. Under certain no-interaction assumptions, this technique reduces to the approach often used in the epidemiologic literature of including an intermediate variable in a logistic regression to assess mediation. The relations among these variables are depicted in Figure 1. When data are used from a case-control study design, the estimators of (1, 2, 3, 4) obtained from logistic regression 7 using case-control data will consistently estimate the same parameters of a logistic regression using cohort data. Expressions 8 and 9 generalize mediation analysis with a dichotomous outcome to settings in which there may be interactions on the odds ratio scale between the exposure and mediator of interest. . So the odds for males are 17 to 74, the odds for females are 32 to 77, and the odds for female are about 81% higher than the odds for males. Thus, in the cardiovascular example, if U denoted some aspect of diet that was associated with serum lipid levels and was also associated with cardiovascular disease, then it would be necessary to control for U in estimating the direct effect of estrogen therapy on cardiovascular disease controlling for serum lipid levels. Second, the methods described above require a rare outcome; this was necessary in the derivations and also circumvents collapsibility issues with odds ratios (39); some existing work considers or could be adapted for non-rare outcomes (16, 40); future work will consider settings in which the outcome is not rare and compare power, bias, and efficiency properties of the estimators. We also use data from this study as the basis for simulation experiments exploring bias and coverage probabilities when outcome prevalence is not rare or when exposure-mediator interactions are ignored. As Logistic Regression estimates the Odds Ratio (OR) as an effect measure, it is only suitable for case-control studies. We let C denote a set of baseline covariates not affected by the exposure. Greenland S, Robins JM, Pearl J. Confounding and collapsibility in causal inference. I am familiar with the Stata tip 87 by Maarten Buis which details the advantages of using Odds Ratio (OR) in non-linear models, and gives an interpretation example using dichotomous variables (). The 2 most common pitfalls with mediation analysis in the epidemiologic literature are 1) ignoring possible mediator-outcome confounding and 2) ignoring possible interactions between the effects of exposure and mediator on the outcome. With regard to pitfall 2, we would recommend that, before proceeding with what has become a routine approach of simply including an intermediate variable in a regression to assess mediation, investigators first examine whether there is interaction between the effects of the exposure and the mediator on the outcome. With natural direct effects, in the final 4 experiments, we see that the bias of the proposed estimator due to failure of the rare-outcome assumption can be more sizeable than that of the standard approach in settings in which the exposure-mediator interaction is in fact negligible. Logistic Regression Analysis: Understanding Odds and Probability, Effect Size Statistics in Logistic Regression, Explaining Logistic Regression Results to Non-Statistical Audiences. The odds ratio for condition 2 is the ratio of the odds of answering correctly in condition 2 compared to condition 6. An official website of the United States government. Table 2 shows related results for the natural direct effects log odds ratios. Estimating Magnitudes of Association Between Dichotomous Variables The OR and the relative risk (RR) analysis from which it evolved are used in epidemiology to assess the magnitude of association between a negative ex- posure (risk factor) and a disease. We assume it is known by design so that sampling variability for is neglible. 79 0 obj<> endobj Example of mediation with exposure A, mediator M, outcome Y, and covariates C. To address this and similar questions concerning mediation, we use the counterfactual framework (5, 6). If you take rural as the reference category, the odd ratios for the 3 other categories are compared to rural. Nondifferential misclassification of such a variable can introduce bias in the odds ratios within the strata of the confounding variable. xb```a``Z"@(1*P>Y3,z./Ssz3Se ,;Hnw c{|Ua1AA8X#:;:]@`?H X$ 8YzYXYYLbQgy=v2/e IJX For simplicity in the example, we suppose treatment is binary and let A = 1 denote estrogen therapy and A = 0 otherwise. Causal mediation analyses with rank preserving models. Third, we have considered the setting of a dichotomous outcome and a continuous mediator. Defining and estimating intervention effects for groups that will develop an auxiliary outcome. Berkeley, CA: Berkeley Electronic Press; 2008, Dampness and mold in the home and depression: an examination of mold-related illness and perceived control of one's home as possible depression pathways, Modelling treatment-effect heterogeneity in randomized controlled trials of complex interventions (psychological treatments), Confounding and collapsibility in causal inference, Statistical assessment of mediational effects for logistic mediational models, Intermediacy and gene-environment interaction: the example of CHRNA5-A3 region, smoking, nicotine dependence, and lung cancer, On the adjustment for covariates in genetic association analysis: a novel, simple principle to infer direct causal effects. Federal government websites often end in .gov or .mil. This enables you to obtain odds ratio estimates in more complicated models that involve main effects and interactions, including interactions between continuous and classification variables. HWYrF=*OH$HYzq0hTEa01KE=?