Stratified Analysis of Probabilities of Causation

Transcription

1 Stratified Analyi of Probabilitie of Cauation Manabu Kuroki Sytem Innovation Dept. Oaka Univerity Toyonaka, Oaka, Japan Zhihong Cai Biotatitic Dept. Kyoto Univerity Sakyo-ku, Kyoto, Japan Abtract Thi paper derive new bound for the probabilitie of cauation defined by Pearl (2), namely, the probability that one oberved event wa a neceary (or ufficient, or both) caue of another. Tian Pearl (2a, 2b) howed how to bound thee probabilitie uing information from experimental obervational tudie,with minimal aumption about the data-generating proce. We derive narrower bound uing covariate meaurement that might be available in the tudie. In addition, we provide identifiable cae under no-prevention aumption dicu the covariate election problem from the viewpoint of etimation accuracy. Thee reult provide more accurate information for public policy, legal determination of reponibility peronal deciion making. 1 Introduction It i an important iue to evaluate the likelihood that one event wa the caue of another in practical cience. For example, epidemiologit are intereted in the likelihood that a particular expoure i the caue of a particular dieae. In order to ae thi likelihood from tatitical data, the probabilitie of cauation have been developed, which can be divided into neceary cauation, ufficient cauation neceary-ufficient cauation. Thee probabilitie of cauation are ued in epidemiology, legal reaoning, artificial intelligence, policy analyi pychology. Pearl (2) Tian Pearl (2a, 2b) developed formal emantic for probabilitie of cauation baed on tructural model of counterfactual. They preented the formal definition for probability of neceity (PN), probability of ufficiency (PS) probability of neceity ufficiency (PNS). In addition, they howed how to bound thee quantitie from data obtained in experimental obervational tudie. Their bound are harp under the minimal aumption concerning the data-generating proce. In thi paper, we call their bound Tian-Pearl bound. When we examine many experimental obervational tudie, we find that there i ome extra information that we can ue in order to bound the quantitie of the probabilitie of cauation. For example, in epidemiological tudie, not only the expoure the outcome are meaured, but alo ome covariate uch a age, gender are meaured. However, Tian- Pearl bound do not provide formula for making full ue of thee information. Therefore, the aim of thi paper i to provide narrower bound for probabilitie of cauation by uing a much a information available from experimental obervational tudie. Our main idea i to ue tratified analyi in order to obtain narrower bound. When we get the tatitical data about the expoure, outcome ome covariate, we firt tratify the data according to the covariate, then we calculate the bound of probabilitie of cauation in each tratum, finally we derive ummarized bound on the probabilitie of cauation. By making ue of the covariate information, we can provide narrower bound than Tian-Pearl bound without making additional aumption. Thi paper i organized a follow. Section 2 give ome preliminary knowledge that will be ued throughout the paper. In Section 3, we introduce the probabilitie of cauation give a new definition of conditional probabilitie of cauation. Then we propoe the nonparametric bound for probabilitie of cauation baed on tratified analyi, invetigate ome propertie of the propoed bound. In addition, we provide identifiable cae under no-prevention aumption dicu the covariate election problem from the viewpoint of etimation accuracy. We do imulation experiment to verify our reult in ection 4. In

