Getting baseline risks for case-control studies
In order to translate measures of relative risk, such as odds ratios or hazard ratios, to the more easily interpretable changes in absolute risk, we need baseline risks: the proportion of people in the ‘unexposed’ group who experience the outcome of interest. This will help make sense of the real magnitude of the risks reported in research and to be able to communicate its relevance to wider audiences.
Unfortunately, baseline risks are not readily obtained for case-control studies: a fairly common design in which people who have experienced the outcome are identified (the cases), and matched to some extent (say same age and sex) with people who have not experienced the outcome (the controls), and then retrospectively look at what proportion in each group had been exposed to the risk factor. This is the opposite to a prospective cohort design, in which exposed and unexposed people are followed up over a long period of time: a case-control study is quicker and more efficient for rare outcomes, but does require good historical data which is not subject to ‘recall bias’ (people who suffer a bad outcome may tend to report exposures that they have heard might be linked to their condition).
Crucially, in a case-control study the number of cases and controls have been chosen rather than observed, and so we cannot estimate the baseline risks. Although, as we shall see, such a design does, rather remarkably, allow us to estimate an odds ratio.
Let’s look at a recent paper by Chalitsios et al (2020) “Risk of osteoporosis and fragility fractures in asthma due to oral and inhaled corticosteroids: two population-based nested case-control studies”
As the title makes clear, this was a case-control study: “We identified 1564 patients with asthma and osteoporosis, and 3313 control subjects as well as 2131 patients with asthma and fractures and 4421 control subjects from a cohort of 69 074 individuals with asthma.”
Tables 3 and 4 provide the crucial information relating to osteoporosis and fragility fractures.
These are reported as follows:
After adjusting for confounders, people receiving more OCS prescriptions (≥9 vs 0) had a 4.50 (95% CI 3.21 to 6.11) and 2.16 (95% CI 1.56 to 3.32) increased risk of osteoporosis and FF, respectively.
You should be able to spot these conclusions in Tables 3 and 4.
Getting odds ratios for case-control studies
First, remember that the odds for an event is the probability of the event occurring, divided by the probability of the event not occurring. So, for example, the odds for a dice coming up ‘six’ is the probability of a ‘six’ divided by the probability of something other than a ‘six’, which is (1/6) / (5/6) = 1/5. In gambling parlance, this is ‘1 to 5 on’, or ‘5 to 1 against’. Note that, for equally likely outcomes, the odds is the number of outcomes that comprise the event of interest (eg 1), divided by the number of outcomes in which teh event does not occur (eg 5).
Now consider Table 3. For No OCS use (baseline), the odds of being a case rather than a control in the total sample, is the number of outcomes that are cases (992) divided by the number which are controls (2,607) = 992/2,607 = 0.38 .
Similarly, or the people exposed to more than 9 OCS prescriptions, the odds of being a case rather than a control is 100/48 = 2.08.
So the (unadjusted) odds ratio is 0.38 / 2.08 = 5.47 (in fact this is reported in Table 3 as 5.37, presumably from a slightly different formula).
So the odds ratios can be estimated from this design, even though the baseline risks cannot.
Getting baseline risks for case-control studies
For case-control studies, it may be necessary to go back to the authors or use external sources of information, but sometimes the baseline risks can be extracted from the paper.
Remember the quote “We identified 1564 patients with asthma and osteoporosis, and 3313 control subjects as well as 2131 patients with asthma and fractures and 4421 control subjects from a cohort of 69 074 individuals with asthma.” If we can assume that these 1564 patients was a complete set of those experiencing osteoporosis, then we can conclude there were 69,074–1,564 = 67,510 patients with asthma who did not experience osteoporosis. We can then use the proportions seen in Table 3 to ‘scale up’ the controls to the whole population, assuming the actual observed controls were a representative sample of all these ‘non-cases’. This is laid out in the table below.
So the average baseline risk of osteoporosis, given no OCS, can be estimated as 992/54,116 = 1.8%.
We could then run this example through the Winton Centre’s RealRisk program and get the output shown here.
Exercise for the reader. Carry out the same analysis to show the average baseline risk of fracture, given no OCS is 3.1%, and run it through RealRisk. You should get something like this.