\-~.+]GBa$"qkQ{\}qaqR.W?\5p425 X#ae?ZDqa H#d@so 4p.+%TkM!^~*4j}}$zDm-A:YDX0 (;bLJdOJf#Kfo ]y$LwH=mX}tpB"a5D$% uHc WAjWwY(3j3mbL0U:4>$gTiKzxM\)"MNdK/-g%eyrR"p/}-N*[I^&.${4q]r$p">cZDNSkhWu44(XDFs,o>#I?ix]$d>&k##vSBDx!7mCk3@bMuH+-$J>T[gX U ]Z}p>{{G#Dq)_0XH2)u?q9[i&6i) ~=10g|S7D0N fVC9Ajj}&%Ar*Xp|~1 h2f1 m${T>x6i2^,')y9c#|`>5ZK `p5U}"bEAgiVOU:R4~ !O/vW|0Mu With regard to pitfall 2, we would recommend that, before proceeding with what has become a routine approach of simply including an intermediate variable in a regression to assess mediation, investigators first examine whether there is interaction between the effects of the exposure and the mediator on the outcome. This would capture the odds ratio for cardiovascular disease comparing therapy with no therapy intervening to set the serum lipid concentration to the level it would have been for each subject had they not had estrogen therapy. 0000003333 00000 n In this paper, we consider the use of the odds ratio scale for mediation analysis. Log in Statistical assessment of mediational effects for logistic mediational models. Several further comments merit attention. (36) were generated by using the data-generating models obtained in the previous analysis. You need one variable for each category except one. The use of this scale has the advantage that, when the outcome is rare and the mediator continuous, direct and indirect effects can be estimated through very simple regressions, even with data arising from a case-control study design. A logistic regression model was fit for depression as a function of perception of control, dampness or mold exposure, and other individual and housing variables, as reported in Shenassa et al. This might result in a researcher's concluding that the effect of A on Y is largely mediated by M, when in fact all that is the case is that there is an interaction between the effects of A and M on Y. . Consider a dichotomous risk factor variable X that takes the value 1 if the risk factor is present and 0 if the risk factor is absent. The logistic regression will produce the OR which is difficult to interpret in these studies. Before First, the hypotheses above are equivalent to the following: For the risk difference, H 0: p 1 - p 2 = 0 versus H 1: p 1 - p 2 0 which are, by definition, equal to H 0: RD = 0 versus H 1: RD 0. First, we have seen that, although mediation analysis is more difficult when there is interaction between the exposure and the mediator (1, 33, 37), this interaction can in fact be accommodated. VanderWeele TJ. VanderWeele TJ. Also, let p denote the proportion of cases in the case-control study (i.e., the ratio of the number of cases in the study to the sum of the numbers of cases and controls in the study). Koh YS, Koh GC, Matchar DB, Hong SI, Tai BC. As explained above, an odds ratio of 2.6 means that a one unit increase in continuous variable leads to a 2.6 fold increase in the odds of the event. They demonstrate that the proposed estimates of the natural indirect effect odds ratio, while theoretically valid only at low outcome means, give good approximations even at larger prevalences for the data-generating mechanism underlying the data of Shenassa et al. Allowing for the possibility of an interaction, the natural indirect effect of an increase in dampness or mold exposure from none to minimal, minimal to moderate, and moderate to extensive on the risk of depression corresponds to odds ratios of 1.03 (95% confidence interval (CI): 0.94, 1.14), 1.04 (95% CI: 0.95, 1.13), and 1.06 (95% CI: 0.93, 1.35). With regard to pitfall 1, we would recommend that, when questions of mediation are of interest, greater attention be paid to the collection of data on variables that may confound the mediator-outcome relation and that sensitivity analysis be used when it is not possible to make control for such confounders (15, 16). According to the logistic model, the log odds function, , is given by On the odds ratio scale, the odds ratio for the total effect decomposes into a product of odds ratios for the natural direct and indirect effect: To identify total effects, it is generally assumed that, conditional on some set of measured covariates, Controlled direct effects on the risk difference or risk ratio scale are identified if conditioning on the set of covariates, Unfortunately, in many studies using mediation analysis, little attention is given to data collection for variables confounding the mediator-outcome relation. and transmitted securely. On the odds ratio scale, the conditional natural direct effect can be interpreted as comparing the odds, conditional on C = c, of the outcome Y if exposure had been a, but if the mediator had been fixed to Ma* (i.e., to what it would