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Causation without manipulation The Mediation Formula and its ramifications Measurement bias and effect restoration.7N76&       @`  s *# cOUTLINE,00   B  N D1?"H  0޽h ? f̙fy___PPT10Y+D=' v = @B +   0  H(    s *~ &3-LEVEL HIERARCHY OF CAUSAL MODELS,'&$$ '  B  N D1?"     0@y|t___PPT9VN vProbabilistic Knowledge P(y | x) Bayesian networks, graphical models Interventional Knowledge P(y | do(x)) Causal Bayesian Networks (CBN) (Agnostic graphs, manipulation graphs) Counterfactual Knowledge P(Yx = y,Yx =y) Structural equation models, physics, functional graphs,  Treatment assignment mechanism "1 %0 '1 H0 +1 [0 ffffff%ffffffffH  fff$fff$ff$fffff[bEdb @h`H  0޽h ? 33___PPT10i.`[+D=' v = @B +0   0N0 ? 7  $  (  $  $ s * PP *TRADITIONAL STATISTICAL INFERENCE PARADIGM++$B $ N D1?"F  t  $  t  " $ B  fZ G H$I@1Ԕ?"y[ $  H@?"  @Data (2 $  B?"> $  E Inference (2 7 $  H ?"[ Q(P) (Aspects of P)z   $  ZޒԔ?"ftU  ^P Joint Distribution.B  $  T DԔ?"X 00B  $ B T DԔ?" 0f0  $ T,#?" 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THE DIFFERENCES,98   9  B , N D1?"?(? , T#?"G / \Note: P (v) P (v | price = 2) P does not tell us how it ought to change Causal knowledge: what remains invariantRx(*  , N#?"rx <What remains invariant when P changes say, to satisfy P (price=2)=1F"  , @  fZ GXH$I+Ԕ?"8l^  , H #?"8 @Data (2  , B#?" T Y  E Inference (2 h  , H|#?"c; *Q(P) (Aspects of P) " , Z&Ԕ?" V  .P Joint Distribution@B , @ T DԔ?"R B , T DԔ?"g , N&Ԕ?"& G  nP Joint Distribution> , B&?"B $  Bchange (2H , 0޽h ?  , , ,  f̙fy___PPT10Y+D=' v = @B +   0N0   @ 0 B (  0  0 s *l&9 ?FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT),@?   @  B 0 N D1?"p~F ?b 0  ?b 0  T&?"O  CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Exogeneity / Ignorability Mediation,,d    0  T&?" b |4STATISTICAL Regression Association / Independence  Controlling for / Conditioning Odd and risk ratios Collapsibility / Granger causality Propensity score, e 0  T&?"?09D<4___PPT9 +Causal and statistical concepts do not mix.",1, @`7 0 T2&?" H@8___PPT9 U "5 @`W  0 H44&?"  t$D 0@8___PPT9 U "1 @`]  0 NE&?"c ]t$D 0@8___PPT9 U "5 @`H 0 0޽h ? f̙fy___PPT10Y+D=' v = @B +6  0N0 E=P4 (  4 F ? 4  ? 4  TpH&?"w  CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Exogeneity / Ignorability Mediation,,f    4  TP&?"  ~6STATISTICAL Regression Association / Independence  Controlling for / Conditioning Odd and risk ratios Collapsibility / Granger causality Propensity score, e 4  TM&?"?09D<4___PPT9 +Causal and statistical concepts do not mix.",1, @`m 4 Nh&?"f `t$D 0@8___PPT9 e 25 @` 4 Hld&?" _ t$D 0@8___PPT9 [Causal assumptions cannot be expressed in the mathematical language of standard statistics."\1\ @`" 4 s *Pv&9,$D 0 8FROM STATISTICAL TO CAUSAL ANALYSIS: 2. MENTAL BARRIERS,98   9  B  4 N D1?"p,$D 0z e   4  e ,$D 0  4  T(t&?" H@8___PPT9 ^No causes in  no causes out (Cartwright, 1989)205 @`  4  TL&?"Q ^  i1statistical assumptions + data causal assumptions22  4  T&?" / e)  Jcausal conclusions 4  T&?"j C  <  4  Td&?" a  N }077H 4 0޽h ? f̙fy___PPT10Y+D=' v = @B +  0N0 `8 (  8 (z f - 8  f -,$D 0 8  N&?"f -ZRJ___PPT9,$  Non-standard mathematics: Structural equation models (Wright, 1920; Simon, 1960) Counterfactuals (Neyman-Rubin (Yx), Lewis (x Y))5o1U @`T  8 # ~X 8  Zܯ& Co?S"?D 2B 8  T Do?"JF ?b 8  ?b 8  Th&?"w  CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Exogeneity / Ignorability Mediation,,f     8  TH&?" b |4STATISTICAL Regression Association / Independence  Controlling for / Conditioning Odd and risk ratios Collapsibility / Granger causality Propensity score, e  8  T&?"