Linear mediation and moderation

# Linear mediation and moderation

**Session 12**

.light[MATH 80667A: Experimental Design and Statistical Methods for Quantitative Research in Management 
HEC Montréal
]

]

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name: outline
class: title title-inv-1

# Outline
--

.box-4.large.sp-after-half[Linear mediation model]

.box-6.large.sp-after-half[Interactions and moderation]

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layout: false
name: linear-mediation
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# Linear mediation

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# Three types of associations

![](12-slides_files/figure-html/confounding-dag-1.png)
.box-inv-2.small.sp-after-half[Common cause]
.box-inv-2.small[Causal forks **X** ← **Z** → **Y**]
]
.pull-middle-3[
.box-2.medium[Causation]
![](12-slides_files/figure-html/mediation-dag-1.png)
.box-inv-2.small.sp-after-half[Mediation]
.box-inv-2.small[Causal chain **X** → **Z** → **Y**]
]
.pull-right-3[
.box-2.medium[Collision]
![](12-slides_files/figure-html/collision-dag-1.png)
.box-inv-2.small.sp-after-half[Selection / endogeneity]
.box-inv-2.small[inverted fork **X** → **Z** ← **Y**]
]

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# Key references

- Imai, Keele and Tingley (2010), [A General Approach to Causal Mediation Analysis](https://doi.org/10.1037/a0020761), *Psychological Methods*.
- Pearl (2014), [Interpretation and Identification of Causal Mediation](http://dx.doi.org/10.1037/a0036434), *Psychological Methods*.
- Baron and Kenny (1986), [The Moderator-Mediator  Variable  Distinction in Social Psychological Research: Conceptual,  Strategic, and Statistical Considerations](https://doi.org/10.1037/0022-3514.51.6.1173), *Journal of Personality and Social Psychology*

Limitations:

- Bullock, Green, and Ha (2010), [Yes, but what’s the mechanism? (don’t expect an easy answer)](https://doi.org/10.1037/a0018933)
- Uri Simonsohn (2022) [Mediation Analysis is Counterintuitively Invalid](http://datacolada.org/103)

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# Sequential ignorability assumption

Define

- treatment of individual `$i$` as `$X_i$`, 
- potential mediation given treatment `$x$` as `$M_i(x)$` and 
- potential outcome for treatment `$x$` and mediator `$m$` as `$Y_i(x, m)$`.

Given pre-treatment covariates `$W$`, potential outcomes for mediation and treatment are conditionally independent of treatment assignment.
$$ Y_i(x', m), M_i(x) \perp\mkern-10mu\perp X_i \mid W_i = w$$
Given pre-treatment covariates and observed treatment, potential outcomes are independent of mediation.
$$ Y_i(x', m) \perp\mkern-10mu\perp  M_i(x) \mid X_i =x, W_i = w$$

Formulation in terms of potential outcomes (what if?) with intervention.

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# Total effect

**Total effect**: overall impact of `$X$` (both through `$M$` and directly)

`$$\begin{align*}\mathsf{TE}(x, x^*) = \mathsf{E}[ Y \mid \text{do}(X=x)] - \mathsf{E}[ Y \mid \text{do}(X=x^*)]\end{align*}$$`

This can be generalized for continuous `$X$` to any pair of values `$(x_1, x_2)$`.

.pull-left[
.box-inv-3[
**X** → **M** → **Y** plus **X** → **Y**
]
]
.pull-right[
<img src="12-slides_files/figure-html/moderation-1.png" width="80%" style="display: block; margin: auto;" />
]
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# Average controlled direct effect

`$$\begin{align*}\textsf{CDE}(m, x, x^*) &= \mathsf{E}[Y \mid \text{do}(X=x, m=m)] - \mathsf{E}[Y \mid \text{do}(X=x^*, m=m) \\&= \mathsf{E}\{Y(x, m) - Y(x^*, m)\} 
\end{align*}$$`
Expected population change in response when the experimental factor changes from `$x$` to `$x^*$` and the mediator is set to a fixed value `$m$`.