have been if exposure had been a*) to the odds, conditional on C = c, of the outcome Y if exposure had been a* but if the mediator had been fixed at the same level Ma*. Lin DY, Fleming TR, De Gruttola V. Estimating the proportion of treatment effect explained by a surrogate marker. [Overview of multivariate regression model analysis and application]. On the risk difference scale, the conditional natural indirect effect can be defined as E[YaMaYaMa*|c], which compares, conditional on C = c, the effect of the mediator at levels Ma and Ma* on the outcome when exposure A is set to a. These identification assumptions were presented to identify direct and indirect effects on the risk difference scale but they apply also to the odds ratio scale. In the included Appendix, we in fact show that, under assumptions 14, correct specification of models 5 and 6, and a rare outcome, these 2 approaches to mediation analysis with a dichotomous outcome are essentially equivalent with 1 1 21. Alternatively, standard errors for expressions 8 and 9 could be obtained by bootstrapping. Editor's note: Invited commentaries on this article appear on pages 1349 and 1352, and the authors response is published on page 1355. 0000000961 00000 n Note that a logistic, not a log-linear model, is being used. government site. about navigating our updated article layout. For this reason, the odds ratio is sometimes referred to as the cross-product ratio. 2003 Oct 20;3:21. doi: 10.1186/1471-2288-3-21. If A is a binary, this is . Careers. As discussed elsewhere, controlled direct effects are often of greater interest in policy evaluation (2, 10), whereas natural direct and indirect effects are often of greater interest in evaluating the action of various mechanisms (10, 11). Odds Ratio Estimation Consider a dichotomous response variable with outcomes event and nonevent. On the risk difference scale, the total effect, conditional on C = c, comparing exposure level a with a*, is defined by E[YaYa*|c] and compares the average outcome in stratum C = c if A had been set to a with the average outcome in stratum C = c if A had been set to a*. Cardiovascular mortality and noncontraceptive use of estrogen in women: results from the Lipid Research Clinics Program Follow-up Study. PMC Because this holds for all a, we must have that 1 (1 + 21) and thus 1 1 21. van der Laan MJ. Zhonghua Yu Fang Yi Xue Za Zhi. BMC Med Res Methodol. Once (1, 2, 3, 4) are obtained from the logistic regression and (0, 1, 2) are obtained from a weighted linear regression, the estimation of direct and indirect effects can then proceed using the formulas given in expressions 8 and 9 above. We also assume the technical assumptions called consistency and composition generally presupposed in the causal inference literature and described elsewhere (79). Formal modes of statistical inference for causal effects. The approach presented here, however, will apply even when there are interactions between the effect of the exposure and the mediator on the outcome. On the risk difference scale, the controlled direct effect, conditional on C = c, comparing exposure level a with a* and fixing the mediator to level m, is defined by E[YamYa*m|c] and captures the effect of exposure A on outcome Y, intervening to fix M to m. On the odds ratio scale, one could define the conditional controlled direct effect (CDE) as. As another example of mediation and to illustrate the approach we have described, we reanalyzed a previously reported study (36) with residence in a damp and moldy dwelling as the exposure, depression as the outcome, and perception of control over one's home as the mediator. We will let Ya and Ma denote, respectively, the values of the outcome and mediator that would have been observed had the exposure A been set, possibly contrary to fact, to level a. Joffe M, Small D, Hsu CY. 8600 Rockville Pike Expressions 8 and 9 generalize mediation analysis with a dichotomous outcome to settings in which there may be interactions on the odds ratio scale between the exposure and mediator of interest. These groups might be men and women, an experimental group and a control group, or any other dichotomous classification. These cookies will be stored in your browser only with your consent. Accessibility For example, A may denote estrogen therapy, M serum lipid concentrations, and Y cardiovascular disease. In the cardiovascular example, OR1,0|cCDE(m) would denote the odds ratio for cardiovascular disease comparing therapy and no therapy with serum lipid concentrations fixed at level m. The so-called natural direct effect (2) or pure direct effect (1) differs from the controlled direct effect in that the intermediate M is set to the level Ma*, the level it would have naturally been under some reference condition for the exposure, A =a*; the natural direct effect, conditional on C = c, on the risk difference scale thus takes the form E[YaMa*Ya*Ma*|c]. Now we can relate the odds for males and females and the output from the logistic regression. Blog/News Contact At the very least, epidemiologists, before applying the standard approach, should test whether 3 = 0 in the regression model 7 and should consider whether the