Letter to the Editor

Thank you to Dr. Valeika for his comments about interpreting odds ratios as risk ratios and for extending the discussion to include case control designs that use incident cases rather than prevalent cases.1, 2, 3 I agree with Dr. Valeikas overall message; however, as presented, it perpetuates a misleading message that the odds ratio from such studies is the risk ratio or rate ratio based on the approach to sampling controls. In addition to Dr. Valeika's comments I would propose a correction and clarification. In presenting this important concept to readers, Dr. Valeika has used a common parlance to infer that the idea that the odds ratio overestimates the risk ratio is not as broadly applicable as stated in “Interpretation of odds and risk ratios”4. I would propose that mathematically it is consistently true that the estimates of the population odds ratio (θ _OR) will overestimate the population risk ratio (θ _RR) and the extent of this overestimation is reduced as the disease becomes rare. Dr. Valeika writes that “if controls are selected correctly, the OR from a case‐control study provides an unbiased estimate of either the RR or IRR”. However, this statement fails to distinguish between the mathematical formulae commonly used to calculate an estimate of the population odds ratio and the actual measure of association; ie, the ratio of the odds of disease in the exposed and unexposed. The prior statement could be more accurately be restated as “when controls are selected from the cohort using either sub‐cohort sampling or density sampling, the cross‐product ratio (or exponent of the logistic regression coefficient) from a case‐control study provides an unbiased estimate of either the population RR or IRR”. The use of the measure of association as a synonym for a mathematical formula is common practice in epidemiology. Such simplifications are certainly useful until more nuanced messages are needed, as is the case with nested study designs. Measures of associations are characteristics of the population we attempt to estimate, and changing the study design cannot change the relationship between those population measures.

To clarify the important difference in the two statements above, consider a population where an exposure‐disease relationship is of interest and observations on individuals are independent. For this population, three metrics can be used to describe the exposure‐disease relationship: a population incidence odds ratio (θ _IOR), a population risk ratio (θ _RR), and a population incidence rate ratio (θ _IRR). If the researcher uses a traditional cohort design, and no loss to follow‐up occurs, the cross‐product ratio from the study population (or exponent of the logistic regression coefficient) will provided an estimate of the θ _IOR, ie, ${\widehat{\mathit{\theta }}}_{IOR}$. In the same study population, the ratio of the disease risks in the exposed and unexposed is ${\widehat{\mathit{\theta }}}_{RR}$. If the researchers misinterpret ${\widehat{\mathit{\theta }}}_{IOR}$ as if it were ${\widehat{\mathit{\theta }}}_{RR}$ then over‐estimation of θ _RR will occur. This was the topic of the prior paper; however, I simplified the notation for a clinically focused audience.4 If the researchers decide, as per Dr. Valeika's example, to conduct a case‐cohort design by sampling controls at the inception of the study, the underlying fact that the population parameters θ _IOR and θ _RR exist and the difference θ _IOR − θ _RR does not change with the study design. What does change when the researcher uses a case‐cohort design is what he or she can estimate. In a design that uses case‐cohort sampling, the cross‐product ratio (or the exponent of the logistic regression coefficient) estimates the risk ratio directly, ie, ${\widehat{\mathit{\theta }}}_{R{R}^2}$. As correctly stated by Dr. Valeika, “In the cohort study, dogs appeared in both the numerator and denominator of the risk calculation. The control group is a substitution for the denominator information, so it makes sense for an individual to appear in the case and control group”. Hence it is misleading to say “the OR from a case‐control study provides an unbiased estimate of either the RR or IRR” and that “The odds ratio is (20/15)/(10/35)=4.6″.” It would however be correct to say that the “cross‐product ratio is an unbiased estimate of the risk ratio” and “The estimated risk ratio is (20/15)/(10/35)=4.6”. The original statement by Dr. Valeika fails to distinguish between the mathematical formula and the population measure of association. I did not include this discussion in the original paper because I considered that it would perpetuate a common misnomer that although interesting was tangential to the aim of the paper which relates to using ${\widehat{\mathit{\theta }}}_{IOR}$ to estimate θ _RR. However, the comments by Dr. Valeika enable me to clarify my opinion of the use of term “odds ratio” as a synonym for mathematical formulae that in some designs does estimate the odds ratio. Similarly, topics such as the interpretation of incidence odds ratio and prevalence odds ratio from cohort studies versus cross‐sectional studies were not included in the manuscript. Again although interesting, these were not germane to the topic discussed.

For case‐cohort designs, ${\widehat{\mathit{\theta }}}_{RR}$ underestimates θ _IOR. Currently, I believe there are very few applications where this bias may be important, as these designs are uncommon and clinicians rarely interpret risk ratios as odds ratios (although the inverse is common). I can envision the very rare circumstance where a meta‐analysis was conducted using traditional cohort designs and case cohort designs and the proposal was to calculate the summary incidence odds ratio. If cross‐product ratios (${\widehat{\mathit{\theta }}}_{RR}$) from case cohort studies were wrongly inferred to be an estimate of θ _IOR and included with valid ${\widehat{\mathit{\theta }}}_{IOR}$ from traditional cohort studies, bias could arise because the ${\widehat{\mathit{\theta }}}_{RR}$ is an underestimate of the θ _IOR.

In summary, I agree that nested designs enable estimation of the risk ratio and rate ratios. However the statement that the odds ratio estimates the risk ratio is misleading because a measure of association is a characteristic of the population and the association between these cannot be changed by the study design. However, what the mathematical formulae (cross‐product ratio or exponent of logistic regression model) estimate does change with study design. Therefore it is inaccurate to equate a mathematical formulae with one measure of association as commonly occurs in epidemiology with use of the odds ratio as a synonym for the cross‐product ratio or the exponent of logistic regression coefficient.

Finally, Dr. Valeika writes, “It is critical to realize that the rare disease assumption does not apply to properly designed case‐control studies.” This may inadvertently imply to the reader that only the incidence case‐control designs discussed that use either case‐cohort sampling or density sampling are “proper case‐control designs”. I would propose that case‐control designs that use cumulative sampling and prevalence case‐control studies are not improper; however, what is estimated should be correctly interpreted.