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H  0޽h ? f̙fy___PPT10Y+D=' = @B +R yq@ (    Z\v Ԕ?F ]QUESTIONS ASKED,   B  N D1?"  H11?"UX^ t Why decompose effects? What are the semantics of direct and indirect effects (in nonlinear and nonparametric models)? What are the policy implications of direct and indirect effects? When can direct and indirect effects be estimated consistently from experimental or nonexperimental data? Can these conditions be verified from accessible causal knowledge, i.e., graphs?B4r5$ VI   PX @`  TD:1?". ,H  0޽h ? f̙fy___PPT10Y+D=' = @B + P (    H|W1?"$ VNF___PPT9(  0Direct (or indirect) effect may be more transportable. Indirect effects may be prevented or controlled. Direct (or indirect) effect may be forbiddenbm0-1m- @`  HP]1?" PWHY DECOMPOSE EFFECTS?,B  N D1?"p(p  TP1?"l :Pill  T;1?" s 5  @ Thrombosis    T`e1?"\ | ? Pregnancy  2   Z o?"   2   Z o?"s 2   Z o?" 6B    ` D>?"/  B    ` D>?"I  B @  `m D>?"; F #  T@j1?"9  Y  9+  To1?" k  L-.  Ttr1?"9 > " Y  9+2  Z o?"Z  2  Z o?" s M 2  Z o?"w  B   ` D>?"  B   ` D>?" A wB @  `m D>?". ; F   Tw1?"- M  >Gender  T{1?"C  <Hiring  T(z1?" /  C QualificationH  0޽h ? f̙fy___PPT10Y+D=' = @B + ` \(    s *,1  {%EFFECT-DECOMPOSITION IN LINEAR MODELS&&,B  N D1?"  Z1?"G"  ;X   Z1?" X  ;Z   Z\1?"h 9   ;Y    Zd1?" n  ;c    Zy?"| ;a    Z01?"W I  <b  B   Z D>?" B   Z D>?"8 hB @ Z D>?"8 h8 X 1  P )   c $A y??X  x wd  Tģ1?" 1  9a  T1?"n 1  Tbc  TD1?"Q  C Definition:  F       c $A r?? x rd  c $A s??"` X x sd  c $A t??"`hR `x tdH  0޽h ? f̙fy___PPT10Y+D=' = @B +&  =5P  (    s *1R o!CAUSAL MODELS AND COUNTERFACTUALS""( "  B  N D1?"{h{V F p  u8  Hȶ1?" ,$D 0 8pDefinition: A causal model is a 3-tuple M = V,U,F9 " @`  N41?",$D 0 (i) V = {V1& ,Vn} endogenous variables, (ii) U = {U1,& ,Um} background variables (unit)TV  N1?"p$D 0LD___PPT9&  F = set of n functions, The sentence:  Y would be y (in unit u), had X been x, denoted Yx(u) = y, is the solution for Y in a mutilated model Mx, with the equations for X replaced by X = x. ( unit-based potential outcome ) 0       $,`3Q8Xx  c $A' ??"`d  x dpH  0޽h ? f̙f___PPT10i.|dž+D=' = @B +  p0L|7(  | | s *]: s%COUNTERFACTUALS: STRUCTURAL SEMANTICS&&( &  w | 0]t Notation: Yx(u) = y Y has the value y in the solution to a mutilated system of equations, where the equation for X is replaced by a constant X=x.  Z&       B | N D1?"hhF  ) | Y   |  BaCDE@F >aM1"%< DOU)/?R _ tUU @     G ; V `Q   |# W W ,$ 0N E l | `Q  Z  | s *E  kN   h  |   lfB  | 6D>  hfB  | 6D>  hfB  | 6D>  hfB | 6D>  hN  l |  l`  l |C  lfB | 6D  h |  BCDE@F D7kxk](V@EX3q @      l |  BCVDEFV @` k |  BCXDE F*(8PXXX@`T  xB |B HZD>:W 1LrB | BD>K G xB | HZD>KM BErB | BD>KK  | <41f)  Wu&    | <1 Yx(u)=yF (     | </]k R WZ&    | < 4]rd   WW&    | <8] 3 OX=x   Z | s *Ef2 | 6Ԕ  ;F o K  | Z xB !