?09D<4___PPT9 +Causal and statistical concepts do not mix.",1, @`  8 H4&?" _ H@8___PPT9 [Causal assumptions cannot be expressed in the mathematical language of standard statistics."\1\ @`  8 s *X&9 8FROM STATISTICAL TO CAUSAL ANALYSIS: 2. MENTAL BARRIERS99  9  B  8 N D1?"pF e  8  e  8  T&?" H@8___PPT9 ^No causes in  no causes out (Cartwright, 1989)205 @` 8  Td&?"Q ^  i1statistical assumptions + data causal assumptions22 8  T&?" / e)  Jcausal conclusions 8  T&?"j C  <  8  T&?" a  N }077Z 8  S  \??pp  \H 8 0޽h ? f̙f___PPT10+ևiuD' = @B DW' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8 +  0 C(    s *& nBAYESIAN NETWORKS,,,   B  N D1?"` `  0(`8B gDefinition P is Markov relative to G iff: Algebraic Representation Graphical Representation *0 0 0 0 0    %@ @h`  c $A A&??"`TPdx A&dA  c $A B&??"`0 \  x B&dc  c $A C&??"`x x C&d! F  Y3!   0 {q `B   0DԔ P`B  B 0DԔPn2   0"`^!    <4 (C"?7 Ic  bViD0   n2   0"`  en2  0"` 3e`B  0DԔ  0`B  0DԔ8(n2  0"`^! n2  0"`  SBN v S  f {Sn2  0"`v  Sn2  0"`FSf2  6Ԕ `B B 0DԔ0n2  0"`^ Q`B  0DԔ  0`B  0DԔ  8`B B 0DԔ   <!(C"? Y  }PAiD0     H((?"0 8V  ?G"0(2H  0޽h ? 33___PPT10i..+D=' v = @B +  0  (  /  H0*(?"xld\___PPT9>6 5 KDefinition Let Px(v) stand for P(v | do(X=x)), X and x arbitrary G is a CBN relative to Px* iff: Px(v) is Markov relative to G; Px(vi)=1 for all if vi agrees with X=x. for all whenever pai agrees with X=x, i.e., each P(vi | pai) remains INVARIANT to interventions not involving Vi. Algebraic representation for all v consistent with x. (Truncated product, G-computation, manipulation theorem) Graphical representation Surgery: remove incoming arrows to X.b1b00 (   ((((( /( ((   (  7              $      $                   0                |               @`   s *( GCAUSAL BAYESIAN NETWORK (CBN) (Agnostic graphs, manipulation graphs)<H!"& H  B  N D1?"x x  c $A D&??"` x D&d  c $A E&??"` x E&d  c $A F&??"`X x F&d  c $A G&??"` l x G&dH  0޽h ? f̙fy___PPT10Y+D=' v = @B +  0    t (   "  @  fZ GH I+Ԕ?"<l^B  N D1?"  H(?"< @Data (2  BL(?" T ]  E Inference (2 /  HН(?"g; Q(M) (Aspects of M)z   Z(Ԕ?" ] PData Generating ModelB  @ T DԔ?"R   Ht(?" j6  DM  Invariant strategy (mechanism, recipe, law, protocol) by which Nature assigns values to variables in the analysis. :wwB  T DԔ?"g  N(Ԕ?"& J o Joint Distribution >B  T DԔ?"   s *x( @[ sTHE STRUCTURAL MODEL PARADIGM$  Z(?"  CM&<  T(?"Ej,$ 0 v> Think Nature, not experiment!     H(?"*J =   2H  0޽h ?    f̙fYQ___PPT101.iK+TsD' v = @B DP' = @BA?%,( < +O%,( < +D' =%(D/' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* %(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +p+0+ 0 ++0+ 0 +  0N0 c [ D  (  D  D s *d(  nSTRUCTURAL CAUSAL MODELS(B D N D1?"" D N?"n vDefinition: A structural causal model is a 4-tuple V,U, F, P(u), where V = {V1,...,Vn} are endogenous variables U = {U1,...,Um} are background variables F = {f1,..., fn} are functions determining V, vi = fi(v, u) P(u) is a distribution over U P(u) and F induce a distribution P(v) over observable variablesI54!5@ Il { tb  D  {tb ,$D 0 D  s *AV ??"` * tb x d D  Zp*?"{ E  =e.g.,H D 0޽h ? f̙f5-___PPT10 +5yD' v = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*D %(D' =-o6Bdissolve*<3<*D +  0 [S 6 (     s *(0 o!CAUSAL MODELS AND COUNTERFACTUALS""$ "  B  N D1?"chc~2  H 33?"![6  H`V?"#] &Definition: The sentence:  Y would be y (in unit u), had X been x, denoted Yx(u) = y, means: The solution for Y in a mutilated model Mx, (i.e., the equations for X replaced by X = x) with input U=u, is equal to y.B         @` l   6 g W,$D 0@  q  5     T9?"BX  =U 0 !  T0>?"]   }X (u)\0 "  TD?"]  }Y (u)\0 #  TL?"6* eq  =M 0B $  T DԔ?"  B % B T DԔ?"  B &  T DԔ?"   -  Z|P?"   =U 0 .  Z0?" h  _X = x>08 /  ZA?"   YX (u)v0  0  Z0?" \   jMx20B 1  Z DԔ?"E   B 3  Z DԔ?"E  H  0޽h ? f̙fE=___PPT10.WV['P+pU[D' v = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*6 %(D' =-o6Bdissolve*<3<*6 +  0   |U (  | | s *0 o!CAUSAL MODELS AND COUNTERFACTUALS""$ "  B | N D1?"chc~2 | H 33?"![6 | H䒓?"#] &Definition: The sentence:  Y would be y (in unit u), had X been x, denoted Yx(u) = y, means: The solution for Y in a mutilated model Mx, (i.e., the equations for X replaced by X = x) with input U=u, is equal to y.B         @`z +9 . |  beM,$D 0  | 0A, &?Ԕ?"`G x &d  | ZX?"+9 .Y  f,The Fundamental Equation of Counterfactuals:-0-H | 0޽h ? MMM̙e]___PPT10=.pA+{ 9D' v = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =-o6Bdissolve*<3<*|+  0 ]U )` (   (  ` ~l ++ : )` + +:,$D 0 @ ++ : (` ++ : `  s *A0 `??  :x `d `   `Ȟ? "++ K  HIn particular:0 `  s *A E?? z  x EdO ` s *0 o!CAUSAL MODELS AND COUNTERFACTUALS""$ "   ` H ?"#] 0Definition: The sentence:  Y would be y (in situation u), had X been x, denoted Yx(u) = y, means: The solution for Y in a mutilated model Mx, (i.e., the equations for X replaced by X = x) with input U=u, is equal to y.B        @`B ` N D1?"chc~2 ` H 33?"![6^l 'V0  &` V'0 ,$D 0 `   `(? "'Vv  =  1  `  s *A# f?? "` 0 x fd  `   `ԓ? "+\|  a'Joint probabilities of counterfactuals:(0(H ` 0޽h ? f̙fPH___PPT10(.Q9ЅO+D' v = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*&` %(D' =-o6Bdissolve*<3<*&` D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*)` %(D' =-o6Bdissolve*<3<*)` +J    0 a Y P   (      s *\ݓ,$D 0 #TWO PARADIGMS FOR CAUSAL INFERENCE,$#$$ $  B  N D1?"'p',$D 0.  T?"KLK $Observed: P(X, Y, Z,...) Conclusions needed: P(Yx=y), P(Xy=x | Z=z)... How do we connect observables, X,Y,Z,& to counterfactuals Yx, Xz, Zy,& ?   P/L\  TX$?"M   NN-R model Counterfactuals are primitives, new variables Super-distribution `O.  T+?"M J \" lStructural model Counterfactuals are derived quantities Subscripts modify the model and distribution lm(.B  N DjJ?" 2  0 NA= ? ?d"`T   d  0 NA> ? ?d"` >   dH  0޽h ? f̙f___PPT10i.p+D=' v = @B +  0  @l(  l l s *7yg v(ARE THE TWO PARADIGMS EQUIVALENT? ))$ )  B l N D1?"_ l s *<.$o Yes (Galles and Pearl, 1998; Halpern 1998) In the N-R paradigm, Yx is defined by consistency: In SCM, consistency is a theorem. Moreover, a theorem in one approach is a theorem in the other.r4 Z               % l c $A =%??"` ) x =%d~H l 0޽h ? MMM̙___PPT10i.`bt+D=' v = @B +  0  ,(  , , HR?".x ADefine: Assume: Identify: Estimate: Test: ,B   , s *8aA/ ~0THE FIVE NECESSARY STEPS OF CAUSAL ANALYSIS11$ 1  B , N D1?" , Hl?". C9Express the target quantity Q as a function Q(M) that can be computed from any model M. Formulate causal assumptions A using some formal language. Determine if Q is identifiable given A. Estimate Q if it is identifiable; approximate it, if it is not. Test the testable implications of A (if any). :%  7" H , 0޽h ? f̙f___PPT10i.hZwm+D=' v = @B +   0   0 (     s *XA/ 4THE FIVE NECESSARY STEPS FOR EFFECT ESTIMATION.5"$$ 5  B  N D1?"   c $AP ??"`b  $D 0dA  H?".x ADefine: Assume: Identify: Estimate: Test: ,B    H?". C9Express the target quantity Q as a function Q(M) that can be computed from any model M. Formulate causal assumptions A using some formal language. Determine if Q is identifiable given A. Estimate Q if it is identifiable; approximate it, if it is not. Test the testable implications of A (if any). :%  7" H  0޽h ? f̙fe]___PPT10=.Wpr+&V"D' v = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +  0  (  B  N D1?"