class: center, middle, inverse, title-slide # L2: DAGs and Confounding ### Jean Morrison ### University of Michigan ### 2022-01-10 (updated: 2022-01-18) --- `\(\newcommand{\ci}{\perp\!\!\!\perp}\)` # Graphical Representations of Casual Effects + We can represent causal effects in a graph, with arrows. + Nodes in the graph are random variables. + Directed edges represent direct causal effects (not mediated by any other variables in the graph). + The absence of an edge indicates the absence of a direct causal effect. -- <div id="htmlwidget-6388aced8af0f82ee9d6" style="width:90%;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-6388aced8af0f82ee9d6">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"1,0!\"] \n \"3\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"2.2,0!\"] \n \"4\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"3.2,0!\"] \n \"5\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"2.7,0.5!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"5\"->\"4\" [color = \"black\"] \n\"5\"->\"3\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> --- # Graph Definitions - A graph, `\(\mathcal{G} = \lbrace V, E\rbrace\)` consists of - A set of nodes (vertices) `\(V = \lbrace V_1, \dots, V_J\rbrace\)` - A set of edges `\(E = \lbrace (V_{1_1}, V_{1_2}), \dots, (V_{K_1}, V_{K_2}) \rbrace\)`, which can be represented as pairs of nodes. - A graph can be either *directed*, in which case elements of `\(E\)` are ordered pairs or *undirected*, in which case elements of `\(E\)` are un-ordered. - Our graphs will almost always be directed. - Two nodes are *adjacent* if they are connected by an edge. + If the edge is directed, the node at the beginning of the edge is the *parent* and the node at the end is the *child*. --- # Graph Definitions - A *path* is a sequence of edges in which each edge contains one node from the previous edge. - In a directed path, all of the edges are oriented in the same direction: i.e. each edge starts at last node of the previous edge. - In this graph: <center> <div id="htmlwidget-88569040d0dd94f31a02" style="width:40%;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-88569040d0dd94f31a02">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1,0!\"] \n \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.5,0.5!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"3\"->\"2\" [color = \"black\"] \n\"3\"->\"1\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> there are two paths from `\(A\)` to `\(Y\)` but only one directed path. --- # Graph Definitions - If there are no paths between two nodes, they are *disconnected* (or *connected* otherwise). - Node `\(j\)` is a descendant of node `\(k\)` if there is a directed path from `\(V_j\)` to `\(V_k\)`. - If a graph contains no directed cycles, it is *acyclic* + We will require all of our graphs to be acyclic. - DAG = Directed Acyclic Graph --- # Example: Different Treatments for Different Patients + Consider this story: - There are two possible treatments for a disease. Treatment `\(A\)` is more effective than `\(B\)`, but has more side effects. - Doctors prefer treatment `\(B\)` for patients who are older or who have more mild disease. - Patient outcome (remission or not) is affected by initial severity, treatment, and treatment adherence. - Conditional on everything else, age has no effect on patient outcome or disease severity. -- + Work with your neighbor to arrange the following variables in a DAG: - Patient outcome - Initial disease severity - Treatment - Treatment Adherence - Age (The DAG is not uniquely determined from the information given.) --- # Disease Treatment Example <center> <div id="htmlwidget-3d1183ad1acd24285734" style="width:90%;height:504px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-3d1183ad1acd24285734">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"Age\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"0.2,1.2!\"] \n \"2\" [label = \"Severity\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"0.2,-1.2!\"] \n \"3\" [label = \"Treatment\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"2,0!\"] \n \"4\" [label = \"Outcome\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"4,0!\"] \n \"5\" [label = \"Adherence\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"3,1.2!\"] \n\"1\"->\"3\" [color = \"black\"] \n\"2\"->\"3\" [color = \"black\"] \n\"2\"->\"4\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"5\"->\"4\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> --- # Temporality - Our definition of causality requires that the exposure occur before the outcome in time. - Under this restriction, a causal DAG must be consistent with at least one strict ordering of nodes. -- - How do we represent a feedback loop? -- + We can create multiple nodes representing unique time points (e.g. `\(A_1, A_2, \dots\)`), with each node only permitted to have causal effects on future nodes. + More on this later. <center> <div id="htmlwidget-3473211cec2660b197ff" style="width:90%;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-3473211cec2660b197ff">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"1,0!