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S d 6TR #" `^`  F*  d 6W #" `^   H*  d 6X\ #" `  H* V d s *h޽h ?#" @@ MMM̙80___PPT10.ݿ 8,  Clouds  0 1)0h(  h h B]P ?#" ` P T Click to edit Master title style! ! h 0T`P " `   P W#Click to edit Master subtitle style$ $ h 6 eP #" `^` P F*  h 6hP #" `^  P H*  h 6mP #" `^  P H* V h s *h޽h ?#" @@ MMM̙80___PPT10.ݿ 8, 0 TLD(   D D N}P|@|@ -  P x*   e44ee D NpP|@|@ 2 - P *   e44eed D c $ ?$  P D NPP|@|@ H ] P 8________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________   D TP|@|@ s  P x*   e44ee D TP|@|@ s2  P z*   e44eeH D 0ηo~ ? ̙3380___PPT10.bUp,Q  ,( t; 4  C xLP|@|@ Ԕ?-  P t*    e44ee>  C xh!Y|@|@ Ԕ?2 - P ~*   e44ee4  C x@P|@|@ Ԕ?s  Y t*    e44ee6  C x8Y|@|@ Ԕ?s2  Y v*    e44eeH  0ηo~ ? ̙3380___PPT10.bUQC  0 ZR (    0dر R CAUSAL INFERENCE: MATHEMATICAL FOUNDATIONS AND METHODOLOGICAL PRINCIPLES Judea Pearl University of California, Los Angeles (www.cs.ucla.edu/~judea/)\Y L8nKZ,W(L .Y3      H  0޽h ? f̙f___PPT10i.:U+D=' = @B +J   0 aY (  >  T?"Qcwx___PPT9ZR  The three-level hierarchy of causal sentences The logical equivalence of structural and counterfactual models The five steps of causal inference methodology 1. Define, 2. Assume, 3. identify, 4. Estimate, 5. Test 4. How to represent causal assumptions Which covariates should be measured? What predictors should enter propensity scores. Why Yes to  Causation without manipulation" The Mediation Formula and effect decomposition Measurement bias and effect restoration. Transportability across studies: "external validity." Y/9)!Y/9  )H`  ZY?"op OTOPICS TO BE COVERED*B  N D1?"rH  0޽h ? f̙f___PPT10i.of`@+D=' = @B +0   0N0 ? 7  $  (  $  $ s *p PP *TRADITIONAL STATISTICAL INFERENCE PARADIGM++$B $ N D1?"F  t  $  t  " $ B  fZ G H$I@1Ԕ?"y[ $  H`?"  @Data (2 $  Bb?"> $  E Inference (2 7 $  H4g?"[ Q(P) (Aspects of P)z   $  ZpԔ?"ftU  ^P Joint Distribution.B  $  T DԔ?"X 00B  $ B T DԔ?" 0f0  $ Tq?" Oc IYe.g., Infer whether customers who bought product A would also buy product B. Q = P(B | A)Z1H $ 0޽h ? $ $ $  f̙fy___PPT10Y+D=' = @B +  0N0 )! ( (  ( B ( N Do?"$ V B ( N Do?"$ V  ( N?" bX 9What happens when P changes? e.g., Infer whether customers who bought product A would still buy A if we were to double the price.; ( s *" 8FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES,98   9  B ( N D1?"?(? ( TT?"?[fs m5Probability and statistics deal with static relations66" ( @  fZ GXH$I+Ԕ?"Bl^  ( Hx?"B @Data (2  ( Bh?" T c  E Inference (2 h  ( H?"m; *Q(P) (Aspects of P) "  ( ZԔ?" ` .P Joint Distribution@B  ( @ T DԔ?"R B ( T DԔ?"g ( NԔ?"& Q nP Joint Distribution>B ( Z D)?"D  ( B?"L$  Bchange (2B ( T DԔ?" H ( 0޽h ?  ( ( (  f̙fy___PPT10Y+D=' = @B +   0N0 0, (  , B , Z D)?"D B , N Do?"$ V B , N Do?"$ V B , T DԔ?"  , s *" 8FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES,98   9  B , N D1?"?(? , TĽ?"G / \Note: P (v) P (v | price = 2) P does not tell us how it ought to change Causal knowledge: what remains invariantRx(*  , NԽ?"rx <What remains invariant when P changes say, to satisfy P (price=2)=1F"  , @  fZ GXH$I+Ԕ?"8l^  , H߽?"8 @Data (2  , B?" T Y  E Inference (2 h  , Hp?"c; *Q(P) (Aspects of P) " , Z0Ԕ?" V  .P Joint Distribution@B , @ T DԔ?"R B , T DԔ?"g , NԔ?"& G  nP Joint Distribution> , BD?"B $  Bchange (2H , 0޽h ?  , , ,  f̙fy___PPT10Y+D=' = @B +   0N0   @ 0 B (  0  0 s *x9 ?FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT),@?   @  B 0 N D1?"p~F ?b 0  ?b 0  Td?"O  CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Exogeneity / Ignorability Mediation,,d    0  TH?" 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 TL)?" H@8___PPT9 U "5 @`W  0 H*?"  t$D 0@8___PPT9 U "1 @`]  0 N@5?"c ]t$D 0@8___PPT9 U "5 @`H 0 0޽h ? f̙fy___PPT10Y+D=' = @B +6  0N0 E=P4 (  4 F ? 4  ? 4  TA?"w  CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Exogeneity / Ignorability Mediation,,f    4  TI?"  ~6STATISTICAL Regression Association / Independence  Controlling for / Conditioning Odd and risk ratios Collapsibility / Granger causality Propensity score, e 4  TW?"?09D<4___PPT9 +Causal and statistical concepts do not mix.",1, @`m 4 N@Z?"f `t$D 0@8___PPT9 e 25 @` 4 H]?" _ t$D 0@8___PPT9 [Causal assumptions cannot be expressed in the mathematical language of standard statistics."