Assumptions - Linear SNMM for mediation analysis

2.3 Linear SNMM for mediation analysis

2.3.1 Assumptions

To identify the causal effects of interest in terms of potential outcomes under the linear SNMM, we first introduce the corresponding assumptions necessary for unbiased inference. The required assumptions are 1) the stable unit treatment value assumption (SUTVA; Rubin, 1986; Angrist et al., 1996); 2) randomization of treatment assignment; and 3) a number of no-interaction assumptions.

We consider the two components of SUTVA. First, the treatment assignment and level of the mediator for one subject does not influence the outcomes of others. That is, there is a single value for each of the potential random outcome variables (Y(r,s)) for a given patientiregardless of the randomization assignment or mediation behavior of any other patienti′

. Notationally, we can denote the potential outcomes for subject ias scalar indices Y(r,s)_{, rather than} _Y(rrr,sss) _{that takes n-dimensional vec-}

tor indices of treatment and tumor response of all subjects into account. For our example, it is likely that this part of SUTVA holds, unless an oncologist sees mul- tiple study patients and alters some aspect (e.g., dose) of the treatment in reaction to the overall response (e.g., side effects) from the previous patients receiving the treatment. Second, only one of the potential outcomes will be the observed outcome for each subject even if the treatment assignment or tumor response occurrence varies (“consistency”; Rubin, 1986). This assumption allows us to link the potential

outcomes to the observed outcome:

Y =rsY(1,1)+ (1−r)sY(0,1)+r(1−s)Y(1,0)+ (1−r)(1−s)Y(0,0).

For the randomization assumption,

E(Y(1,1), Y(1,0), Y(0,1), Y(0,0) |R=r,X=x) = E(Y(1,1), Y(1,0), Y(0,1), Y(0,0) |X=x).

Under this assumption, the first order moments of the potential outcomes and by definition observed and unobserved baseline covariates should be equal between the randomized intervention groups.

To achieve identifiability of the causal effects of R and S under the proposed mediation SNMM, certain no-interaction assumptions are also needed (e.g., Ten Have et al., 2007; Hern´an and Robins, 2006). The first no-interaction assumption is that there is no causal linear interaction between the randomized intervention and mediator. That is, there is no structural interaction term between R and S in the proposed SNMM which would correspond to:

E(Y(1,1)|XXX =xxx, R =r, S =s)−E(Y(1,0)|XXX =xxx, R=r, S =s) =E(Y(0,1)|XXX =xxx, R =r, S =s)−E(Y(0,0)|XXX =xxx, R=r, S =s)

The second set of no-interaction assumptions requires that the causal linear main effects of R and S must not differ across observed groups of subjects defined by the observed levels of R and/or S. A representative no-interaction indicating no difference of the causal main effect of R between two groups of subjects defined by R= 1 andR = 0 is presented as follows:

E(Y(1,s)|XXX =xxx, R= 1, S=s′ )−E(Y(0,s)|XXX =xxx, R= 1, S =s′ ) =E(Y(1,s)|XXX =xxx, R= 0, S=s′ )−E(Y(0,s)|XXX =xxx, R= 0, S =s′ )

for allxxx,s(which is the set indices ofS),s′

(which is the observed value ofS defining the sub-group).

Similar no-interaction equalities are needed when the sub-groups are differenti- ated by different values ofSrather thanR. These equalities are not testable because of the necessity of stratifying on S and thus requiring that the randomization assumption for R hold for sub-groups defined by S, which is unfeasible under the current set of assumptions. Hence, these interactions that are assumed to be zero are not identifiable without further assumptions. For the second set of no-interaction assumptions, the main causal effects of R and S are assumed not to vary between different groups distinguished by the levels of the elements ofXXX. We can theoret- ically stratify the sample byXXX and estimate the above interaction and then test if the causal main effects of R and S are statistically different across these strata, if

of course there are sufficient subjects. Moreover, one can code the specific variable elements of XXX such that the causal effects are interpreted as a weighted average across the levels of these variable elements (e.g., Bond et al. 2007).