``  s *ĸH6 )FORMULATING ASSUMPTIONS THREE LANGUAGES@*$$$ *    T?"L,$ 0 =1. English: Smoking (X), Cancer (Y), Tar (Z), Genotypes (U)>  8  Tʖ?" L# ,$ 0 r<Not too friendly: consistent? complete? redundant? arguable?==z \ +~   \ L~,$D 07N 7! ~    7+~b   3 r GuHI5#p?"7 8   T8Ж?" 7 ~ 9ZT  7~  # 7! ~  TӖ?" 7 ~ 9X  T<ؖ?""7~ 9YB  T D>?" B  T D>?" T   Zܖ?"\ | S3. Structural:(jl L  L ,$D 0  0A z ? ?"`- x z d  Z0?"LA X2. Counterfactuals:(H  0޽h ?   f̙f  ___PPT10h .+D ' v = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +p+0+0 ++0+0 +   0   ( (  ( ( H?".x L4Define: Assume: Identify: Estimate: 5   ( s *A =IDENTIFYING CAUSAL EFFECTS IN POTENTIAL-OUTCOME FRAMEWORK>>" >  B ( N D1?" ( HP?". ,Express the target quantity Q as a counterfactual formula, e.g., E(Y(1)  Y(0)) Formulate causal assumptions using the distribution: Determine if Q is identifiable using P* and Y=x Y (1) + (1  x) Y (0). Estimate Q if it is identifiable; approximate it, if it is not.6   7p |     G ( c $A ?%??"` v x ?%dAH ( 0޽h ? f̙f___PPT10i.TC)+D=' v = @B +p   0   @  (     s *$n JGRAPHICAL  COUNTERFACTUALS SYMBIOSIS&&$ &  B  N D1?"h  T)?"IU @Every causal graph expresses counterfactuals assumptions, e.g., X Y ZzKA @`y  HLC?" ] yconsistent, and readable from the graph. Express assumption in graphs Derive estimands by graphical or algebraic methods**PzN # %   s *A ? ?"`uBx d  ZF?"/ D<4___PPT9 ,Missing arrows Y Z` @`  ZlI?"( ?o  n2. Missing arcs, @`  TJ?"1 f W x  9Y  Tb?"1 n _ x  9Zb  3 r GϸH4I@p?"i j    s *A ? ?"`J  x dH  0޽h ?   f̙f___PPT10i.zOi+D=' v = @B +  0 W (   B  N D1?"00 = Th?"5p QIDENTIFICATION IN SCM(3 ? Tq?"o {Find the effect of X on Y, P(y|do(x)), given the causal assumptions shown in G, where Z1,..., Zk are auxiliary variables.|&,<L @0 0  @ # [  K B A B Z Do?"@ B B B Z Do?"@ ` B C  Z Do?"@0 B D B Z Do?"`@0 B E B Z Do?"@ 0 B F  Z Do?"0  B G  Z Do?"  B H Z Do?"  B I B Z Do?"@0 P B J B Z Do?"  z K  <  ?" 0  L H ?" B LZ6.  M H ?" +B  LZ3.  N H ?"N b  LZ2.  O H, ?" N 2  LZ5.  P H螠 ?"7   LZ1.  Q H0 ?"  9X  R HЖ ?"  B 9Y  S H ?" + 2  LZ4. F \ T  T p S U  T?"}3 S G 4 V  N ?"\ W T|?" gCCan P(y|do(x)) be estimated if only a subset, Z, can be measured? D8H  0޽h ? f̙f___PPT10i.)+D=' v = @B +>/  0N0 ((7:T "((  T  T HŠ?"8& r8ELIMINATING CONFOUNDING BIAS THE BACK-DOOR CRITERION99$ T T͠?"h 'c P(y | do(x)) is estimable if there is a set Z of variables such that Z d-separates X from Y in Gx.d  B T N D1?"8h8L @0 0  T # 0 B T B Z Do?"@ B T B Z Do?"@ ` B T  Z Do?"@0 B  T B Z Do?"`@0 B  T B Z Do?"@ 0 B  T  Z Do?"0  B  T  Z Do?"  B  T Z Do?"  B T B Z Do?"@0 P B T B Z Do?"  z T  <  ?" 0  T Hp ?" %D  HZ6*   T H| ?" Ut  HZ3*   T H\ ?"  HZ2*   T H ?"   HZ5*   T HH ?"a HZ1*   T H ?"    7X  T H ?" uT  7Y  T H ?" Ut  HZ4*  B T @ Z Do?"  B T @ Z Do?"  B T Z Do?" p B T @ Z Do?" p B T @ Z Do?" p B T Z Do?" p B T Z Do?"  B  T @ Z Do?" p B !T @ Z Do?"N  x "T B  ?"   #T H ?" e  HZ6*   $T H ?"   HZ3*   %T H ?"  HZ2*   &T H$ ?"   HZ5*   'T H" ?" HZ1*   (T H' ?" ` I  7X  )T B8,?"   7Y  *T H1 ?"8 X  HZ4*  2 +T N o?"8 @  ,T H5 ?"8pOX  GZ*  B -T @ N Do?"(  F h 1T  X 2T  H:?"? DGx&  3T  N ?"hF \ 4T  X 5T  Tx??"}3 S G 4 6T  N ?"\ 7T HpE?" &, =   26l 3 v :T  3v,$D 0 /T  HH ?"3?h BfMoreover, ( adjusting for Z) Ignorability4  '  9T  c $A '??"` vAx 'dAT T <3v޽h @ ? f̙f$___PPT10+M4~D' v = @B Dc' = @BA?%,( < +O%,( < +D' =%(%(D"' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*7T %(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*:T %(D' =-o6Bdissolve*<3<*:T +8+0+7T 0 +:.  0 (  L  C $AS a7004A  N0ZԔ?"+ c,$D  0 D Front Door (2   s *_A/ =EFFECT OF WARM-UP ON INJURY (After Shrier & Platt, 2008)>>$&(      B  N D1?"  