\"] \n \"3\" [label = <A<FONT POINT-SIZE=\"8\"><SUB>1<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"2,0.3!\"] \n \"4\" [label = <Y<FONT POINT-SIZE=\"8\"><SUB>1<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"2.5,-0.3!\"] \n \"5\" [label = <A<FONT POINT-SIZE=\"8\"><SUB>2<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"3,0.3!\"] \n \"6\" [label = <Y<FONT POINT-SIZE=\"8\"><SUB>2<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", fontcolor = \"black\", color = \"black\", fillcolor = \"#FFFFFF\", pos = \"3.5,-0.3!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"2\"->\"1\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"4\"->\"5\" [color = \"black\"] \n\"4\"->\"6\" [color = \"black\"] \n\"5\"->\"6\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> <!-- - What if measurements are taken at the same time? --> <!-- -- --> <!-- + We are generally forced to assume a temporal ordering even if our observations are taken at the same time. --> <!-- + For example, if we measure height and blood pressure, we can assume that height is stable over a long period of time while blood pressure is more variable. --> --- # Causal Markov Property - The causal Markov property translates graph structure into probability statements. + It states that, conditional on it's parents, each node is independent of all nodes that are not not it's descendants. + This implies that the joint probability distribution of all nodes can be factored as $$ P(V) = \prod_{j = 1}^{J} P(V_j \vert pa_j). $$ --- # Example Conditional independence statements in the disease tratment graph: `$$S\ci A \qquad S\ci Ad \qquad A \ci Ad$$` `$$T \ci Ad\ \vert A, S \qquad O \ci A \ \vert S, T, Ad$$` <center> <div id="htmlwidget-4589c4dd10c2cf99f6fa" style="width:432px;height:360px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-4589c4dd10c2cf99f6fa">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"Age\nA\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"0.2,1.2!\"] \n \"2\" [label = \"Severity\nS\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"0.2,-1.2!\"] \n \"3\" [label = \"Treatment\nT\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"2,0!\"] \n \"4\" [label = \"Outcome\nO\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"4,0!\"] \n \"5\" [label = \"Adherence\nAd\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"3,1.2!\"] \n\"1\"->\"3\" [color = \"black\"] \n\"2\"->\"3\" [color = \"black\"] \n\"2\"->\"4\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"5\"->\"4\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> --- # Example In our example, we can factor the joint probability as `$$P(A, S, T, Ad, O) = P(A)P(S)P(Ad)P(T \vert A, S)P(O \vert S, T, Ad)$$` <center> <div id="htmlwidget-52f09353f6317b950938" style="width:432px;height:360px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-52f09353f6317b950938">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"Age\nA\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"0.2,1.2!\"] \n \"2\" [label = \"Severity\nS\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"0.2,-1.2!\"] \n \"3\" [label = \"Treatment\nT\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"2,0!\"] \n \"4\" [label = \"Outcome\nO\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"4,0!\"] \n \"5\" [label = \"Adherence\nAd\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"3,1.2!\"] \n\"1\"->\"3\" [color = \"black\"] \n\"2\"->\"3\" [color = \"black\"] \n\"2\"->\"4\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"5\"->\"4\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> --- # Exchangeability - Is the outcome O exchangeable with respect to the treatment? `\(O(t) \ci T\)`? <center> <div id="htmlwidget-ac60e05356fa6323f1da" style="width:432px;height:360px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-ac60e05356fa6323f1da">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"Age\nA\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"0.2,1.2!\"] \n \"2\" [label = \"Severity\nS\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"0.2,-1.2!\"] \n \"3\" [label = \"Treatment\nT\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"2,0!\"] \n \"4\" [label = \"Outcome\nO\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"4,0!\"] \n \"5\" [label = \"Adherence\nAd\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"3,1.2!\"] \n\"1\"->\"3\" [color = \"black\"] \n\"2\"->\"3\" [color = \"black\"] \n\"2\"->\"4\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"5\"->\"4\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> --- # Recognizing Lack of Exchangeability in a DAG - Informally, there are two sources of lack of exchangeability: + The presence of common causes (confounders) that have not been conditioned on. + Common effects (colliders) that have been conditioned on. - We will see how to formalize these statements and how to use a DAG to identify a sufficient conditioning set to remove confounding. --- # Common Causes (Confounders) - The presence of a common cause introduces association between two variables that is not due to a causal effect. <center> <div id="htmlwidget-3d303e935f1a9fce6cf6" style="width:432px;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-3d303e935f1a9fce6cf6">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1,0!