Conceptual issues concerning mediation, interventions and composition


 Clementine Osborne
 1 years ago
 Views:
Transcription
1 Statistics and Its Interface Volue 2 (2009) Conceptual issues concerning ediation, interventions and coposition Tyler J. VanderWeele and Stijn Vansteelandt Concepts concerning ediation in the causal inference literature are reviewed. Notions of direct and indirect effects fro a counterfactual approach to ediation are copared with those arising fro the standard regression approach to ediation of Baron and Kenny (1986), coonly utilized in the social science literature. It is shown that concepts of direct and indirect effect fro causal inference generalize those described by Baron and Kenny and that under appropriate identification assuptions these ore general direct and indirect effects fro causal inference can be estiated using regression even when there are interactions between the priary exposure of interest and the ediator. A nuber of conceptual issues are discussed concerning the interpretation of identification conditions for ediation, the notion of counterfactuals based on hypothetical interventions and the so called consistency and coposition assuptions. Keywords and phrases: Causal inference, Counterfactuals, Direct and indirect effects, Intervention, Mediation. The notion of ediation concerns the extent to which the effect of one variable on another is ediated by soe possible interediate variable. As such, notions of ediation concern causality. The use of ediation analysis has becoe quite coon in the social sciences. An approach based on regression analysis advocated by Baron and Kenny (1986) is now utilized routinely, especially within the literature on psychology. More recently, an approach to ediation arising fro the causal inference literature and based on the notion of counterfactuals has been proposed (Robins and Greenland, 1992; Pearl, 2001). The current paper is structured in two parts. In the first part we will review notions of direct and indirect effects fro the causal inference literature on ediation and will relate these notions to the approach advocated by Baron and Kenny (1986). We will show how the notions fro causal inference generalize those arising fro the Baron and Kenny approach. In particular, under certain identification conditions, the approach based on causal inference allows for the definition of direct and indirect effects and for effect decoposition of a total effect into a direct and indirect effect even in settings with interaction and nonlinearities. Under appropriate identification conditions, the Corresponding author. direct and indirect effects defined in the causal inference literature can also be estiated in a regression fraework. In the second part of the paper we will consider in soe detail a nuber of conceptual issues raised by the causal inference or counterfactual approach to ediation. We will discuss the constraints iposed by identification conditions for ediation; we will discuss the extent to which an approach to ediation based on interventions is possible and we will finally consider the interpretation of a conceptual assuption, soeties referred to as coposition, which is ade in the causal inference work on ediation when decoposing total effects into direct and indirect effects. DIRECT AND INDIRECT EFFECTS IN CAUSAL INFERENCE To foralize the eaning of a direct effect, we conceptualize for each subject the existence of a socalled counterfactual outcoe Y (a), which denotes the outcoe that we would possibly contrary to fact have observed for that subject had the exposure A been set to the value a through soe intervention or anipulation (Rubin, 1978; Hernán, 2004). Variables such as Y (a) are referred to as potential outcoes or counterfactual outcoes. If the exposure A is dichotoous (e.g., taking values 0 for no exposure and 1 otherwise), then we can think of each subject having 2 such counterfactual outcoes, Y (0) and Y (1). The (average) causal effect of the exposure on the outcoe can then be defined as the expected difference E[Y (1) Y (0)] between both counterfactual outcoes for the sae study population. This is to be contrasted with the ore usual expected difference E[Y A = 1] E[Y A = 0], where Y denotes the observed outcoe, which ay not carry the interpretation of a causal effect when the subgroups of exposed and unexposed subjects are not inherently coparable. More generally, we define the conditional causal effect of exposure level a versus 0 (where we let 0 denote an arbitrary reference level) on the outcoe, given preexposure covariates C, as the expected contrast E[Y (a) Y (0) C]. To be able to identify total causal effects, the following 2 assuptions ust be ade. The socalled consistency assuption states that aongst subjects with observed exposure level A = a, the observed outcoe Y isequaltothe potential outcoe Y (a) (i.e., Y (a) =Y when A = a). Under this assuption, we can observe one of the potential
2 outcoes for each subject, naely the one corresponding to the observed exposure level (i.e., Y = Y (A)). We return to this assuption below. We need one further assuption for the identification of total causal effects; we also need soe additional notation. For rando variables A, B and C, let A B C denote that A is conditionally independent of B, given C. For the identification of total causal effects, we will assue that subjects with different observed exposure levels A, but the sae preexposure characteristics C, are coparable in the sense that Y (a) A C for all exposure levels a. This assuption states that a subject s choice of exposure level A, while possibly related to preexposure characteristics C, has no residual dependence on how that subject would fare under an arbitrary, fixed exposure level. It is usually referred to as the no uneasured confounders assuption as it effectively states that the variables in C are the only confounders of the association between exposure and outcoe. Both these assuptions cannot be tested on the basis of the observed data, but, in cobination, are sufficient for identifying the conditional causal effect as E[Y (a) Y (0) C] = E[Y (a) C] E[Y (0) C] = E[Y (a) A = a, C] E[Y (0) A =0,C] = E[Y A = a, C] E[Y A =0,C]. We could take averages over the distribution of C to obtain average causal effects, E[Y (1) Y (0)]. By extending the previous concepts to a joint exposure (A, M) where M is the potential ediator, definitions of direct and indirect effects can be constructed. For each subject, let us define Y (a, ) to be the outcoe that we would possibly contrary to fact have observed for that subject had the exposure A been set to the value a and, likewise, M to the value, through soe intervention or anipulation. Siilarly, we can consider counterfactual variables M(a) which denote the value of the ediator if possibly contrary to fact the exposure A were set to a. For a dichotoous exposure, the controlled direct effect of the exposure on the outcoe, controlling for M, can then be defined as the expected contrast E[Y (1,) Y (0,)] (Robins and Greenland, 1992; Pearl, 2001). It expresses the exposure effect that would be realized if the ediator were controlled at level uniforly in the population. For instance, in accordance with Figure 1, let A be the father s occupation, let Y be the respondent s incoe, let M be the respondent s occupation, let C 1 be the father s education, and let C 2 be the respondent s education. Then E[Y (a, ) Y (0,)] expresses the average change in incoe that would be realized in a subgroup of respondents if their father changed occupation (fro 0 to a), but their own occupation were kept Figure 1. Exaple of the effect of A on Y ediated by M with both exposureoutcoe confounders and ediatoroutcoe confounders. uniforly at level. More generally, we will define the conditional controlled direct effect of exposure level a versus 0 on the outcoe (controlling for M), given covariates C, as the expected contrast E[Y (a, ) Y (0,) C]. The consistency assuption for joint exposure (A, M) is then that aongst the subgroup with observed exposure A = a and observed ediator M =, the observed outcoe Y is equal to Y (a, ). The consistency assuption for the effect of the exposure on the ediator is that aongst the subgroup with observed exposure A = a the observed ediator M is equal to M(a). The assuption of nouneasuredconfounders for the exposureoutcoe relationship can then be expressed as (1) Y (a, ) A C for all levels of a and. However, controlled direct effects in general require stronger conditions for identification than do total causal effects. This is because the definition of a controlled direct effect requires evaluating the ipact of holding the ediator M fixed. For this purpose, one ust know all confounders of the association between ediator and outcoe, which we forally express as (2) Y (a, ) M A, C for all levels of a and. To identify controlled direct effects, the set C ust contain all of the confounders of both the exposureoutcoe relationship and the ediatoroutcoe relationship i.e. in Figure 1 control ust be ade for both C 1 and C 2 to identify controlled direct effects. If control is not ade for the variables that confound the relationship between the ediator and the outcoe (the variables C 2 in Figure 1) then estiates of direct effects will generally be biased (Cole and Hernán, 2002). In the early ediation literature, this point about controlling for the ediatoroutcoe confounders was ade by Judd and Kenny (1981) but was not pointed out by Baron and Kenny (1986) and was also subsequently ignored by uch of the social science literature on ediation. The iportance of controlling for the confounders of the ediatoroutcoe relationship has been 458 T. J. VanderWeele and S. Vansteelandt