Te?" ) ,2 @ Z Ԕ?" F ,$D  0  B@h?"$ . 2z h% hW    % kkW ,$D  0   Z?"` h z l    ( ,$D  02   T Ԕ?"lz   NpԔ?"J D Watch out! (2 B  T D>?"I  Bu?"zG iWarm-up Exercises (X)8 2  Bxz?"{N n Injury (Y)H  2  NTԔ?"% ],$D   0 =???(22  Z 0 Ԕ?"~ VG,$@   0B  T D>?"[@ 7 ,$@  0H  0޽h ? f̙f___PPT10..jŰ~+'HWD' v = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(Dc' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* Dq' =%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(Da' =4@BBBB%()))D' =-o6Bwipe(up)*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(Da' =4@BBBB%()))D' =-o6Bwipe(up)*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D^' =%(D' =%(DR' =A@BBB B0B%(D' =-6B'blinds(horizontal)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*++0+0 ++0+0 ++0+0 +Z&  0 ..(  XB  0DԔ999 f2  0"`Y\XB  0DԔw? f2  0"`< [ XB @ 0DԔ f2  0"`  ? f2  0"`Y XB  @ 0DԔ9 f2  @ 0"`YAXB  @ 0DԔw  f2  @ 0"`< B f2  @ 0"` G ? XB  0DԔ S f2 @ 0"` G . f2  0"`  . XB  0DԔ] ?] XB @ 0DԔ EE XB  0DԔ E XB  0DԔ Y   <KB PW120    <8K   PW420    <Ģ9 | OX20    < 9  OY20    <P 7 @  PV220    <İ o @  PV120    <4 V `  PW220    <P P `  PW320   z P   hP ,$D  0f2  6ԔPP   < ?Z"0 ZB   s *DԔ@ z @   ! `@  ,$D  0f2 " 6Ԕ@ P  #B < Ť  ?T"0 ZB $B s *DԔ [ x % s *ɤ`3 HCONFOUNDING EQUIVALENCE WHEN TWO MEASUREMENTS ARE EQUALLY VALUABLE,IH   I  B & N D1?"P Pz 8Y ' 8p,$D  0n2 ( 0"` 4 y ) <Ϥ Y  OL20  fB *B 6DԔ8X fB + 6DԔ h5z =  , = ,$D 0 - <8դh =  | Z ~ TV0      . <ۤw  ??"0 H  0޽h ? 33  ___PPT10 .{+mD ' v = @B Dm ' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bwipe(up)*<3<*!D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-o6Bdissolve*<3<*,D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-o6Bwipe(up)*<3<*'+   0 L(    s *`3 HCONFOUNDING EQUIVALENCE WHEN TWO MEASUREMENTS ARE EQUALLY VALUABLE,IH   I  B  N D1?"P P  0 \TL___PPT9.& TDefinition: T and Z are c-equivalent if Definition (Markov boundary): Markov boundary Sm of S (relative to X) is the minimal subset of S that d-separates X from all other members of S. Theorem (Pearl and Paz, 2009) Z and T are c-equivalent iff Zm=Tm, or Z and T are admissible (i.e., satisfy the back-door condition) )0 0 0 J1 0      1>XV@h`  c $A$  '??"`x  'dH  0޽h ? 33___PPT10i.@+D=' v = @B +  0  ..(  XB  0DԔ999 f2  0"`Y\XB  0DԔw? f2  0"`< [ XB @ 0DԔ f2  0"`  ? f2  0"`Y XB  @ 0DԔ9 f2  @ 0"`YAXB  @ 0DԔw  f2  @ 0"`< B f2  @ 0"` G ? XB  0DԔ S f2 @ 0"` G . f2  0"`  . XB  0DԔ] ?] XB @ 0DԔ EE XB  0DԔ E XB  0DԔ Y   <)KB PW120    <h.K   PW420    <39 | OX20    <9  OY20    < C @  RV240 33 33  <4 o @  RV140    <T V `  RW240    < P `  PW320  F P   hP f2  6ԔPP   <P ?Z"0 ZB   s *DԔ@F @   ! `@  f2 " 6Ԕ@ P  #B <T  ?T"0 ZB $B s *DԔ [ x % s *Y`3 HCONFOUNDING EQUIVALENCE WHEN TWO MEASUREMENTS ARE EQUALLY VALUABLE,IH   I  B & N D1?"P P ' <,_h =  | Z ~ TV0     _z 88x  ( 88x ,$D  0sN 8Y ) 8pn2 * 0"` 4 y + <Xf Y  OL20  fB ,B 6DԔ8X fB - 6DԔ hZB .B s *Do 8x H  0޽h ? 33E=___PPT10.{+e>8D' v = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bwipe(up)*<3<*(+ '  0 xp ++(  XB  0DԔ99 f2  0"`\XB  0DԔ? f2  0"` [ XB @ 0DԔ u f2  0"`r  f2  0"` XB  @ 0DԔ f2  @ 0"`AXB  @ 0DԔ  f2  @ 0"` B f2  @ 0"`r G XB  0DԔ S u f2 @ 0"`a G f2  0"`a  XB  0DԔ ? XB @ 0DԔ E XB  0DԔ  XB  0DԔ Y   <tsB  PW120    <Lx   PW420    <h} |  OX20    <   OY20    <q 7  PV220    <q o  PV120    <ؑ V  PW220    < P  PW320    s *`3 HCONFOUNDING EQUIVALENCE WHEN TWO MEASUREMENTS ARE EQUALLY VALUABLE,IH   I  B  N D1?"P Pz =   = ,$D 0N 0=    h =  ! <40=  | Z ~ TV0     ZB "B s *Do8@  # <詫  6"0 z    $   ,$D  0 % <ܬC"? ?Z"0 ZB & s *DԔ  '  BHCDELFԔSS5_h= [ c93N!]5L$s?1HH-M <.fu?WW6{V 6I 'NHnOh !!KXd}tx@             `S" d  z   (  ,$D  0 )B <౫C"?  ?T"0 ZB *B s *DԔc   +B  BHCDELFԔSS5_h= [ c93N!]5L$s?1HH-M <.fu?WW6{V 6I 'NHnOh !!KXd}tx@             `S" aL H  0޽h ? 33; 3 ___PPT10 . +!?D ' v = @B D ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bwipe(up)*<3<*$D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bwipe(up)*<3<*(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+$  0 0''X(  XB  0DԔ99 f2  0"`\.XB  0DԔ? f2  0"` [ XB @ 0DԔ f2  0"`  f2  0"` .XB  @ 0DԔ f2  @ 0"`A.XB  @ 0DԔ  f2  @ 0"` B f2  @ 0"` G XB  0DԔ S f2 @ 0"`y G f2  0"`y  XB  0DԔ ? XB @ 0DԔ E XB  0DԔ  XB  0DԔ Y   <B" PW120    <ū  " PW420    <ʫ | OX20    <ϫ  OY20    <ԫ -  PV220    <٫ o  PV120    < ߫ V  PW220    <( P  PW320    <D = ,$ 0 | Z ~ TV0       s *t`3 HCONFOUNDING EQUIVALENCE WHEN TWO MEASUREMENTS ARE EQUALLY VALUABLE,IH   I  B  N D1?"P PCz L      ,$D  0 ! <( G ?Z"0 ZB " s *DԔ  #  FBClDE4FԔMMP,K-E.@0<14444ppDx[ 8`<nPP,@[ll\\xK;(4`8Y{LLp'X 4lT4 Y|LpDT@T@,Uxh P,PT@                   `S" L cz 8  $  ,$D  0 %B <C"? G ?T"0 ZB &B s *DԔc   'B  FBClDE4FԔMMP,K-E.@0<14444ppDx[ 8`<nPP,@[ll\\xK;(4`8Y{LLp'X 4lT4 Y|LpDT@T@,Uxh P,PT@                   `S" 8 H  0޽h ? 33 x ___PPT10X .س+TڣD ' v = @B D ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bwipe(up)*<3<*$D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+0 +  0 @f(    s *`M /BIAS AMPLIFICATION BY INSTRUMENTAL VARIABLES<0(   0  B  N D1?" XB  0DԔ!!1 f2  0"`QDf2  0"`4 [y f2  0"`Q f2 @ 0"`4 By XB   0DԔ ;- f2   0"` . XB   0DԔU ?U XB  @ 0DԔ E =    <(C* PW220    <C   PW120    <1 |x  OX20    <81 x  OY20  XB  0DԔ =   <#! G ] h  OU20  XB @ 0DԔy5 - 2  6ԔP @ 8,$D  02  6ԔH ( ,$D  0&L Ha # Hah  <)Ha W1 ~ {W1, W2}0  $    $ $$ZB B s *Do$x   <4( ,$ 0 lAdding W2 to Propensity Score increases bias (if such exists) (Wooldridge, 2009) In linear systems  always In non-linear systems  almost always (Pearl, 2010) Outcome predictors are safer than treatment predictors>1 0 1  5 H  0޽h ? 33 x ___PPT10X .PI0+VuCD ' v = @B D ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+0 +)  0 PHX(  X  X ZO Ԕ?* 3EFFECT DECOMPOSITION (direct vs. indirect effects)44( 4  B X N D1?"C X HDS?"q:D<4___PPT9 qWhy decompose effects? What is the definition of direct and indirect effects? What are the policy implications of direct and indirect effects? When can direct and indirect effect be estimated consistently from experimental and nonexperimental data?"5 @` X T(b?"i. ,H X 0޽h ? f̙fy___PPT10Y+D=' v = @B +8  0 OG\(  \ \ Ze Ԕ?L fWHY DECOMPOSE EFFECTS?(   B \ N D1?"# \ H4k?"PH@___PPT9" ETo understand how Nature works To comply with legal requirements To predict the effects of new type of interventions: Signal routing, rather than variable fixingnz5,4z @`H \ 0޽h ? f̙f___PPT10i.!Wh+D=' v = @B +L  0N0 `(  ` ` T,}?"G'$ 9X ` TQ?"Gw h  9Z ` T?"_ H 9  9YB `  ` Dp?" B `  ` Dp?"Y B `@  ` Dp?"Y  ` s *莮 $ |&LEGAL IMPLICATIONS OF DIRECT EFFECT''$B  ` N D1?"u  ` H?"T GK& 9What is the direct effect of X on Y ? (averaged over z):Z  ` S   ??pp   Z  ` S   ??pp     ` c $A  ??"` Yx  d ` T؛?"]h } F(Qualifications) ` T@?" .  >(Hiring)   ` T?"] } >(Gender)   ` H|?"G' q;Can data prove an employer guilty of hiring discrimination?<<\l 18 `18,$D  0 `  `?"18 jAdjust for Z? No! No!8  R ` s  GJH%zIJԔ?"9   H ` 0޽h ? ` f̙f5-___PPT10 +D' v = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =-o6Bwipe(up)*<3<*`+L  0N0 [Sd(  d| d T?"2 D z = f (x, u) y = g (x, z, u) d T?"wt  9X d T,Į?"  9Z d TXǮ?"%  9YB d  ` Dp?"Q B d  ` Dp?"! B d@  ` Dp?"!    d s *ˮ  4NATURAL INTERPRETATION OF AVERAGE DIRECT EFFECTS55$B  d N D1?"44  d Hۮ?"