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Explaining odds ratios as conditional risk ratios

  • Fred M. Hoppe
    Correspondence
    Corresponding author. Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S4K1, Canada. Tel.: +1-905-525-9140x24688; fax: +1 905-522-0935.
    Affiliations
    Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S4K1, Canada
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  • Author Footnotes
    1 Present address: 2401 Yonge St., Suite LL01, Toronto, ON M4P3H1, Canada.
    Daniel J. Hoppe
    Footnotes
    1 Present address: 2401 Yonge St., Suite LL01, Toronto, ON M4P3H1, Canada.
    Affiliations
    University of Toronto Orthopaedic Sports Medicine, 76 Grenville St., Toronto, ON M5S1B2, Canada
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  • Stephen D. Walter
    Affiliations
    Department of Clinical Epidemiology and Biostatistics, McMaster University, Hamilton, ON L8N3Z5, Canada
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  • Author Footnotes
    1 Present address: 2401 Yonge St., Suite LL01, Toronto, ON M4P3H1, Canada.
      We thank Dr. Tedeschi [
      • Tedeschi F.
      A clarification on odds ratios as conditional risk ratios.
      ] for his commentary on our paper [
      • Hoppe F.M.
      • Hoppe D.J.
      • Walter S.D.
      Odds ratios deconstructed: a new way to understand and explain odds ratios as conditional risk ratios.
      ]. Before responding, we recall that our original motivation was to provide a derivation of the odds ratio, OR (as a parameter), that does not involve odds and which has an interpretation in terms of patient numbers as a conditional risk ratio given discordant pairs. This led us to then consider matched pairs, with dependence within pairs.
      We have never claimed that the common and general odds ratios (given by p1(1 − p2)/(p2(1 − p1)) and p10/p01, respectively) would be the same, rather that both have an interpretation as a risk ratio conditional on discordant outcomes. Then assuming independence within pairs, a ratio of odds obtains. Tedeschi refers to these two situations as marginal (common) and conditional (general) and indicates (by the equation at the end of his Section 1) that they will agree when collapsibility pertains. He also mentions that the OR “has no epidemiologic interpretation” if the data are noncollapsible. Here, Tedeschi is essentially extending the discussion to allow for potential heterogeneity in the OR, implicitly including the possibility of additional variables (other than treatment) which affect the outcome.
      We have no disagreement with these points. However, one should note that we simply presented the definitions of the two ORs, but we did not discuss their potential validity, with heterogeneity in particular being beyond the scope of our paper. We remark that noncollapsibility would actually invalidate the interpretation of both ORs. Finally, note also that both the marginal and conditional estimators relate to the same underlying parameter (the population OR), even though their definitions and numerical values in data will (in general) be different.
      Regarding Tedeschi's Section 2, the third paragraph in the right column of p89 of the article by Hoppe et al. [
      • Hoppe F.M.
      • Hoppe D.J.
      • Walter S.D.
      Odds ratios deconstructed: a new way to understand and explain odds ratios as conditional risk ratios.
      ] should show the percentage as 62.18% not 61%. We thank him for noticing our typographical error. We had in fact given the correct percentage two paragraphs earlier, writing “whereas in a fraction 148/238 = 0.6218 the adverse event would occur only with the control” and our intention here was to express this statement in an equivalent way. Since 0.6218 is the conditional risk of a control event given discordant pairs and events are adverse, then in 62.18% of situations where there is a difference in outcomes the treatment will be better. So while 61% is the conditional risk ratio of a treatment vs. control event, 62.18% is the conditional risk of a control event. We refer the reader to our Equation (7) in the article by Hoppe et al. [
      • Hoppe F.M.
      • Hoppe D.J.
      • Walter S.D.
      Odds ratios deconstructed: a new way to understand and explain odds ratios as conditional risk ratios.
      ] showing that the conditional risk of a treatment event given discordant outcomes may also be expressed as OR/(1 + OR). Events being adverse, this conditional risk is the probability of a better result with control, and therefore, the conditional probability, given discordant pairs, of a better result with treatment becomes 1 − OR/(1 + OR) = 1/(1 + OR) = 0.6211 (or 0.6218, without rounding, as above).
      In the Appendix, there is a typo with the subscripts reversed in the second term defining c, which should clearly be X11 + X01.
      In our Section 4, parentheses should be included as Tedeschi indicates, but in his letter, he has reversed the subscripts 1 and 2.

      References

        • Tedeschi F.
        A clarification on odds ratios as conditional risk ratios.
        J Clin Epidemiol. 2018; 97: 133-134
        • Hoppe F.M.
        • Hoppe D.J.
        • Walter S.D.
        Odds ratios deconstructed: a new way to understand and explain odds ratios as conditional risk ratios.
        J Clin Epidemiol. 2017; 82: 87-93

      Linked Article

      • A clarification on odds ratios as conditional risk ratios
        Journal of Clinical EpidemiologyVol. 97
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          Hoppe et al. [1] propose a new interpretation of the odds ratio (OR) for situations where individuals are paired and then randomized to either the treatment or the control condition. They show that the OR is equivalent to the risk ratio (RR) conditionally on pairs having different outcomes and describe two cases, calling them “parallel groups” and “matched pairs,” respectively. In the former case (authors define it as unmatched case, given that pairs are matched randomly), the OR is called “common OR,” and it is equal to: p1*(1−p2)/p2*(1−p1) (defining p1 and p2 as the probability of event for treated and controls, respectively), and in the latter case (defined as matched case, given within-pair independence does not hold anymore), the OR is called “general OR” and is equal to: p10/p01 (defining p10 and p01 as the probability of discordant pairs, with event for the treated in the former case and for the control in the latter one).
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