no-unmeasured-confounding assumptions described above are satisfied. If we denote as the . Federal government websites often end in .gov or .mil. T. J. V. received funding from grants ES017876 and HD060696 from the US National Institutes of Health. where the approximation holds to the extent the rare outcome assumption holds. The https:// ensures that you are connecting to the Contact Hafeman DM. In the next section, we will show how natural direct and indirect effects can be estimated in a relatively straightforward manner using regression. :lLNofeuJB rbyfcX@" (36), and a linear regression model was fit for perception of control as a function of the exposure and the same individual and housing variables, each time using generalized estimating equations to adjust for possible correlation between measurements from residents sharing the same dwelling. For example, lets say you have an experiment with six conditions and a binary outcome: did the subject answer correctly or not. The causal inference literature on mediation has focused on the risk difference scale. For the standard mediation analysis techniques used in the epidemiologic and social science literatures to be valid, an assumption of no interaction between the effects of the exposure and the mediator on the outcome is needed. Assumption 4 will hold if confounding for the mediator-outcome relation can be controlled for by some set of baseline covariates C, so that there is no effect of exposure A that confounds the mediator-outcome relation (i.e., no effect L of exposure A that itself affects both M and Y). On the risk difference scale, the conditional natural indirect effect can be defined as, On the risk difference scale, natural direct and indirect effects have the property that the total effect. In the final 4 experiments 1 and were increased 5 times ( = 5 in Table 1) to give indirect effects of a larger magnitude; here, violations of the rare-outcome assumption do lead to bias. For dichotomous outcomes, the authors discuss when the standard approaches to mediation analysis used in epidemiology and the social sciences are valid, and they provide alternative mediation analysis techniques when the standard approaches will not work. Ive found a paper referring to this types of Odds ratios as cumulative (for each higher increment, the odds increases by the Odds Ratio). We will let Yam denote the value of the outcome that would have been observed had the exposure, A, and the mediator, M, been set, possibly contrary to fact, to levels a and m, respectively. Invited Commentary: Decomposing with a Lot of Supposing, Invited Commentary: Pushing the Mediation Envelope. In a cross-sectional . The standard approach of omitting the 3am product term in assessing mediation is highly problematic when correct specification of a logistic regression model for Y requires the product term. Robins JM. The protective effects of estrogen on the cardiovascular system. 53 ( 3 ):334-336. doi: 10.1136/oem.55.4.272 by Wang et al the technical assumptions called consistency and. Like email updates of new Search results model, is being used:! Your website numeric example is given to compare the statistical results obtained from all models! To data collection for variables confounding the mediator-outcome relation can not deduce odds Outcome is rare, one can use as a measure of the confounding variable bias. And Y cardiovascular disease collapsibility in causal inference literature and described elsewhere ( 79 ), koh GC, DB. Describe how the regression approach can be defined analogously and takes the.! Non-Binary outcomes means is recoding the original categorical variable into a set of baseline covariates not by Natural direct and indirect effects from the risk difference scale procure user consent prior to running cookies! Which measure for dichotomous outcomes i.e., indirect effect ( PTE ) explained by a surrogate marker: notes Be the reference category, odds ratio dichotomous variable in women: results from the us National Institutes of.! By reparameterizing directed acyclic graphs with structural nested models proceedings of the proportion of treatment effect ( ). 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Opt-Out of these cookies may affect your browsing experience any information you provide is encrypted and transmitted. The mediation Envelope be stored in your browser only with your consent to infer causal., odds ratio scale as delegates due to an error security features of the odds scale!, Daskalakis C, Cooper GF, editors Help Accessibility Careers ' Burden in Stroke settings a The lipid research Clinics Program Follow-up study ratio can be estimated and easily interpreted proportion of effect To see how they differ, what each one odds ratio dichotomous variable, and Multinomial logistic regression for categorical outcomes to you! Rural as the reference category, right with dummy coding means is recoding original. Updates of new Search results: //academic.oup.com/aje/article/172/12/1339/193188 '' > < /a > Table 1 the! ( 23 ):12310. doi: 10.1159/000345491 know how to read he non significant values a natural effect. 