3 Figure 2. Exaple of an effect of the exposure confounding the ediatoroutcoe relationship. ephasized in the causal inference literature on ediation (Robins and Greenland, 1992; Pearl, 2001; Cole and Hernán, 2002). If assuptions (1) and (2) hold, then controlled direct effects are identified (Pearl, 2001) by E[Y (a, ) Y (0,) C] = E[Y A = a, M =, C] E[Y A =0,M =, C]. We could take averages over the distribution of C to obtain average controlled direct effects, E[Y (a, ) Y (0,)]. Note that, because the ediator M arises later in tie than the exposure A, it is possible that the exposure itself ay affect one or ore of the confounders of the ediatoroutcoe relationship. Such an exaple is given in Figure 2 with L being both an effect of exposure A and a confounder of the ediatoroutcoe relationship. Note that in Figure 2, L could also be considered to be a ediator of the effect of A on Y ; which variable is taken to be the ediator of interest will depend on the context. In this paper we will use M to denote the ediator of interest and L to denote any confounders of the ediatoroutcoe relationship that are affected by the exposure A. When there are confounders of the ediatoroutcoe relationship that are affected by the exposure, condition (2) can be odified to Y (a, ) M A, C, L for all levels of a and and then controlled direct effects can still be identified (Robins and Greenland, 1992; Pearl, 2001; VanderWeele, 2009a). For ost of this paper, however, we will restrict our attention to settings in which the confounders of the ediatoroutcoe relationship are not affected by exposure. We will consider the scope of such settings further below. There are a nuber of liitations to the concept of a controlled direct effect. First, as in the previous exaple, it is often not realistic to iagine scenarios where one would consider forcing the ediator to be the sae for all subjects in the population (e.g. forcing all subjects to have the sae occupation). Second, indirect effects cannot be defined in a siilar anner as controlled direct effects because it is ipossible to hold a set of variables fixed, in such a way that the effect of exposure on outcoe would circuvent the direct pathway. In particular, the total causal effect, say E[Y (a) Y (0)], inus the controlled direct effect, say E[Y (a, ) Y (0,)], ay not carry the interpretation of an indirect effect. Because of potential exposureediator interactions, the difference ay be nonzero even if the expsoure has no effect on the ediator so that none of the effect of the exposure is ediated by M (Kaufan et al., 2004; VanderWeele 2009b). Both liitations can be overcoe by considering socalled natural direct effects (Pearl, 2001; Robins, 2003) which ay be defined as the expected contrast E[Y (a, M(0)) Y (0,M(0))]. Robins and Greenland (1992) refer to this quantity as the pure direct effect to distinguish it fro the total direct effect, E[Y (a, M(a)) Y (0,M(a))]; both are instances of what Pearl (2001) calls natural direct effects and we will thus use the ters pure natural direct effect and total natural direct effect. The pure natural direct effect, E[Y (a, M(0)) Y (0,M(0))], expresses the effect that would be realized if the exposure were adinistered, but its effect on the ediator were soehow blocked, or equivalently, if the ediator were kept at the level it would have taken in the absence of the exposure. More generally, we will define the conditional pure natural direct effect of exposure level a versus 0 on the outcoe (other than through odifying M), given preexposure covariates C, as the expected contrast E[Y (a, M(0)) Y (0,M(0)) C]. In the context of the exaple, this expresses the average change in incoe that would be realized in a subgroup of respondents (all of whose fathers had the sae education) if their father changed occupation (fro 0 to a), but they kept their own occupation. While controlled direct effects are often of greater interest in policy evaluation (Pearl, 2001; Robins, 2003), natural direct and indirect effects ay be of greater interest in evaluating the action of various echaniss (Robins, 2003; Joffe et al., 2007). In considering natural direct effects, one typically akes not only the consistency assuption but also a coposition assuption that Y (a) = Y (a, M(a)) i.e. that the potential outcoe Y (a) intervening to set A to a is equal to the potential outcoe Y (a, M(a)) intervening to set A to a and to set M to the value it would have been if A had been a. Under this coposition assuption the pure natural direct effect E[Y (1,M(0)) Y (0,M(0)) C] can also be expressed as E[Y (1,M(0)) Y (0) C]. We return to the interpretation of the coposition assuption below. The use of natural direct effects overcoes the previously listed liitations of controlled direct effects. First, this is so because the level M(0) at which the ediator is controlled allows for natural variation between subjects. Second, this is so because the difference between the total causal effect and a pure natural direct effect E[Y (a) Y (0) C] E[Y (a, M(0)) Y (0) C] = E[Y (a, M(a)) Y (a, M(0)) C] Conceptual issues concerning ediation, interventions and coposition 459
4 expresses how uch the outcoe would change on average if the exposure were controlled at level a, but the ediator were changed fro level M(0) to M(a). It thus carries the interpretation of an indirect effect and will be tered the total natural indirect effect (Robins and Greenland, 1992; Robins, 2003). Iportantly, the above effect decoposition does not assue that the functional for relating A, M, and Y is linear nor that there is no interaction between the effects of A and M on Y. Likewise, the difference between the total effect and the total natural direct effect, E[Y (a, M(a)) Y (0,M(a)) C] =E[Y (a) Y (0,M(a)) C], gives E[Y (a) Y (0) C] E[Y (a) Y (0,M(a)) C] = E[Y (0,M(a)) Y (0,M(0)) C], which expresses how uch the outcoe would change on average if the exposure were controlled at level 0, but the ediator were changed fro its natural level M(0) to the level M(a) which it would have taken at exposure level a. This is tered the pure natural indirect effect (Robins and Greenland, 1992; Robins, 2003). In the context of the exaple, this expresses the average change in incoe that would be realized in a subgroup of respondents (all of whose fathers had the sae education) if their father s occupation were uniforly controlled at the reference level 0, but they changed their occupation to what they would have had if their father had a different occupation a. Controlled and natural direct and indirect effects can all be defined so as to copare levels of exposure a and a rather than a and 0. Thus, the controlled direct effect under this coparison is E[Y (a, ) Y (a,) C]; the pure natural direct effect is E[Y (a, M(a )) Y (a,m(a )) C]; the total natural indirect effect is E[Y (a, M(a)) Y (a, M(a )) C]; the total natural direct effect is E[Y (a, M(a)) Y (a,m(a)) C]; and the pure natural indirect effect is E[Y (a,m(a)) Y (a,m(a )) C]. Identification of natural direct effects, like controlled direct effects, requires assuptions (1) and (2) but relies on additional assuptions as well. First, to be able to assess what values the ediator would take if the exposure were controlled, one ust have easured all confounders of the association between exposure and ediator, which we forally express as (3) M(a) A C for all levels of a. In addition to assuptions (1) (3) the identification of natural direct and indirect effects is generally ade by iposing an additional assuption. Pearl (2001), for exaple, effectively identifies natural direct and indirect effects by also assuing (4) Y (a, ) M(a ) C for all levels of a, a and. For exaple, for the natural direct effect Y (a, M(0)) Y (0,M(0)) we would need (4) to hold for a = 0. Condition (4) is soewhat difficult to interpret but will generally hold if condition (2) holds and if there are no variables L that are effects of exposure and that confound the ediatoroutcoe relationship (Pearl, 2001). Under assuptions (1) (4), the natural direct effect can be identified (Pearl, 2001) by (5) E[Y (a, M(0)) Y (0,M(0)) C] = {E[Y A = a, M =, C] E[Y A =0,M =, C]} f(m = A =0,C)d. Natural indirect effects can be obtained by subtracting a natural direct effect fro a total effect as explained above. Note that even in settings in which the identification conditions for controlled and natural direct and indirect effects are not satisfied, data can soeties still be used to obtain bounds on these effects (Kaufan et al., 2005, 2009; Cai et al., 2008; Sjölander, 2009). Alternatively, instead of assuption (4), one can identify natural direct and indirect effects by aking the nointeraction assuption that Y (a, ) Y (0,) is a rando variable that does not depend on. This assuption states that the effect of changing the exposure fro 0 to a while holding the ediator fixed, is the sae no atter what value to which one intervenes to fix the ediator. Under assuptions (1) and (2) and the nointeraction assuption, natural direct effects can be identified (Robins, 2003). Petersen et al. (2006) consider an assuption that essentially allows for identification of natural direct and indirect effects under assuptions (1) (3) along with a disjunction of assuption (4) and the nointeraction assuption. In the next section, we will show how the approach of Baron and Kenny (1986) of using regression to assess ediation can be extended to settings including interactions. Instead of the nointeraction assuption, we will thus assue that conditions (1) (4) hold. That is, we will restrict our consideration to settings in which there are no variables L that are effects of exposure and that confound the ediatoroutcoe relationship and we will assue the set C contains all confounders of the exposureoutcoe, ediatoroutcoe and exposureediator relationships. DIRECT AND INDIRECT EFFECTS USING REGRESSION In assessing ediation, Baron and Kenny (1986) proposed using regression odels such as (6) E[M A = a, C = c] =β 0 + β 1 a + β 2c and (7) E[Y A = a, M =, C = c] =θ 0 + θ 1 a + θ 2 + θ 4c 460 T. J. VanderWeele and S. Vansteelandt