2.3.2 Model

The proposed mediation model belongs to the general class of linear structural nested mean models for longitudinal data (Robins, 1994, 1997; Robins et al., 1999). The model is “nested” in the sense that at each time point, we only condition on information that is available according to the current measured “treatment” history. In the mediation context, “treatment” pertains to the randomized intervention at time 1 and then the mediator at time 2. Apart from baseline covariates, we condition on the randomized intervention at time 2 when assessing the effect of the mediator on outcome. Whereas when assessing the effect of the randomized intervention at time 1, we do not condition on any previous “treatment”. In terms of the lung cancer data, since tumor response occurs after randomization, the effect of the randomized treatment under the linear SNMM is not conditional on tumor response, whereas the effect of tumor response is conditional on the randomized treatment assignment.

In general, the mean of the observed outcome for a patient with assignment r and mediator levels can be parameterized as the reference potential outcome plus a series of effects related toR and S, where Γ1(xxx) is the bias parameter reflecting the

how the control group is different from the overall population;q1(xxx, r) andq2(xxx, r, s)

are selection bias and unmeasured confounding parameters; γ1(xxx, r) is the causal

direct effect of R; Γ2(xxx, r) is the effect of R, adjusting for S versus not adjusting

for S; and γ2(xxx, r, s) is the causal effect of S fixing R. The functions γ1(xxx, r) and

γ2(xxx, r, s) are known as Robins’ blip functions (Robins, 1994, 2008) and the functions

Γ1(xxx), q1(xxx, r), Γ2(xxx, r), and q2(xxx, r, s) are defined such that the right hand side of

the equation is equal to the left hand sideE(Y(r,s)|XXX =xxx, R=r, S =s). It directly follows from the model thatγ1(xxx,0) =γ2(xxx, r,0) = 0.

The parameter Γ2(XXX, R) + q2(XXX, R, S) denotes the effect of unmeasured con-

founders on outcome. According to the definition of Γ2(XXX, R) and q2(XXX, R, S)

for linear SNMM (Robins et. al., 1999), we can obtain an explicit expression for Γ2(XXX, R) +q2(XXX, R, S) by noting the following relationship

Γ2(xxx, r) =−ES(q2(XXX, R, S)|XXX=xxx, R =r).

functionq2(XXX, R, S) can be expressed as q2(xxx, r, S) = E(Y(r,0)|XXX =xxx, R =r, S)−E(Y(r,0)|XXX =xxx, R=r, S = 0) = (δδδXxxx+δRr+δSS)−(δδδXxxx+δRr) = δSS. Therefore, Γ2(xxx, r) +q2(xxx, r, S) = q2(xxx, r, S)−ES(q2(X, R, SXX )|XXX =xxx, R=r) = δS[S−E(S|XXX =xxx, R=r)].

Under the randomization assumption, Γ1(xxx) =q1(xxx, r) = 0. By SUTVA,

E(Y(r,s)|XXX =xxx, R =r, S =s) =E(Y|XXX =xxx, R =r, S =s).

Under these assumptions and the expression for Γ2(XXX, R) +q2(XXX, R, S), the linear

SNMM in equation (2.3.1) can be rewritten as

E(Y|XXX =xxx, R, S)

whereγ1(xxx, R) is parameterized asθRRandγ2(xxx, R, S) is parameterized asθSS. The

parameterθR represents the direct effect of randomized treatment on outcome; and

θS represents the effect of mediator on outcome holding randomized treatment fixed

at any level r. We will show in the next section that unbiased parameter estimates can be obtained using the two-stage IV approaches under the linear SNMM defined in equation (2.3.2).

In document Causal Effect Estimation Under Linear and Log-Linear Structural Nested Mean Models in the Presence of Unmeasured Confounding (Page 36-43)