2 ection 5, we give an example to illutrate how the propoed formula can narrow the bound o a to provide more evidence for health care policy making. Finally, ection 6 conclude thi paper. 2 Preliminary In thi ection, we introduce the potential outcome variable that will be ued to define the probabilitie of cauation. We conider the cae where an expoure variable an outcome variable are dichotomou. We denote X a an expoure variable (x: true; x : fale) Y a an outcome variable (y: true; y : fale). In addition, let P(x, y) be a trictly poitive joint probability of (X, Y ) =(x, y) P(y x) the conditional probability of Y = y given X = x. Similar notation are ued for other ditribution. In principle, the ith of the N ubject ha both a potential outcome Y x (i) that have reulted if X had been x, a potential outcome Y x (i) that have reulted if X had been x. Then Y x (i) Y x (i) i called the unit-level caual effect (Rubin, 25). When the N ubject in the tudy are conidered a a rom ample from ome population, ince Y x (i) Y x (i) can be referred a the value of rom variable Y x Y x repectively, the average caual effect can be defined a P(y x ) P(y x ), (1) where y x indicate the counterfactual entence Variable Y would have the value y, had X been x. Similar notation are ued for other potential outcome. The outcome Y x i oberved only if X i x, Y x i oberved only if X i x. Thi property i called the conitency (Robin, 1989), which i formulated a follow: (X = x) (Y x = Y ). (2) Thu, when a romized experiment i conducted compliance i perfect, the average caual effect i P(y x x) P(y x x )=P(y x) P(y x ), (3) which i equal to equation (1). On the other h, when a romized experiment i difficult to conduct only obervational data i available, we can till etimate the average caual effect according to the trong-ignorable-treatment-aignment (SITA) condition (Roenbaum Rubin, 1983). That i, for the treatment variable X, if there exit uch a et S of oberved covariate that X i conditionally independent of (Y x,y x ) given S, denoted a X (Y x,y x ) S, we hall ay treatment aignment i trongly ignorable given S, ors atifie the SITA condition. Thu, equation (1) i etimable by uing S E {P(y x, ) P(y x,)}. (4) 3 Nonparametric bound baed on tratification 3.1 Formulation The PN i defined a the expreion PN =Pr(y x x, y), (5) which t for the probability that event y would not have occurred in the abence of x, given that x y did in fact occur. In thi paper, we define a conditional PN given S = a PN() = Pr(y x x, y, ), which t for the probability that event y would not have occurred in the abence of x, given that x y did in fact occur in tratum. The PS i defined a the expreion PS = P (y x x,y ), which t for the probability that event y would have occurred in the preence of x, given that x y did in fact occur. In addition, we define a conditional PS given S = a PS() =P (y x x,y,) which t for the probability in tratum that event y would have occurred in the preence of x, given that x y did in fact occur in tratum. In thi paper, we do not dicu the PS ince it ha the ame propertie a the PN by changing (x, y) to(x,y ). The PNS i defined a the expreion PNS = P (y x,y x ), (6) which meaure both the ufficiency neceity of x to produce y. In thi paper, the conditional PNS given S = i defined a PNS() =P (y x,y x ), which meaure both the ufficiency neceity of x to produce y in tratum. When an experimental tudy or an obervational tudy i conducted, it i the uual cae that not only the expoure the outcome variable are meaured, but alo ome information on background factor intermediate factor are available. Then, making full ue of thee available information will give narrower bound on the PN the PNS, which i the aim of thi ection. The bound can be obtained by applying the linear programming technique (Tian Pearl, 2a, 2b) to conditional caual effect (Tian, 24). In thi paper, we provide a imple proof by uing equation (2) recurive factorization of probabilitie. Letting S be a et of oberved variable, when we tratify the ubject according to the level of S, we can derive new bound on the PN the PNS by (i) calculating the lower upper bound of the conditional

