class: center, middle, inverse, title-slide .title[ # Lecture 22. Mediation ] .subtitle[ ## Introduction to Causal Inference ] .author[ ### Jonathan Platt ] .institute[ ### University of Iowa ] .date[ ### 4/6/23 ] --- # Announcements 1. Sign up for final project presentation times - https://docs.google.com/document/d/1QSZZfRmY1TSdHW-6zU8p53Wx8vwKQZSMVPo220TMnDo/edit?usp=sharing - Presentations will be 10 min including questions 2. Please read H&R chapter 23 for Tuesday --- # Principle Stratification review - What is an example of a type of research question principal stratification can help answer? -- -Causal effects for treatment noncompliance, survival average causal effects in the setting of truncation by death, and surrogate marker evaluation for clinical trials. Generally, this strategy can help address questions that must address post-treatment confounders --- # Learning objectives - Understand the purpose of mediation - Understand the approach and limitations of 'traditional' mediation methods - Describe the counterfactual approach to mediation, its advantages and assumptions - Define and distinguish the controlled vs. natural indirect and direct effects --- # What is mediation? <div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-54c5e54ec9f65d0050c3" style="width:288px;height:144px;"></div> <script type="application/json" data-for="htmlwidget-54c5e54ec9f65d0050c3">{"x":{"diagram":"\ndigraph {\n graph [ranksep = 0.2]\n node [shape = circle]\n A Y M\n edge [minlen = 2]\n A->Y\n A->M\n M->Y\n \n{ rank = same; A; M ; Y }\n}\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> - Often we are interested in the extent to which the effect of treatment `\(A\)` on outcome `\(Y\)` is mediated by an intermediate variable M, and to what extent it is not -- - These estimates correspond to the indirect and direct effects -- - Mediation is a part of causal explanation, it tells us something about **how** the treatment causes the outcome -- - We want to account for it, not to remove it in our analysis -- - Conceptually, mediation is always present, although we may or may not be interested in estimating it --- # Why Conduct A Mediation Analysis? ### Strengthening main effect causal hypothesis - _Do genetic variants affect lung cancer through smoking or independently?_ -- ### Testing pathway-specific hypotheses - _Does low early SES affect adult health through adult SES?_ -- ### Evaluating and improving interventions. - _Will refining an educational intervention to better target classroom quality improve educational outcomes?_ - _Does a CBT intervention improve depressive symptoms only through antidepressant use?_ - _Why did this intervention fail? Did the intervention not affect the mediator, or does the mediator not affect the outcome, or was the direct effect in the opposite direction of the mediated effect?_ --- # Approaches to mediation analysis ### "Traditional" approaches - Difference method - Product method (Baron & Kenny, 1986) ### Counterfactual approaches --- # Difference method - In regression analysis we control for all variables associated with an treatment and outcome, except those in the causal pathway. - Using the difference method, we examine the regression models with and without the mediator -- `\(E[Y|A=a, C=c] = \alpha_0 + \alpha_1a + \alpha_2c\)` `\(E[Y|A=a, C=c, M=m] = \beta_0 + \beta_1a + \beta_2c+ \beta_3m\)` -- - `\(\alpha_1\)` = the Total Effect - `\(\beta_1\)` = Direct Effect not occurring through `\(M\)` - `\(\alpha_1 - \beta_1\)` = the indirect effect - If `\(\alpha_1 \ne \beta_1\)`, then we conclude that there is mediation by `\(M\)` --- # Product method (Baron & Kenny) -- `\(E[M|A=a, C=c] = \gamma_0 + \gamma_1a + \gamma_2c\)` `\(E[Y|A=a, C=c, M=m] = \beta_0 + \beta_1a + \beta_2c+ \beta_3m\)` -- - `\(\beta_1\)` = Direct effect - `\(\gamma_1*\beta_3\)` = Indirect effect -- <div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-271c460c6b56c42698a6" style="width:288px;height:288px;"></div> <script type="application/json" data-for="htmlwidget-271c460c6b56c42698a6">{"x":{"diagram":"\ndigraph {\n graph [ranksep = 0.5]\n node [shape = circle]\n A Y M\n edge [minlen = 2]\n A->Y [label=\"β1\"][alpha=\".2\"]\n A->M [label=\"γ1\"][color = red] [fontcolor=red]\n M->Y [label=\"β3\"][color = blue] [fontcolor=blue]\n \n{ rank = same; A; M; Y }\n}\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> --- # Do these methods yield the same answer? - Assuming no model misspecification, methods will yield similar estimates for continuous outcomes and rare binary outcomes - see [Vanderweele, 2016](https://www.annualreviews.org/doi/10.1146/annurev-publhealth-032315-021402) - Logistic regression models will be biased when the outcome is common, due to OR non-collapsibility - Can use log-binomial regression models as an alternative --- # Mediation Assumptions <div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-db54ec32d64eab926377" style="width:288px;height:288px;"></div> <script type="application/json" data-for="htmlwidget-db54ec32d64eab926377">{"x":{"diagram":"\ndigraph {\n graph [ranksep = 0.2]\n node [shape = circle]\n A Y M C1\n edge [minlen = 2]\n A->Y [color=lightgray]\n A->M \n M->Y \n C1->{A Y}[color=blue]\n C2->{M Y}[color=red]\n C3->{A M}[color=green]\n A->C2 [color=darkTurquoise]\n{ rank = same; A; M; Y }\n}\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> -- 1. No treatment-outcome confounding -- 2. No mediator-outcome confounding -- 3. No treatment-mediator confounding -- 4. No mediator-outcome confounder that is itself affected by the treatment. -- - Which of these are guaranteed in an RCT? --- # Mediator-Outcome Confounding - Even if the treatment is randomized or if all of the treatment-outcome confounders are included in the model, there may be confounders of the mediator-outcome relationship --- # Strong et al., 2008 - Randomized cognitive behavioral therapy intervention on depressive symptoms at three months follow-up. - Beneficial effect: Those who had received the therapy had lower depressive symptoms in follow-up. - However, treatment arm reported higher rates of antidepressant use - So what should the treatment be? CBT or antidepressants? <div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-83d1e39d402c50380896" style="width:504px;height:360px;"></div> <script type="application/json" data-for="htmlwidget-83d1e39d402c50380896">{"x":{"diagram":"\ndigraph {\n graph [ranksep = 0.2]\n node [shape = plaintext]\n A [label=\"CBT\"]\n Y [label=\"Dep sx\"]\n M [label=\"Anti-Dep\"]\n edge [minlen = 2]\n A->Y [label=\"(-)\"]\n A->M [label=\"(+)\"]\n M->Y \n \n{ rank = same; A; M; Y }\n}\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> --- # Unmeasured mediator-outcome confounding - We have to consider confounding of antidepressants `\(\rightarrow\)` depression - In their naive analysis, antidepressant use was positively associated with depression - Those who take antidepressants, probably differ on other risk factors for depression; more difficult contexts, stress/trauma, etc. <div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-db7d61b0b43eed8c9d9e" style="width:504px;height:360px;"></div> <script type="application/json" data-for="htmlwidget-db7d61b0b43eed8c9d9e">{"x":{"diagram":"\ndigraph {\n graph [ranksep = .2]\n node [shape = plaintext]\n A [label=\"CBT\"]\n Y [label=\"Dep sx\"]\n M [label=\"Anti-Dep\"]\n edge [minlen = 2]\n A->Y [label=\"(-)\"]\n A->M [label=\"(+)\"]\n M->Y [label=\"(+)\"]\n C->{M Y}\n{ rank = same; A; M; Y }\n}\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> --- # Exposure-Mediator interaction - Typically, we assume no interaction between the exposure and the mediator on the outcome - But, if the following model were true `\(E[Y|A=a, C=c, M=m] = \beta_0 + \beta_1a + \beta_2m + \beta_3am + \beta_4c\)` with `\(\beta_1 = 0.5\)` and `\(\beta_3=-1.0\)`, so that the sign of the effect of the treatment differed in the presence (-0.5) vs. absence (0.5) of the mediator. If we ignore `\(\beta_3\)` and fit the model `\(\beta_0 + \beta_1a + \beta_2m + \beta_4c\)`, then `\(\beta_1 \approx 0\)` Using the ‘difference method’, the effect of the treatment on the outcome appears to be fully mediated --- # Counterfactual mediation approach #### Two sets of potential outcomes - Let `\(Y^a\)` be the potential outcome `\(Y\)` for each individual when intervening to set `\(A\)` to `\(a\)` - Let `\(M^a\)` be the potential outcome `\(M\)` for each individual when intervening to set `\(A\)` to `\(a\)` - Let `\(Y^{am}\)` be the potential outcome `\(Y\)` for each individual when intervening to set `\(A\)` to `\(a\)` and `\(M\)` to `\(m\)` -- - Can think of these as sequential target trials, in order to clearly define the causal effects of interest --- # Defining direct and indirect effects (Robins and Greenland, 1992; Pearl, 2001) - **Controlled direct effect**: the effect comparing treatment level A=1 to A=0, while fixing M=m $$ `\begin{aligned} CDE= Y^{1m}-Y^{0m} \end{aligned}` $$ -- - **Natural direct effect**: the effect comparing treatment level `\(A=1\)` to `\(A=0\)`, while fixing `\(M\)` to its value under one treatment condition $$ `\begin{aligned} NDE = Y^{1M^0}-Y^{0M^0} (Pure)\\ = Y^{1M^1}-Y^{0M^1} (Total) \end{aligned}` $$ -- - **Natural indirect effect**: the effect comparing mediator level `\(M=M^1\)` to `\(M=M^0\)`, while fixing treatment to one level $$ `\begin{aligned} NIE = Y^{0M^1}-Y^{0M^0} (Pure)\\ = Y^{1M^1}-Y^{1M^0} (Total) \end{aligned}` $$ --- # Controlled vs. Natural effects CDE: what is the change in outcome due to the treatment, after controlling for the mediator when fixing mediator at some value M=m? - The treatment effect that would be realized