\1\ @`" 4 s *Hf9,$D 0 8FROM STATISTICAL TO CAUSAL ANALYSIS: 2. MENTAL BARRIERS,98   9  B  4 N D1?"p,$D 0z e   4  e ,$D 0  4  T|?" H@8___PPT9 ^No causes in  no causes out (Cartwright, 1989)205 @`  4  T~?"Q ^  i1statistical assumptions + data causal assumptions22  4  T?" / e)  Jcausal conclusions 4  T|?"j C  <  4  TT?" a  N }077H 4 0޽h ? f̙fy___PPT10Y+D=' = @B +  0N0 `8 (  8 (z f - 8  f -,$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  Z( Co?S"?D 2B 8  T Do?"JF ?b 8  ?b 8  T?"w  CAUSAL Spurious correlation Randomization / Intervention Confounding / Effect Instrumental variable Exogeneity / Ignorability Mediation,,f     8  Tб?" 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 H?" _ H@8___PPT9 [Causal assumptions cannot be expressed in the mathematical language of standard statistics."\1\ @`  8 s *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  T?"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 +m   0 | (    s *~ &3-LEVEL HIERARCHY OF CAUSAL MODELS,'&$$ '  B  N D1?"   l  0˱@l|t___PPT9VN BProbabilistic Knowledge P(y | x) Bayesian networks, graphical models Interventional Knowledge P(y | do(x)) Causal Bayesian Networks (CBN) (Manipulation graphs) Counterfactual Knowledge P(Yx = y,Yx =y) Structural equation models, physics, functional graphs, regret and introspection"1 %0 '1 70 +1 R0 ffffff%ffffffff7  fff$fff$ff$fffffRbESY @h`H  0޽h ? 33___PPT10i.`[+D=' 7= @B +  0 C(    s * nBAYESIAN NETWORKS,,,   B  N D1?"` `  0`8B gDefinition P is Markov relative to G iff: Algebraic Representation Graphical Representation *0 0 0 0 0    %@ @h`  c $A A&??"`TPdx A&dA  c $A B&??"`0 \  x B&dc  c $A C&??"`x x C&d! F  Y3!   0 {q `B   0DԔ P`B  B 0DԔPn2   0"`^!    <$C"?7 Ic  bViD0   n2   0"`  en2  0"` 3e`B  0DԔ  0`B  0DԔ8(n2  0"`^! n2  0"`  SBN v S  f {Sn2  0"`v  Sn2  0"`FSf2  6Ԕ `B B 0DԔ0n2  0"`^ Q`B  0DԔ  0`B  0DԔ  8`B B 0DԔ   <C"? Y  }PAiD0     H ?"0 8V  ?G"0(2H  0޽h ? 33___PPT10i..+D=' 8= @B +   0    (     H,&?" `Definition Let Px(v) stand for P(v | do(X=x)), X and x arbitrary G is a CBN relative to Px* iff: each P(vi | pai) remains INVARIANT to interventions not involving Vi. Algebraic representation for all v consistent with x. (Truncated product, G-computation, manipulation theorem) Graphical representation Surgery: remove incoming arrows to X.bH000 (   (((  (7$$             0                   @`   s *d'- 6CAUSAL BAYESIAN NETWORK (CBN) (Manipulation graphs)<7!" 7  B   N D1?", ,  c $A {'??"` l  x {'dH  0޽h ? f̙fy___PPT10Y+D=' = @B +  0    t (   "  @  fZ GH I+Ԕ?"<l^B  N D1?"  H;?"< @Data (2  B??" T ]  E Inference (2 /  H`C?"g; Q(M) (Aspects of M)z   ZKԔ?" ] PData Generating ModelB  @ T DԔ?"R   HQ?" j6  DM  Invariant strategy (mechanism, recipe, law, protocol) by which Nature assigns values to variables in the analysis. :wwB  T DԔ?"g  NB  T DԔ?"   s *+ @[ sTHE STRUCTURAL MODEL PARADIGM$  Zę?"  CM&<  T?"Ej,$ 0 v> Think Nature, not experiment!     Hܢ?"*J =   2H  0޽h ?    f̙fYQ___PPT101.iK+TsD' = @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 *   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 ,$D 0 D  s *AV ??"` * tb x d D  Z?"{ E  =e.g.,H D 0޽h ? f̙f5-___PPT10 +5yD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*D %(D' =-o6Bdissolve*<3<*D +  0 [S 6 (     s *D0 o!CAUSAL MODELS AND COUNTERFACTUALS""$ "  B  N D1?"chc~2  H 33?"![6  HX ?"#] &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     T?"BX  =U 0 !  T ?"]   }X (u)\0 "  T?"]  }Y (u)\0 #  T ?"6* eq  =M 0B $  T DԔ?"  B % B T DԔ?"  B &  T DԔ?"   -  Z?"   =U 0 .  Z?" h  _X = x>08 /  ZM?"   YX (u)v0  0  ZDJ?" \   jMx20B 1  Z DԔ?"E   B 3  Z DԔ?"E  H  0޽h ? f̙fE=___PPT10.WV['P+pU[D' = @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 *HZ0 o!CAUSAL MODELS AND COUNTERFACTUALS""$ "  B | N D1?"chc~2 | H 33?"![6 | H_?"#] &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,$D 0  | 0A, &?Ԕ?"`G x &d  | Z+B#style.visibility<*|%(D' =-o6Bdissolve*<3<*|+   0 e] +` 0   (  ` ~l ++ : +` + +:,$D 0 `   `q? "++ K  HIn particular:0 @  : *`  : `  s *A 2(??  :x 2(d `  s *A E??   x EdO ` s *v0 o!CAUSAL MODELS AND COUNTERFACTUALS""$ "   ` H8|?"