti-hypertensives
0.91
Anti-psychotic agents
1.31**
Anxiolytics/sedatives/hypnotics
1.35**

20. Interaction
Interaction occurs between two risk factors when the effect of one risk factor upon disease is different at different levels of the second risk factor.
Also know as ‘effect modification’.
When there is no interaction the effects of each risk factor are constant across levels of other risk factors (homogeneous).