. The objective of this paper is to develop a new estimator of the same odds ratio parameters through regression analysis on the original continuous outcome without the inherent loss of information caused by . Your email address will not be published. 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. As theoretically expected, ignoring exposure-mediator interactions when they are present can generate a substantial bias in the indirect effect estimates. Outcome and mediator data conditional on the observed exposure and covariates in the study by Shenassa et al. Ive found a paper referring to this types of Odds ratios as cumulative (for each higher increment, the odds increases by the Odds Ratio). But opting out of some of these cookies may affect your browsing experience. Table 2 shows related results for the natural direct effects log odds ratios. Standard analyses, ignoring such interactions, gave corresponding natural indirect effect odds ratios of 1.04 (95% CI: 0.99, 1.10), 1.04 (95% CI: 0.99, 1.09), and 1.04 (95% CI: 0.99, 1.19), respectively. If this assumption holds, then the odds ratio for the total causal effect, ORa,a*|cTE, is identified and can be estimated from the data using. The causal inference literature has made a considerable contribution to mediation analysis by providing definitions for direct and indirect effects that allow for the effect decomposition of a total effect into a direct and an indirect effect even in settings involving nonlinearities and interactions (1, 2), thereby circumventing an important limitation to the concepts and methods for mediation that have been used in the social sciences (2). 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. An official website of the United States government. endstream endobj 80 0 obj<> endobj 82 0 obj<> endobj 83 0 obj<>/XObject<>/ProcSet[/PDF/Text/ImageB]>> endobj 84 0 obj<> endobj 85 0 obj<> endobj 86 0 obj<>stream Koh YS, Koh GC, Matchar DB, Hong SI, Tai BC. We assume it is known by design so that sampling variability for is neglible. Measures and models for causal inference in cross-sectional studies: arguments for the appropriateness of the prevalence odds ratio and related logistic regression. The formula for the natural indirect effect odds ratio requires that assumptions 14 hold, that models 6 and 7 are correctly specified, and that the outcome Y is rare. 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. Berkeley, CA: Berkeley Electronic Press; 2008, Testing and estimation of direct effects by reparameterizing directed acyclic graphs with structural nested models, Causal mediation analyses with rank preserving models, Related causal frameworks for surrogate outcomes, Estimating direct effects in cohort and case-control studies, An Introduction to Statistical Mediation Analysis, The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations, Statistical validation of intermediate endpoints for chronic diseases, Estimating the proportion of treatment effect explained by a surrogate marker, A method to assess the proportion of treatment effect explained by a surrogate endpoint, Proportion of treatment effect (PTE) explained by a surrogate marker, A measure of the proportion of treatment effect explained by a surrogate marker, Counterfactual links to the proportion of treatment effect explained by a surrogate marker, A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation [electronic article]. Can i say that all of them are not significant or what should i say that? Clipboard, Search History, and several other advanced features are temporarily unavailable. If estrogen therapy were randomized, then its effect on serum lipid concentrations only or on cardiovascular disease only could be estimated without control for U but, when the direct effect of estrogen therapy on cardiovascular disease controlling for serum lipid concentrations is of interest, data on U would be needed. In epidemiologic research for which questions of mediation are of interest, greater effort should be made to collect data on potential mediator-outcome confounders. 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. 0000001153 00000 n We assume it is known by design so that sampling variability for is neglible. All dummy coding means is recoding the original categorical variable into a set of binary variables that have values of one and zero. On the odds ratio (OR) scale, the total effect (TE), conditional on C = c, comparing exposure level a with a*, is defined by. Causality: Models, Reasoning, and Inference. On the odds ratio scale, the natural direct and indirect effects also have a decomposition property. Example of mediation with exposure A, mediator M, outcome Y, covariates C, and a mediator-outcome confounder L that is itself affected by the exposure. Wang Y, Taylor JM. How can I confirm this? Note that none of assumptions 14 can be tested by using data; a researcher will have to rely on subject matter knowledge in evaluating them. VanderWeele TJ. On the risk difference scale, natural direct and indirect effects