5 in order to assess ediation. They proposed that the direct effect be assessed by estiating θ 1 and that the indirect effect be assessed by estiating β 1 θ 2 ; subsequent literature has discussed a variety of estiation procedures (MacKinnon, 2008). Iportantly, the regression odels Baron and Kenny proposed do not include an interaction ter θ 3 a in the regression odel for Y. In this section we show how the notions of direct and indirect effects fro causal inference introduced in the previous section can extend the Baron and Kenny approach in settings in which there is an interaction ter θ 3 a in the regression odel for Y i.e. in settings in which the exposure and the ediator interact in their effects on the outcoe. We assue that the ediator M and outcoe Y are continuous; the results will apply to arbitrary exposures A. When the ediator M is dichotoous rather than continuous a soewhat siilar approach to the one described here could potentially be used; however, if the odel for M is logistic, rather than linear, then the analytic forulas for the natural direct and indirect effects no longer take quite as siple a for. Suppose then that, instead of using regression odels (6) and (7), one were to use regression odel (6) and the following regression odel (8) E[Y A = a, M =, C = c] = θ 0 + θ 1 a + θ 2 + θ 3 a + θ 4c i.e. one were to use the regression odels of Baron and Kenny (1986) but to include a product ter θ 3 a in the regression odel for Y. In the appendix we show that under assuptions (1) (4) and under correct specification of the regression odels, the controlled direct effect and the natural direct and indirect effects are given by: (9) E[Y (a, ) Y (a,) C] =(θ 1 + θ 3 )(a a ) E[Y (a, M(a )) Y (a,m(a )) C] =(θ 1 + θ 3 β 0 + θ 3 β 1 a + θ 3 β 2C)(a a ) E[Y (a, M(a)) Y (a, M(a )) C] =(θ 2 β 1 + θ 3 β 1 a)(a a ). Note that the expression for the controlled direct effect only requires assuptions (1) and (2) and that the regression odel (8) is correctly specified; the expressions for natural direct and indirect effects requires assuptions (1) (4) and correct specification of odels (6) and (8). Several points erit attention. First, if there were indeed no product ter θ 3 a in the regression odel, i.e. if θ 3 =0, then the expressions in (9) reduce to E[Y (a, ) Y (a,) C] =θ 1 (a a ) E[Y (a, M(a )) Y (a,m(a )) C] =θ 1 (a a ) E[Y (a, M(a)) Y (a, M(a )) C] =θ 2 β 1 (a a ) and thus the controlled and the natural direct effects coincide and are equal to θ 1 for a one unit change in a, whichis the direct effect given by the Baron and Kenny approach; the natural indirect effect is equal to θ 2 β 1 for a one unit change in a, which thus coincides with the indirect effect given by the Baron and Kenny approach. Thus when θ 3 =0,the notions of direct and indirect effects fro causal inference reduce to the effects given in Baron and Kenny s proposal. Second, when there is interaction so that θ 3 0, one can still define direct and indirect effects as described above and one can still decopose a total effect into a natural direct effect and a natural indirect effect; one can oreover, still use regression odels to estiate natural direct and indirect effects and doing so gives the expressions given in (9) provided assuptions (1) (4) hold. The definitions of natural direct and indirect effects given in the causal inference literature thus usefully extend concepts of direct and indirect effects fro the social science literature to include settings with interactions. The definitions fro causal inference given above furtherore apply also to settings with nonlinear odels (cf. Pearl, 2001; van der Laan and Petersen, 2008; VanderWeele, 2009a). Third, the expressions given in (9) are for the pure natural direct effect, E[Y (a, M(a )) Y (a,m(a )) C], and the total natural indirect effect, E[Y (a, M(a)) Y (a, M(a )) C]. One can siilarly obtain expressions for the total natural direct effect, E[Y (a, M(a)) Y (a,m(a)) C], and the pure natural indirect effect, E[Y (a,m(a)) Y (a,m(a )) C]. Using calculations siilar to those in the appendix one obtains E[Y (a, M(a)) Y (a,m(a)) C] =(θ 1 + θ 3 β 0 + θ 3 β 1 a + θ 3 β 2C)(a a ) E[Y (a,m(a)) Y (a,m(a )) C] =(θ 2 β 1 + θ 3 β 1 a )(a a ). The expression for the total natural direct effect differs fro pure natural direct effect given in (9) in that it has the ter θ 3 β 1 a, rather than θ 3 β 1 a. The expression for the pure natural indirect effect differs fro total natural indirect effect given in (9) in that it has the ter θ 3 β 1 a rather than θ 3 β 1 a. Finally, if the forulas in (9) are used to estiate direct and indirect effects by using estiates fro linear regressions (6) and (8) then standard errors for the estiates of these direct and indirect effects could be obtained either through bootstrapping (Efron and Tibshirani, 1993; MacKinnon, 2008) or by the delta ethod as described in the appendix. In this section we have restricted attention to the setting in which there are no effects of exposure that confound the ediatoroutcoe relationship. Soe progress can be ade even in settings in which there are such exposureinduced ediatoroutcoe confounders but estiation in such cases requires techniques beyond siple regression analysis (Robins, 1999; van der Laan and Petersen, 2008; Conceptual issues concerning ediation, interventions and coposition 461
6 Goetgeluk et al., 2008; VanderWeele, 2009a; Vansteelandt, 2009). In the next section we will consider in greater detail this assuption that there are no effects of exposure that confound the ediatoroutcoe relationship. CONCEPTUAL ISSUES CONCERNING THE IDENTIFICATION OF NATURAL DIRECT AND INDIRECT EFFECTS In the previous section, to derive the expressions for direct and indirect effects given in (9) we have assued that condition (4) would hold. Condition (4) allowed for the identification of natural direct and indirect effects but condition (4) is a strong assuption. Pearl (2001) gives a graphical interpretation of condition (4) essentially showing that it requires that there be no consequence of exposure that confounds the relationship between the ediator and the outcoe. Condition (4) will be violated if there are such exposureinduced confounders, irrespective of whether or not data is available on the. Condition (4) contrasts with conditions (1) (3). Conditions (1) (3) could potentially be satisfied, at least approxiately, by collecting data on ore and ore confounding variables C; conditions (1) (3) are nouneasuredconfounding assuptions. Condition (4) however requires that there be no effect of exposure that confounds the relationship between the ediator and the outcoe; if there are such variables condition (4) will be violated irrespective of whether data is available for all such variables. In any contexts, condition (4) will not be reasonable. Thus in the exaple concerning the effects of a father s occupational choice, A, on a respondent s incoe, Y, as ediated by respondent s occupation, M, if the father s occupational choice affects the respondent s education level, which we ight denote by L in Figure 2, then condition (4) will be violated. If the father s occupational choice does not affect the respondent s education level, as in Figure 1 where we would denote respondent s education level by C 2,thencondition (4) ay be satisfied. In general one can iagine any variables on the pathway between the exposure of interest and the ediator. Condition (4), that there are no effects of exposure that confound the ediatoroutcoe relationship, essentially then requires that of all the variables that are on the pathway fro the exposure to the ediator, none of these also affects the outcoe. As noted above, this assuption will often be violated and in such cases, natural direct and indirect effects will not in general be identified. There are exceptions; for exaple, natural direct and indirect effects can be identified as in (5) without assuption (4) if the nointeraction assuption holds (Robins, 2003); see Petersen et al. (2006), Hafean and VanderWeele (2009) and Iai et al. (2009) for other exceptions. However, it has been shown that whenever there is a consequence of the exposure that also affects both the ediator and the outcoe then natural direct and indirect effects are not in general identified (Avin et al., 2005). Perhaps one notable exception in which assuption (4) is ore reasonable are settings in which the ediator is easured iediately after, or very shortly after, the exposure takes place. For exaple, Nelson et al. (1997) exaine the extent to which the fraing of political issues in news edia affected subjects tolerance of a Ku Klux Klan rally as ediated by affecting subjects general political attitudes. Subjects in the study were randoized to watch one of two news clips, one of which presented the rally as a free speech issue and the other of which as a public order issue. Following the clip, survey questions were used to assess general political attitudes towards the right to free speech and the aintenance of public order and two further questions were used to assess subjects tolerance for Klan speeches and rallies. The analysis is presented in Nelson et al. (1997) and is reanalyzed using conteporary ideas fro causal inference by Iai et al. (2009). The point here, however, is that since political attitudes are assessed iediately after the exposure (the watching of the video clip) it is less likely that there is anything that is an effect of exposure that confounds the ediatoroutcoe relationship. Thus assuption (4) ight be reasonable in this case; the exposure (video clip) was randoized so assuptions (1) and (3) would hold. Thus provided that control is adequately ade for variables that confound the ediatoroutcoe relationship so that assuption (2) holds, one could proceed with the estiation of natural direct and indirect effects. We will consider another siilar exaple. In a study by Seesters et al. (2003), the investigators used a onetrial prisoner s dilea gae with a fictitious partner to study the extent to which the effect of orality or ight pries on cooperative versus copetitive behavior was ediated by expectations about a partner. Expectations about the partner s behavior was assessed after exposure to either the orality or the ight prie. As with the Nelson et al. (1997) study, the ediator, in this case expectations about the behavior of the partner, could be assessed iediately after the exposure and it thus ay be plausible that there are no variables that are effects of exposure but confound the ediatoroutcoe relationship. Assuption (4) ight thus be reasonable, assuptions (1) and (3) hold by randoization of the exposure and thus, provided adequate control is ade for ediatoroutcoe confounders (assuption (2)), natural direct and indirect effects could be identified. These two exaples exhibit a siilar structure with the ediator being easured iediately after the exposure occurs thereby potentially rendering condition (4) ore plausible. One can potentially conceive of a whole class of psychological experients that follow a siilar structure to these two exaples for which assuption (4) ay be ore reasonable than in any other settings. In suary, if there is an exposureinduced confounder of the ediatoroutcoe relationship, natural direct and indirect effects are not in general identified. Controlled direct 462 T. J. VanderWeele and S. Vansteelandt