\"] \n \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.5,0.5!\"] \n\"3\"->\"2\" [color = \"black\"] \n\"3\"->\"1\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> -- - In the disease treatment example, disease severity is a confounder: - Sicker patients are more likely to receive drug `\(B\)` and sicker patients are also more likely to have a poor outcome. - So there would be an association between treatment and outcome, even if the two drugs worked equally well. --- # Confounding - Confounding as a concept is quite old and therefore has been given many definitions. - We will define confounding as the lack of exchangeability that results from common causes. - *Confounders* are variables which can be used to adjust for confounding. + In the graph below, `\(L_1\)` and `\(L_2\)` are both confounders, even though only `\(L_1\)` is a common cause of `\(A\)` and `\(Y\)`. <center> <div id="htmlwidget-124c8786d3225b09bb89" style="width:432px;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-124c8786d3225b09bb89">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.5,0!\"] \n \"3\" [label = <L<FONT POINT-SIZE=\"8\"><SUB>1<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.5,0.8!\"] \n \"4\" [label = <L<FONT POINT-SIZE=\"8\"><SUB>2<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1,0.4!\"] \n\"3\"->\"1\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"4\"->\"2\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> --- # Common Effects (Colliders) + A variable `\(L\)` is a collider relative to `\(A\)` and `\(Y\)` if `\(L\)` is a descendent of both `\(A\)` and `\(Y.\)` + Conditioning on a collider introduces an association between `\(A\)` and `\(Y\)`. - This is not necessarily as intuitive as the bias introduced by a common cause. <center> <div id="htmlwidget-dda6a70facae73743de9" style="width:40%;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-dda6a70facae73743de9">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1,0!\"] \n \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.5,-0.5!\"] \n\"2\"->\"3\" [color = \"black\"] \n\"1\"->\"3\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> --- # Collider Example: Routes to Stardom + Suppose that in order to become a movie star, one must either be talented or beautiful. + Suppose that in the population, talent and beauty are uncorrelated. - For simplicity, suppose `\(P(\text{talent}) = P(\text{beauty}) = 0.1\)`. + For simplicity, assume that anyone with talent or beauty has the same chance of becoming a star. <center> <div id="htmlwidget-e45b2ebca144cad04f6b" style="width:504px;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-e45b2ebca144cad04f6b">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"Talent\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.5\", color = \"black\", shape = \"ellipse\", fixedsize = \"FALSE\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Beauty\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.5\", color = \"black\", shape = \"ellipse\", fixedsize = \"FALSE\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1,0!\"] \n \"3\" [label = \"Stardom\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.5\", color = \"black\", shape = \"ellipse\", fixedsize = \"FALSE\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.5,-0.75!\"] \n\"2\"->\"3\" [color = \"black\"] \n\"1\"->\"3\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> --- # Collider Example: Routes to Stardom + Tables below show the proportions of talent and beauty in the general population and among stars: <center> <table class="table" style="width: auto !important; float: left; margin-right: 10px;"> <caption>Population</caption> <thead> <tr> <th style="text-align:left;"> </th> <th style="text-align:right;"> \(B = 0\) </th> <th style="text-align:right;"> \(B = 1\) </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> \(T = 0\) </td> <td style="text-align:right;"> 0.81 </td> <td style="text-align:right;"> 0.09 </td> </tr> <tr> <td style="text-align:left;"> \(T = 1\) </td> <td style="text-align:right;"> 0.09 </td> <td style="text-align:right;"> 0.01 </td> </tr> </tbody> </table> <table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;"> <caption>Stars</caption> <thead> <tr> <th style="text-align:left;"> </th> <th style="text-align:right;"> \(B = 0\) </th> <th style="text-align:right;"> \(B = 1\) </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> \(T = 0\) </td> <td style="text-align:right;"> 0.00 </td> <td style="text-align:right;"> 0.47 </td> </tr> <tr> <td style="text-align:left;"> \(T = 1\) </td> <td style="text-align:right;"> 0.47 </td> <td style="text-align:right;"> 0.05 </td> </tr> </tbody> </table> </center> -- - In the population `\(P[T=1] = P[ T = 1 \vert B = 1] = 0.1\)` - Among stars `\(P[T=1] = 0.1/0.19 = 0.53\)`, and `\(P[T = 1 \vert B = 