\  pNatural 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  d S   ??pp   Z  d S   ??pp    d c $A  ??9Jx  d d c $A  ??"`h x x  d  d NX?"Pp ~HRobins and Greenland (1992)   Pure %% d c $AZ  ??"`ex  dH d 0޽h ? f̙fy___PPT10Y+D=' v = @B +   0N0   0l^ (  lj l T?" DN z = f (x, u) y = g (x, z, u) l Tl?"wt 9X l T?"  9Z l T ?"  9YB l  ` Dp?"XQ XB l  ` Dp?"!B l@  ` Dp?"!   l s *   v DEFINITION OF INDIRECT EFFECTS!!$B  l N D1?"44  l H?"\  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   `  l c $A  ?? |     l c $A  ??yFx  d l c $A  ??"`i b y x  dAH l 0޽h ? f̙fy___PPT10Y+D=' v = @B +  0N0 @p(  p p H2?"P f,POLICY IMPLICATIONS OF INDIRECT EFFECTS--( p Z7o?"; o m  M f ( Ez B! c p ! Bc,$D 0 p T;?"B! A  NGENDER, p T@?"Z ! A  S QUALIFICATION*  p TE?" C c <HIRINGB p T D>?" } B  p T D>?" _ % B  p T D>?"w . . V B  p@ T D>?" % B  p N D1?"^  p HV?"G &What is the indirect effect of X on Y?'   @`G p ZlM?"},$ 0 {CThe effect of Gender on Hiring if sex discrimination is eliminated.DD p TQ?"! A  ;X p T`?"! j S A  IZ, p Te?"C c GY*B p@ N DԔ?"8   p Tj?" _  <IGNORE4 p H`d?"B `Deactivating a link  a new type of intervention11 @`H p 0޽h ? f̙f___PPT10p+D' o= @B D' = @BA?%,( < +O%,( < +D' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-o6Bdissolve*<3<*p+8+0+p0 +  0 p(  ,  H|?" D<4___PPT9 ZThe natural direct and indirect effects are identifiable in Markovian models (no confounding), And are given by: Applicable to linear and non-linear models, continuous and discrete variables, regardless of distributional form.<  @`B  N D1?";;  H<?" LMEDIATION FORMULAS(  c $A &??"`- x &dH  0޽h ? f̙f___PPT10i.++D=' v = @B +   0 ` p#(  p p s *  |.MEDIATION FORMULAS IN UNCONFOUNDED MODELS //$ /  B p N D1?"QhQ8 J)   py, p TL?"JTG  9X p Tt?")p 9Z p T?" T  9YB p Z Dp?"Q B  p Z Dp?"!XB  p  f D>?"!X   p c $A A'??"`> x A'd p c $A C%??"`Yx C%dH p 0޽h ? MMM̙___PPT10i.`bt+D=' v = @B +J  0 IIp`uI(  t t s *`o q#COMPUTING THE MEDIATION FORMULA$$$ $  B t N D1?"QhQ t Zh?"bmV 7X t Z?"  7Z  t Z<?"b 7YB  t Z Dp?"B  t Z Dp?"bbB  t  f D>?"wb  t c $A &??"`\ x &d; L  u #"6* L  su NԾ?"_ A L  D @` ou Ndz?"_ LA  D @` mu NԳ?"_ L  E(Z|x)=hx $ @` `u N|ϳ?" _  D @` \u N³?"A _   D @` Xu N8?"_ A  D @` Tu N?"_  D @` Ru N|?"_   E(Y|x,z)=gxz $ @` 3t Nx?"\   M1 @` 2t N?" \  M1 @` 1t Np?"   M1 @` 0t Nd(?"   dn80 @` /t N'?"\   M0 @` .t N@?" \  M1 @` -t NJ?"   M1 @` ,t NS?"   dn70 @` +t N]?"\`   M1 @` *t Ne?"` \  M0 @` )t Nn?"`   M1 @` (t Ndx?"`   dn60 @` 't NPr?"\A `  M0 @` &t N?"A \`  M0 @` %t Nȓ?"A `  M1 @` $t N?"A `  dn50 @` #t Nh?"\"A  M1 @` "t N?""\A  M1 @` !t N?""A  M0 @`  t N(û?""A  dn40 @` t N?"\" M0 @` t N|ջ?"\" M1 @` t N޻?"" M0 @` t N?"" dn30 @` t N,?"\ M1 @` t NH?"\ M0 @` t N?" M0 @` t N ?" dn20 @` t N?"\ M0 @` t N!?"\ M0 @` t NT?" M0 @` t NT4?" dn10 @` t NL=?"\ QY @` t NHF?"\ QZ @` t NO?" QX @` t NX?" D @`B 4t B 1?"B =t B 1?"  B >t B 1?"B Bt B 1?"LLB t B 1 ?" B t B 1 ?" B t B 1 ?"\\ B t B 1 ?"LB t B 1 ?"B t B 1 ?"_ B t B 1 ?"""B t B 1 ?"A LA B t B 1 ?"` ` B t B 1 ?" _  B t B 1 ?"  B t B 1?"LLA B t B 1?"LA L B t B 1?"B t B 1?"B t B 1?""B t B 1?""A B t B 1?"A ` B t B 1?"`  B t B 1?"  B t B 1?"  B t B 1?"B t B 1?"\B t B 1?"\LB t B 1?"  B t B 1?" \ B t B 1?"\ L B Su B 1 ?" B nu B 1 ?"_ _  u c $A J%??"`  x J%dAi u c $A R%??"`7   x R%dAi u c $A P%??"`j  O x P%dAi u c $A U%??"`  x U%dAi u c $A [%??"` hx [%dAi u c $A a%??"` [~ x a%dAi~B u H D?" B u N DԔ?"I @I H t 0޽h ? MMM̙___PPT10i.! 7F+D=' v = @B +S  0 jb`(    s *` (RAMIFICATION OF THE MEDIATION FORMULA,)((  )  B  N D1?" P  <,ES 8DE should be averaged over mediator levels, IE should NOT be averaged over exposure levels. TE-DE need not equal IE TE-DE = proportion for whom mediation is necessary IE = proportion for whom mediation is sufficient TE-DE informs interventions on indirect pathways IE informs intervention on direct pathways.