3 PN the conditional PNS within each tratum, (ii) ummarizing the lower upper bound in all the trata. Firt, regarding the PN, P(y x ) = P(y x w, z, )P(w, z ), (7) P(y x x,y,) = P(y x x,y,) = 1 hold true from equation (2). Thu, the bound of the conditional PN given S = can be given a min P(y x ) P(y ) P(x, y ) PN() 1 P(y x ) P(x,y ) P(x, y ). Here, letting PN be the PN when a variable S i ued to evaluate it, by noting PN = PN()P( x, y), we can derive the new bound baed on tratification P(y x ) P(y ) P() PN P(x, y) P(x, y ) P(x, y) min P(y x ) P(x,y P(). (8) ) P(x, y) Regarding the PNS, it i trivial that P (y x,y x ) min{p (y x ),P(y x )} hold true. In addition, we can obtain P (y x,y x ) = P (y x,y x z,w,)p(z,w ), P (y x,y x ) =P (y x ) P (y x,y x ) =P (y x ) P (y x,y x z,w,)p(z,w ), P (y x,y x ) =P (y x ) P (y x,y x ) =P (y x ) P (y x,y x z,w,)p(z,w ), P (y x,y x ) =P (y x ) P (y x )+P(y x,y x ) = P (y x ) P (y x ) + P (y x,y x z,w,)p(z,w ). Thu, by uing P(y x x,y,) = P(y x x,y,)= 1, the bound of the conditional PNS given S = can be given a min P (y x ) P (y ) P (y x ) P (y ) P (y x ) P (y x ) PNS() P (y x ) P (y x ) P (x, y )+P (x,y ) P (y x ) P (y x )+P (x,y )+P (x, y ) Thu, letting PNS be the PNS when a variable S i ued to evaluate it, by noting PNS = PNS()P(), we can obtain P (y x ) P (y ) P (y x ) P P () PNS (y ) P (y x ) P (y x ) P (y x ) P (y min x ) P (x, y )+P (x,y ) P (y x ) P (y x )+P(x,y )+P(x, y ) P (). (9) The propoed bound (8) (9) are narrower than Tian-Pearl bound (Tian Pearl, 2a, 2b). An intuitive explanation i that the propoed bound alway elect the imal value in lower bound minimal value in upper bound within every tratum of S, while Tian-Pearl bound alway elect a fixed one from all the probabilitie acro every tratum of S. In addition, it i obviou that the propoed bound reduce to the Tian-Pearl bound if there i no tratified analyi on S. 3.2 Property of the propoed bound An intereting property of the propoed bound i that the oberved variable S need not be confounder between an expoure variable X an outcome variable Y. Fig.1 how ix cae that the oberved variable S may be. Here, U, U 1 U 2 in Fig.1 indicate et of unoberved variable. An arrow ( ) indicate that a variable of it tail ha an effect on another variable of it head, a bidirected arc ( ) indicate that two variable connected by the arc have an aociation with each other. Fig.1 (a) how that the variable S are confounder that have an effect on both X Y ; Fig.1 (b) how that the variable S are prognotic factor that have an effect only on Y ; Fig.1 (c) how that the variable S are intermediate variable that are affected by X have an effect on Y ; Fig.1 (d) how that the variable S are variable that atify the intrumental variable condition (e.g. Bowden Turkington, 1984; Greenl, 2); Fig.1 (e) how.

4 (a): Confounder (b): Prognotic Factor 1 P(y x y,x,) = 1, we can obtain PN() = P(y x ) P(y ). P(x, y ) Thu, the PN can be identified a PN = PN()P(), which require both obervational data experimental data. When only obervational data i available, if we can oberve ome covariate that atify the SITA condition (Roenbaum Rubin, 1983), we can till identify the PN: (c): Intermediate Variable (d): Intrumental Variable PN = P(y x,) P(y ) P(). (1) P(x, y) The aymptotic variance of the PN i given by (e): M-tructure (f): No-aociation Fig.1. Six Cae of Oberved Variable S that the covariate S are in a M-tructure (Greenl et al., 1999; Greenl, 23); Fig.1 (f) how that the variable S are aociated with neither X nor Y. When data are available from both an experimental tudy an obervational tudy, equation (8) (9) are applicable to the former five cae, that i, we can obtain narrower bound than Tian-Pearl bound under thee cae. Hence, when we obtain information on oberved variable, they may belong to cae (a), (b), (c), (d) or (e). Whichever they belong to, if we tratify the population according to the level of S, we can obtain narrower bound on probabilitie of cauation. 