7 effects are identified in such settings (Robins and Greenland, 1992; Pearl, 2001) but require special techniques for estiation (Robins, 1999; van der Laan and Petersen, 2008; Goetgeluk et al., 2008; VanderWeele, 2009a; Vansteelandt, 2009). Unfortunately, however, controlled direct effects are of liited use in assessing ediation because the difference between a total effect and a controlled direct effect cannot in general be interpreted as an indirect effect (Kaufan et al., 2004; VanderWeele, 2009b) except in cases in which there is no interaction between the effects of the exposure and the ediator on the outcoe (Robins, 2003). Natural direct and indirect effects are desirable in that they allow for effect decoposition even in settings with interaction and nonlinearities but as we have seen above, the identification conditions required for these effects are in general quite stringent. The possibility of designing an experient so that the ediator is easured or occurs iediately after the exposure, so that assuption (4) ay be plausible, suggests a range of settings in which the assuptions required to identify natural direct and indirect effects ay be reasonable. CONCEPTUAL ISSUES CONCERNING INTERVENTIONS Thus far, we have not discussed the kind of intervention or anipulation that would enable setting M to soe given value. As iplied by the consistency assuption, the kind of interventions or anipulations that we consider are noninvasive in the sense that their only effect is to set the ediator at soe predeterined value such that the intervention has no effect aongst those for who ediator level was naturally observed. Any conclusion that we draw in ters of controlled direct effects holds for interventions satisfying this assuption. In practice, however, it is often difficult to conceive of such noninvasive interventions, either because the ediator is not anipulable, or because any conceivable anipulation would affect the outcoe in ways other through setting the ediator at soe predeterined value. Thus in the Nelson et al. (1997) study described above, the ediator was general political attitudes towards the right to free speech and the aintenance of public order. Clearly one cannot intervene on these political attitudes directly. One ight conceive of soe type of intervention to change these political attitudes such as the watching of another video clip; however, such interventions ight well also affect the outcoe in ways other than through the ediator. Different conceivable interventions to affect M so that it is set to level ight then result in different counterfactual outcoes Y (a, ); the counterfactual outcoe Y (a, ) would then not be well defined. Clearly in any psychological experients, siilar probles will arise; when the ediator is a psychological construct, such as a particular attitude, intervening directly will not in general be possible. In soe cases, however, hypothetical interventions on the ediator ay be conceivable. For exaple, in the study of Seesters et al. (2003) described above, the ediator was the subjects expectations about the behavior of the fictitious partner. One could iagine a study in which one could intervene on the subjects expectations by having the study investigators telling the subject the behavioral decision of the fictitious partner before the subject decides on his or her own action (or alternatively telling the subject a distribution of possible actions for the fictitious partner). Once the subject is told the behavior of the partner, the expectations are effectively fixed. One thus has a potential intervention on the ediator. One still ay have to be concerned with violations of the consistency assuption as to whether an individual will select the sae action under a particular set of expectations as the subject would select if the subject were told the partner s behavior was that which he or she was expecting. Further discussion of the consistency assuption and of potential violations is given in the appendix. Extending the corresponding discussion of total effects in VanderWeele (2009c), we discuss in the appendix the consistency assuptions for ediation. We clarify that essentially two things are being assued about the interventions under consideration: first, that different conceivable interventions to fix the ediator M to soe level all give rise to the sae counterfactual outcoes and, second, that interventions to set the ediator to its naturally ocurring level will give the sae outcoe as not intervening. The first assuption ay be referred to as an assuption of treatentvariation irrelevance (or noultipleversionsoftreatent) which is needed for counterfactuals of the for Y (a, ) to be welldefined and the second as an assuption of consistency. Both types of assuptions are subsued by Rubin s stable unit treatent value assuption, abbreviated SUTVA (Rubin, 1990). See also VanderWeele (2009d) for discussion of potential violations of the nointerference coponent of SUTVA within a ediation context. Even in cases in which it is not possible to conceive of a noninvasive intervention to change the ediator, the definition of a controlled direct effect can be eaningful because it effectively counicates what the exposure effect would be if nothing happened over and above fixing the ediator at soe level uniforly in the population. For instance, while there are no conceivable interventions that would fix the occupation of a given subject at soe predeterined choice, and do nothing on top of that, the question of whether the father s occupation affects the respondent s incoe through pathways other than by influencing the respondent s occupation, is arguably a eaningful one. The fact that any conceivable intervention on the respondent s occupation would do ore than just deterine the occupation iplies that there are no realistic interventions that would bring about exactly the controlled direct exposure effect. Nonetheless, estiates of the controlled direct exposure effect can be eaningful in this setting because they give a ore pure reflection Conceptual issues concerning ediation, interventions and coposition 463
8 of the direct exposure effect than could be attained through realistic interventions on the ediator. The interpretability of natural direct and indirect effects arguably hinges to a lesser extent on the ability to control or anipulate the ediator. First, the definitions of natural direct and indirect effects only require anipulations to set the ediator to levels which are naturally occurring e.g. M(0) or M(a) rather than to soe arbitrary level which, for certain individuals, ight never occur for any exposure a. Second, rather than conceiving of a natural direct effect such as E[Y (1,M(0)) Y (0,M(0))] as the effect of exposure intervening to fix the ediator at level M(0), one ight alternatively think about natural direct effects as requiring interventions that would block the exposure s effect on the ediator (Pearl, 2001; Robins, 2003). Such interventions are soeties ore easily conceivable (as well as ore natural by not necessarily requiring that the ediator be uniforly fixed at the sae level). Consider for instance a study on the effect of gender, A, on graduate adissions, Y, for graduate school applicants. Suppose that the goal of the study was to assess whether gender differences in adissions to a particular departent are entirely explained by the adission coittee s perception of gender, M, and thus potential discriination based on gender. In principle, blinding of gender at the tie of application (e.g. having all applicants in the study list either ale as gender, or having all applicants in the study list feale as gender) would bring about the natural direct effect of gender on adission because letting say A =1denote feale and A = 0 denote ale, we would have that E[Y (1,M(0)) Y (0,M(0))] = E[Y (1, 0) Y (0, 0)]. In this case the natural direct effect and the controlled direct effect in fact coincide. Robins (2003) notes that generally natural direct effects can be only interpreted as the effect of exposure on the outcoe intervening to block the effect of the exposure on the ediator if the intervention were to block the first conceivable link on the pathway fro exposure to ediator. The issue of the applicability of hypothetical interventions arise in causal inference outside the ediation context, when one is only exaining total effects (Hernán, 2005; van der Laan et al., 2005). In the occupation exaple, it is clearly as