Methods Lunch Talk: Causal Mediation Analysis

Transcription

1 Methods Lunch Talk: Causal Medaton Analyss Taeyong Park Washngton Unversty n St. Lous Aprl 9, 2015 Park (Wash U.) Methods Lunch Aprl 9, / 1

2 References Baron and Kenny The Moderator-Medator Varable Dstncton n Socal Psychologcal Research: Conceptual, Strategc, and Statstcal Consderatons. Journal of Personalty and Socal Psychology. MacKnnon Introducton to Statstcal Medaton Analyss. NY: Routledge. Ima et al. 2010a. Identfcaton, Inference, and Senstvty Analyss for Causal Medaton Effects. Socal Scence. Ima et al. 2010b. A General Approach to Causal Medaton Analyss. Psychologcal Methods. Ima et al Unpackng the Black Box of Causalty: Learnng about Causal Mechansms from Expermental and Observatonal Studes. Amercan Poltcal Scence Revew. Park (Wash U.) Methods Lunch Aprl 9, / 1

3 Overvew 1. What does a causal mechansm mean? 2. Methods for nvestgatng causal mechansms Causal medaton analyss based on the structural equaton modelng framework based on the potental outcomes framework 3. The potental outcomes approach: dentfyng and estmatng causal mechansms 4. Emprcal applcatons Park (Wash U.) Methods Lunch Aprl 9, / 1

4 What does a causal mechansm mean? Researchers are nterested n whether an varable has an mpact on another varable (causal effects) how the mpact operates (causal mechansm) Example: A job tranng program Job seekers mental health How, or why, does the program affect the mental health? One possble explanaton: attendng the program enhances partcpants confdence n ther ablty to search for a job better mental health job search self-effcacy ntermedates between the job program and job seekers mental health Park (Wash U.) Methods Lunch Aprl 9, / 1

5 What does a causal mechansm mean? Self effcacy Job program Mental health Park (Wash U.) Methods Lunch Aprl 9, / 1

6 What does a causal mechansm mean? Total effect = Medaton effect + Drect effect Medator Medaton effect Treatment Outcome Drect effect Park (Wash U.) Methods Lunch Aprl 9, / 1

7 Does a standard multvarate regresson estmate causal mechansms? Y = α + βt + γm + ɛ M γ T β Y Park (Wash U.) Methods Lunch Aprl 9, / 1

8 Methods for nvestgatng causal mechansms Causal medaton analyss Lnear Structural Equaton Modelng (LSEM) wth a sngle medator Y = α 1 + ζt + ɛ 1, M = α 2 + δt + ɛ 2, Y = α 3 + βt + γm + ɛ 3 Total effect = ζ Drect effect = β Medaton effect = ζ β = δγ Park (Wash U.) Methods Lunch Aprl 9, / 1

9 Methods for nvestgatng causal mechansms Causal medaton analyss M M δ γ γ T β LSEM Y T β Regresson Y Park (Wash U.) Methods Lunch Aprl 9, / 1

10 Methods for nvestgatng causal mechansms Causal medaton analyss What assumptons are requred for δγ to be an asymptotcally consstent estmate? Lnearty Zero correlaton between ɛ 2 and ɛ 3 The potental outcomes framework: an alternatve approach to causal medaton analyss (Ima et al. 2011). A general method that s not dependent on any statstcal model applcable to lnear and nonlnear models n the same way. Clarfes the requred assumptons and provdes a senstvty tool. Park (Wash U.) Methods Lunch Aprl 9, / 1

11 The potental outcomes approach to medaton analyss Potental outcomes (Holland 1986; Neyman 1923; Rubn 1974) Gven a unt and a set of actons ncludng control and treatment T = t, for t = 0 or 1, two outcomes reman potental untl one s actually realzed: Y = Y (t = 0) and Y = Y (t = 1), where Y s the realzed outcome. Y (0) s s mental health that would be realzed f does not attend the job program. Y (1) s s mental health that would be realzed f attend the job program. Park (Wash U.) Methods Lunch Aprl 9, / 1