#] 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 ,$D 0 `   `? "'Vv  =  1  `  s *A# f?? "` 0 x fd  `   `? "+\|  a'Joint probabilities of counterfactuals:(0(H ` 0޽h ? f̙fPH___PPT10(.Q9ЅO+4 D' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*&` %(D' =-o6Bdissolve*<3<*&` D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*+` %(D' =-o6Bdissolve*<3<*+` +J    0 a Y P   (      s *ش,$D 0 #TWO PARADIGMS FOR CAUSAL INFERENCE,$#$$ $  B  N D1?"'p',$D 0.  T?"KLK $Observed: P(X, Y, Z,...) Conclusions needed: P(Yx=y), P(Xy=x | Z=z)... How do we connect observables, X,Y,Z,& to counterfactuals Yx, Xz, Zy,& ?   P/L\  T ?"M   NN-R model Counterfactuals are primitives, new variables Super-distribution `O.  T?"M J \" lStructural model Counterfactuals are derived quantities Subscripts modify the model and distribution lm(.B  N DjJ?" 2  0 NA= ? ?d"`T   d  0 NA> ? ?d"` >   dH  0޽h ? f̙f___PPT10i.p+D=' = @B +  0 5-@l(  l l s *H   v(ARE THE TWO PARADIGMS EQUIVALENT? ))$ )  B l N D1?"/_/G l s *d+.$ Yes (Galles and Pearl, 1998; Halpern 1998) In the N-R paradigm, Yx is defined by consistency: In SCM, consistency is a theorem. Moreover, a theorem in one approach is a theorem in the other. Difference: Clarity of assumptions and their implicationsZ,4"?;,   " ? ;Z               % l c $A =%??"`)x =%d~H l 0޽h ? MMM̙___PPT10i.`bt+D=' = @B +h#  0 #+PA(  B  N D1?"``  s *H6 )FORMULATING ASSUMPTIONS THREE LANGUAGES@*$$$ *  l L\ ~ +\ L~,$D 0A@ |7~ *|7~b   C x GuHI5#p?"78   Z?" 7 ~ 9ZT  7~  # |7~  T\#?" 7 ~ 9X  T ?""7~ 9YB  Z D>?"N  B  Z D>?"     `B?"L\ 5| S3. Structural:(jl L`  L` ,$D 0  0A }'? ?"`-` x }'d  ZH?"LA  X2. Counterfactuals:(   `M?"y "  7 m l L )L,$D 0  TQ?"L =1. English: Smoking (X), Cancer (Y), Tar (Z), Genotypes (U)>  P @ E (E  Z|]?" =X 02  T ?"B B Z D>?"O B  Z D>?"1 v   Zlb?"  =Y 0B  Z D>?" O| 2  T ?"i     Zf?"   =Z 02 ! T ?" I T  "#  2 # N ?"2 $ N ?"i2 % Z o?" 08 zB & Z D>?"  ' Tk?" E"  =U 0H  0޽h ?   f̙f  ___PPT10 .+ ^D ' = @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<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-o6Bdissolve*<3<*++  0 '  @  (     s *tn JGRAPHICAL  COUNTERFACTUALS SYMBIOSIS&&$ &  B  N D1?"h  T(y?"IU @Every causal graph expresses counterfactuals assumptions, e.g., X Y ZzKA @`y  H?" ] yconsistent, and readable from the graph. Express assumption in graphs Derive estimands by graphical or algebraic methods**PzN # %   s *A ? ?"`uBx d  Z?"/ D<4___PPT9 ,Missing arrows Y Zt @`  Zt?"( ?o  n2. Missing arcs, @`  T ?"1 f W x  9Y  T`?"1 n _ x  9Zb  3 r GϸH4I@p?"i j    s *A ? ?"`J  x dB  @ N Do?"R  H  0޽h ?   f̙f___PPT10i.zOi+D=' = @B +  0 `"\(  \ \ HD?".x ADefine: Assume: Identify: Estimate: Test: ,B   \ s *A/ ~0THE FIVE NECESSARY STEPS OF CAUSAL ANALYSIS11$ 1  B \ N D1?" \ 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 ? MMM̙___PPT10i.hО+D=' = @B +c  0 k c 0 (     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" A  s * A/,$D 0 5THE FIVE NECESSARY STEPS FOR EFFECT ESTIMATIONN6$$$$ 6  B  N D1?"   c $A '??"`b  '$@ 0dA  H?".x ADefine: Assume: Identify: Estimate: Test: ,B  H  0޽h ? f̙f___PPT10x.Wpr+28DL' = @B D' = @BA?%,( < +O%,( < +D>' =%(%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* 6%(D' =-o6Bdissolve*<3<* 6+  0 K C "P(  P P H,'?". C9Express the target quantity Q as a function Q(M) that can be computed from any model M. Formulate causal assumptions A using some formal language. Determine if Q is identifiable given A. Estimate Q if it is identifiable; approximate it, if it is not. Test the testable implications of A (if any). :%  7" ! P s *8CA ITHE FIVE NECESSARY STEPS FOR EFFECT OF TREATMENT ON THE TREATEDNJ$$$'$ J  B P N D1?" P c $A '??"`b  '$D 0dA P HHM?".x ADefine: Assume: Identify: Estimate: Test: ,B  H P 0޽h ? f̙fe]___PPT10=.Wpr+!7D' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-o6Bdissolve*<3<*P+"2  0 9111 >>0(  B @ T DԔ?" X B  T DԔ?" X   Z Ԕ?"x D$   N]Ԕ?#" ` L  CAUSAL MODEL (MA)\0(2    s *|e| oTHE LOGIC OF SEM,$$   B  N D1?"ll  NjԔ?c"$`?( fA - CAUSAL ASSUMPTIONS40(2   NpԔ?c"$`?X` eQ Queries of interest40(2g   N\tԔ?c"$`?  Q(P) - Identified estimandsd0(2    N|Ԕ?c"$`?H U  hData (D)D 0(2`   NԔ?c"$`? @8 Q - Estimates of Q(P) 0(2   N̉Ԕ?c"$`?P] PCausal inference$0(2Y  NhԔ?c"$`?  