have the property that the total effect E[YaYa*|c] decomposes into a natural direct and indirect effect: The decomposition holds even when there are nonlinearities and interactions. You need one variable for each category except one. Concerning the consistency assumption in causal inference. Multivariable analysis in cerebrovascular research: practical notes for the clinician. We will adopt the definitions and nomenclature of Pearl (, One can similarly define a natural indirect effect. Cardiovascular mortality and noncontraceptive use of estrogen in women: results from the Lipid Research Clinics Program Follow-up Study. This tells us that an increase of one year in age is . Standard analyses, ignoring such interactions, gave corresponding natural indirect effect odds ratios of 1.04 (95% CI: 0.99, 1.10), 1.04 (95% CI: 0.99, 1.09), and 1.04 (95% CI: 0.99, 1.19), respectively. In this paper, we consider the use of the odds ratio scale for mediation analysis. We have described a similar approach for continuous outcomes elsewhere (9). In a case-control study, estimation of model 7 is thus straightforward. Hi, The next 4 experiments evaluate the impact of exposure-mediator interactions. We will follow the exposition of VanderWeele (12) and VanderWeele and Vansteelandt (9) on the identification assumptions proposed by Pearl (2). Note that throughout this paper we will consider all effects conditional on the covariates C, and we will thus use expressions such as natural direct effect and conditional natural direct effect interchangeably. Considering that no significant evidence of an interaction between dampness or mold exposure and perception of control was found (P = 0.91, 0.89, and 0.22 for minimal, moderate, and extensive dampness or mold exposure, respectively, relative to no exposure), the fact that these results are very similar is not surprising. As theoretically expected, ignoring exposure-mediator interactions when they are present can generate a substantial bias in the indirect effect estimates. thank you for those explanations. A related approach, common in both the epidemiologic literature and the social science literature, consists of regressing Y on A, M, C as in model 5 and then examining whether the coefficient for A is different from that obtained when Y is regressed on A and C alone, such as the folllowing: The difference between coefficients for A, 1 1, is sometimes interpreted as an indirect effect. If there is interaction, then the routine approach of omitting the product term from the regression model should be avoided; instead, the product term can be included and, provided that the outcome is rare, the approach we have described in this paper can be used. Disclaimer, National Library of Medicine Fallibility in estimating direct effects. You need to control for a number of covariates, so you cant just use a chi-square test. There is no association between condition and event occurrence. Unable to load your collection due to an error, Unable to load your delegates due to an error. 0000007063 00000 n One can similarly define a natural indirect effect. Controlled direct effects on the risk difference or risk ratio scale are identified if conditioning on the set of covariates C suffices to control for confounding of both the exposure-outcome and the mediator-outcome relations. official website and that any information you provide is encrypted If assumptions 1 and 2 hold, then the controlled direct effect on the risk difference scale and on the odds ratio scale is identified, and ORa,a*|cCDE(m) is then given by. On the log scale, this is . The objective of this paper is to develop a new estimator of the same odds ratio parameters through regression analysis on the original continuous outcome without the inherent loss of information caused by dichotomizing. I want to ask you about condition that all of the dummy variables are not significant. Understanding Probability, Odds, and Odds Ratios in Logistic Regression. Formal modes of statistical inference for causal effects. 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. Fourth, in genetic epidemiology, the extent to which genetic variants affect an outcome (e.g., lung cancer) through intermediate phenotypes (e.g., nicotine addiction) has recently been a topic of interest (4143); the approach we have described here for case-control studies can be applied to address such questions in genetics research. On the odds ratio scale, the natural direct and indirect effects also have a decomposition property. 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. All rights reserved. In some settings, assumption 4 may be plausible if the mediator M occurs shortly after the exposure A (9). 