|B HZD>  rB "| BD>  xB #| HZD>  rB $| BD>  %| <>]8\  Wu&    &| <A]K YY(    '| <F]= o  WZ&    (| <J]D,   WW&    )| <O] + MX   Z *| s * ]xB +| HZD>+ "  ,| TR]?" H  WProbability of Counterfactuals:   -| TV]?"   nFunctional Bayes Net  .| s *A ??dx d` /| c $A 7??, d  7 L| c $A! ??"` x dH | 0޽h ? f̙fy___PPT10Y+D=' = @B +K0  (    c $A 8??T x 8dAL  s *Xj] 4< TTOTAL, DIRECT, AND INDIRECT EFFECTS HAVE CONTROLLED-BASED SEMANTICS IN LINEAR MODELSfU$$$ $9$B  N D1?"tt  Tc]?"wt 9X  To]?"  9Z  T0t]?"5  9Y  T x]?"F  9c   Tv]?"6W;} 9a   T@~]?"n :b B   T D>?"Q B   T D>?"! B  @ T D>?"!    T]?"6 , b8z = bx + e1 y = ax + cz + e2,*  T]?"   a + bcT  Z]?"-0 t i bc(  Z`]?" 6  L 9aH  0޽h ? f̙fy___PPT10Y+D=' = @B + K0   y (    T@]?" J <<z = f (x, e1) y = g (x, z, e2)  c $A 9??p $ x 9dA  T]?"wt 9X  T`]?"  9Z  T]?"  9YB  T D>?"XQ XB  T D>?"!B  @ T D>?"! -   s *h] LT kCONTROLLED-BASED SEMANTICS NONTRIVIAL IN NONLINEAR MODELS (even when the model is completely specified)0l>$. B   N D1?"   T|]?". g  ,   T]?"  1a cDependent on z?8   T,]?"p G RVoid of operational meaning?H  0޽h ? f̙fy___PPT10Y+D=' = @B +y    0 (    N e?"p  ``The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of different race (age, sex, religion, national origin etc.) and everything else had been the same  [Carson versus Bethlehem Steel Corp. (70 FEP Cases 921, 7th Cir. (1996))] x = male, x = female y = hire, y = not hire z = applicant s qualificationsj 0!8      @`  s *<] M BLEGAL DEFINITION OF DIRECT EFFECT (FORMALIZING DISCRIMINATION),C%& B  N D1?""  T]?"V t   HNO DIRECT EFFECT  0A ?Ԕ? 0 @(x dH  0޽h ? f̙fy___PPT10Y+D=' = @B +AK0 PH (  |  Te?"2 D z = f (x, u) y = g (x, z, u)  Te?"wt  9X  T]?"  9Z  Tde?"%  9YB  T D>?"Q B  T D>?"! B @ T D>?"!     s *`$e  /NATURAL SEMANTICS OF AVERAGE DIRECT EFFECTS00$B   N D1?"44   H2e?"\  pAverage Direct Effect of X on Y: The expected change in Y, when we change X from x0 to x1 and, for each u, we keep Z constant at whatever value it attained before the change. In linear models, DE = Controlled Direct Effectt   <Z   S  :??pp  :Z   S  ;??pp  ;  c $A <??9Jx <d  c $A* ??"`h x x d   NTFe?"Pp ~HRobins and Greenland (1992)   Pure %%H  0޽h ? f̙fy___PPT10Y+D=' = @B +]  ]U` (    HLe?"P k!POLICY IMPLICATIONS (Who cares?)."( (  ZOeo?"; o m  M f ( Ez B! c  ! Bc,$D 0  TUe?"B! A  NGENDER,   TxZe?"Z ! A  S QUALIFICATION*    TL_e?" C c <HIRINGB   T D>?" } B   T D>?" _ % B   T D>?"w . . V B @ T D>?" % B  N D1?"\  H8oe?"G $What is the direct effect of X on Y?%   @`f  Z,fe?"},$ 0 @Is employer guilty of sex-discrimination given data on (X,Y,Z)?>A9  Txe?"! A  ;X  Tie?"! j S A  IZ,  Tze?"C c GY*B @ N DԔ?"8    Te?" xK  QCAN WE IGNORE THIS LINK?  c $A" ??ix dH  0޽h ? f̙f___PPT10.|P/<+kVD' = @B D' = @BA?