T(MA) - Testable implicationsv0(2  NԔԔ?c"$`? X  UStatistical inference$0(2  NԔ?c"$`? OGoodness of fit$0(2  Z Ԕ?"D$   T Ԕ?"H   T Ԕ?" dB  T DԔ?"4 4 X"  # l G~HeIԔ?",   T Ԕ?" H   T Ԕ?" d "  # l GHĿIԔ?"` < ` 2  N ?"  H 2  N ?" d B  T DԔ?", , B  T DԔ?"| | B  T DԔ?" , , B  T DԔ?" B  T DԔ?" , , B   T DԔ?" B ! T DԔ?"x xB " T DԔ?"X@XB # T DԔ?"   $ N4Ԕ?c"$`?  M Model testing$0(2 % N4Ԕ?c"$`?  RConditional claims$0(2 & c $A ~'??"`8G x ~'d3 ' c $A '??"`$ cx 'dA53B ( T DԔ?"P $ ) T8?" A* - Logical implications of AX0B * T DԔ?" vF H +  7 B , N Do?"HB - N Do?"H2 . N ?"H H  / N Ԕ?#" ` L  CAUSAL MODEL (MA)\0(2  F @  0 @  1 T Ԕ?"@  2 T Ԕ?"@  B 3 T DԔ?"$ $ B 4 T DԔ?" $ $ B 5 T DԔ?" $ $  6 T Ԕ?" '/ T2 7 N ?" d vF H 8 `PB 9 N D3o?"HB : N D3o?"H ; T Ԕ?"  T <  0e0e    BCDEF Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||Gpd@  "<  0GQ  =  0e0e    BCDEF Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||Gpd@  "< P0G!2 > N ?"  H  0޽h ?/@ MMM̙___PPT10i.~.Jl+D=' = @B +  0 W (   B  N D1?"00 = T?"5p QIDENTIFICATION IN SCM(3 ? T?"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 HL ?" B LZ6.  M H ?" +B  LZ3.  N H ?"N b  LZ2.  O Hx ?" N 2  LZ5.  P HL ?"7   LZ1.  Q H$  ?"  9X  R H  ?"  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=' = @B +>/  0N0 ((7:T "((  T  T H" ?"8& r8ELIMINATING CONFOUNDING BIAS THE BACK-DOOR CRITERION99$ T T(+ ?"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 HhA  ?" %D  HZ6*   T H+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  N Ԕ?"+ c,$D  0 D Front Door (2   s * A/ =EFFECT OF WARM-UP ON INJURY (After Shrier & Platt, 2008)>>$&(      B  N D1?"  T ?" ) ,2 @ Z Ԕ?" F ,$D  0  B$ ?"$ . 2z h% hW    % kkW ,$D  0   Z o?"h% YW  =No, no!B   T D>?"` h z l    ( ,$D  02   T Ԕ?"lz   Nl Ԕ?"J D Watch out! (2 B  T D>?"I  B ?"zG iWarm-up Exercises (X)8 2  B ?"{N n Injury (Y)H  2  N Ԕ?"% ],$D   0 =???(22  Z 0 Ԕ?"~ VG,$@   0B  T D>?"[@ 7 ,$@  0H  0޽h ? f̙f___PPT10..jŰ~+'HWD' = @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 +   0   (    H ?".xf .6Define: Assume: Identify: Estimate: 7    s *t A/ #FROM IDENTIFICATION TO ESTIMATION.$$ $ $  B  N D1?")  H\&?".f ;Express the target quantity Q as a function Q(M) that can be computed from any model M. Formulate causal assumptions using ordinary scientific language and represent their structural part in graphical form. Determine if Q is identifiable. Estimate Q if it is identifiable; approximate it, if it is not.L<&x  5  c $A '??"` '$D 0dAH  0޽h ? f̙fe]___PPT10=.TC)+< D' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+   0 # ##(    s *&A/ 6PROPENSITY SCORE ESTIMATOR (Rosenbaum & Rubin, 1983)77$ 7  B  N D1?"B @ Z Do?";a iB @ Z Do?"xM B  Z Do?";a  /B @ Z Do?";= )B @ Z Do?"?_( B   Z Do?"?(  B   Z Do?" D B   Z Do?"  B  @ Z Do?";QB  @ Z Do?" E r  <  ?" E   Hh& ?"  %  HZ6*    H@"& ?"Xx HZ3*    H\'& ?"?^; HZ2*    Hx,& ?"w?^ HZ5*    H1& ?"  HZ1*    H6& ?" >(  7X   H;& ?"   7Y   H>& ?" %  HZ4*  z @   @ ,$D  0B  T DԔ?"` R B  T DԔ?" @B B T DԔ?". @B B T DԔ?"F :L  TD&?"v _  7Lz $   $ ,$D 0  c $A '??"`Y$ t x 'dLN K y  K y$   TI&?"Yp *Adjustment for L replaces Adjustment for ZD+ ! c $A* '??"`YL yx 'd " TdQ&?"K Rk >Theorem:  ~ # TU&?"E P(y | do(x)) = ?$H  0޽h ? f̙f0(___PPT10.LL+D' = @B D' = @BA?%,( < +O%,( < +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 ! {(      s *d&P ;WHAT PROPENSITY SCORE (PS) PRACTITIONERS NEED TO KNOW<<  <  B   N D1?"h   Hh&?" vp$D 0<4___PPT9 0The asymptotic bias of PS is EQUAL to that of ordinary adjustment (for same Z). Including an additional covariate in the analysis CAN SPOIL the bias-reduction potential of others. In particular, instrumental variables tend to amplify bias. Choosing sufficient set for PS, requires knowledge of the model.1*5WH @`   c $A/ '??"`jWobx 'dd   c $A1 '??"`2x 'dH   0޽h ? f̙f  ___PPT10 .Ug+%:D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* P%(D' =-o6Bdissolve*<3<* PD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* 1%(D' =-o6Bdissolve*<3<* 1+  0 @f(    s * &`M /BIAS AMPLIFICATION BY INSTRUMENTAL VARIABLES<0(   0  B  N D1?" XB  0DԔ!!1 f2  0"`QDf2  0"`4 [y f2  0"`Q f2 @ 0"`4 By XB   0DԔ ;- f2   0"` . XB   0DԔU ?U XB  @ 0DԔ E =    <8u&C* PW220    <$&C   PW120    <Ė&1 |x  OX20    < &1 x  OY20  XB  0DԔ =   <P&! 