3.3 Identification under the no-prevention In the dicuion above, we propoed narrower bound on the probabilitie of cauation. When the monotonicity aumption P(y x,y x ) = i added, we can derive the point etimator baed on covariate adjutment. In epidemiology, thi aumption i often expreed a no-prevention, which mean that no individual in the population can be helped by expoure to a rik factor, that i, a hazardou expoure i either harmful or indifferent to every member of the population. The generalization of monotonicity aumption i offered through the ue of the conditional monotonicity aumption P(y x,y x ) =. Thi aumption implie that both P(y x y x,x,)=p(y x y, x,)= P(y x y x,x,)= P(y x y,x,)= hold true. Then, from equation (7), by noting that P(y x y,x,) = a.var( PN ˆ ) = ( (1 PN ) 2 P(y x, )P(y x, ) NP(x, ) + P(y x,)p(y x )( ) 2,) P(x, ) NP(x.,) P(x, y) (Cai Kuroki, 25). By the imilar procedure, ince we can obtain PNS() =P(y x ) P(y x ), the PNS can be identified a PNS = PNS()P(), which require obervational data or experimental data. When only obervational data i available, if we can oberve ome covariate that atify the SITA condition (Roenbaum Rubin, 1983), we can till identify PNS: PNS = (P(y x, ) P(y x,))p(), (11) which i conitent with the average caual effect provided in equation (4). The aymptotic variance of the PNS i given by a.var( PNS ˆ ) = ( P(y x, )P(y x, ) NP(x, ) + P(y x,)p(y x ),) NP(x P() 2.,) (Cai Kuroki, 25). Moreover, when two different ubet S T of the covariate atify the SITA condition, the covariate election problem occur: whether it i better to ue both of them than to ue one of them in order to obtain a point etimator with maller variance. Regarding thi problem, we provide the following reult: (1) if Y T {X, S} hold true from data, we can obtain a.var( ˆ PN ) a.var( ˆ PN,t ),

5 a.var( PNS ˆ ) a.var( PNS ˆ,t ); (2) if X S T hold true from data, we can obtain a.var( ˆ PN,t ) a.var( ˆ PN t ) a.var( PNS ˆ,t ) a.var( PNS ˆ t ). The proof i provided in Appendix. We can decribe uch ituation a the graph hown in Fig.2, which indicate that both X S T Y T {X, S} hold true. Thee conditional independence relationhip can be read off from the graph by the d-eparation criterion (Pearl, 2). Since both Fig. 2: Graphical repreentation of covariate election T S atify the SITA condition relative to (X, Y ) both Y T {X, S} X S T hold true, we can obtain a.var( ˆ PN ) a.var( ˆ PN t, ) a.var( ˆ PN t ) a.var( PNS ˆ ) a.var( PNS ˆ t, ) a.var( PNS ˆ t ). That i, the aymptotic variance when S i ued i maller than that when T i ued under the condition above. Thee reult provide qualitative relationhip between different covariate that atify the SITA condition, which indicate that it i not alway better to ue a much a covariate information in order to etimate the probabilitie of cauation. 4 Simulation experiment We compare the variance in ection 3.3 through imulation experiment. For implicity, we only conider the cae in Fig. 2, where there are two oberved dichotomou covariate S T. We conider four cenario of the odd ratio between X T between S T, which i hown in Table 1. The value in each cell repreent the probability P (x,, t), which atifie X S T. Setting 1 repreent the cae where both the odd ratio between X T between S T are larger than 1. Setting 2 repreent the cae where the odd ratio between X T i larger than 1 but that between S T i cloe to 1. Setting 3 repreent the cae where the odd ratio between X T i cloe to 1 but that between S T i larger than 1. Setting 4 repreent the cae where both the odd ratio between X T between S T are cloe to 1. In addition, the etting of conditional probabilitie of Y given X S are fixed at (P(y x, 1 ), P(y x, 2 ), P(y x, 1 ), P(y x, 2 )) = (.7,.3,.8,.4). In order Table 1: Four Parameter Setting Setting 1 Setting 2 t 1 t 2 t 1 t x x Setting 3 Setting 4 t 1 t 2 t 1 t x x to verify the propertie of the variance in Section 3.3, we did imulation experiment baed on the four etting in ample ize N = 5, 1, 15 2, repectively. Table 2 report the variance etimate from 5 replication in variou ample ize. S mean the variance when a covariate S i ued to evaluate the PN the PNS, T mean the variance when a covariate T i ued to evaluate the PN the PNS, {S, T } mean the variance when both S T are ued to evaluate the PN the PNS. The firt line how the value of variance obtained from imulation experiment, denoted