1] = \frac{0.01/0.19}{0.1/0.19} = 0.1\)` -- - Among stars, talent and beauty are negatively correlated. Talent is more rare among beautiful stars than among stars as a whole. - This is because we have conditioned on the collider "Becoming a star". --- # Colliders Can Block Confounding - In the graph below, there is no confounding because there is no common cause of `\(A\)` and `\(Y\)`. - The collider `\(L_2\)` is "blocking" the path from `\(A\)` to `\(Y\)`. - Causal Markov properties show us that `\(A\)` and `\(Y\)` are independent in this graph. - d-Separation formalizes the rules for identifying pairs of independent variables based on graphical rules. <div id="htmlwidget-a27c445d1a15733e5c7b" style="width:432px;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-a27c445d1a15733e5c7b">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.5,0!\"] \n \"3\" [label = <L<FONT POINT-SIZE=\"8\"><SUB>1<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.5,0.8!\"] \n \"4\" [label = <L<FONT POINT-SIZE=\"8\"><SUB>2<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1,0.4!\"] \n\"3\"->\"1\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"2\"->\"4\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> --- # No Statistical Definition of Confounding - One commonly given characterization of a confounder is a variable which + Is associated with the exposure. + Is associated with the outcome. + Is not on the pathway of interest between exposure and outcome. -- - Note that `\(L_2\)` satisfies all of these criteria but is not a confounder. <center> <div id="htmlwidget-934d31e7d29688b6b4ca" style="width:432px;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-934d31e7d29688b6b4ca">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.5,0!\"] \n \"3\" [label = <L<FONT POINT-SIZE=\"8\"><SUB>1<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.5,0.8!\"] \n \"4\" [label = <L<FONT POINT-SIZE=\"8\"><SUB>2<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1,0.4!\"] \n\"3\"->\"1\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"2\"->\"4\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> - Determining confounding requires a causal model. + The data cannot tell you if confounding is present. --- # d-Separation + A path is *blocked* if: 1. Two arrowheads on the path collide ( `\(\rightarrow W \leftarrow\)` ) at a variable that is not being conditioned on *and* which has no descendants in the conditioning set. OR 1. It contains a non-collider that is being conditioned on. + A path is *open* if it is not blocked: - It does not contain a collider and no variables on the path are being conditioned on. OR - All colliders are conditioned on and no non-colliders are conditioned on. + Two variables are *d-separated* if all paths between them are blocked. --- # d-Separation and the Causal Markov Property Let `\(A\)`, `\(B\)`, and `\(C\)` be sets of variables. Verma and Pearl (1988) proved that: If `\(A\)` and `\(B\)` are d-separated given `\(C\)` then `\(A \ci B \vert\ C\)`. - We will not prove this in class. --- # Faithfulness Faithfulness is the property that, for three sets of variables `\(A\)`, `\(B\)`, and `\(C\)` `$$A \ci B \vert C \Rightarrow\ A \text{ is } d\text{-separated from }B\text{ given }C$$`. - Violations of faithfulness occur when confounding effects perfectly cancel each other. + Pearl calls these "incidental cancellations". + And then defines "stable" vs "unstable" unbiasedness. Unstable unbiasedness occurs when faithfulness is violated. - Practically, we can assume that faithfulness is never violated except when it is violated by design. - Matching studies intentionally violate faithfulness. --- # Example Which pairs of variables are d-Separated? <center> <div id="htmlwidget-74f0f52f6288636a9e8b" style="width:432px;height:360px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-74f0f52f6288636a9e8b">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"Age\nA\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"0.2,1.2!\"] \n \"2\" [label = \"Severity\nS\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"0.2,-1.2!\"] \n \"3\" [label = \"Treatment\nT\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"2,0!\"] \n \"4\" [label = \"Outcome\nO\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"4,0!\"] \n \"5\" [label = \"Adherence\nAd\", fontname = \"Helvetica\", fontsize = \"12\", color = \"black\", fontcolor = \"black\", fixedsize = \"TRUE\", width = \"1\", fillcolor = \"#FFFFFF\", pos = \"3,1.2!\"] \n\"1\"->\"3\" [color = \"black\"] \n\"2\"->\"3\" [color = \"black\"] \n\"2\"->\"4\" [color = \"black\"] \n\"3\"->\"4\" [color = \"black\"] \n\"5\"->\"4\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> -- - The causal Markov property allowed us to conclude that `\(T \ci Ad \vert S, A\)`. - Because `\(T\)` and `\(Ad\)` are d-separated, we can also conclude that `\(T \ci Ad\)` unconditionally. --- # Backdoor Path - A backdoor path from `\(A\)` to `\(Y\)` is a path from `\(A\)` to `\(Y\)` that begins with an edge going *into* `\(A\)`. <center> <div id="htmlwidget-cdf5a2fc5021f728a698" style="width:60%;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-cdf5a2fc5021f728a698">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1,0!