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$x   <,&( ,$ 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 ' 8= @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 Z& Ԕ?* 3EFFECT DECOMPOSITION (direct vs. indirect effects)44( 4  B X N D1?"C X H &?"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<&?"i. ,H X 0޽h ? f̙fy___PPT10Y+D=' 8= @B +8  0 OG\(  \ \ Z& Ԕ?L fWHY DECOMPOSE EFFECTS?(   B \ N D1?"# \ H8&?"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=' Y= @B +L  0N0 `(  ` ` TT&?"G'$ 9X ` T&?"Gw h  9Z ` TD.?"_ H 9  9YB `  ` Dp?" B `  ` Dp?"Y B `@  ` Dp?"Y  ` s * . $ |&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 $A  ??"` Yx  d ` TP.?"]h } F(Qualifications) ` T.?" .  >(Hiring)   ` T.?"] } >(Gender)   ` H#.?"G' q;Can data prove an employer guilty of hiring discrimination?<<\l 18 `18,$D  0 `  `@).?"18 jAdjust for Z? No! No!8  R ` s  GJH%zIJԔ?"9   H ` 0޽h ? ` f̙f5-___PPT10 +D' Y= @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@1.?"2 D z = f (x, u) y = g (x, z, u) d T<.?"wt  9X d T03.?"  9Z d ThD.?"%  9YB d  ` Dp?"Q B d  ` Dp?"! B d@  ` Dp?"!    d s * Q.  4NATURAL INTERPRETATION OF AVERAGE DIRECT EFFECTS55$B  d N D1?"44  d HW.?"\  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 N|k.?"Pp ~HRobins and Greenland (1992)   Pure %% d c $AZ  ??"`ex  dH d 0޽h ? f̙fy___PPT10Y+D=' Y= @B +   0N0   0l^ (  lj l Ts.?" DN z = f (x, u) y = g (x, z, u) l T}.?"wt 9X l T\.?"  9Z l T8.?"  9YB l  ` Dp?"XQ XB l  ` Dp?"!B l@  ` Dp?"!   l s *.  v DEFINITION OF INDIRECT EFFECTS!!$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 $A  ??"`i b y x  dAH l 0޽h ? f̙fy___PPT10Y+D=' Y= @B +  0N0 @p(  p p HH.?"P f,POLICY IMPLICATIONS OF INDIRECT EFFECTS--( p Z.o?"; o m  M f ( Ez B! c p ! Bc,$D 0 p T.?"B! A  NGENDER, p Th.?"Z ! A  S QUALIFICATION*  p T0.?" 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 Ht.?"G &What is the indirect effect of X on Y?'   @`G p Z(.?"},$ 0 {CThe effect of Gender on Hiring if sex discrimination is eliminated.DD p T.?"! A  ;X p T.?"! j S A  IZ, p T.?"C c GY*B p@ N DԔ?"8   p T.?" _  <IGNORE4 p H$.?"B `Deactivating a link  a new type of intervention11 @`H p 0޽h ? f̙f___PPT10p+D' .= @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(  -  H6?"D<4___PPT9 [The 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 6?" LMEDIATION FORMULAS(  c $A '??"`x- x 'dH  0޽h ? f̙f___PPT10i.++D=' 8= @B +$  0 !$$(  B  N D1?"  H(6?" ?Z"0 22  N ?"2  N ?"v'2 @ N ?"v    Tx6?"* J Lm2.0 B @ T DԔ?"B   T DԔ?"    T,6?".N ;X0   T|6?". N ;Y0B   T DԔ?"    T 6?"*J Lm1.0   c $A$ '??"`Px 'd  c $A2 '??"`4 x 'd  c $A3 '??"` H x 'dAN  c $A4 '??"` h '$D 0d  c $A5 '??"`z~x 'd  c $A6 '??"`2 x 'd z  '   ',$D 0B  T DԔ?"88 8 B  T DԔ?"x  B  T DԔ?"( ( B  T DԔ?"8( ( B  N DԔ?"p 0p B  N DԔ?" h B   ` DԔ?" `   T,+6?" !' NDisabling mediation0  T/6?" PDisabling direct path0  T46?"+ 1  <DE0  Tt86?" 1 Q  ]TE - DE:0   T<6?"   <TE0 ! T @6?"   <IE0B " N DԔ?"8   # HC6?" KIn linear systems0 $ TtH6?" ~g ,$ 0 JIs NOT equal to:0H  0޽h ? f̙f  ___PPT10x +4D$ ' 8= @B D' = @BA?%,( < +O%,( < +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<*$D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+$0 +   0 ` p#(  p p s *Q6  |.MEDIATION FORMULAS IN UNCONFOUNDED MODELS //$ /  B p N D1?"QhQ8 J)   py, p T$W6?"JTG  9X p T0[6?")p 9Z p T_6?" 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=' 8= @B +  0 0!#G@T(    s *Lf6 < _TRANSPORTABILITY -- WHEN CAN WE EXTRPOLATE EXPERIMENTAL FINDINGS TO DIFFERENT POPULATIONS?``B  N D1?"HrH  HTq6?"5C /Experimental study in LA Measured: Problem: We find (LA population is younger) What can we say about Intuition: t0' @`  c $A: '??"`l  l x 'd  c $A '??"` x 'd %  `xx6?"h . , Nȅ6?"5 ;  h$Observational study in NYC Measured:(%  - c $A< '??"`P H x 'd . c $A? '??"`  x 'dA / c $A7 '??"`x 'dF   0 ? T ` 1# 2 2 N ?"OS2 3 N ?"B 4B T DԔ?"n7B 5 T DԔ?" 6 TxS6?" ;X0 7 Tw6?" ;Y0B 8 T DԔ?"+n2 9 N ?"  : T@6?"` OZ = age,0 ; Z$6?"  .F   <  ;? T ` =# 2 > N ?"OS2 ? N ?"B @B T DԔ?"n7B A T DԔ?" B T6?" ;X0 C T6?" ;Y0B D T DԔ?"+n2 E N ?"  F TȨ6?"` OZ = age,0 G ZP6?"  .H  0޽h ? f̙f___PPT10i.f+D=' 8= @B +,:  0   C";" !00!(    s *к6  ,TRANSPORT FORMULAS DEPEND ON THE STORY-- B  N D1?"rz     ,$D 0}T ` # 2  N ?"OS2  N ?"B B T DԔ?"n7B   T DԔ?"   Th6?" ;X0   T6?" ;Y0B   T DԔ?"+n2   N ?"   T6?"` ;Z0  Z6?"  9(a)z ^   ^ ,$D 02  N ?" k 0B B T DԔ?"O  B  T DԔ?"^   Tx6?"^G  ;X0  T6?".  ;Y0B  T DԔ?"  2  N ?" A<  T6?"   ;Z0  ZL6?" e  9(b)2  T fo?"z  2  N ?"{ 7 B  T DԔ?"  B   T$6?"U ,$@ 0 E_a) Z represents age b) Z represents language skill c) Z represents a bio-marker `0 @`  c $A7 '??"`4   '$@ 0d  c $A8 '??"`   '$@ 0d   c $A9 '??"`  '$@  0d> z   ! s,$D  0 " T6?" ;X02 # N ?"4B $B T DԔ?"B % T DԔ?"1v & T6?" ;Y0B ' T DԔ?"|2 ( N ?"i@ ) Z\F?"y  9(c) * TF?"! ;Z02 + N ?"I<~N  , B2 - N ?"2 . N ?"i2 / T fo?"8B 0 T DԔ?"H  0޽h ? f̙f}u___PPT10U.GVz+)>D)' 8= @B D' = @BA?%,( < +O%,( < +DY' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*DY' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-o6Bdissolve*<3<*;D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*DY' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*A`%(D' =-o6Bdissolve*<3<*A`D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bdissolve*<3<*!+   0   @!(    s *F   -TRANSPORTABILITY (Pearl and Bareinboim, 2010)0.(  B  N D1?"prp  N,'F?",` @Definition 1 (Transportability) Given two populations, denoted P and P*, characterized by probability distributions P and P*, and causal diagrams G and G*, respectively, a causal relation R is said to be transportable from P to P* if 1. R(P) is estimable from the set I of interventional studies on P, and 2. R(P*) is identified from I, P, P*, G, and G*.:| .#  @`H  0޽h ? f̙f___PPT10i.bfxQ+D=' F= @B +E  0 --!AA,,(  , , s *+F  ,TRANSPORT FORMULAS DEPEND ON THE STORY-- B , N D1?"r  , T8FF?"U -,$D 0 C]a) Z represents age b) Z represents language skill c) Z represents a bio-marker ^0 @` , c $A7 '??"`D 0  '$@ 0d , c $AB '??"` 0   '$@ 0d , c $AC '??"`0P '$D  0dm z ^  ,  ,$@ 02  , N ?" k 0B  ,B T DԔ?"=  B  , T DԔ?"^   , TxwF?"^G  ;X0  , T{F?".  ;Y0B , T DԔ?" 2 , N ?" A< , T:F?" # C ;Z0 , Z~F?" e  9(b)2 , T fo?"z  2 , N ?"{ 7 B , T DԔ?"  B2 ,B Z Ԕ?") i 2 ,B  `L Ԕ?" m  , T, T ?"D ?, TF?"2Ll ;S0B @, T DԔ?"B A, T DԔ?"1vH , 0޽h ?o`',,;,,(,<, f̙f}u___PPT10U.\f(t+:D)' F= @B D' = @BA?%,( < +O%,( < +DY' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-o6Bdissolve*<3<*,D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-o6Bdissolve*<3<*,D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-o6Bdissolve*<3<*,DY' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-o6Bdissolve*<3<*,D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-o6Bdissolve*<3<*,D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,:%(D' =-o6Bdissolve*<3<*,:DY' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,?^%(D' =-o6Bdissolve*<3<*,?^D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*-,%(D' =-o6Bdissolve*<3<*-,D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-o6Bdissolve*<3<*,+^  0 umP! 0 (      s *F  ?WHICH MODEL LICENSES THE TRANSPORT OF THE CAUSAL EFFECT@@"B   N D1?"ri'F      yT    #   N A    A B   T DԔ?"! n    TF?"A * ;X0   TF?"  ;Y02   N ?" $8    ZF?" H 9(e)   TF?" c ;Z02   N ?" N- 2   N ?"n - 2   N ?" $: 2  B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"  B  B Z DԔ?"     T ?"p     TF?"D vd  ;S02  B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"@  B   T DԔ?"   N  h    h   T`F?"  ;X0   TF?" h ;Y02   N ?"i 8    ZF?"y  9(c)   TF?"  ;Z02   N ?" - 2   N ?" I- 2   N ?"i : 2  B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"  B  B Z DԔ?"" "  !  T ?" D  "  TF?"2D d  ;S02 #  N ?" - 2 $ B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"X   %  TF?"  ;W0B &  T DԔ?" f B '  T DԔ?"  B (  T DԔ?"   T )  ) #    *  TDK?")   ;X0 +  TK?"  ;Y02 ,  N ?" 8  -  ZK?" 0  9(d) .  T K?"l K  ;Z02 /  N ?" 6 - 2 0  N ?"V - 2 1  N ?" : 2 2 B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T" B 3 B Z DԔ?"z z  4  T ?"X  5  TK?" D ^ d  ;S02 6  N ?" k -  7  T4K?" %  ;W02 8 B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"+ h s  B 9  T DԔ?"y   B :  T DԔ?") f B ;  T DԔ?" 6 s L  pa  < #  i B =  T DԔ?" ll >  TK?" N n ;X0 ?  TK?"Npn ;Y02 @  N ?"qN A  Z K?"Aa  9(b) B  T#K?"Nn ;Z02 C  N ?" C 2 D  N ?"CQ2 E  N ?"qPB F B Z DԔ?"