a var, the econd line how the value of aymptotic variance calculated from the formula in Section 3, denoted a a.var. From Table 2, we draw the following concluion. (1) The ratio of variance to aymptotic variance i cloe to 1., which how that the aymptotic variance are ufficient approximation of the variance. (2) In all cae, the variance when S i elected i maller than the variance when T or {S, T } i elected, which i conitent with the reult in ection 3.3. In addition, the variance when T i elected i larger than the variance when {S, T } i elected, which i alo conitent with the reult in ection 3.3. (3) The variance vary in each cae, which may reult from the different parameter etting. 5 Example The above reult are applicable to analyze the data from the Northern Alberta Breat Cancer Regitry (Newman, 21). Thi data, which i hown in Table 3, wa collected to invetigate the effect of receptor level on breat cancer urvival. It wa alo reanalyzed by Greenl (24), with the purpoe of dicuing the attributable fraction rik ratio. The ize of

6 Table 2: Simulation Reult on the Variance PN Setting 1 Setting 2 S T {S, T } S T {S, T } N = 5 var a.var N = 1 var a.var N = 15 var a.var N = 2 var a.var PN Setting 3 Setting 4 S T {S, T } S T {S, T } N = 5 var a.var N = 1 var a.var N = 15 var a.var N = 2 var a.var PNS Setting 1 Setting 2 S T {S, T } S T {S, T } N = 5 var a.var N = 1 var a.var N = 15 var a.var N = 2 var a.var PNS Setting 3 Setting 4 S T {S, T } S T {S, T } N = 5 var a.var N = 1 var a.var N = 15 var a.var N = 2 var a.var the ample i 192 the variable of interet are the following: X: Receptor Level (x: high; x : low). Y : Survival Indicator (y: death; y : urvive). S: Stage at Diagnoi ( 1 : tage 1; 2 : tage 2; 3 : tage 3). Table 3: Receptor Level-Breat Cancer Study (Newman, 21) x x x x x x y y In thi example, we aume that S atifie the SITA condition. Since only obervational data i available, the propoed bound of equation (8) (9) can be written a { P (y x,) P (y ) P (x, y) min P () PN P (x, y ) P (x, y) P (y x,) P (x,y ) P (x, y) P (y x, ) P (y x,) } P () PNS P ()

7 { P (y x, ) min P (y x,) } P (). t P(y x,, t)p(y x,, t) P(x,, t) 2 =. P(x,, t) Regarding the lower bound of the PN the PNS, the conditional rik difference take the ame ign in all the trata (tage 1:.76; tage 2:.179; tage 3:.257), which indicate that the propoed lower bound i equal to Tian-Pearl lower bound. On the other h, regarding the upper bound of the PN the PNS, in tage 1 tage 2, the ign of P(y x, ) P(y x,) are the ame (tage 1:.742; tage 2:.361), but are different from that in tage 3 (tage 3:.457), which indicate that the propoed upper bound i maller than Tian-Pearl upper bound. By calculation, the propoed bound of the PN i (.,.778), Tian-Pearl bound i (., 1.), which how that the propoed bound can provide more information for judging probability of cauation. In addition, the propoed bound of the PNS i (.,.168), which i alo narrower than Tian-Pearl bound (.,.237). 6 Concluion Probabilitie of cauation are widely ued in epidemiology, artificial intelligence public policy analyi. Therefore, bounding identifying the probabilitie of cauation i an important problem. In many experimental obervational tudie, uually not only the expoure outcome variable are meaured, but alo ome covariate intermediate variable are meaured. Thi information enable u to narrow the bound of the probabilitie of cauation. In thi paper, we defined conditional probabilitie of cauation, ued thi definition to propoe narrower bound than Tian-Pearl bound by tratifying on ome meaured covariate intermediate variable. We alo conider the identification of the probabilitie of cauation baed on tratified analyi. Since covariate election problem occur in thi ituation, we further compared different cae of covariate election from the viewpoint of etimation accuracy. Finally, we gave imulation reult analyzed an empirical data by uing our formula. With thee new reult added to the framework of Tian Pearl (2a, 2b), the probabilitie of cauation hould find wider ue