\"] \n \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.5,0.5!\"] \n \"4\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.5,0!\"] \n \"5\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"2.5,0!\"] \n \"6\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"2,0.5!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"3\"->\"2\" [color = \"black\"] \n\"3\"->\"1\" [color = \"black\"] \n\"4\"->\"5\" [color = \"black\"] \n\"6\"->\"5\" [color = \"black\"] \n\"4\"->\"6\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> <div id="htmlwidget-6645108e6cf396a36068" style="width:60%;height:180px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-6645108e6cf396a36068">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.8,0!\"] \n \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.9,0.4!\"] \n \"4\" [label = <U<FONT POINT-SIZE=\"8\"><SUB>1<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.45,0.9!\"] \n \"5\" [label = <U<FONT POINT-SIZE=\"8\"><SUB>2<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.35,0.9!\"] \n\"4\"->\"1\" [color = \"black\"] \n\"4\"->\"3\" [color = \"black\"] \n\"5\"->\"3\" [color = \"black\"] \n\"5\"->\"2\" [color = \"black\"] \n\"1\"->\"2\" [color = \"black\"] \n\"3\"->\"2\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> --- # Backdoor Criterion and Exchangeability Theorem: If a set of variables, `\(L\)`, + blocks every backdoor path between `\(A\)` and `\(Y\)` + contains no descendants of `\(A\)`, then `\(Y(a) \ci A \vert L\)`. -- - The two conditions in the theorem are referred to as the *backdoor criterion*. -- - To justify this, we need a connection between DAGs and counterfactuals. --- # Examples What set of variables makes `\(A\)` and `\(Y\)` conditionally exchangeable? <center> <div id="htmlwidget-ad7f0b05af880108fea5" style="width:60%;height:216px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-ad7f0b05af880108fea5">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.8,0!\"] \n \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.9,0.4!\"] \n \"4\" [label = <U<FONT POINT-SIZE=\"8\"><SUB>1<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.45,0.9!\"] \n \"5\" [label = <U<FONT POINT-SIZE=\"8\"><SUB>2<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.35,0.9!\"] \n\"4\"->\"1\" [color = \"black\"] \n\"4\"->\"3\" [color = \"black\"] \n\"5\"->\"3\" [color = \"black\"] \n\"5\"->\"2\" [color = \"black\"] \n\"1\"->\"2\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> -- - `\(A\)` and `\(Y\)` are exchangeable unconditionally. - Conditioning on `\(L\)` induces bias. (M-Bias). --- # Examples What set of variables makes `\(A\)` and `\(Y\)` conditionally exchangeable? <center> <div id="htmlwidget-81ccb3571ebfe3aacd71" style="width:60%;height:216px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-81ccb3571ebfe3aacd71">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.8,0!\"] \n \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.9,0.4!\"] \n \"4\" [label = <U<FONT POINT-SIZE=\"8\"><SUB>1<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.45,0.9!\"] \n \"5\" [label = <U<FONT POINT-SIZE=\"8\"><SUB>2<\/SUB><\/FONT>>, fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.35,0.9!\"] \n\"4\"->\"1\" [color = \"black\"] \n\"4\"->\"3\" [color = \"black\"] \n\"5\"->\"3\" [color = \"black\"] \n\"5\"->\"2\" [color = \"black\"] \n\"1\"->\"2\" [color = \"black\"] \n\"3\"->\"2\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> -- - If `\(U_1\)` and `\(U_2\)` are unobserved, there is no available set of variables we can condition on to remove bias. - If we can measure either, `\(\lbrace U_1, L\rbrace\)` and `\(\lbrace U_2, L \rbrace\)` are both sufficient adjustment sets. --- # Examples What set of variables makes `\(A\)` and `\(Y\)` conditionally exchangeable? <center> <div id="htmlwidget-e6a13c9735ee0ce7b4f0" style="width:60%;height:216px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-e6a13c9735ee0ce7b4f0">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0,0!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1.8,0!\"] \n \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.9,0!\"] \n \"4\" [label = \"U\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.45,0.5!\"] \n\"1\"->\"3\" [color = \"black\"] \n\"3\"->\"2\" [color = \"black\"] \n\"4\"->\"1\" [color = \"black\"] \n\"4\"->\"3\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> -- - Conditioning on `\(U\)` blocks all backdoor paths. - Could we condition on `\(L\)` instead? -- - No. `\(L\)` is a descendant of `\(A\)`, so it does not satisfy the backdoor criterion. --- # Single World Intervention Graphs (SWIGs) - SWIGs are a method of including counterfactuals in DAG. - To create a SWIG, the node representing the intervened on variable is split into two nodes. - One node represents the natural value of of the variable. - The other represents the fixed value due to the intervention. <center> <img src="img/2_dag1.png" width="30%" /> <img src="img/2_swig1.png" width="80%" /> </center> <!