if the mediator were controlled at level m uniformly in the population - What is the causal effect of eliminating the treatment, while also changing the mediator? -- _Controlled indirect effect: Difficult to conceptualize_ - Effect of treatment that is prevented if mediator is blocked for everyone - Effect of treatment that is prevented if mediator is assigned to everyone --- # Controlled vs. Natural effects - NDE: what is the effect that would be realized if the treatment were administered, but its effect on the mediator were blocked - Pure: Effect of treatment if treatment does not cause the mediator - Total: Effect of treatment if lack of treatment does not prevent the mediator -- - NIE: the effect on the outcome if the treatment were controlled at A=0 (Pure) or A=1 (Total), but the mediator were changed from its natural level M(0) to the level M(a) which it would have taken at treatment level a - Pure: Effect of treatment if only action of treatment is to cause mediator - Total: Effect of treatment due to treatment-induced changes in the mediator --- # Controlled vs. Natural effects - In absence of interaction, all direct effects are equal - In absence of interaction, all indirect effects are equal - In presence of interaction, all direct effects are different - In presence of interaction, all indirect effects are different --- # Controlled vs. Natural effects e.g., Controlled direct effect: - The difference in depression risk among those who received CBT (vs. no CBT), holding levels of antidepressant use constant Natural direct effect: - The difference in depression risk among those who received CBT (vs. no CBT), holding levels of antidepressant use at its expected level under no CBT Natural indirect effect: - The difference in depression risk among those who used antidepressants (vs. did not use antidepressants), with everyone being untreated (i.e., no CBT) <div class="grViz html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-ab09528407589a4fdc83" style="width:504px;height:216px;"></div> <script type="application/json" data-for="htmlwidget-ab09528407589a4fdc83">{"x":{"diagram":"\ndigraph {\n graph [ranksep = 0.2]\n node [shape = plaintext]\n A [label=\"CBT\"]\n Y [label=\"Dep sx\"]\n M [label=\"Anti-Dep\"]\n edge [minlen = 2]\n A->Y \n A->M \n M->Y \n \n{ rank = same; A; M; Y }\n}\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> --- # Checking-in: Which of these quantities are unobservable? $$ `\begin{aligned} CDE= Y^{1m}-Y^{0m} \end{aligned}` $$ $$ `\begin{aligned} NDE = Y^{1M^0}-Y^{0M^0} (Pure)\\ = Y^{1M^1}-Y^{0M^1} (Total) \end{aligned}` $$ $$ `\begin{aligned} NIE = Y^{0M^1}-Y^{0M^0} (Pure)\\ = Y^{1M^1}-Y^{1M^0} (Total) \end{aligned}` $$ --- # Cross-world counterfactuals - Quantities that cannot be empirically confirmed by any intervention on A and M, not even in principle - But using potential outcomes methods we are familiar with, we can estimate these quantities with weighting to estimate population averages --- # Effect decomposition #### Total Effect = Direct + Indirect `\((Y^{1}-Y^{0})\)` `\(= (Y^{1M^1}-Y^{0M^0})\)` -- `\(= (Y^{1M^0}-Y^{0M^0}) + (Y^{1M^1}-Y^{1M^0})\)` - = Pure Direct + Total Indirect `\(= (Y^{1M^1}-Y^{0M^1}) + (Y^{0M^1}-Y^{0M^0})\)` - = Total Direct + Pure Indirect = NDE + NIE --- # With Regression Modeling - 2 models 1. `\(E[M|A=a, C=c] = \beta_0 + \beta_1a + \beta'_2c\)` 2. `\(E[Y|A=a, C=c, M=m] = \theta_0 + \theta_1a + \theta_2m + \theta_3am+ \theta'_4c\)` -- `\(CDE = (\theta_1 + \theta_3m)(a-a*)\)` `\(NDE = (\theta_1 + \theta_3(\beta_0 + \beta_1a + \beta'_2c))(a-a*)\)` `\(NIE = (\theta_2\beta_1+\theta_3\beta_1a)(a-a*)\)` --- # Which effects to use? - When there is no treatment-mediator interaction, controlled and natural effects will be equal (assuming other assumptions are met) #### Controlled effects Strengths - Prescriptive RCT-like approach - More policy relevant? (Naimi et al, 2014) -- Weaknesses - Unrealistic mediator assignment - Indirect effect has limited interpretability --- # Which effects to use? #### Natural effects Strengths - Descriptive - Effect decomposition works in the presence of interaction -- Weaknesses - Stronger assumptions - Cross-world counterfactual assumption --- # In summary - Traditional methods used for mediation analysis are subject to problems of no unmeasured mediator-outcome confounding and the assumption of no interactions - Counterfactual definitions make the assumptions for causal interpretation clear and explicit - Counterfactual definitions of direct and indirect effects are model agnostic --- # Learning objectives - Understand the purpose of mediation - Understand the approach and limitations of 'traditional' mediation methods - Describe the counterfactual approach to mediation, its advantages and assumptions - Define and distinguish the various indirect and direct effects