~-1 10 1 e0 21 ,0 +/$ $ -  *   hhH  0޽h ? 33___PPT10i.0%+D=' v = @B +    0 x:P(   $ Tئ?" * ;W0  s *Ъn x*MEASUREMENT BIAS AND EFFECT RESTORATION++$ +  B  N D1?"h2  T ?"2  T ?"FB  Z DԔ?"@2  T ?"6B @ Z DԔ?"02 @ T ?"6B   T DԔ?"hh ! Tp?"(~ U Unobserved Z,0  " T`?" *  ;X0 # T䏼?" *  ;Y0B , T DԔ?"0\  W 7#  W,$@  0 (  0e0e    BCDEpF0A Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||xHx/PFEzXk1@Oa@8Txg,zx9@         "< hW4 ) T?"^ 2 P(w|z)^0 5 Nܽ?"w ,$ 0 nP(y | do(x)) is identifiable from measurement of W, if P(w | z) is given (Selen, 1986; Greenland & Lash, 2008),o0 2%0H- l  :,$D 0@ b 1 4 ( *2 ZA- &? ?d"`b  & d. 12  `A& &? ?d"`Pq1 & d 9 T,м?"   N(local independence)0H  0޽h ? MMM̙  ___PPT10 .+]f%D$ ' v = @B D' = @BA?%,( < +O%,( < +D+' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-o6Bwipe(up)*<3<*7D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-o6Bdissolve*<3<*5D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-o6Bdissolve*<3<*:+8+0+50 +/*   0 %%,T"%(  B ; T D>?"P, , 0 B < T D>?", , (  Z A!?"(    s *PZ  v(EFFECT RESTORATION IN BINARY MODELS))  )  B  N D1?"h   0 ZA' d&? ?d"`  d& dB  Z DԔ?"8 B  T DԔ?"4 <4 B  N DԔ?" B  T DԔ?" B  N D>?" 8 B  T DԔ?" 0   Z A!?"@  0  , s *A( l&? ?"` i x l&d - T?"Zz 910 . T,?" j  910 0 c $A% m&??"`\  x m&dAg 1 c $A) p&??"`\ m  x p&dAgH 3  0e0e    B9C_DE@F A Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||9-!adiA /> W _ @     "< W ( N 4@  0e0e    B9C_DE@F  Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||9-!adiA /> W _ @     "<  (  5 T?"OI  E undefined 0  6 T ?"OAI  E undefined 0 8    9 X ( 2 ZA" ^&? ?d"`   ^& d2 7 N ?"V  >l w  :G ,$D 0.  2  `A+ v&? ?d"`w  v& d2 8 N ?">  B A T<?"'  To cell (x,y,Z = 0)^0  @ B T?"@  To cell (x,y,Z = 1)\0 > D T?"$ *Weight distribution from cell (x,y,W = 1)D+0  F T(?"" ;W02 G N ?"d2 H N ?">dB I T DԔ?"82 J N ?".BxB K@ T DԔ?"h(2 L@ N ?".xB M T DԔ?"`p` N TH ?"X7 ;Z0 O TR ?"N7 ;X0 P T\ ?" ;Y0B Q T DԔ?"(H  0޽h ? MMM̙E=___PPT10..+MD' v = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-o6Bdissolve*<3<*:+,  0 (% %` 3=P$(    s *c t&EFFECT RESTORATION IN LINEAR MODELS''$ '  B  N D1?" h   Zh?"Zz ;W02  T ?"P2  T ?"B   Z DԔ?"(2   T ?"B  @ Z DԔ?" 2  @ T ?"8 B   Z DԔ?"  Zn?"*J ;Z0  Zde?"Zz ;X0  Zu?"Z&  z ;Y0  Zz?" Lc1.0B  Z DԔ?"    ZP?" Lc2.0  Zp?"2R Lc3.0  Z?"r Lc0.0 . T?"f  ?(a) 0l *&\ =*&\,$D 0Z@  *  < *   Z?"^G ;V02  T ?"/P2  T ?"DB  Z DԔ?" (i2  T ?"  B B Z DԔ?" 2  B T ?"PB ! Z DԔ?"  " Z8?"*J ;Z0 # Z|?" Z z ;X0 $ Z?">Zz ;Y0 % Z8?" + Lc1.0B & Z DԔ?" ( ' Z?" Lc2.0 ( N?"&:;Z Lc3.0 ) Z̰?"r Lc0.02 * N ?"\ + T?"^ r ;W0B ,B T DԔ?"(  - Tȹ?" :Z Lc4.0 / T<?"  ?(b) 0@ \ D ;$ &\ 0 s *A. z&??"` Dx z&d2 1 T ?"\  4 s *A '??"` J x 'd2 5 T ?"6  3l TYy :|,$D 02 8 T ?"T 9 TTĔ?"nYy 'Correlated proxies (Cai & Kuroki, 2008)(0(H  0޽h ? MMM̙0(___PPT10.6+I=iD' v = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-o6Bdissolve*<3<*=D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-o6Bdissolve*<3<*:+2*  0N0 91p|(  | | Hh?"],$@ 0 I TOLD YOU CAUSALITY IS SIMPLE Formal basis for causal and counterfactual inference (complete) Unification of the graphical, potential-outcome and structural equation approaches Friendly and formal solutions to century-old problems and confusions."0}1}%0}&"% @` | H?"i>,$@ 0 He is wise who bases causal inference on an explicit causal structure that is defensible on scientific grounds. (Aristotle 384-322 B.C.) 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