inconceivable to fix the occupation of a respondent s father (the exposure A) at soe predeterined choice, and do nothing on top of that, as it is to do so for that of the respondent (the ediator M). Many exposures or causes of interest are not obviously anipulable; exaples of such exposures ight include gender, race or psychological disposition. Without clear interventions to change these variables, counterfactual contrasts are not necessarily welldefined or are illdefined to the extent to which the intervention is not specified (Lewis, 1973; Robins and Greenland, 2000). However, in general we still are interested in exaining the effects of these variables. The counterfactual or potential outcoes fraework is perhaps not ideally suited, at least in its present for, to conceptualize the effects of such nonanipulable exposures. Nevertheless, as discussed above, the consistency assuption suggests that the inferred effects can perhaps be interpreted as what would be realized under noninvasive interventions. Soe further progress can perhaps be ade by relaxing or refining the consistency assuption and allowing for ultiple versions of treatent (van der Laan et al., 2005; Tauban et al., 2008; Vander Weele, 2009c). However, an alternative approach, which perhaps ore closely ties causation to physical laws governing systes, ay be desirable (Coenges and GegoutPetit, 2009). Work has been done in causal inference on direct and indirect effects outside of the counterfactual fraework (Didelez et al., 2007; Geneletti, 2008) but this work also conceptualizes direct and indirect effects through various potential interventions. The point here, however, is that the issue of the anipulability of variables is not unique to the context of ediation but arises even in the context of exaining total effects. We conclude this section by noting that in cases in which interventions on the exposure of interest are conceivable but interventions on the ediator are not, an alternative approach to assessing ediation is possible using concepts of principal stratification (Frangakis and Rubin, 2002; Rubin, 2004; VanderWeele, 2008; Gallop et al., 2009). Principal strata are strata defined by the joint counterfactual outcoes M(a) for all possible values of a. One ight assess direct effects, for exaple, by exaining the effect of exposure on the outcoe for individuals for who the exposure does not affect the value of the ediator e.g. for who M(0) = M(a). Such an approach is, in a certain sense, advantageous in that it only requires counterfactuals M(a)and Y (a), rather than also counterfactuals of the for Y (a, ) i.e. it does not require hypothetical interventions on the ediator. Unfortunately, however, the utility of the approach based on principal stratification is liited because of the inability to identify which individuals fall into which principal strata, because the probability of falling within the principal stratu M(a) =M(0) will be zero in any realistic applications (Robins et al., 2007), and because the notions of ediation based on principal direct effects do not correspond in any clear way to echaniss of scientific interest (Joffe et al., 2007). CONCEPTUAL ISSUES CONCERNING COMPOSITION In this section we would like to discuss briefly the assuption that is soeties referred to as coposition (Pearl, 2000) that Y (a) =Y (a, M(a)) i.e. that the potential outcoe Y (a) intervening to set A to a is equal to the potential outcoe Y (a, M(a)) intervening to set A to a and to set M to the value it would have been if A had been a. This assuption allows one to express the natural direct effect, E[Y (1,M(0)) Y (0,M(0)) C], as E[Y (1,M(0)) Y (0) C] 464 T. J. VanderWeele and S. Vansteelandt
9 and allows one to decopose a total effect into a natural direct and indirect effect: E[Y (a) Y (0) C]=E[Y (a, M(a)) Y (a, M(0)) C] + E[Y (a, M(0)) Y (0,M(0)) C]. The coposition assuption contrasts with the consistency assuption for Y (a, ) which states that when A = a and M = then Y (a, ) =Y i.e. that aongst the subgroup with observed exposure A = a and observed ediator M =, the observed outcoe Y is equal to the value of Y that would have been obtained intervening to set A to a and M to. Because the coposition assuption is used for effect decoposition, it is an iportant assuption in the context of ediation. The assuption is, however, often not explicitly stated but erely iplicitly assued. Perhaps this is in part because the coposition assuption arguably does not involve substantial conceptual issues above and beyond those entailed by the consistency and treatentvariation irrelevance assuptions. The consistency and treatentvariation irrelevance assuptions effectively presuppose hypothetical interventions on the exposure and the ediator; the coposition assuption also presupposes hypothetical interventions on the exposure and the ediator but only iposes restrictions concerning hypothetical interventions on the ediator for levels which naturally occur under various interventions on the exposure. The consistency assuption requires that when A = a and M = then Y (a, ) =Y and thus that the value of Y under two interventions to set A and M to their natural levels siply equals the observed outcoe; the coposition assuption requires that Y (a) = Y (a, M(a)) and thus that, under interventions on a, interventions on M to set it to its naturally occurring level M(a) have no further effect on the outcoe obtained. The consistency and treatentvariation irrelevance assuptions thus essentially presuppose that interventions on both A and M are in soe sense noninvasive, as discussed above and also in the appendix, while the coposition assuption essentially just presupposes that interventions on M are noninvasive. The conceptual issues involved in the coposition assuption thus do not see to entail considerably ore than that which was required with the consistency assuption. The consistency assuption does not atheatically entail the coposition assuption but we find it difficult to iagine cases in which a researcher would be willing to ake the consistency assuption but unwilling to ake the coposition assuption. CONCLUDING REMARKS In this paper we have related concepts of direct and indirect effects fro causal inference to concepts arising fro the approach of Baron and Kenny (1986), popular in the social sciences. We have also considered a nuber of iportant conceptual issues concerning ediation. We have seen that in certain psychological experients the assuptions required to identify natural direct and indirect effects ay be rendered ore plausible. This is iportant because it is notions of natural direct and indirect effects which allow for effect decoposition even in settings involving interactions and nonlinearities. We have also considered at length the issues of the relation of hypothetical interventions on the ediator and the exposure to the counterfactual fraework. We have seen that often it is difficult to conceive of interventions that noninvasively fix the ediator and have discussed potential violations of the socalled consistency assuption. We have furtherore seen that in soe cases interventions on the ediator are conceivable and that, oreover, for natural direct and indirect effects, one only need conceive of interventions on the ediator to set the ediator to naturally occurring levels. In addition, natural direct and indirect effects can in soe cases be conceived of as effects that would result by blocking the effect of an exposure on the ediator. Nevertheless, the counterfactual fraework is, at least at present, tied quite closely to the notion of a hypothetical intervention and in soe cases this can be probleatic, even when total effects are in view. The foralization of the counterfactual approach is still arguably of use even in settings in which conceiving of interventions is difficult. Questions of total effects and of ediation are likely to be of interest in practice even when interventions on the considered exposures and ediators are not possible; the counterfactual approach clarifies at least the assuptions and identification conditions required for assessing direct and indirect effects and akes clear when violations of these assuptions will lead to bias. The counterfactual fraework is a siplification of a coplex reality but it arguably oves us one step closer to the quantities of interest when we think about ediation. Finally, we have extended our discussion of interventions in the ediation context to the coposition assuption, an assuption that is not always explicitly stated but is often utilized when natural direct and indirect effects are in view. We have seen that when the various consistency and treatentvariation irrelevance assuptions hold, the coposition assuption does not ipose considerable conceptual challenges above and beyond those already iplicit in the consistency assuptions. The notions of ediation using the counterfactual approach are soeties subtle, but we hope that the contributions in this paper will help clarify the conceptual issues involved in utilizing the counterfactual fraework to address these questions of ediation in the social sciences and beyond. ACKNOWLEDGEMENTS We would like to thank the participants at a session on Mediation and Causal Inference at the 2009 ENAR International Bioetric Society eetings for the interesting discussion at that session that propted the writing of this paper. We also thank Haiqun Lin for the invitation to write Conceptual issues concerning ediation, interventions and coposition 465