-- --- --> <!-- # SWIG Procedure --> <!-- - To represent the counterfactual `\(Y(a)\)`, we split the node `\(A\)` in the original graph into two nodes: --> <!-- - `\(A\)` (in the SWIG) represents the naturally occurring treatment -- what would have occurred with no intervention. --> <!-- + All of the arrows which entered into `\(A\)` in the original DAG will enter into `\(A\)` in the SWIG. --> <!-- - `\(a\)` represents the intervened on value of the treatment. --> <!-- + This variable is fixed at the value `\(a\)` (i.e. it is deterministic rather than random). --> <!-- + All arrows which exited `\(A\)` in the original DAG now exit the `\(a\)` intervention node. --> <!-- + All variables which were downstream of `\(A\)` in the original DAG are replaced by their counterfactual values. --> --- # Templates (SWITs) - Each SWIG can represent only a single intervention, i.e. the world in which everyone receives treatment `\(A = a\)`. - A template is graph valued function `\(x \rightarrow \mathcal{G}(x)\)` + Input is the value the intervened on variable is set to. + Output is a SWIG. <center> <img src="img/2_swig1.png" width="80%" /> <img src="img/2_swig4.png" width="40%" /> </center> --- # Single World + `\(Y(0)\)` and `\(Y(1)\)` never appear on the same graph. + SWIGs cannot represent relationships between counterfactuals "across worlds" (i.e. `\(Y(0)\)` and `\(Y(1)\)` ). + From the SWIGs below, we can conclude $ Y(0) \ci X$ and `\(Y(1) \ci X\)` but not `\(Y(1), Y(0) \ci X\)` or `\(Y(1) \ci Y(0)\)`. <center> <img src="img/2_swig1.png" width="80%" /> </center> --- # SWIG Procedure - Step 1: Split nodes <center> <img src="img/2_split_nodes.png" width="40%" /> </center> - Split every intervention node into - `\(A\)` the random component; what `\(A\)` would have been without intervention. - `\(a\)` a fixed component representing the intervention. - Incoming arrows go into `\(A\)` and outgoing arrows go out of `\(a\)`. --- # SWIG Procedure - Step 2: Re-label downstream nodes as counterfactuals. <center> <img src="img/2_label_nodes.png" width="80%" /> </center> --- # d-Separation in SWIGs - In SWIGs we have an additional rule for d-separation. - Any path containing an intervention node is blocked. - Alternatively, `\(A\)` and `\(Y\)` are d-separated in `\(\mathcal{G}(a)\)` if they are d-separated in the subgraph of `\(\mathcal{G}(a)\)` achieved by removing all of the fixed nodes. - Remember, intervention nodes are fixed at a single value, so no information can propogate through them. - We are heading toward the result that (conditional) d-separation of `\(Y(a)\)` and `\(A\)` in a SWIG `\(\mathcal{G}(a)\)` implies (conditional) exchangeability of `\(Y(a)\)` and `\(A\)`. --- # Applying d-separation in SWIGs <center> <img src="img/2_fig16.png" width="80%" /> </center> - This is the M-Bias example. - We can see that `\(Y(a)\)` is d-separated from `\(A\)` unconditionally. - Conditional on `\(Z\)`, `\(Y(a)\)` is not d-separated from `\(A\)`. --- # Linking `\(\mathcal{G}\)` and `\(\mathcal{G}(a)\)` - Let `\(P(\mathbf{V})\)` be a probability distribution over nodes `\(\mathbf{V}\)` that factorizes according to the DAG `\(\mathcal{G}\)`. - Goal: conditional d-separation in SWIG `\(\Rightarrow\)` conditional exchangeability - To get there, we need to know that `\(P(\mathbf{V}(\mathbf{a}))\)`, the probability distribution over counterfactuals subject to the intervention `\(\mathbf{a},\)` factorizes according to the SWIG `\(\mathcal{G}(\mathbf{a})\)`. - To get this, we need to be more specific about defining a model to go with our DAGs. --- # Non-Parametric Structural Equation Models - The models we have expressed so far as DAGs represent underlying counterfactual models. We can now give a formal definition of that model. - Start with a set of variables `\(\mathbf{V}\)` represented by a DAG `\(\mathcal{G}\)`. -- Counterfactual Existance Assumption: - Every variable `\(V \in \mathbf{V}\)` can be intervened on. - Let `\(\mathbf{R}\)` be a set of variables to be intervened on and set to `\(\widetilde{\mathbf{r}}\)`. We define the counterfactual value of `\(V(\mathbf{r})\)` recursively as a function of the counterfactual values of it's parents. Let `\(\widetilde{\mathbf{r}}_{pa(V)}\)` be the subset of `\(\widetilde{\mathbf{r}}\)` which correspond to parents of `\(V\)`. `$$V(\widetilde{\mathbf{r}}) = V \left(\widetilde{\mathbf{r}}_{pa(V)}, (\mathbf{PA}_V \backslash \mathbf{R})(\widetilde{\mathbf{r}}) \right)$$` --- # Equivalent Equation Version The previous definition is equivalent to: - Assume the existence of unmeasured errors `\(\epsilon_V\)`. - Let `\(\widetilde{\mathbf{pa}}_V\)` be an intervention on the parents of `\(V\)`. Then we assume the existence of deterministic functions `$$V(\widetilde{\mathbf{pa}}_V) = f_V(\widetilde{\mathbf{pa}}_V, \epsilon_V).