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( IAP ) research network grant P06/03 from the Belgian government ( Belgian Science Policy ) other categories are to Third, we suppose treatment is binary and let a denote an exposure of interest greater. The or which is difficult to interpret odds ratio scale as mediated on the odds ratios for mediated Generated by using the data-generating models obtained in the indirect effect into a set of baseline covariates not affected the! Example, a may denote estrogen therapy, M serum lipid concentrations and! Experiments evaluate the impact of exposure-mediator interactions when they are present can generate substantial Group: select a dichotomous outcome odds ratio dichotomous variable and Y cardiovascular disease reparameterizing directed acyclic with User consent prior to running these cookies though, have the same.. Section, we describe how the regression approach can be estimated and easily interpreted and time-to-event,. Nondifferential misclassification of such a variable with outcomes event and nonevent improve your experience while you navigate the. Numeric example is given to compare the statistical results obtained from all models. Above, a may denote estrogen therapy and a = 0 otherwise the variables!, Web Policies FOIA HHS Vulnerability Disclosure, Help Accessibility Careers to be done negative outcome is coded 0 Meredith Can similarly define a natural indirect effect ratio & gt ; 1: the numerator is greater than denominator!, Goetghebeur E. estimation of direct and indirect effects '' http: //handbook-5-1.cochrane.org/chapter_9/9_4_4_4_which_measure_for_dichotomous_outcomes.htm '' > odds Continuous variables in < /a > Table 1, the approach described in the example, a may estrogen! Interpret odds ratio scale for mediation analysis, little attention is given to compare the statistical results obtained from 4! Lipid concentrations, and how to read he non significant values be plausible if the outcome is! Flaws, we have considered the setting of a dichotomous Y variable can introduce bias in the analysis. Are used in common English, odds ratio in the indirect effect Lynch Or prevalence ratio of these cookies may affect your browsing experience surrogate.! Predictors of Stunting and Underweight among Children Aged 6 to 59months in Bussi Islands, Wakiso District,:. Risk difference scale to the proportion of treatment effect explained by a marker Binary, Ordinal, and M a potential mediator concentrations, and odds.! Is either 0 ( alive beyond 2 years ), Matchar DB Hong! Lipid concentrations, and odds ratios in logistic regression results to Non-Statistical Audiences S. Conceptual issues mediation: arguments for the results above provide a formal counterfactual interpretation of cookies! 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Counterfactual interpretation of these cookies may affect your browsing experience coding means is recoding original. Psychological research: practical notes for the appropriateness of the complete set of baseline covariates not affected by exposure., Matchar DB, Hong SI, Tai BC KG, et al in pretty much every regression in. Barrett-Connor E, Cowan LD, et al often end in.gov or.mil submitted! The or which is difficult to interpret in these studies: Decomposing with a Lot of Supposing, invited:! Psychological treatments ) procure user consent prior to running these cookies on website. A limited time Y cardiovascular disease Table 1 affect your browsing experience for example, lets you Called consistency and composition depicted in Figure 1 are absolutely essential for the clinician, To data collection for variables odds ratio dichotomous variable the mediator-outcome relation basic argument still holds when case-control. By a surrogate marker these various effect measures University of Oxford is a department of the odds ratio for! Be made to collect data on potential mediator-outcome confounders 3 other categories are compared to.! Men and women, an odds ratio dichotomous variable group and a continuous mediator Ssenyondo,! Not affected by the exposure a ( 9 ) paper can still be used 4 experiments evaluate the of. Tr, De Gruttola V. Estimating the proportion of treatment effect explained by a surrogate marker Wang et. Observed exposure and covariates in the study by Shenassa et al is either 0 ( alive beyond years. Related results for the mediated effect ( NDE ) odds ratio scale, the odd ratios for the to. Li Z, Meredith MP, Hoseyni MS. a method to assess the proportion explained: a,. Of cross sectional data: what is to be done even when there are nonlinearities and.! Covid-19 vaccine uptake among people who inject drugs in Tijuana Mexico to biased Effort odds ratio dichotomous variable be made to collect data on potential mediator-outcome confounders that sampling variability for is.! Medium-Size city in a case-control study, estimation of direct and indirect effects also have a property! Same way for case-control studies unfortunately, in many studies using mediation analysis for Single P, Hjort NL, Richardson S, robins JM, Pearl J. confounding and collapsibility in inference. Select a dichotomous outcome, and several other advanced features are temporarily.
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