12 The potental outcomes approach to medaton analyss Defne the potental outcomes as a functon of treatment and medator such as Y (T = t, M (T ) = m) for ndvdual. Y : an outcome; T : a treatment varable; M : a medator; t and m: specfc values for T and M. For the sake of smplcty, suppose two alternatve values of the treatment. t = 0 represents the control condton and t = 1 represents the treatment condton. The potental medator values for are denoted by M (0) and M (1). For nstance, Y (0, M (1)) refers to a potental outcome that would be realzed under the control condton wth the medator takng a counterfactual value that would be realzed under the treatment condton. Y (0, M (0))? Park (Wash U.) Methods Lunch Aprl 9, / 1

13 The potental outcomes approach to medaton analyss Identfyng the medaton effect: By holdng the treatment constant and changng only the medator, thereby controllng for the drect causal pathway between the treatment and outcome, we can solate the medaton effect from other alternatve causal mechansms. The medaton effect s a purely counterfactual quantty that s nherently unobservable. Medator Medaton effect Treatment Outcome Drect effect Park (Wash U.) Methods Lunch Aprl 9, / 1

14 The potental outcomes approach to medaton analyss Explots predcted counterfactual values of the treatment and the medator. Clarfes what assumptons are requred to dentfy the unobservable medaton effect. Average Medaton Effect (AME) ME Y (t, M (1)) Y (t, M (0)), for t = 0 or 1 AME E[Y (t, M (1)) Y (t, M (0))], for t = 0 or 1 How much the outcome would change f the medator changes whle the treatment remans constant. Park (Wash U.) Methods Lunch Aprl 9, / 1

15 The potental outcomes approach to medaton analyss Average Drect Effect (ADE) DE Y (1, M (t)) Y (0, M (t)), for t = 0 or 1 ADE E[Y (1, M (t)) Y (0, M (t))], for t = 0 or 1 How much the outcome would change f the treatment changes whle the medator remans constant. Average Total Effect (ATE) TE Y (1, M (1)) Y (0, M (0)) ATE E[Y (1, M (1)) Y (0, M (0))] The total effect of the treatment on the outcome. Park (Wash U.) Methods Lunch Aprl 9, / 1

16 The potental outcomes approach to medaton analyss Requred Assumptons The sequental gnorablty assumpton (Ima et al. 2010a; 2010b; 2011). (1) The treatment s assumed to be ndependent of potental outcomes and potental medators, so that t s gnorable, gven the observed pretreatment covarates. (2) The medator s assumed to be ndependent of potental outcomes, so that t s gnorable, gven the observed pretreatment covarates and the gnorable treatment. Perfect complance wth treatment assgnment. SUTVA (stable unt treatment value assumpton): Non-nterference and excluson restrcton. Park (Wash U.) Methods Lunch Aprl 9, / 1

17 The potental outcomes approach to medaton analyss Estmaton procedure (Three steps): 1. Fttng models 2. Smulatng potental values of the medator and the outcome 3. Computng the medaton effect and the drect effect 1. Fttng models: Runnng two regressons. A regresson model for the medator as a functon of the treatment and pretreatment covarates f θm (M T, X ) A regresson model for the outcome as a functon of the treatment, medator and pretreatment covarates f θy (Y T, M, X ) That s, M = α 2 + δt + ɛ M Y = α 3 + βt + γm + ɛ Y Park (Wash U.) Methods Lunch Aprl 9, / 1