in more more area that require the evaluation of caual effect. Appendix Regarding the PN, when Y T {X, S} hold true, we can obtain P(y x, )P(y x, ) P(x, ) 2 P(x, ) In addition, ince we can obtain P(x ) t = t P(x t, ) 2 P(t ) P(x t, ) P(x t, )P(t ) t P(x t, ) 2 P(t ) P(x P(x ) 2 t, ) by the Cauchy-Schwarz Inequality, P(y x,)p(y x,) P(x P(x, ) 2,) P(y x,,t)p(y x,,t) P(x P(x,, t) 2,,t) t = P(y x,)p(y x,)p() ( P(x ) 2 P(x ) ) P(x t, ) 2 P(t ) P(x. t, ) t Thu, by noting that PN = PN,t hold true, we can obtain a.var( PN ˆ ) a.var( PN ˆ,t ). Next, when X S T hold true, by the variance baic formula, we can obtain P(y x, t)p(y x, t) P(x, t) 2 P(x, t) P(y x,, t)p(y x,, t) P(x,, t) 2 P(x,, t) P(y x,, t)p(y x,, t) (P(x, t)p(x,, t) P(x,, t) P(x,, t) 2 ) P(y x,t)p(y x,t) P(x P(x, t) 2,t) P(y x,,t)p(y x,,t) P(x P(x,, t) 2,,t) P(y x,,t)p(y x,,t) P(x (P(x, t)p(x,, t),,t) P(x,, t) 2). Thu, by noting that PN t = PN,t hold true, we can obtain a.var( PN ˆ,t ) a.var( PN ˆ t ). Regarding the PNS, when Y T {X, S} hold true, we can obtain a.var( PNS ˆ t, ) a.var( PNS ˆ ) = { P(y x, )P(y x, ) P() N P(x ) x ( P(x ) )} P(t ) P(x t, ) 1. t

8 Here, by the Cauchy-Schwarz Inequality, P(x ) t = t P(t ) P(x t, ) P(x t, )P(t ) t P(t ) P(x t, ) 1. Thu, we can obtain a.var( PNS ˆ ) a.var( PNS ˆ t, ). When X S T hold true, by comparing a.var( PNS ˆ t ) with a.var( PNS ˆ t, ), we can obtain a.var( PNS ˆ t ) a.var( PNS ˆ t, ) = { P(t) ( P(y x, t)p(y x, t) NP(x t) x t )} P(y x,, t)p(y x,, t)p( t). By uing the variance baic formula, we can obtain a.var( PNS ˆ,t ) a.var( PNS ˆ t ). Acknowledgement The author would like to thank Judea Pearl of UCLA for hi helpful dicuion about the paper. Thi reearch wa upported by the Minitry of Education, Culture, Sport, Science Technology of Japan, the Sumitomo Foundation, the Murata Overea Scholarhip Foundation, the Kayamori Foundation of Information Science Advancement, the College Women Aociation of Japan the Japan Society for the Promotion of Science. REFERENCES Bowden, R. J., Turkington, D. A. (1984). Intrumental Variable, Cambridge Univerity Pre. Cai, Z. Kuroki, M. (25). Variance Etimator for three Probabilitie of Cauation, Rik Analyi, 25, Greenl, S. (1987). Variance etimator for attributable fraction etimate conitent in both large trata pare data. Statitic in Medicine, 6, Greenl, S. (2). An introduction to intrumental variable for epidemiologit. International Journal of Epidemiology, 29: Greenl, S. (23). Quantifying biae in caual model: Claical confounding veru collidertratification bia. Epidemiology, 14, Greenl, S. (24). Model-baed Etimation of Relative Rik Other Epidemiologic Meaure in Studie of Common Outcome in Cae-Control Studie. American Journal of Epidemiology, 16, Greenl S., Pearl J. Robin J. M. (1999). Caual diagram for epidemiologic reearch. Epidemiology, 1, Newman, S. C. (21). Biotatitical method in epidemiology, Wiley. Pearl, J. (2). Cauality: Model, Reaoning, Inference, Cambridge Univerity Pre. Robin, J. M. (1989). The analyi of romized non-romized AIDS treatment trial uing a new approach to caual inference in longitudinal tudie. Health Service Reearch Methodology: A Focu on AIDS. Ed: Sechret L., Freeman H., Mulley A. Wahington, D.C.: U.S. Public Health Service, National Center for Health Service Reearch., Robin, J. M. Greenl, S. (1989). The probability of cauation under a tochatic model for individual rik. Biometric, 45, Roenbaum, P. Rubin, D. (1983). The central role of propenity core in obervational tudie for caual effect. Biometrika, 7, Rubin, D. B. (25). Caual Inference Uing Potential Outcome: Deign, Modeling, Deciion. Journal of the American Statitical Aociation, 1, Tian, J. Pearl, J. (2a). Probabilitie of cauation: Bound identification. Annal of Mathematic Artificial Intelligence, 28, Tian, J. Pearl, J. (2b). Probabilitie of cauation: Bound identification. Proceeding of 16th Conference on Uncertainty in Artificial Intelligence, Tian, J. (24). Identifying conditional caual Effect. Proceeding of 2th Conference on Uncertainty in Artificial Intelligence,