*P*W G  T ?"LS H  T(K?": ;S02 I B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"K{vB J  T DԔ?"!lnl L  Z,K?"V)v ;X0 M  Zt1K?"V v ;Y02 N  T ?"V  O   `/K?"I0i  9(a) P  Z8K?"VlKv ;Z02 Q  T ?"K62 R  T ?"KV2 S  T ?"X  U  Z ?"X[ V  Z=K?"^ ;S02 W  T ?"K k X  ZXAK?"V%v ;W02 Y @ c  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"+snB Z  Z DԔ?"|y |B [  Z DԔ?"t)ftB \  Z DԔ?"t6ti'F   ]   yT   ^ #   N A  _  A B `  T DԔ?"! n  a  TGK?"A * ;X0 b  TKK?"  ;Y02 c  N ?" $8  d  ZOK?" H 9(e) e  TTK?" c ;Z02 f  N ?" N- 2 g  N ?"n - 2 h  N ?" $: 2 i B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"  B j B Z DԔ?"   k  T ?"p   l  TXK?"D vd  ;S02 m B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"@  B n  T DԔ?"   N  h o   h p  T]K?"  ;X0 q  TbK?" h ;Y02 r  N ?"i 8  s  ZTeK?"y  9(c) t  TiK?"  ;Z02 u  N ?" - 2 v  N ?" I- 2 w  N ?"i : 2 x B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"  B y B Z DԔ?"" "  z  T ?" D  {  T\!K?"2D d  ;S02 |  N ?" - 2 } B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"X   ~  TqK?"  ;W0B   T DԔ?" f B   T DԔ?"  B   T DԔ?"   T )   #      TvK?")   ;X0   TT{K?"  ;Y02   N ?" 8    Z~K?" 0  9(d)   T܂K?"l K  ;Z02   N ?" 6 - 2   N ?"V - 2   N ?" : 2  B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T" B  B Z DԔ?"z z    T ?"X    TK?" D ^ d  ;S02   N ?" k -    TK?" %  ;W02  B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"+ h s  B   T DԔ?"y   B   T DԔ?") f B   T DԔ?" 6 qL    #   N A    A B   T DԔ?"! n    TXK?"A * ;X0   TK?"  ;Y02   N ?" $8    ZęK?" H 9(e)   TK?" c ;Z02   N ?" N- 2   N ?"n - 2   N ?" $: 2  B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"  B  B Z DԔ?"     T ?"p     TK?"D vd  ;S02  B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"@  B   T DԔ?"   N  h    h   TlK?"  ;X0   TK?" h ;Y02   N ?"i 8    Z(K?"y  9(c)   TlK?"  ;Z02   N ?" - 2   N ?" I- 2   N ?"i : 2  B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"  B  B Z DԔ?"" "    T ?" D    TpK?"2D d  ;S02   N ?" - 2  B S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"X     TK?"  ;W0B   T DԔ?" f B   T DԔ?"  B   T DԔ?"     ZK?"  ;X0   ZHK?"  ;Y02   T ?" 8     `lK?"   9(d)   ZDK?" L +  ;Z02   T ?"  - 2   T ?" 6 - 2   T ?" : 2  @ c  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"     Z ?" 8 |    ZK?"D j > d  ;S02   T ?" K -    ZK?"   ;W02  @ c  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"h S  B   Z DԔ?" Y  B   Z DԔ?" F B   Z DԔ?"  B  @  ` DԔ?" Z Z B T @  ` DԔ?"Xzz_H   0޽h ?O@ $ 2   8 I Y  x  }   i  m         f̙f___PPT10i.df +D=' F= @B +$  0 ##`!66$"#(  $ $ TK?"*J ;U0&2 $@ c L CbUENGjtHܳJYTQԔ? `T$^TbU`T$^TbUYT$^TbUYT"ylE2 $@ c ҽ BCeE$GɔI`TQԔ? e`Te`T`T`Te`T`T`T"B $ T DԔ?"0 $ TK?"  ;W0 $ s *pK ($ 5DETERMINE IF THE CAUSAL EFFECT IS TRANSPORTABLE66(B $ N D1?"8r8  $ TK?"iR  ;X0  $ TK?"  ;Y02  $ N ?" %   $ TTK?"D#   ;Z0vF n 4   $ .% 2 $ N ?"n 2 $ N ?" 4 2 $ N ?"c92 $@ S  BC|VE$GbI`TQԔ? |V`T|V`T`T`T|V`T`T`T"kB $ T DԔ?" i B $ T DԔ?" > 2 $ N ?"1|% B $ T DԔ?"  2 $ c  BCTE$GHI`TQԔ? TxTTxT`T`TTxT`T`T" C  $ TM?"Dd ;V0B $@ Z DԔ?"RR $ T ?"0t $ TM?",b6 L ;S0vF n 4  $ .% 2 $ N ?"n 2 $ N ?" 4 2 $ N ?"kMB $@ T DԔ?"08vF n 4   $ .% 2 !$ N ?"n 2 "$ N ?" 4 2 #$ N ?"kM|L n 4  $$# .% 2 %$ N ?"n 2 &$ N ?" 4 |L n 4  '$# .% 2 ($ N ?"n 2 )$ N ?" 4 2 *$ N ?"1|M +$ TpM?"zI( ;T0B ,$@ Z DԔ?" -$ T ?"p .$ TM?"` NS.0B /$ T DԔ?"P B 0$@ T DԔ?"(hX 1$ ZhM?" '8  ZThe transport formula (` 2$ c $A@ '??$ ' 3$ NM?"   OWhat measurements need to be taken in the study and in the target population? PP 4$ c $AA '??"`H (hXx 'dB 5$ T DԔ?"P B 6$@ T DԔ?"(hXH $ 0޽h ?O@$$$$ f̙f___PPT10i.df+D=' F= @B +@%  0N0 !0f(  0 0 H%M?",$@ 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. No other method can do better (theorem)"0}1}%0}(1}&"%( @` 0 s *xAM  q CONCLUSIONS.  , B 0 N D1?"H 0 0޽h ? f̙f___PPT10+ {D~' Y= @B D9' = @BA?