$$` - With consistancy, this also implies that $$ V = f_V(PA_V, \epsilon_V)$$ --- # Example This graph <center> <img src="img/2_swig6_fig9.png" width="40%" /> </center> corresponds to equations `$$Z = f_Z(\epsilon_Z)$$` `$$M(z) = f_M(z, \epsilon_M)$$` `$$Y(z, m) = f_Y(z, m, \epsilon_Y)$$` - Our model also says that intervening on the future, does not affect past variables: `\(Z(m, y) = Z\)`, `\(M(z, y) = M(z)\)`, `\(M(y) = M(Z)\)`. - For partial interventions, we have `\(Y(m) = Y(m, Z)\)`, `\(Y(z) = Y(z, M(z))\)`, `\(Y = Y(Z, M(Z))\)`. --- # Independence Assumptions - From this model alone, we can't conclude the causal Markov property. We need an additional assumption. - FFRCISTG Assumption: Let `\(\mathbf{v}^\dagger\)` be an intervention on every variable in `\(V\)`. The FFRCISTG independence assumption says that all of the counterfactual variables `\(\lbrace V(\mathbf{v}^\dagger) \rbrace\)` are mutually independent after this intervention. - NPSEM-IE assumption: The NPSEM-IE assumption says that all of the errors `\(\epsilon_V\)` are independent. - NPSEM-IE is strictly stronger than FFRCISTG. - NPSEM-IE implies cross-world independences while FFRCISTG does not. - We will generally always assume FFRCISTG. --- # FFRCISTG vs NPSEM-IE <center> <img src="img/2_swig6_fig9.png" width="40%" /> </center> - FFRCISTG says $$ Z\ci M(z) \ci Y(z, m)$$ for any `\(z\)` and `\(m\)`. - NPSEM-IE says `$$Z \ci \lbrace M(z) \text{ for all } z \rbrace \ci \lbrace Y(z, m)\ \text{for all } z, m \rbrace$$` - For example, `\(M(0) \ci Y(Z = 1, M = 0)\)`. --- # Factorization Proof Sketch - The result that `\(P(\mathbf{V})\)` factorizes according to `\(\mathcal{G} \Rightarrow P(\mathbf{V}(\mathbf{a}))\)` factorizes according to the SWIG `\(\mathcal{G}\)` follows from the FFRCISTG independence assumption. Proof idea: Work by reverse induction - FFRCISTG tells us that the property holds for interventions on all variables. - This follows trivially (in a mathematical sense) - CI properties for of the SWIG for an intervention on all variables say that `\(P(\mathbf{V}(\mathbf{a})) = \prod_j P(V_j(\mathbf{a}))\)`. - This is exactly the FFRCISTG independence assumption. --- # Factorization Proof Sketch - Suppose we have an intervention on a set of variables `\(A \subset \mathbf{V}\)` and that the property holds for all (strict) supersets of `\(A\)`. - We can find a single variable `\(C\)` to add to `\(A\)` and prove the factorization result only for terms involving `\(C\)`. <center> <img src="img/2_factorization.png" width="75%" /> </center> - See appendix B1 of Richardson and Robins (2013). --- # SWIGs, Exchangeability, and d-Separation - The factorization result allows us to conclude that d-separation in the SWIG implies conditional exchangeability. - Recall: Conditional exchangeability says that `\(Y(a) \ci A \vert L\)`, which is exactly what is implied by `\(d\)`-separation in the SWIG. - This is an alternative formulation of the backdoor criterion. --- # Descendents of `\(A\)` in the Backdoor Criterion + Using SWIGs helps show why the backdoor criterion excludes descendants of `\(A\)`. <center> <img src="img/2_swig3.png" width="50%" /> </center> + The SWIG on the right shows that, `\(Y(x) \ci X \vert L_1, L_2(x)\)`. + However, we cannot conclude that `\(Y(x) \ci X \vert L_1, L_2\)` because `\(L_2\)` is not on the graph. + If there is a causal effect of `\(X\)` on `\(Y\)`, then conditioning on `\(L_2\)` introduces a type of selection bias (more on this in the future). --- # More Examples: Confounder <center> <img src="img/2_fig9iii.png" width="40%" /> </center> - `\(Y(m)\)` is not unconditionally independent of `\(M\)` but, `\(Y(m) \ci M \vert Z\)` - The factorization result gives us `$$P(Z, M, Y(m)) = P(Z)P(M \vert Z)P(Y(m) \vert Z)$$` - Note that we left the fixed node out of the probability calculation. --- # Mediator <center> <img src="img/2_fig9ii.png" width="40%" /> </center> - Here we do have unconditional exchangeability because `\(Y(z) \ci Z\)`. - From factorization, `\(P(Z, M(z), Y(z)) = P(Z)P(M(z))P(Y(z) \vert M(z))\)` --- # Mediator with Two Interventions <center> <img src="img/2_fig9iv.png" width="40%" /> </center> - From this graph we can get `\(Y(z, m) \ci M(z)\)`. - We have to use the new rule that fixed nodes block paths. - This is saying that intervening on `\(M\)` blocks the effect of `\(Z\)` that is propogated through `\(M(z)\)`. - From factorization, `\(P(Z, M(z), Y(z)) = P(Z)P(M(z))P(Y(z))\)` --- # Mediation Effects <center> <div id="htmlwidget-b5747f0d5c4dca829d08" style="width:40%;height:144px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-b5747f0d5c4dca829d08">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"-0.1,0.5!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1,0!\"] \n \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.2,0!