10 the paper and we thank an anonyous referee for helpful coents. APPENDIX Controlled and natural direct and indirect effects using regression If the regression odels (6) and (8) are correctly specified and assuptions (1) and (2) hold then we could copute the controlled direct effect as follows: E[Y (a, ) (a,) C = c] = E[Y C = c, A = a, M = ] E[Y C = c, A = a,m = ] =(θ 0 + θ 1 a + θ 2 + θ 3 a + θ 4c) (θ 0 + θ 1 a + θ 2 + θ 3 a + θ 4c) =(θ 1 a + θ 3 a θ 1 a θ 3 a ) = θ 1 (a a )+θ 3 (a a ) If the regression odels (6) and (8) are correctly specified and assuptions (1) (4) hold, we could copute natural direct effects by E[Y (a, M(a )) Y (a,m(a )) C = c] = {E[Y C = c, A = a, M = ] E[Y C = c, A = a,m = ]} P (M = C = c, A = a ) = {(θ 0 + θ 1 a + θ 2 + θ 3 a + θ 4c) (θ 0 + θ 1 a + θ 2 + θ 3 a + θ 4c)} P (M = C = c, A = a ) = {(θ 1a + θ 2 + θ 3 a) (θ 1 a + θ 2 + θ 3 a )} P (M = C = c, A = a ) = {θ 1 a + θ 2 E[M A = a,c = c] + θ 3 ae[m A = a,c = c]) (θ 1 a + θ 2 E[M A = a,c = c] + θ 3 a E[M A = a,c = c])} = {θ 1 a + θ 2 (β 0 + β 1 a + β 2c) + θ 3 a(β 0 + β 1 a + β 2c)) (θ 1 a + θ 2 (β 0 + β 1 a + β 2c) + θ 3 a (β 0 + β 1 a + β 2c))} = {θ 1 a + θ 3 a(β 0 + β 1 a + β 2c) (θ 1 a + θ 3 a (β 0 + β 1 a + β 2c))} =(θ 1 + θ 3 β 0 + θ 3 β 1 a + θ 3 β 2c)(a a ) If the regression odels (6) and (8) are correctly specified and assuptions (1) (4) hold, we could copute natural indirect effects by E[Y (a, M(a)) Y (a, M(a )) C = c] = E[Y C = c, A = a, M = ] P (M = C = c, A = a) E[Y C = c, A = a, M = ] P (M = C = c, A = a ) = (θ 0 + θ 1 a + θ 2 + θ 3 a + θ 4c) P (M = C = c, A = a) (θ 0 + θ 1 a + θ 2 + θ 3 a + θ 4c) P (M = C = c, A = a ) =(θ 0 + θ 1 a + θ 2 E[M A = a, C = c] + θ 3 ae[m A = a, C = c]+θ 4c) (θ 0 + θ 1 a + θ 2 E[M A = a,c = c] c + θ 3 ae[m A = a,c = c]+θ 4c) =(θ 0 + θ 1 a + θ 2 (β 0 + β 1 a + β 2c) + θ 3 a(β 0 + β 1 a + β 2c)+θ 4c) (θ 0 + θ 1 a + θ 2 (β 0 + β 1 a + β 2c) + θ 3 a(β 0 + β 1 a + β 2c)+θ 4c) = θ 2 β 1 (a a )+θ 3 β 1 a(a a ). Standard errors of controlled and natural direct and indirect effects using regression Suppose that odels (6) and (8) have been fit using standard linear regression software and that the resulting estiates ˆβ of β (β 0,β 1,β 2) and ˆθ of θ (θ 0,θ 1,θ 2,θ 3,θ 4) have covariance atrices Σ β and Σ θ, which can be obtained fro ost offtheshelf statistical software packages. Then the covariance atrix of ( ˆβ, ˆθ ) is ( ) Σβ 0 Σ, 0 Σ θ which can be seen upon noting that Cov( ˆβ, ˆθ) =E { } Cov( ˆβ, ˆθ M,A,C) +Cov { E( ˆβ M,A,C),E(ˆθ M,A,C) } =0+Cov ( ) ˆβ,θ =0, whereweusethefactthat ˆβ is a function of M,A and C only. Standard errors of the controlled and natural direct and indirect effects in (9) can then be obtained (using the Delta ethod) as ΓΣΓ a a with Γ (0, 0, 0, 0, 1, 0,,0 ) for the controlled direct effect in (9), Γ (θ 3,θ 3 a,θ 3 C, 0, 1, 0,β 0 + β 1 a + β 2C, 0 )forthe pure natural direct effect in (9) (the sae expression holds for the total natural direct effect upon substituting a and a ) 466 T. J. VanderWeele and S. Vansteelandt
Sharp sensitivity bounds for mediation under unmeasured mediatoroutcome confounding
Bioetrika (2016), 103,2,pp. 483 490 doi: 10.1093/bioet/asw012 Printed in Great Britain Advance Access publication 29 April 2016 Sharp sensitivity bounds for ediation under uneasured ediatoroutcoe confounding
More informationBlock designs and statistics
Bloc designs and statistics Notes for Math 447 May 3, 2011 The ain paraeters of a bloc design are nuber of varieties v, bloc size, nuber of blocs b. A design is built on a set of v eleents. Each eleent
More informationCasual Mediation Analysis
Casual Mediation Analysis Tyler J. VanderWeele, Ph.D. Upcoming Seminar: April 2122, 2017, Philadelphia, Pennsylvania OXFORD UNIVERSITY PRESS Explanation in Causal Inference Methods for Mediation and Interaction
More informationModel Fitting. CURM Background Material, Fall 2014 Dr. Doreen De Leon
Model Fitting CURM Background Material, Fall 014 Dr. Doreen De Leon 1 Introduction Given a set of data points, we often want to fit a selected odel or type to the data (e.g., we suspect an exponential
More informationChapter 6: Economic Inequality
Chapter 6: Econoic Inequality We are interested in inequality ainly for two reasons: First, there are philosophical and ethical grounds for aversion to inequality per se. Second, even if we are not interested
More informationFeature Extraction Techniques
Feature Extraction Techniques Unsupervised Learning II Feature Extraction Unsupervised ethods can also be used to find features which can be useful for categorization. There are unsupervised ethods that
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search
Quantu algoriths (CO 781, Winter 2008) Prof Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search ow we begin to discuss applications of quantu walks to search algoriths
More information3.3 Variational Characterization of Singular Values
3.3. Variational Characterization of Singular Values 61 3.3 Variational Characterization of Singular Values Since the singular values are square roots of the eigenvalues of the Heritian atrices A A and
More informationProbability Distributions
Probability Distributions In Chapter, we ephasized the central role played by probability theory in the solution of pattern recognition probles. We turn now to an exploration of soe particular exaples
More informationGraphical Models in Local, Asymmetric MultiAgent Markov Decision Processes
Graphical Models in Local, Asyetric MultiAgent Markov Decision Processes Ditri Dolgov and Edund Durfee Departent of Electrical Engineering and Coputer Science University of Michigan Ann Arbor, MI 48109
More informationAn Extension to the Tactical Planning Model for a Job Shop: ContinuousTime Control
An Extension to the Tactical Planning Model for a Job Shop: ContinuousTie Control Chee Chong. Teo, Rohit Bhatnagar, and Stephen C. Graves SingaporeMIT Alliance, Nanyang Technological Univ., and Massachusetts
More informationNonParametric NonLineofSight Identification 1
NonParaetric NonLineofSight Identification Sinan Gezici, Hisashi Kobayashi and H. Vincent Poor Departent of Electrical Engineering School of Engineering and Applied Science Princeton University, Princeton,
More informationI. Understand get a conceptual grasp of the problem
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departent o Physics Physics 81T Fall Ter 4 Class Proble 1: Solution Proble 1 A car is driving at a constant but unknown velocity,, on a straightaway A otorcycle is
More informationSoft Computing Techniques Help Assign Weights to Different Factors in Vulnerability Analysis
Soft Coputing Techniques Help Assign Weights to Different Factors in Vulnerability Analysis Beverly Rivera 1,2, Irbis Gallegos 1, and Vladik Kreinovich 2 1 Regional Cyber and Energy Security Center RCES
More information3.8 Three Types of Convergence
3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to
More informationProc. of the IEEE/OES Seventh Working Conference on Current Measurement Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES
Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES Belinda Lipa Codar Ocean Sensors 15 La Sandra Way, Portola Valley, CA 98 blipa@pogo.co
More informationCOS 424: Interacting with Data. Written Exercises
COS 424: Interacting with Data Hoework #4 Spring 2007 Regression Due: Wednesday, April 18 Written Exercises See the course website for iportant inforation about collaboration and late policies, as well
More informationChaotic Coupled Map Lattices
Chaotic Coupled Map Lattices Author: Dustin Keys Advisors: Dr. Robert Indik, Dr. Kevin Lin 1 Introduction When a syste of chaotic aps is coupled in a way that allows the to share inforation about each
More informationA Simple Regression Problem
A Siple Regression Proble R. M. Castro March 23, 2 In this brief note a siple regression proble will be introduced, illustrating clearly the biasvariance tradeoff. Let Y i f(x i ) + W i, i,..., n, where
More informationThe Transactional Nature of Quantum Information
The Transactional Nature of Quantu Inforation Subhash Kak Departent of Coputer Science Oklahoa State University Stillwater, OK 7478 ABSTRACT Inforation, in its counications sense, is a transactional property.
More informationCausal Mechanisms Short Course Part II:
Causal Mechanisms Short Course Part II: Analyzing Mechanisms with Experimental and Observational Data Teppei Yamamoto Massachusetts Institute of Technology March 24, 2012 Frontiers in the Analysis of Causal
More information16 Independence Definitions Potential Pitfall Alternative Formulation. mcsftl 2010/9/8 0:40 page 431 #437
csftl 010/9/8 0:40 page 431 #437 16 Independence 16.1 efinitions Suppose that we flip two fair coins siultaneously on opposite sides of a roo. Intuitively, the way one coin lands does not affect the way
More information12 Towards hydrodynamic equations J Nonlinear Dynamics II: Continuum Systems Lecture 12 Spring 2015
18.354J Nonlinear Dynaics II: Continuu Systes Lecture 12 Spring 2015 12 Towards hydrodynaic equations The previous classes focussed on the continuu description of static (tieindependent) elastic systes.
More informationModule #1: Units and Vectors Revisited. Introduction. Units Revisited EXAMPLE 1.1. A sample of iron has a mass of mg. How many kg is that?
Module #1: Units and Vectors Revisited Introduction There are probably no concepts ore iportant in physics than the two listed in the title of this odule. In your firstyear physics course, I a sure that
More informationPattern Recognition and Machine Learning. Artificial Neural networks
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3  MMIS Fall Seester 2017 Lessons 7 20 Dec 2017 Outline Artificial Neural networks Notation...2 Introduction...3 Key Equations... 3 Artificial
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Notes for EE227C (Spring 2018): Convex Optiization and Approxiation Instructor: Moritz Hardt Eail: hardt+ee227c@berkeley.edu Graduate Instructor: Max Sichowitz Eail: sichow+ee227c@berkeley.edu October
More informationOcean 420 Physical Processes in the Ocean Project 1: Hydrostatic Balance, Advection and Diffusion Answers
Ocean 40 Physical Processes in the Ocean Project 1: Hydrostatic Balance, Advection and Diffusion Answers 1. Hydrostatic Balance a) Set all of the levels on one of the coluns to the lowest possible density.