%,( < +O%,( < +D>' =%(%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*0 %(D' =-o6Bdissolve*<3<*0 D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*0 "%(D' =-o6Bdissolve*<3<*0 "DY ' =%(D ' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*0"b%(D' =-o6Bdissolve*<3<*0"bD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*0b%(D' =-o6Bdissolve*<3<*0bD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bdissolve*<3<*0D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bdissolve*<3<*0D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*0$%(D' =-o6Bdissolve*<3<*0$D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*0 %(D ' =%(Dq ' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*0 %(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*0 "%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*0"b%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*0b%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*0%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*0%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*0$%(+8+0+00 +   0 "`-(  ` ` HJM?"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.) $ ` s *YM  q CONCLUSIONS.  , B ` N D1?"H ` 0޽h ? MMM̙___PPT10x.h 8$+sS4DL' Y= @B D' = @BA?%,( < +O%,( < +D>' =%(%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*`q%(D' =-o6Bdissolve*<3<*`qD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*`q%(D' =-o6Bdissolve*<3<*`q+  0 !4x(  4 4 T^M?"< N They will be answered( 4 s *cM  r QUESTIONS???.  , B 4 N D1?"b 4 <v޽h @ ?"` f̙f___PPT10i.s+D=' Y= @B +'  0 "H(  H0 H ZDoM?"  ,$ 0 dY = 2XB H s *D}M V x"PHYSICS DESERVES A NEW ALGEBRA## B H N D1?"rN H Z$M?"[ XW G,$ 0 Had X been 3, Y would be 6. If we raise X to 3, Y would be 6. Must  wipe out X = 1.U  z  R  H | R ,$D 02 H TM?"g   X = 1 Y = 2p  H T@M?" R  D The solution    H ThM?"  ,$ 0 KProcess information |id\   H# |d ,$D  0  H TM?"d\  Y := 2Xb$  H T(M?"|i| ICorrect notation:  H ZM?"  ,$ 0 QX = 10 H Z0M?"|(,$ 0 ;e.g., Length (Y) equals a constant (2) times the weight (X)^<*D H ZM?"4|7T tScientific Equations (e.g., Hooke s Law) are non-algebraic;;H H 0޽h ? f̙f___PPT10f.GVz+Z{D' Y= @B D' = @BA?%,( < +O%,( < +D ' =%(Dp ' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bdissolve*<3<*HD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-o6Bdissolve*<3<* HD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-o6Bdissolve*<3<* HD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bdissolve*<3<*HD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bdissolve*<3<*HD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bdissolve*<3<*HD)' =%(D' =%(D>' =A@BB BB0B%(D' =-o6Bdissolve*<3<*HD' =1:Bhidden*o3>+B#style.visibility<*H%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-o6Bdissolve*<3<* H+P+0+H0 ++0+H0 ++0+H0 ++0+ H0 ++0+ H0 ++0+H0 +  0 " Li(  LB L N D1?"h L s *DMH BREGRESSION VS. STRUCTURAL EQUATIONS (THE CONFUSION OF THE CENTURY)CC  C   L H4P?">2H  Regression (claimless, nonfalsifiable): Y = ax + eY Structural (empirical, falsifiable): Y = bx + uY ( := assignment) Claim: (regardless of distributions): E(Y | do(x)) = E(Y | do(x), do(z)) = bx T( %  ']W  @`F L HM?"L^y,$ 0 Q. When is b a partial regression? b = bYX " Z A. Shown in the diagram, Slide 36.U $ @`z ^ 6 L  ^6,$D 0  L Z P?"^ 6 icThe mothers of all questions: Q. When would b equal a? A. When (uY X), read from the diagramd B  @` T    L#  %+  L ZM?"   :^  L ZM?"   :^H L 0޽h ? MMM̙`X___PPT108.4pX+p7D ' Y= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bdissolve*<3<*LD' =%(D' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bdissolve*<3<*L+   0N0   P" X (  XB X N D1?"$$ X s *0P0 &AXIOMS OF STRUCTURAL COUNTERFACTUALS0'&$$ '   X Hl@P?"_PH@___PPT9" ^Definiteness Uniqueness Effectiveness Composition (generalized consistency) Reversibility <^^ @` X 0A '? ??"`P x 'd X 0A '? ??"`PP x 'd X 0A '? ??"` P  x 'd X 0A '? ??"`6 P3x 'd  X s *A '???"`hPx 'd  X TKP?"^g TYx(u)=y: Y would be y, had X been x (in state U = u) (Galles, Pearl, Halpern, 1998):xU  ",6  H X 0޽h ? f̙f___PPT10i.zI?+D=' 8= @B + 0 ((  ^  S D$   P  c $PDH ]  P  H  0ηo~ ? ̙3380___PPT10.0$ 0 p!(4(  (d ( c $D$   P ( s *PDH ]  P  H ( 0ηo~ ? ̙3380___PPT10.pfpI"xX]lTE>wf;nZ .EjPJ U%ei(J($_K"!$&9wl[rքʞ33wΙ3gΝ]6sy!6ɩ`R&H5&-(A/A7B8J9d9<â^;~btί5W-]KL: e,p:㖟Dy'9i3\9\~׏Ԙ$:/ɤY8 {>n1g2/)`{>? xAcoys /&{4 A1=CtsO4;oOu57zSUQkS-lKyHuwz|!K<3Iyc<}Gȍ=A^[hx¸Rxkz;Z`W΃*!bg\X>>+~/FY(~L#ʮTP p4gƎۃnnBOͶX%q l|6r-6ߒ|Rwټ|Omܵ:㲞QX3#e,m6e @uُVrκrB/9?tfzZhƝs|ɂLӃed9 5t03qeh" rӦۀ5? 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