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"1\"->\"3\" [color = \"black\"] \n\"3\"->\"2\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> - In the graph above, `\(L\)` is *mediating* part of the effect of `\(A\)` on `\(Y\)`. - Using our machinery so far, we can define the total effect (TE) of `\(A\)` on `\(Y\)` as `$$E[Y(A = 1)] - E[Y(A = 0)]$$` - We might be interested in the effect of `\(A\)` that is not mediated through `\(L\)`. - This would be the effect of `\(A\)` on `\(Y\)` if we intervened on `\(L\)` and prevented `\(L\)` from changing. - For example, we intervene on `\(A\)` and set it to 1. - But we intervene on `\(L\)` and set it to `\(L(0)\)`. --- # Natural Direct and Indirect Effect <center> <div id="htmlwidget-564c7422c091440b05ea" style="width:40%;height:144px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-564c7422c091440b05ea">{"x":{"diagram":"digraph {\n\ngraph [layout = \"neato\",\n outputorder = \"edgesfirst\",\n bgcolor = \"white\"]\n\nnode [fontname = \"Helvetica\",\n fontsize = \"10\",\n shape = \"circle\",\n fixedsize = \"true\",\n width = \"0.5\",\n style = \"filled\",\n fillcolor = \"aliceblue\",\n color = \"gray70\",\n fontcolor = \"gray50\"]\n\nedge [fontname = \"Helvetica\",\n fontsize = \"8\",\n len = \"1.5\",\n color = \"gray80\",\n arrowsize = \"0.5\"]\n\n \"1\" [label = \"A\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"-0.1,0.5!\"] \n \"2\" [label = \"Y\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"1,0!\"] \n \"3\" [label = \"L\", fontname = \"Helvetica\", fontsize = \"10\", width = \"0.3\", color = \"black\", fillcolor = \"#FFFFFF\", fontcolor = \"#000000\", pos = \"0.2,0!\"] \n\"1\"->\"2\" [color = \"black\"] \n\"1\"->\"3\" [color = \"black\"] \n\"3\"->\"2\" [color = \"black\"] \n}","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </center> - The *natural direct effect*(NDE) is `$$E[Y(A = 1, L = L(0))] - E[Y(A = 0, L = L(0)]$$` - The *natural indirect effect*(NIE) is $$ E[Y(A = 1, L= L(1))] - E[Y(A = 1, L = L(0))] $$ - So TE = NDE + NIE - Note that both of these involve "cross-world" counterfactuals. --- # Identifying NIE and NDE - The FFRCISTG independence assumption does not allow identification of the NIE and NDE. - See Robins and Greenland (1992) for a good example. - If we further assume the NPSEM-IE model, NIE and NDE are identifiable. --- # Identifying the NDE under NPSEM-IE Assumptions - Under NPSEM-IE assumptions, cross-world independcies hold. - Specifically, we have $$ Y(A = 1, l) \ci L(A = 0) \qquad \forall\ l $$ - We want to estimate `\(E[Y(A = 1, L(A = 0))]\)`. $$ E[Y(A = 1, L(A = 0))] = \sum_l E[Y(A = 1, L = l)\vert L(0) = l]P[L(0) = l] $$ $$ =\sum_l E[Y(A = 1, L=l)]P[L(0) = l] $$ - The second equality comes from the independence assumption above. --- # Identifying the NDE under NPSEM-IE Assumptions - We now only need to estimate `\(E[Y(A = 1, L = l)]\)` and `\(P[L(0) = l]\)` which can be estimated as $$ E[Y(A = 1, L = l)] = E[Y \vert A = 1, L = l] $$ $$ P[L(0) = l ] = P[L=l \vert A = 0] $$ - Both equalities come from exchangeability - So $$ E[Y(A = 1, L(A = 0))] = \sum_l E[Y \vert A = 1, L = l]P[L =l \vert A = 0] $$ --- # Non-Identifiability of NDE under FFRCISTG Intuition - Consider two types of units: + Both have `\(L(0) = 0\)` and `\(L(1) = 1\)` - In both cases `\(Y(1, 1) = 1\)` and `\(Y(0, 0) = 0\)`. - For type 1, `\(Y(1, 0) = 0\)` and for type 2 `\(Y(1, 0) = 1\)`. - For type 1, if `\(A\)` had not caused `\(L\)`, they would not have experienced `\(Y\)`. - For type 2, `\(L\)` did not matter. - Even if we could observe both types under both treatments of `\(L\)`, we could not tell them apart. - But type 1 individuals represent indirect effects while type 2 individuals represent direct effects. <!-- --- --> <!-- # Non-Identifiability of NDE under FFRCISTG Intuition --> <!-- - Type 1: `\(L(0) = 0\)`, `\(L(1) = 1\)`, `\(Y(1, 1) = 1\)`, `\(Y(0, 0) = 0\)`, `\(Y(1, 0) = 0\)` --> <!-- - Type 2: `\(L(0) = 0\)`, `\(L(1) = 1\)`, `\(Y(1, 1) = 1\)`, `\(Y(0, 0) = 0\)`, `\(Y(1, 0) = 1\)` --> <!-- - NPSEM-IE assumes that the proportion of type 1 individuals is equal to the proportion of --> <!-- - Type 3: `\(L(0) = 1\)`, `\(L(1) = 1\)`, `\(Y(1, 1) = 1\)`, `\(Y(0, 0) = 0\)`, `\(Y(1, 0) = 0\)` --> <!-- - And that the proportion of type 2 individuals is equal to the proportion --> <!-- - Type 4: `\(L(0) = 1\)`, `\(L(1) = 1\)`, `\(Y(1, 1) = 1\)`, `\(Y(0, 0) = 0\)`, `\(Y(1, 0) = 1\)` --> --- # Controlled Direct and Indirect Effects - Controlled effects are an alternative to natural effects. - These are the effect of `\(Y\)` when we control `\(L\)` at some pre-determined value. `$$E[Y(A = 1, L = l)] - E[Y(A = 0, L = l)]$$` - This definition only requires a joint intervention and doesn't require cross-world relationships between counterfactuals. - Therefore CDE and CIE are identifiable under the FFRCISTG model. <!-- --- --> <!-- # Extras: Collapsibility and Confounding --> <!-- --- --> <!-- # Causal Identification --> <!-- - Suppose I know the joint distribution `\(P(V)\)` over a set of variables. --> <!-- - Suppose I also assume: --> <!-- - The distribution factorizes according to a DAG. --> <!-- - We have observed all of the nodes on the DAG. --> <!-- - Faithfulness --> <!-- - When is it possible to infer the structure of the DAG? --> <!-- --- --> <!-- # Markov Equivalence --> <!-- - Multiple graphs can imply the same set of conditional independence relations. --> <!-- - Two graphs `\(\mathcal{G}_1\)` and `\(\mathcal{G}_2\)` are Markov equivalent if they imply the same set of d-separation relations. -->