%,( < +O%,( < +D' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+e +     p  8 (    0m(b p6SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS7 7(B  N D1?"hh   0 m Consider the quantity Given M, P(u), Q is well defined Given u, Zx*(u) is the solution for Z in Mx*, call it z is the solution for Y in Mxz Can Q be estimated from data? Experimental: nest-free expression Nonexperimental: subscript-free expressionB $$ 4$$:  @`  c $A# ?? x d   c $A$ ??( V x d   c $A% ?? !Q9 x dH  0޽h ? f̙f___PPT10i.|\@+D=' = @B + K0   B (  j  Te?" DN z = f (x, u) y = g (x, z, u)  T8e?"wt 9X  Te?"  9Z  Te?"  9YB  T D>?"XQ XB  T D>?"!B @ T D>?"!    s *e  ~(NATURAL SEMANTICS OF INDIRECT EFFECTS))$B   N D1?"44   He?"\  YIndirect Effect of X on Y: The expected change in Y when we keep X constant, say at x0, and let Z change to whatever value it would have attained had X changed to x1. In linear models, IE = TE - DE t  5   `   c $A =?? |  =   c $A >??yFx >d  c $A) ??"`i b y x dAH  0޽h ? f̙fy___PPT10Y+D=' = @B +_K0 WO(    Hde?"P k!POLICY IMPLICATIONS (Who cares?)."( (  Z\eo?"; o m  M f ( Ez B! c  ! Bc,$D 0  Te?"B! A  NGENDER,  TTe?"Z ! A  S QUALIFICATION*   Te?" C c <HIRINGB  T D>?" } B   T D>?" _ % B   T D>?"w . . V B  @ T D>?" % B   N D1?"^   He?"G &What is the indirect effect of X on Y?'   @`G  Ze?"},$ 0 {CThe effect of Gender on Hiring if sex discrimination is eliminated.DD  Te?"! A  ;X  Te?"! j S A  IZ,  T0e?"C c GY*B @ N DԔ?"8    Txe?" _  <IGNOREH  0޽h ? f̙f___PPT10p+D'  = @B D' = @BA?%,( < +O%,( < +D' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+m + p0<(  0 0 Hm?" !QTheorem 5: The total, direct and indirect effects obey The following equality In words, the total effect (on Y) associated with the transition from x* to x is equal to the difference between the direct effect associated with this transition and the indirect effect associated with the reverse transition, from x to x*. R C' g @` 0 Hx#m?"@z o5RELATIONS BETWEEN TOTAL, DIRECT, AND INDIRECT EFFECTS66(B 0 N D1?" 0 c $A ??x dAH 0 0޽h ? f̙fy___PPT10Y+D=' = @B +  F>  (    T*m?"   n6Is identifiable from experimental data and is given by77B  N D1?"  Hnm?"(c 4Theorem: If there exists a set W such that 5  @`   Hgm?"p :EXPERIMENTAL IDENTIFICATION OF AVERAGE DIRECT EFFECTS.;    c $A ?? zx d   T81m?" 2  VThen the average direct effect   c $A& ?? D4 x d   c $A ??x dH  0޽h ? f̙fy___PPT10Y+D=' = @B +g  0 (    s *m p iHOW THE PROOF GOES?(B  N D1?"  s *A L??v:x Ld  s *A x??):9x xd   Tԇm?"> _Proof: <    c $A W??-:] x Wd  c $A ?? Jx d  TLm?"[ 0Each factor is identifiable by experimentation. <1/ T  <v޽h @ ? f̙fy___PPT10Y+D=' = @B +  /(   T   C ,Ar273-fig2S :p   Tm?"   @Example:  B   N D1?"   Nm?"c_ 4Theorem: If there exists a set W such that 5  @`   H m?" LGRAPHICAL CONDITION FOR EXPERIMENTAL IDENTIFICATION OF DIRECT EFFECTS.M 9    c $A o??r x od   c $A y??h0x yd   Tm?"