More informationPattern Recognition and Machine Learning. Learning and Evaluation for Pattern Recognition
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3  MMIS Fall Seester 2017 Lesson 1 4 October 2017 Outline Learning and Evaluation for Pattern Recognition Notation...2 1. The Pattern Recognition
More informationPhysically Based Modeling CS Notes Spring 1997 Particle Collision and Contact
Physically Based Modeling CS 15863 Notes Spring 1997 Particle Collision and Contact 1 Collisions with Springs Suppose we wanted to ipleent a particle siulator with a floor : a solid horizontal plane which
More informationIn this chapter, we consider several graphtheoretic and probabilistic models
THREE ONE GRAPHTHEORETIC AND STATISTICAL MODELS 3.1 INTRODUCTION In this chapter, we consider several graphtheoretic and probabilistic odels for a social network, which we do under different assuptions
More informationThe Weierstrass Approximation Theorem
36 The Weierstrass Approxiation Theore Recall that the fundaental idea underlying the construction of the real nubers is approxiation by the sipler rational nubers. Firstly, nubers are often deterined
More informationChapter 6 1D Continuous Groups
Chapter 6 1D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well defined range. This chapter explores:
More informationE0 370 Statistical Learning Theory Lecture 6 (Aug 30, 2011) Margin Analysis
E0 370 tatistical Learning Theory Lecture 6 (Aug 30, 20) Margin Analysis Lecturer: hivani Agarwal cribe: Narasihan R Introduction In the last few lectures we have seen how to obtain high confidence bounds
More informationA Simplified Analytical Approach for Efficiency Evaluation of the Weaving Machines with Automatic Filling Repair
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber 4, 006 0 A Siplified Analytical Approach for Efficiency Evaluation of the eaving
More information26 Impulse and Momentum
6 Ipulse and Moentu First, a Few More Words on Work and Energy, for Coparison Purposes Iagine a gigantic air hockey table with a whole bunch of pucks of various asses, none of which experiences any friction
More informationDistributed Subgradient Methods for Multiagent Optimization
1 Distributed Subgradient Methods for Multiagent Optiization Angelia Nedić and Asuan Ozdaglar October 29, 2007 Abstract We study a distributed coputation odel for optiizing a su of convex objective functions
More informationEnsemble Based on Data Envelopment Analysis
Enseble Based on Data Envelopent Analysis So Young Sohn & Hong Choi Departent of Coputer Science & Industrial Systes Engineering, Yonsei University, Seoul, Korea Tel) 822223404, Fax) 822 3647807
More informationDataDriven Imaging in Anisotropic Media
18 th World Conference on Non destructive Testing, 16 April 1, Durban, South Africa DataDriven Iaging in Anisotropic Media Arno VOLKER 1 and Alan HUNTER 1 TNO Stieltjesweg 1, 6 AD, Delft, The Netherlands
More informationC na (1) a=l. c = CO + Clm + CZ TWOSTAGE SAMPLE DESIGN WITH SMALL CLUSTERS. 1. Introduction
TWOSTGE SMPLE DESIGN WITH SMLL CLUSTERS Robert G. Clark and David G. Steel School of Matheatics and pplied Statistics, University of Wollongong, NSW 5 ustralia. (robert.clark@abs.gov.au) Key Words: saple
More informationA proposal for a FirstCitationSpeedIndex Link Peerreviewed author version
A proposal for a FirstCitationSpeedIndex Link Peerreviewed author version Made available by Hasselt University Library in Docuent Server@UHasselt Reference (Published version): EGGHE, Leo; Bornann,
More information2 Q 10. Likewise, in case of multiple particles, the corresponding density in 2 must be averaged over all
Lecture 6 Introduction to kinetic theory of plasa waves Introduction to kinetic theory So far we have been odeling plasa dynaics using fluid equations. The assuption has been that the pressure can be either
More informationStatistical Methods for Causal Mediation Analysis
Statistical Methods for Causal Mediation Analysis The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Accessed Citable
More informationAn Introduction to MetaAnalysis
An Introduction to MetaAnalysis Douglas G. Bonett University of California, Santa Cruz How to cite this work: Bonett, D.G. (2016) An Introduction to Metaanalysis. Retrieved fro http://people.ucsc.edu/~dgbonett/eta.htl
More informationEfficient Filter Banks And Interpolators
Efficient Filter Banks And Interpolators A. G. DEMPSTER AND N. P. MURPHY Departent of Electronic Systes University of Westinster 115 New Cavendish St, London W1M 8JS United Kingdo Abstract:  Graphical
More informationSexually Transmitted Diseases VMED 5180 September 27, 2016
Sexually Transitted Diseases VMED 518 Septeber 27, 216 Introduction Two sexuallytransitted disease (STD) odels are presented below. The irst is a susceptibleinectioussusceptible (SIS) odel (Figure 1)
More information1 Bounding the Margin
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #12 Scribe: Jian Min Si March 14, 2013 1 Bounding the Margin We are continuing the proof of a bound on the generalization error of AdaBoost
More informationIntelligent Systems: Reasoning and Recognition. Artificial Neural Networks
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley MOSIG M1 Winter Seester 2018 Lesson 7 1 March 2018 Outline Artificial Neural Networks Notation...2 Introduction...3 Key Equations... 3 Artificial
More informationModified Systematic Sampling in the Presence of Linear Trend
Modified Systeatic Sapling in the Presence of Linear Trend Zaheen Khan, and Javid Shabbir Keywords: Abstract A new systeatic sapling design called Modified Systeatic Sapling (MSS, proposed by ] is ore
More informationma x = bv x + F rod.
Notes on Dynaical Systes Dynaics is the study of change. The priary ingredients of a dynaical syste are its state and its rule of change (also soeties called the dynaic). Dynaical systes can be continuous
More informationPattern Recognition and Machine Learning. Artificial Neural networks
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3  MMIS Fall Seester 2016 Lessons 7 14 Dec 2016 Outline Artificial Neural networks Notation...2 1. Introduction...3... 3 The Artificial
More informationRevision list for Pearl s THE FOUNDATIONS OF CAUSAL INFERENCE
Revision list for Pearl s THE FOUNDATIONS OF CAUSAL INFERENCE insert p. 90: in graphical terms or plain causal language. The mediation problem of Section 6 illustrates how such symbiosis clarifies the
More informationEstimating Parameters for a Gaussian pdf
Pattern Recognition and achine Learning Jaes L. Crowley ENSIAG 3 IS First Seester 00/0 Lesson 5 7 Noveber 00 Contents Estiating Paraeters for a Gaussian pdf Notation... The Pattern Recognition Proble...3
More informationAnisotropic reference media and the possible linearized approximations for phase velocities of qs waves in weakly anisotropic media
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 5 00 007 04 PII: S007708676 Anisotropic reference edia and the possible linearized approxiations for phase
More informationP (t) = P (t = 0) + F t Conclusion: If we wait long enough, the velocity of an electron will diverge, which is obviously impossible and wrong.
4 Phys520.nb 2 Drude theory ~ Chapter in textbook 2.. The relaxation tie approxiation Here we treat electrons as a free ideal gas (classical) 2... Totally ignore interactions/scatterings Under a static
More informationMSEC MODELING OF DEGRADATION PROCESSES TO OBTAIN AN OPTIMAL SOLUTION FOR MAINTENANCE AND PERFORMANCE
Proceeding of the ASME 9 International Manufacturing Science and Engineering Conference MSEC9 October 47, 9, West Lafayette, Indiana, USA MSEC98466 MODELING OF DEGRADATION PROCESSES TO OBTAIN AN OPTIMAL
More information4 = (0.02) 3 13, = 0.25 because = 25. Simi
Theore. Let b and be integers greater than. If = (. a a 2 a i ) b,then for any t N, in base (b + t), the fraction has the digital representation = (. a a 2 a i ) b+t, where a i = a i + tk i with k i =
More informationAn Introduction to Causal Mediation Analysis. Xu Qin University of Chicago Presented at the Central Iowa R User Group Meetup Aug 10, 2016
An Introduction to Causal Mediation Analysis Xu Qin University of Chicago Presented at the Central Iowa R User Group Meetup Aug 10, 2016 1 Causality In the applications of statistics, many central questions
More informationExperimental Design For Model Discrimination And Precise Parameter Estimation In WDS Analysis
City University of New York (CUNY) CUNY Acadeic Works International Conference on Hydroinforatics 812014 Experiental Design For Model Discriination And Precise Paraeter Estiation In WDS Analysis Giovanna
More informationIdentification, Inference, and Sensitivity Analysis for Causal Mediation Effects
Identification, Inference, and Sensitivity Analysis for Causal Mediation Effects Kosuke Imai Luke Keele Teppei Yamamoto First Draft: November 4, 2008 This Draft: January 15, 2009 Abstract Causal mediation
More informationGeneral Properties of Radiation Detectors Supplements
Phys. 649: Nuclear Techniques Physics Departent Yarouk University Chapter 4: General Properties of Radiation Detectors Suppleents Dr. Nidal M. Ershaidat Overview Phys. 649: Nuclear Techniques Physics Departent
More informationPh 20.3 Numerical Solution of Ordinary Differential Equations
Ph 20.3 Nuerical Solution of Ordinary Differential Equations Due: Week 5 v20170314 This Assignent So far, your assignents have tried to failiarize you with the hardware and software in the Physics Coputing
More informationESTIMATING AND FORMING CONFIDENCE INTERVALS FOR EXTREMA OF RANDOM POLYNOMIALS. A Thesis. Presented to. The Faculty of the Department of Mathematics
ESTIMATING AND FORMING CONFIDENCE INTERVALS FOR EXTREMA OF RANDOM POLYNOMIALS A Thesis Presented to The Faculty of the Departent of Matheatics San Jose State University In Partial Fulfillent of the Requireents
More informationOBJECTIVES INTRODUCTION
M7 Chapter 3 Section 1 OBJECTIVES Suarize data using easures of central tendency, such as the ean, edian, ode, and idrange. Describe data using the easures of variation, such as the range, variance, and
More informationPhysics 2107 Oscillations using Springs Experiment 2
PY07 Oscillations using Springs Experient Physics 07 Oscillations using Springs Experient Prelab Read the following bacground/setup and ensure you are failiar with the concepts and theory required for
More informationMachine Learning Basics: Estimators, Bias and Variance
Machine Learning Basics: Estiators, Bias and Variance Sargur N. srihari@cedar.buffalo.edu This is part of lecture slides on Deep Learning: http://www.cedar.buffalo.edu/~srihari/cse676 1 Topics in Basics
More informationBootstrapping Dependent Data
Bootstrapping Dependent Data One of the key issues confronting bootstrap resapling approxiations is how to deal with dependent data. Consider a sequence fx t g n t= of dependent rando variables. Clearly
More informationBirthday Paradox Calculations and Approximation
Birthday Paradox Calculations and Approxiation Joshua E. Hill InfoGard Laboratories March v. Birthday Proble In the birthday proble, we have a group of n randoly selected people. If we assue that birthdays
More informationNONCOMPLIANCE BIAS CORRECTION BASED ON COVARIATES IN RANDOMIZED EXPERIMENTS
NONCOMPLIANCE BIAS CORRECTION BASED ON COVARIATES IN RANDOMIZED EXPERIMENTS YVES ATCHADE AND LEONARD WANTCHEKON Noveber 1, 2005 Abstract We propose soe practical solutions for causal effects estiation
More informationFigure 1: Equivalent electric (RC) circuit of a neurons membrane
Exercise: Leaky integrate and fire odel of neural spike generation This exercise investigates a siplified odel of how neurons spike in response to current inputs, one of the ost fundaental properties of
More informationUSEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS. By: Ian Blokland, Augustana Campus, University of Alberta
1 USEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS By: Ian Bloland, Augustana Capus, University of Alberta For: Physics Olypiad Weeend, April 6, 008, UofA Introduction: Physicists often attept to solve
More informationSpine Fin Efficiency A Three Sided Pyramidal Fin of Equilateral Triangular CrossSectional Area
Proceedings of the 006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miai, Florida, USA, January 180, 006 (pp1318) Spine Fin Efficiency A Three Sided Pyraidal Fin of Equilateral Triangular
More informationA SemiParametric Approach to Account for Complex. Designs in Multiple Imputation
A SeiParaetric Approach to Account for Coplex Designs in ultiple Iputation Hanzhi Zhou, Trivellore E. Raghunathan and ichael R. Elliott Progra in Survey ethodology, University of ichigan Departent of
More informationare equal to zero, where, q = p 1. For each gene j, the pairwise null and alternative hypotheses are,
Page of 8 Suppleentary Materials: A ultiple testing procedure for ultidiensional pairwise coparisons with application to gene expression studies Anjana Grandhi, Wenge Guo, Shyaal D. Peddada S Notations
More information13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 01 Matrices
CS71 Randoness & Coputation Spring 018 Instructor: Alistair Sinclair Lecture 13: February 7 Disclaier: These notes have not been subjected to the usual scrutiny accorded to foral publications. They ay
More informationSampling How Big a Sample?
