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I've really tried to understand why you can't use relative risk (RR) in retrospective studies but for some reason I can't understand it. Has anyone been able to convey this simply to their students?

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  • $\begingroup$ As an exercise, try to develop an estimator for RR for retrospective studies. This must be well explained in some epidemiology text, so look there. $\endgroup$ May 15, 2020 at 16:38
  • $\begingroup$ Let's say I look through 1000 patient charts for outcome Z. I find 100 patients have outcome Z. I'm interested in whether gender increases the risk of outcome Z. The prevalence in my cohort for outcome Z is 10%. 30 of 500 males have outcome Z while 70 of 500 females have outcome Z. The risk of outcome Z for males is 30/500 = 6%, the risk for outcome Z for females is 70/500 = 14%. The RR for females is 14/6 = 2.3. I'm obviously doing something wrong but I can't see what exactly as this makes perfect sense to me? $\endgroup$
    – Paze
    May 15, 2020 at 17:13

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I have been able to explain the need for the use of the odds ratio (cross-product ratio) to estimate effect size in a case-control study to students with a simplified example of two hypothetical studies of the same (hypothetical) people—one a cohort study and the other a case-control study.

The hypothetical people are 300,000 adults. 100,000 of these adults smoke cigarettes and 200,000 don’t smoke. The distributions of age and sex in the smokers and non-smokers are identical.

Prospective Cohort Study

One researcher does a prospective cohort study of these 300,000 adults to evaluate the relationship between smoking and lung cancer. The 100,000 smokers and 200,000 non-smokers are recruited and followed for 5 years. Over 5 years, 1,000 lung cancers occur in smokers and 200 occur in non-smokers. The lung cancer incidence in smokers is 0.010 (1,000/100,000) and the lung cancer incidence in non-smokers is 0.001 (200/200,000). The true relative risk (RR) of lung cancer in smokers is:

               RR = 0.010/0.001  = 10.0

Case-Control Study Based on the Same People

Another researcher does a case-control study to evaluate the relationship between smoking and lung cancer based on the same 300,000 people. (Of course, this wouldn’t happen—this is hypothetical.) S/he identifies the 1,200 cases of lung cancer that occurred in these 300,000 people and selects 1,200 people without lung cancer as controls. The controls are a random sample of the 298,800 people who did not develop lung cancer. The smoking history of the cases and controls is determined. In the 1,200 cases of lung cancer, 1,000 people smoke (0.8333) and 200 do not smoke (0.1667). In the 1,200 controls, 400 smoke (0.3333) and 800 do not smoke (0.6667).

The data are given in the familiar case-control 2x2 table.

                           Case       Control

    Smoker

    Yes                     1,000         400

    No                        200         800

    Total                   1,200        1,200

To assess the strength of the association of smoking with lung cancer, a first instinct is to calculate the ratio of the proportion of smokers in the cases to the proportion of smokers in the controls. This is:

               0.8333/0.3333 = 2.50

Very far away from the true relative risk of 10.0.

In a generic 2x2 table for data from a case-control study, the numbers of cases and controls according to exposure is as follows:

                         Case      Control

  Exposed
    
  Yes                      A            B

  No                       C            D
    

The odds ratio—is calculated as the cross-product ratio:

                [(A*D)]/[(B*C)]  

For the hypothetical case-control study of smoking and lung cancer, the odds ratio is:

                [(1,000*800)]/[(200*400)] = 10.0. 

The odds ratio is very close to the relative risk—in this hypothetical example, they are exactly the same but this is not generally true.

Generalizing

Replacing the numbers in the 2 x 2 tables with proportions where p1 is the proportion exposed among cases and p2 is the proportion exposed in the controls, gives the following

                          Case        Control

        Exposed

        Yes                   p1          p2

        No                 1 - p1        1- p2

    

The odds ratio (cross-product ratio) is:

               [(p1)*(1 – p2)] / [(p2)*(1 – p1)] 

The odds ratio approximates the relative risk when the incidence of the disease is “rare”—rule of thumb, less than 0.10. Using the odds ratio from a case-control to estimate relative risk also requires that the cases and the controls both come from the same defined population.

Explanation

In a 1951 publication, Cornfield explains why the cross-product ratio from a case-control approximates the relative risk mathematically.

(https://pubmed.ncbi.nlm.nih.gov/14861651/

This is an example meant to help students being introduced to epidemiology understand better why the odds ratio needs to be used to estimate the magnitude of the association of exposure with disease in a case-control study. No further detail will be given here.

Historical Note

Of historical interest, the use of the term—odds ratio—to describe the cross-product ratio as an estimate of relative risk did not become widespread until the early 1990’s. Until then, estimates of the relative from case-control studies were variously described as “relative risks” (without qualification), “cross-product ratios,” “estimated relative risks,” or “risk ratios.”

Cornfield J. A method of estimating comparative rates from clinical data; applications to cancer of the lung, breast, and cervix. J Natl Cancer Inst. 1951 Jun;11(6):1269-75. PMID: 14861651.

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  • $\begingroup$ Hi Diana sorry for accepting your answer late, it was a wonderful write-up and very detailed. I did read it a while ago but I must not have clicked accept when I did! Thank you! $\endgroup$
    – Paze
    Oct 24, 2020 at 19:10
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    $\begingroup$ @Paze Thx. I've been using this set-up in teaching for decades and hope others might find it useful. $\endgroup$ Oct 24, 2020 at 21:13

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