(OR =then,H   0޽h ? f̙fy___PPT10Y+D=' = @B + 0$U(  $9 $ HTm?" YGRAPHICAL CONDITION FOR NONEXPERIMENTAL IDENTIFICATION OF AVERAGE NATURAL DIRECT EFFECTS>Z$T $ C ,Ar273-fig2 jF $ Nܼm?"Phx ~v___PPT9XP ,Identification conditions There exists a W such that (Y Z | W)GXZ There exist additional covariates that render all counterfactual terms identifiable. Z,31#0   3  # @` $ s *A ?? l dx dB $ N D1?" $ c $A q??x qdH $ 0޽h ? f̙fy___PPT10Y+D=' = @B +    @X (  XK X Hm?"1  Corollary 3: The average direct effect in Markovian models is identifiable from nonexperimental data, and it is given by where S stands for any sufficient set of covariates.    T  @` X Hm?" \"IDENTIFICATION IN MARKOVIAN MODELS##,B X N D1?"2 X N o ?"?  N c 2 X N o ?" 2 X N o ?"  J B X T D>?"Z = B X T D>?"R = *  X Tlm?" L@  9X X Tm?" 1 4  9ZB X@ T D>?" - * X T|?"X   Example: S = `  X Tl|?" w  9Y X c $A D??4x Dd X c $A F??8x FdH X 0޽h ? f̙fy___PPT10Y+D=' = @B +& ++T[(  T2 T N Ԕ ?"><2 T N Ԕ ?"x m B T@ T DԔ?"W B T T D?"2 T N Ԕ ?"J2 T N Ԕ ?"}nB T T D?"Y YB  T@ T D?"f l  T T|?"b  7Y  T T|?"4 7Z  T T<|?"<kT\ 7X  T T|?" 4 7W2 T N Ԕ?"3B T@ T DԔ?" )2 T N Ԕ?"c/z . J T J. ,$D  0B T N D?"A  T T |?". J=p Hx** ~z Y T Y,$D  0B T N D?"Y` T T0%|?"  z* = Zx* (u)|    ( T Hx/|?"}nn (Nonidentifiable even in Markovian models))$  @` T H9|?" ^$GENERAL PATH-SPECIFIC EFFECTS (Def.)%%$B T N D1?"(( T c $A ?? x x d  T c $A ??x d2 T N Ԕ ?">A2 T N Ԕ ?"x,$D 0B T@ T DԔ?" ,$D 0B T T D?"22,$D  02  T N Ԕ ?"O,$D 02 !T N Ԕ ?"}sB "T T DԔ?">@,$D 0B #T T D?"  ,$D  0B $T@ T D?",$D 0 %T T@|?"3  7Y &T T0D|?"j 4 7Z 'T TG|?"<\ 7X (T TJ|?"4 7W2 )T TM|?" nU0  6Form a new model, , specific to active subgraph g,75, *T c $A ?? D T x d +T TT|?" n  Definition: g-specific effectL H T 0޽h ? f̙f___PPT10+AcD' = @B DC' = @BA?%,( < +O%,( < +Dz' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-o6Bwipe(up)*<3<*TD' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-s6Bwipe(down)*<3<*T+3 ZR(    Za Ԕ? `SUMMARY OF RESULTS(   B  N D1?"hhn  Hhf?"UD<4___PPT9 Formal semantics of path-specific effects, based on signal blocking, instead of value fixing. Path-analytic techniques extended to nonlinear and nonparametric models. Meaningful (graphical) conditions for estimating direct and indirect effects from experimental and nonexperimental data. Estimability conditions hold in Markovian models. Graphical techniques of inferring effects of policies involving signal blocking.54d0">   Z @`  Tz?"i. ,H  0޽h ? f̙fy___PPT10Y+D=' = @B +C 0 (  ^ S 0P    3 r1 Kp   ; The subject of my lecture this evening is CAUSALITY. It is not an easy topic to speak about, but it is a fun topic to speak about. 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