C. G. G. Aitken, 1 Ph.D. Sapling How Big a Saple? REFERENCE: Aitken CGG. Sapling how big a saple? J Forensic Sci 1999;44(4):750 760. ABSTRACT: It is thought that, in a consignent of discrete units, a certain
More informationWhat is Probability? (again)
INRODUCTION TO ROBBILITY Basic Concepts and Definitions n experient is any process that generates welldefined outcoes. Experient: Record an age Experient: Toss a die Experient: Record an opinion yes,
More informationRevealed Preference and Stochastic Demand Correspondence: A Unified Theory
Revealed Preference and Stochastic Deand Correspondence: A Unified Theory Indraneel Dasgupta School of Econoics, University of Nottingha, Nottingha NG7 2RD, UK. Eail: indraneel.dasgupta@nottingha.ac.uk
More informationConstrained Consensus and Optimization in MultiAgent Networks arxiv: v2 [math.oc] 17 Dec 2008
LIDS Report 2779 1 Constrained Consensus and Optiization in MultiAgent Networks arxiv:0802.3922v2 [ath.oc] 17 Dec 2008 Angelia Nedić, Asuan Ozdaglar, and Pablo A. Parrilo February 15, 2013 Abstract We
More informationOn Poset Merging. 1 Introduction. Peter Chen Guoli Ding Steve Seiden. Keywords: Merging, Partial Order, Lower Bounds. AMS Classification: 68W40
On Poset Merging Peter Chen Guoli Ding Steve Seiden Abstract We consider the follow poset erging proble: Let X and Y be two subsets of a partially ordered set S. Given coplete inforation about the ordering
More informationecompanion ONLY AVAILABLE IN ELECTRONIC FORM
OPERATIONS RESEARCH doi 10.1287/opre.1070.0427ec pp. ec1 ec5 ecopanion ONLY AVAILABLE IN ELECTRONIC FORM infors 07 INFORMS Electronic Copanion A Learning Approach for Interactive Marketing to a Custoer
More informationMultiDimensional HegselmannKrause Dynamics
MultiDiensional HegselannKrause Dynaics A. Nedić Industrial and Enterprise Systes Engineering Dept. University of Illinois Urbana, IL 680 angelia@illinois.edu B. Touri Coordinated Science Laboratory
More informationDeflation of the IO Series Some Technical Aspects. Giorgio Rampa University of Genoa April 2007
Deflation of the IO Series 19592. Soe Technical Aspects Giorgio Rapa University of Genoa g.rapa@unige.it April 27 1. Introduction The nuber of sectors is 42 for the period 19652 and 38 for the initial
More informationWBASED VS LATENT VARIABLES SPATIAL AUTOREGRESSIVE MODELS: EVIDENCE FROM MONTE CARLO SIMULATIONS
WBASED VS LATENT VARIABLES SPATIAL AUTOREGRESSIVE MODELS: EVIDENCE FROM MONTE CARLO SIMULATIONS. Introduction When it coes to applying econoetric odels to analyze georeferenced data, researchers are well
More informationPhysics 215 Winter The Density Matrix
Physics 215 Winter 2018 The Density Matrix The quantu space of states is a Hilbert space H. Any state vector ψ H is a pure state. Since any linear cobination of eleents of H are also an eleent of H, it
More informationNow multiply the lefthandside by ω and the righthand side by dδ/dt (recall ω= dδ/dt) to get:
Equal Area Criterion.0 Developent of equal area criterion As in previous notes, all powers are in perunit. I want to show you the equal area criterion a little differently than the book does it. Let s
More informationSUR LE CALCUL DES PROBABILITÉS
MÉMOIRE SUR LE CALCUL DES PROBABILITÉS M. le MARQUIS DE CONDORCET Histoire de l Acadéie des Sciences des Paris, 783 Parts 4 & 5, pp. 539559 FOURTH PART Reflections on the ethod to deterine the Probability
More informationHybrid System Identification: An SDP Approach
49th IEEE Conference on Decision and Control Deceber 1517, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA Hybrid Syste Identification: An SDP Approach C Feng, C M Lagoa, N Ozay and M Sznaier Abstract The
More informationWhen Short Runs Beat Long Runs
When Short Runs Beat Long Runs Sean Luke George Mason University http://www.cs.gu.edu/ sean/ Abstract What will yield the best results: doing one run n generations long or doing runs n/ generations long
More informationLower Bounds for Quantized Matrix Completion
Lower Bounds for Quantized Matrix Copletion Mary Wootters and Yaniv Plan Departent of Matheatics University of Michigan Ann Arbor, MI Eail: wootters, yplan}@uich.edu Mark A. Davenport School of Elec. &
More informationarxiv: v1 [math.nt] 14 Sep 2014
ROTATION REMAINDERS P. JAMESON GRABER, WASHINGTON AND LEE UNIVERSITY 08 arxiv:1409.411v1 [ath.nt] 14 Sep 014 Abstract. We study properties of an array of nubers, called the triangle, in which each row
More informationTactics Box 2.1 Interpreting PositionversusTime Graphs
1D kineatic Retake Assignent Due: 4:32p on Friday, October 31, 2014 You will receive no credit for ites you coplete after the assignent is due. Grading Policy Tactics Box 2.1 Interpreting PositionversusTie
More informationCHAPTER ONE. Physics and the Life Sciences
Solution anual for Physics for the Life Sciences 2nd Edition by Allang Link download full: http://testbankair.co/download/solutionanualforphysicsforthelifesciences2ndeditionbyallang/ CHAPTER
More informationCombining Classifiers
Cobining Classifiers Generic ethods of generating and cobining ultiple classifiers Bagging Boosting References: Duda, Hart & Stork, pg 475480. Hastie, Tibsharini, Friedan, pg 246256 and Chapter 10. http://www.boosting.org/
More informationASSUME a source over an alphabet size m, from which a sequence of n independent samples are drawn. The classical
IEEE TRANSACTIONS ON INFORMATION THEORY Large Alphabet Source Coding using Independent Coponent Analysis Aichai Painsky, Meber, IEEE, Saharon Rosset and Meir Feder, Fellow, IEEE arxiv:67.7v [cs.it] Jul
More informationA note on the multiplication of sparse matrices
Cent. Eur. J. Cop. Sci. 41) 2014 111 DOI: 10.2478/s135370140201x Central European Journal of Coputer Science A note on the ultiplication of sparse atrices Research Article Keivan Borna 12, Sohrab Aboozarkhani
More informationStatistical Analysis of Causal Mechanisms
Statistical Analysis of Causal Mechanisms Kosuke Imai Princeton University April 13, 2009 Kosuke Imai (Princeton) Causal Mechanisms April 13, 2009 1 / 26 Papers and Software Collaborators: Luke Keele,
More informationThe Characteristic Planet
The Characteristic Planet Brano Zivla, bzivla@gail.co Abstract: I have calculated a relation significant for planets fro a logical starting point that a whole and its parts are ianently depandant on each
More informationSome Perspective. Forces and Newton s Laws
Soe Perspective The language of Kineatics provides us with an efficient ethod for describing the otion of aterial objects, and we ll continue to ake refineents to it as we introduce additional types of
More informationTEST OF HOMOGENEITY OF PARALLEL SAMPLES FROM LOGNORMAL POPULATIONS WITH UNEQUAL VARIANCES
TEST OF HOMOGENEITY OF PARALLEL SAMPLES FROM LOGNORMAL POPULATIONS WITH UNEQUAL VARIANCES S. E. Ahed, R. J. Tokins and A. I. Volodin Departent of Matheatics and Statistics University of Regina Regina,
More information