18 The potental outcomes approach to medaton analyss 2. Smulatng potental values of the medator and the outcome. Use the estmated parameters from the two regresson models (ˆθ M and ˆθ Y ) to smulate potental values of the medator and the outcome. M (k) Y (k) Y (k) (t X ) from fˆθ(k) (M t, X ) for k = 1, 2,..., K (t, M (k) (t X ) X ) from fˆθ (k) (t, M (k) (t X ) X ) from fˆθ (k) M Y Y (Y t, M (k) (t), X ) for k = 1, 2,..., K (Y t, M (k) (t ), X ) for k = 1, 2,..., K Example: t = 0 or 1 K copes of M (0 X ) and M (1 X ) K copes of Y (0, M (k) (0 X ) X ) and Y (0, M (k) (1 X ) X ) for condton t = 0 K copes of Y (1, M (k) (0 X ) X ) and Y (1, M (k) (1 X ) X ) for condton t = 1 Park (Wash U.) Methods Lunch Aprl 9, / 1

19 The potental outcomes approach to medaton analyss 3. Computng the medaton and drect effects ÂME = nk 1 n K [Y (k) t=0 =1 k=1 (t, M (k) (1 X ) X ) Y (k) (t, M (k) (0 X ) X )] 1 2 [{Y (0, M (1 X ) X ) Y (0, M (0 X ) X )} + {Y (1, M (1 X ) X ) Y (1, M (0 X ) X )}] How much the outcome would change f the medator changes whle the treatment remans constant. Park (Wash U.) Methods Lunch Aprl 9, / 1

20 The potental outcomes approach to medaton analyss 3. Computng the medaton and drect effects ÂDE = nk 1 n K [Y (k) t=0 =1 k=1 (1, M (k) (t X ) X ) Y (k) (0, M (k) (t X ) X )] How much the outcome would change f the treatment changes whle the medator remans constant. ÂTE = 1 nk n K [Y (k) =1 k=1 (1, M (k) (1 X ) X ) Y (k) (0, M (k) (0 X ) X )] The total effect of the treatment on the outcome. Park (Wash U.) Methods Lunch Aprl 9, / 1

21 Emprcal Applcaton Local economc condtons Presdental electons n the U.S.? Hypothess: Voters observe local economc condtons (gas prces, foreclosures, etc) shape subjectve evaluatons of the natonal economy use these evaluatons to cast a presdental vote at the ballot box. Subjectve evaluatons of the economy: the medatng role. M: Evaluatons Medaton effect T: Local economy Y: Vote choce Drect effect Park (Wash U.) Methods Lunch Aprl 9, / 1

22 Emprcal Applcaton How to mplement causal medaton analyss? Use medaton package (STATA, R). Wrte your own code. Contnuous treatment, ordered categorcal medator, bnary outcome wth multlevel data structure. Want to run a fully Bayesan multlevel model. Park (Wash U.) Methods Lunch Aprl 9, / 1

23 Emprcal Applcaton Predcted Effects of Gas Prce on John McCan's Vote Share n ADE AME ATE [Fgures enlarged: The dotted lnes ndcate the predcted ATE and AME due to an $1 ncrease n gas prce] ATE AME Change n Gas Prce Park (Wash U.) Methods Lunch Aprl 9, / 1

24 Emprcal Applcaton Predcted Effects of Foreclosure on John McCan's Vote Share n ATE AME ADE Number of Foreclosures per 1000 Households [Fgures enlarged: The dotted lnes ndcate the predcted ATE and AME due to a 70 unts ncrease n foreclosures per 1000 households] ATE AME Park (Wash U.) Methods Lunch Aprl 9, / 1

25 Senstvty Analyss (1) An objectve local economc condton s ndependent of potental outcomes for vote choce and subjectve evaluatons of the economy gven the observed pretreatment covarates. tem (2) A subjectve economc evaluaton s ndependent of potental outcomes for vote choce gven the observed pretreatment covarates and the observed values for objectve economc condtons. To see how much a volaton of the requred assumpton would affect the estmates If nference s senstve, a slght volaton of the assumpton may lead to substantvely dfferent conclusons (Ima et al. 2011). ρ = Corr(ɛ M, ɛ Y ) Non-zero values of ρ mply departures from (2). Use medaton package to fgure out the value of ρ that makes our estmates equal to zero or nsgnfcant. Park (Wash U.) Methods Lunch Aprl 9, / 1