consider that the cohort have been categorised according to their exposure status exposed, , and non-exposed, , and also by their disease status after follow-up: (: diseased/ : non-diseased)
the results of the study can then be illustrated via the tabular representation:
Disease status | |||
---|---|---|---|
Diseased | Non-diseased | Total | |
Exposed | a | b | a+b |
Not exposed | c | d | c+d |
Total | a+c | b+d | n |
what is the estimated risk of disease in the exposed group?
what is the estimated risk of disease in the unexposed group?
how might we compare the risk of disease for the exposed and non-exposed?
:
Let and denote the respective disease risks (event probabilities) for the exposed and non-exposed groups.
The risk difference, , is estimated:
The relative risk, , is estimated:
The disease odds, , provides the ratio of success (here death) to failure (here survival). The odds ratio comparing the exposed and non-exposed groups is estimated:
Intervals are based upon the Normal approximation CI:
risk difference:
with , , and
relative risk (natural log scale):
odds ratio (natural log scale):
Study of all cause mortality at 7-years in smoking and non-smoking doctors
Disease status | |||
---|---|---|---|
Died | Alive | Total | |
Smokers | 133 | 25636 | 25769 |
Non-smokers | 3 | 5436 | 5439 |
Total | 136 | 31072 | 31208 |
Study aim: to compare the ‘all cause’ mortality risk amongst smokers and non-smokers
Cohort study prospective: we can investigate association using the either the risk difference, the relative risk or the odds ratio
Example: relative risk (smokers compared to non-smokers)
RR on the natural log scale:
In addition to estimating the relative risk we should construct a confidence interval for the true relative risk
first compute the standard error of the natural logarithm of the relative risk:
then compute a confidence interval for the logarithm of the relative risk:
then back transform (exponentiate) to the relative risk scale:
Conclusion: on average the ‘all cause’ mortality risk amongst smokers was found to be 9.36 times that of non-smokers. We are 95% confident that the true relative risk lies between 2.98 and 29.40 and since the said interval does not contain one we have significant evidence (at the 5% level) of increased mortality risk amongst those who smoke compared to those who do not smoke.
Note that the interval is very wide and thus we have large uncertainty in the true magnitude of the relative risk