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. In logistic regression, the odds ratios for a dummy variable is the factor of the odds that Y=1 within that category of X, compared to the odds that Y=1 within the reference category. Tyler J. VanderWeele, Stijn Vansteelandt, Odds Ratios for Mediation Analysis for a Dichotomous Outcome, American Journal of Epidemiology, Volume 172, Issue 12, 15 December 2010, Pages 13391348, https://doi.org/10.1093/aje/kwq332. Hernn MA. Note that a logistic, not a log-linear model, is being used. Prevalence odds ratio or prevalence ratio in the analysis of cross sectional data: what is to be done? Despite the way the terms are used in common English, odds and probability are not interchangeable. Statistical validation of intermediate endpoints for chronic diseases. 0000000616 00000 n 0000005814 00000 n When there is in fact such interaction between A and M, ignoring this (as is often done) can result in highly misleading inferences concerning mediation. Under certain assumptions that the set of covariates C contains all relevant confounding variables, the direct and indirect effects can be identified with observed data. Vansteelandt S, Goetgeluk S, Lutz S, et al. Accessibility Defining and estimating intervention effects for groups that will develop an auxiliary outcome. The calculation of Relative risk & Odds ratio requires two categorical variables, one for outcome and one for group. So the odds ratio for condition 1 is a ratio of the odds of answering correctly in condition 1 compared to the odds of answering correctly in condition 6. MeSH document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Quick links Many analyses in epidemiology, however, use the odds ratio scale because the outcome is dichotomous and the data arise from a case-control study design. The 2 measures will often not coincide. If, however, there is a variable . 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. 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 (). For simplicity in the example, we suppose treatment is binary and let A = 1 denote estrogen therapy and A = 0 otherwise. This category only includes cookies that ensures basic functionalities and security features of the website. The formula for the controlled direct effect odds ratio requires that assumptions 1 and 2 hold and that model 7 is correctly specified; no rare outcome assumption is required. A measure of the proportion of treatment effect explained by a surrogate marker. Y is either 0 (alive beyond 2 years) or 1 (death within 2 years). Marginal structural models for the estimation of direct and indirect effects. As noted above, a dichotomous Y variable can have only 2 values. 81 0 obj<>stream Table 1. 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. American Journal of Epidemiology The Author 2010. Performance of Existing and Novel Symptom- and Antigen TestingBased COVID-19 Case Definitions in a Community Setting, Peripheral Neuropathy and Vision and Hearing Impairment in US Adults With and Without Diabetes, Physical Activity Trends Among Adults in a National mHealth Program: A Population-Based Cohort Study of 411,528 Adults, Estimating the Long-Term Causal Effects of Attending Historically Black Colleges or Universities on Depressive Symptoms, Are We Undercounting the True Burden of Mortality Related to Suicide, Alcohol-Related, or Drug Use? See: Opposite Results in Ordinal Logistic RegressionSolving a Statistical Mystery and Binary, Ordinal, and Multinomial Logistic Regression for Categorical Outcomes. The estimation technique for natural direct and indirect effect odds ratios will require assumptions 14 above and will combine the results of a linear and logistic regression to obtain the effects of interest; the estimation technique for natural direct and indirect effects will also require that the outcome Y is rare so that odds ratios approximate risk ratios, which allows one to obtain particularly simple formulae. 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. 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*. According to the logistic model, the log odds function, , is given by The odds ratio is defined as the ratio of the odds for those with the risk factor () to the . Dummy coding, interactions, quadratic termsthey all work the same way. Taylor JM, Wang Y, Thibaut R. Counterfactual links to the proportion of treatment effect explained by a surrogate marker. The left-hand side is the odds ratio for the total causal effect, ORa,a*|cTE; the right-hand side is an expression that can be estimated from the data. 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. One advantage of dichotomizing is that it allows estimation of odds ratio parameters through a logistic regression analysis. With these regression models, there are then 2 approaches to estimation typically used for the mediated effect (i.e., indirect effect). T. J. V. received funding from grants ES017876 and HD060696 from the US National Institutes of Health. In this section, we describe how the above approach can be adapted when using case-control data. Because this holds for all a, we must have that 1 (1 + 21) and thus 1 1 21. Author affiliations: Departments of Epidemiology and Biostatistics, Harvard School of Public Health, Boston, Massachusetts (Tyler J. VanderWeele); and Department of Applied Mathematics and Computer Sciences, Ghent