Third-variable Effect Analysis With Multilevel Additive Models - PLOS

2.2 Definitions of third-variable effects with data of two levels

The unique data structure in multilevel models raises the potential problem of confounding TVEs from different levels. As pointed out in [13], the relationship between two level-one variables can be decomposed into between-group and within-group components. In particular, the aggregated variables at the second level can be highly related while the relationship may be very weak or even at an opposite direction when considered at the individual level [23]. For example, [24] points out that the proportion of black residents may be an important variable for the census tract, while it is different from the meaning of ethnicity as an individual-level variable. [25] discussed the difference of the two components extensively. It is important to differentiate the between-group and within-group components in third-variable analysis, where the TVEs can be decomposed to level 1 and level 2 effects. To identify the level 1 and level 2 TVEs separately, [13] proposed the group-mean centering method (CWC), where they subtracted the group means from individual level variables and added group means as level 2 covariates. In their paper, [13] showed that the CWC method efficiently separated level 1 and level 2 TVEs and resulted in less bias and more power compared with non-centering methods. In this paper, we use a different way to estimate the level 1 and level 2 TVEs: extend the definitions of TVEs with single level models by [7] to multilevel models.

With the generalized definition of TVEs, [22] has shown that a third-varaible analysis can involve multiple exposure variables and multivariate outcomes. The purpose of third-variable analysis is to differentiate the direct effect and indirect effect from each third-variable for each pair of the exposure-outcome relationship. If the outcome is at level 2, all exposure and mediators have to be level 2 as discussed in Section 1. Therefore, a single-level third-variable analysis works. If the outcome is at level 1, the exposure variable can be a level 1 or level 2 variable. The third-variables can be level 1 or 2 for a level 2 exposure, but have to be level 1 for a level 1 exposure variable. In this paper, we focus on level 1 outcomes.

Denote Mij = (Mij1, …, MijK) as the vector of K potential level 1 third-variables for the ith object at the jth group. Mij,−k is the vector Mij excluding the kth element. Denote M.j = (M.j1, …, M.jL) as the vector of the L potential level 2 third-variables or level 1 third-variables aggregated at level 2 within group j. Let Mijk(xij) be a random variable that has a conditional distribution given Xij = xij. For an exposure variable X at any level, let u* be the minimum unit of X, such that if x ∈ domain(X), then x + u* ∈ domain(X). For now, we ignore other covariates Z. Assume effects of exposures and third-variables on the outcome are additive, we have the general definitions of TVEs, following [7], for level 1 (Definition 1) and level 2 exposure variables (Definition 2). Note that a level 1 exposure can have only level 1 mediators while a level 2 exposure can have both level 1 and level 2 mediators.

Definition 1. For a level 1 exposure variable X, the level 1 total effect (TE1) of X on Y, the level 1 direct effect () of X on Y not from level 1 third-variable Mk and the level 1 indirect effect of X on Y through Mk at X = xij () are defined as: (1) (2) (3)

The average level one TVEs are the mean value of the TVEs defined by Definition 1: ATE1 = Eij[TE1(xij)], ADE1,\k = Eij[DE1,\k(xij)] and AIE1,k = ATE1 − ADE1,\k.

Definition 2. For a level 2 exposure variable X, the level 2 total effect (TE2) of X on Y, the level 2 direct effect () of X on Y not from the level 1 third variable Mk and level 2 third variable Ml, and the level 2 indirect effect of X on Y through Mk and Ml at X = x.j () are defined as: (4) (5) (6) (7) (8)

The average level 2 TVEs are the TVEs defined by Definition 2 averaged at the group level: ATE2 = Ej[TE2,j(x.j)], AIE21,k = Ej[IE21,jk(x.j)] and AIE22,l = Ej[IE22,jl(x.j)].