Fixing versus conditioning Mediation analysis quantifies the extent to which a variable participates in the transmittance of change from a cause to its effect. It is inherently a causal notion, hence it cannot be defined in statistical terms. Traditionally, however, the bulk of mediation analysis has been conducted within the confines of linear regression, with statistical terminology masking the causal character of the relationships involved. This led to difficulties, biases, and limitations that have been alleviated by modern methods of causal analysis, based on causal diagrams and counterfactual logic. The source of these difficulties lies in defining mediation in terms of changes induced by adding a third variables into a regression equation. Such statistical changes are epiphenomena which sometimes accompany mediation but, in general, fail to capture the causal relationships that mediation analysis aims to quantify. The basic premise of the causal approach is that it is not always appropriate to "control" for the mediator
M when we seek to estimate the direct effect of
X on
Y (see the Figure above). The classical rationale for "controlling" for
M" is that, if we succeed in preventing
M from changing, then whatever changes we measure in Y are attributable solely to variations in
X and we are justified then in proclaiming the effect observed as "direct effect of
X on
Y." Unfortunately, "controlling for
M" does not physically prevent
M from changing; it merely narrows the analyst's attention to cases of equal
M values. Moreover, the language of probability theory does not possess the notation to express the idea of "preventing
M from changing" or "physically holding
M constant". The only operator probability provides is "Conditioning" which is what we do when we "control" for
M, or add
M as a regressor in the equation for
Y. The result is that, instead of physically holding
M constant (say at
M =
m) and comparing
Y for units under
X = 1' to those under
X = 0, we allow
M to vary but ignore all units except those in which
M achieves the value
M =
m. These two operations are fundamentally different, and yield different results, except in the case of no omitted variables. Improperly conditioning mediated effects can be a type of
bad control. To illustrate, assume that the error terms of
M and
Y are correlated. Under such conditions, the structural coefficient
B and
A (between
M and
Y and between
Y and
X) can no longer be estimated by regressing
Y on
X and
M. In fact, the regression slopes may both be nonzero even when
C is zero. This has two consequences. First, new strategies must be devised for estimating the structural coefficients
A, B and
C. Second, the basic definitions of direct and indirect effects must go beyond regression analysis, and should invoke an operation that mimics "fixing
M", rather than "conditioning on
M."
Definitions Such an operator, denoted do(
M =
m), was defined in Pearl (1994) or "structural counterfactuals". These new variables provide convenient notation for defining direct and indirect effects. In particular, four types of effects have been defined for the transition from
X = 0 to
X = 1: (a) Total effect – : TE = E [Y(1) - Y(0)] (b) Controlled direct effect - : CDE(m) = E [Y(1,m) - Y(0,m) ] (c) Natural direct effect - : NDE = E [Y(1,M(0)) - Y(0,M(0))] (d) Natural indirect effect : NIE = E [Y(0,M(1)) - Y(0,M(0))] Where
E[ ] stands for expectation taken over the error terms. These effects have the following interpretations: •
TE measures the expected increase in the outcome
Y as
X changes from
X=0 to
X =1, while the mediator is allowed to track the change in
X as dictated by the function
M = g(X, ε2). • CDE measures the expected increase in the outcome
Y as
X changes from
X = 0 to
X = 1, while the mediator is fixed at a pre-specified level
M = m uniformly over the entire population •
NDE measures the expected increase in
Y as
X changes from
X = 0 to
X = 1, while setting the mediator variable to whatever value it
would have obtained under
X = 0, i.e., before the change. •
NIE measures the expected increase in
Y when the
X is held constant, at
X = 1, and
M changes to whatever value it would have attained (for each individual) under
X = 1. • The difference
TE-NDE measures the extent to which mediation is
necessary for explaining the effect, while the
NIE measures the extent to which mediation is
sufficient for sustaining it. A controlled version of the indirect effect does not exist because there is no way of disabling the direct effect by fixing a variable to a constant. According to these definitions the total effect can be decomposed as a sum : TE = NDE - NIE_r where
NIEr stands for the reverse transition, from
X = 1 to
X = 0; it becomes additive in linear systems, where reversal of transitions entails sign reversal. The power of these definitions lies in their generality; they are applicable to models with arbitrary nonlinear interactions, arbitrary dependencies among the disturbances, and both continuous and categorical variables.
The mediation formula In linear analysis, all effects are determined by sums of products of structural coefficients, giving : \begin{align} TE & = C + AB \\ CDE(m) & = NDE = C, \text{ independent of } m\\ NIE & = AB. \end{align} Therefore, all effects are estimable whenever the model is identified. In non-linear systems, more stringent conditions are needed for estimating the direct and indirect effects. For example, if no confounding exists, (i.e., ε1, ε2, and ε3 are mutually independent) the following formulas can be derived: and have become the target of estimation in many studies of mediation. They give distribution-free expressions for direct and indirect effects and demonstrate that, despite the arbitrary nature of the error distributions and the functions
f,
g, and
h, mediated effects can nevertheless be estimated from data using regression. The analyses of
moderated mediation and
mediating moderators fall as special cases of the causal mediation analysis, and the mediation formulas identify how various interactions coefficients contribute to the necessary and sufficient components of mediation.
Example Assume the model takes the form : \begin{align} X & = \varepsilon_1 \\ M & = b_0 + b_1X + \varepsilon_2 \\ Y & = c_0 + c_1X + c_2M + c_3XM + \varepsilon_3 \end{align} where the parameter c_3 quantifies the degree to which
M modifies the effect of
X on
Y. Even when all parameters are estimated from data, it is still not obvious what combinations of parameters measure the direct and indirect effect of
X on
Y, or, more practically, how to assess the fraction of the total effect TE that is
explained by mediation and the fraction of TE that is
owed to mediation. In linear analysis, the former fraction is captured by the product b_1 c_2 / TE, the latter by the difference (TE - c_1)/TE, and the two quantities coincide. In the presence of interaction, however, each fraction demands a separate analysis, as dictated by the Mediation Formula, which yields: : \begin{align} NDE & = c_1 + b_0 c_3 \\ NIE & = b_1 c_2 \\ TE & = c_1 + b_0 c_3 + b_1(c_2 + c_3) \\ & = NDE + NIE + b_1 c_3. \end{align} Thus, the fraction of output response for which mediation would be
sufficient is : \frac{NIE}{TE} = \frac{b_1 c_2}{c_1 + b_0 c_3 + b_1 (c_2 + c_3)}, while the fraction for which mediation would be
necessary is : 1- \frac{NDE}{TE} = \frac{b_1 (c_2 +c_3)}{c_1 + b_0c_3 + b_1 (c_2 + c_3)}. These fractions involve non-obvious combinations of the model's parameters, and can be constructed mechanically with the help of the Mediation Formula. Significantly, due to interaction, a direct effect can be sustained even when the parameter c_1 vanishes and, moreover, a total effect can be sustained even when both the direct and indirect effects vanish. This illustrates that estimating parameters in isolation tells us little about the effect of mediation and, more generally, mediation and moderation are intertwined and cannot be assessed separately. ==See also==