University, Ghent, Belgium (Stijn Vansteelandt). The following are the hypothesis for testing for a difference in proportions using the risk difference, the risk ratio and the odds ratio. One great thing about logistic regression, at least for those of us who are trying to learn how to use it, is that the predictor variables work exactly the same way as they do in linear regression. The https:// ensures that you are connecting to the Under certain assumptions that the set of covariates C contains all relevant confounding variables, the direct and indirect effects can be identified with observed data. As noted in the text, the approach often used in the social sciences (25) involves using regressions such as models 5 and 6, along with a regression of Y on just A (and C): Potential confounding variables are often ignored in many of the analyses in the social sciences in which the exposure is randomized (even though the mediator is not randomized), and thus the set C is sometimes assumed to be empty. These cookies will be stored in your browser only with your consent. The protective effects of estrogen on the cardiovascular system. Rubin DB. When the mediator M is dichotomous, rather than continuous, a somewhat similar approach to the one described here could potentially be used, but the analytic formulas for mediated effects no longer take quite as simple a form. T. J. V. received funding from grants ES017876 and HD060696 from the US National Institutes of Health. 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. An Analysis Using Veteran Colorado Death Certificate Data, About the Johns Hopkins Bloomberg School of Public Health, REGRESSION ANALYSIS FOR DIRECT AND INDIRECT EFFECT ODDS RATIOS, ODDS RATIOS FOR MEDIATION ANALYSIS IN CASE-CONTROL STUDIES, Receive exclusive offers and updates from Oxford Academic, Assistant or Associate Professors in Orthodontics, Open Rank Informatics Research Faculty Position, Postdoctoral Fellowship Infections and Immunoepidemiology Branch, Assistant Professor in the Department of Psychiatry and Human Behavior, Copyright 2022 Johns Hopkins Bloomberg School of Public Health. The odds ratio for condition 2 is the ratio of the odds of answering correctly in condition 2 compared to condition 6. They extend definitions of controlled direct effects and natural direct and indirect effects from the risk difference scale to the odds ratio scale. In this case, the odd ratios refer to the odds compared to the reference category, but not to the other categories, right? 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 An Introduction to Statistical Mediation Analysis. The causal inference literature has made a considerable contribution to mediation analysis by providing definitions for direct and indirect effects that allow for the effect decomposition of a total effect into a direct and an indirect effect even in settings involving nonlinearities and interactions (1, 2), thereby circumventing an important limitation to the concepts and methods for mediation that have been used in the social sciences (2). Dear all, I am struggling with the interpretation of interacted odds ratio in a conditional logit. 2021 Nov 23;18(23):12310. doi: 10.3390/ijerph182312310. For example, lets say you have an experiment with six conditions and a binary outcome: did the subject answer correctly or not. We have described a similar approach for continuous outcomes elsewhere (9). Predictors of Stunting and Underweight Among Children Aged 6 to 59months in Bussi Islands, Wakiso District, Uganda: A Cross-Sectional Study. Semantics of causal DAG models and the identification of direct and indirect effects. 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. The site is secure. Barry Kurt Moser, Barry Kurt Moser. trailer The left-hand side is the odds ratio for the total causal effect, ORa,a*|cTE; the right-hand side is an expression that can be estimated from the data. Nondifferential misclassification of such a variable can introduce bias in the odds ratios within the strata of the confounding variable. Oxford University Press is a department of the University of Oxford. (2009) who provides formulas to convert log odds ratio to d and d to r. For the last step (d to r), you are supposed to use a correction factor . Greenland S, Robins JM, Pearl J. Confounding and collapsibility in causal inference. The case-control setting is of particular importance in mediation analysis with a dichotomous outcome because often, if the outcome is rare, it will be infeasible to conduct a cohort study with a sufficient number of individuals with the outcome. Outcome: select a dichotomous variable where a positive outcome is coded 1 and a negative outcome is coded 0. Search Dunn G, Bentall R. Modelling treatment-effect heterogeneity in randomized controlled trials of complex interventions (psychological treatments). If, however, the outcome is not rare or if the error term in regression model 6 is heteroscedastic or not normally distributed, then the 2 quantities 12 and 1 1 need not be approximately equal. It is nevertheless possible to adapt the approach to the estimation of direct and indirect effects described above in a relatively straightforward manner if the prevalence of the outcome Y is known. A simple technique to estimate direct and indirect effect odds ratios by combining logistic and linear regressions is described that applies when the outcome is rare and the mediator continuous. where ij is the covariance between and in model 6, and ij is the covariance between and in model 7; these covariances are given in the regression output of standard statistical software. Thank you, Your email address will not be published. HHS Vulnerability Disclosure, Help If, however, there is a variable L that is an effect of A and affects both M and Y, then assumption 4 is violated and natural direct and indirect effects will not in general be identified (17), irrespective of whether data are available on L. In such settings, it may still be possible to identify controlled direct effect odds ratios, but alternative statistical approaches such as marginal structural models (12, 18, 19) or structural nested models (2024) will generally be needed. To identify total effects, it is generally assumed that, conditional on some set of measured covariates C, the effect of exposure A on outcome Y is unconfounded; in counterfactual notation, this is YaA|C, where we use the independence symbol to denote that Ya is independent of A conditional on C. In practice, to make this assumption more plausible, a researcher will attempt to collect data on a sufficiently rich set of covariates C to try to control for confounding of the exposure-outcome relation. Precise measurement of confounding or effect modifying variables is seldom possible. 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 Our simple formulae did, however, assume no interaction between the confounders and the treatment or mediator; other estimation techniques (16) could be used if there are confounder-exposure interactions; other identification approaches are also possible when such interactions are present in their effects on the mediator (21, 38). They extend definitions of controlled direct effects and natural direct and indirect effects from the risk difference scale to the odds ratio scale. Here, the magnitude 3 = 0.22 was chosen to equal 22/3 and thus to generate a potentially substantial bias in the natural indirect effect odds ratio at a = 3, which was the largest observed exposure value. We extend the definitions of direct and indirect effects (1, 2) in causal inference from the risk difference to the odds ratio scale. 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). The conditional natural indirect effect (NIE) can be defined analogously on the odds ratio scale as. Joffe M, Small D, Hsu CY. The use of models 5 and 6 along with the expressions above is often referred to as the Baron-Kenny approach to mediation (26). Proceedings of the Seventeenth Conference on Uncertainty and Artificial Intelligence. The relations among these variables are depicted in Figure 1. Natural direct and indirect effects will be identified if, in addition to assumptions 1 and 2, the following 2 assumptions hold, that for all, Consider the use of the following 2 models, a logistic regression for the outcome, However, a limitation of all of the standard approaches is that they presuppose that there is no statistical interaction on the odds ratio scale between, Ninety-five percent confidence intervals for the controlled direct effect odds ratio in expression 8 and the natural indirect effect odds ratio in expression 9 can be computed by using standard regression output and are given, respectively, by, As noted in the text, the approach often used in the social sciences (, Identifiability and exchangeability for direct and indirect effects, Proceedings of the Seventeenth Conference on Uncertainty and Artificial Intelligence, The protective effects of estrogen on the cardiovascular system, Cardiovascular mortality and noncontraceptive use of estrogen in women: results from the Lipid Research Clinics Program Follow-up Study, Formal modes of statistical inference for causal effects, A definition of causal effect for epidemiological research, Causality: Models, Reasoning, and Inference, Concerning the consistency assumption in causal inference, Conceptual issues concerning mediation, interventions and composition, Semantics of causal DAG models and the identification of direct and indirect effects, Defining and estimating intervention effects for groups that will develop an auxiliary outcome, Marginal structural models for the estimation of direct and indirect effects, Process analysis: estimating mediation in treatment evaluations, Bias formulas for sensitivity analysis for direct and indirect effects, A general approach to causal mediation analysis, Proceedings of the International Joint Conferences on Artificial Intelligence, Marginal structural models and causal inference in epidemiology, article 23.
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