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# Direct and indirect effects

**Natural direct effect**: `$\textsf{NDE}(x, x^*) = \mathsf{E}[Y\{x, M(x^*)\} - Y\{x^*,  M({x^*})\}]$`    
   - expected change in `$Y$` under treatment `$x$` if `$M$` is set to whatever value it would take under control `$x^*$`

**Natural indirect effect**: `$\textsf{NIE}(x, x^*) = \mathsf{E}[Y\{x^*, M(x)\} - Y\{x^*,  M(x^*)\}]$`  
   - expected change in `$Y$` if we set `$X$` to its control value and change the mediator value which it would attain under `$x$`

.small[
Counterfactual conditioning reflects a physical intervention, not mere (probabilistic) conditioning.
]

Total effect is `$\mathsf{TE}(x, x^*) = \textsf{NDE}(x, x^*) - \textsf{NIE}(x^*, x)$`
???

# Necessary and sufficiency of mediation

From Pearl (2014):

> The difference `$\textsf{TE}-\textsf{NDE}$` quantifies the extent to which the response of `$Y$` is owed to mediation, while `$\textsf{NIE}$` quantifies the extent to which it is explained by mediation. These two components of mediation, the necessary and the sufficient, coincide into one in models void of interactions (e.g., linear) but differ substantially under moderation

- In linear systems, changing the order og arguments amounts to flipping signs
- This definition works under temporal reversal and gives the correct answer (the regression-slope approach of the linear structural equation model does not).

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# Linear structural equation modelling and mediation

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# The Baron−Kenny model

Given **uncorrelated** unobserved noise variables `$U_M$` and `$U_Y$`, consider linear regression models
`$$\begin{align}
M &= c_M + \alpha x + U_M\\
Y &=  c_Y + \beta x + \gamma m + U_Y
\end{align}$$`
Plugging the first equation in the second, we get the marginal model for `$Y$` given treatment `$X$`,
`$$\begin{align}
\mathsf{E}_{U_M}(Y \mid x) &= \underset{\text{intercept}}{(c_Y + \gamma c_M)} + \underset{\text{total effect}}{(\beta + \alpha\gamma)}\cdot x + \underset{\text{error}}{(\gamma U_M + U_Y)}\\
&= c_Y' + \tau X + U_Y'
\end{align}$$`

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# The old method
Baron and Kenny recommended running regressions and estimating the three models with

1. whether `$\mathscr{H}_0: \alpha=0$`
2. whether `$\mathscr{H}_0: \tau=0$` (total effect)
3. whether `$\mathscr{H}_0: \gamma=0$`

The conditional indirect effect `$\alpha\gamma$` and we can check whether it's zero using Sobel's test statistic.

Problems?
???

- Type I errors
- The total effect can be zero because `$\alpha\gamma = - \beta$`
- The method has lower power for small mediation effect
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# Sobel's test

Based on estimators `$\widehat{\alpha}$` and `$\widehat{\gamma}$`, construct a Wald-test
`$$S  = \frac{\widehat{\alpha}\widehat{\gamma}-0}{\sqrt{\widehat{\gamma}^2\mathsf{Va}(\widehat{\alpha}) + \widehat{\alpha}^2\mathsf{Va}(\widehat{\gamma}) + \mathsf{Va}(\widehat{\gamma})\mathsf{Va}(\widehat{\alpha})}} \stackrel{\cdot}{\sim}\mathsf{No}(0,1)$$` 
where the point estimate `$\widehat{\alpha}$` and its variance `$\mathsf{Va}(\widehat{\alpha})$` can be estimated via SEM, or more typically linear regression (ordinary least squares).

???

Without interaction/accounting for confounders, `$\alpha\gamma = \tau - \beta$` and with OLS we get exactly the same point estimates. The derivation of the variance is then relatively straightforward using the delta method.

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# Null distribution for the test

The large-sample normal approximation is poor in small samples.

The popular way to estimate the _p_-value and the confidence interval is through the nonparametric **bootstrap** with the percentile method.

Repeat `$B$` times, say `$B=10\ 000$`
1. sample **with replacement** `$n$` observations from the database 
 - tuples `$(Y_i, X_i, M_i)$`
2. recalculate estimates `$\widehat{\alpha}^{(b)}\widehat{\gamma}^{(b)}$`

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# Boostrap _p_-values and confidence intervals
.pull-left[
.box-inv-3[Confidence interval]

Percentile-based method: for a equi-tailed `$1-\alpha$` interval and the collection
`$$\{\widehat{\alpha}^{(b)}\widehat{\gamma}^{(b)}\}_{b=1}^B,$$`
compute the `$\alpha/2$` and `$1-\alpha/2$` empirical quantiles.

]
.pull-right[
.box-inv-3[Two-sided _p_-value]

Compute the sample proportion of bootstrap statistics `$S^{(1)}, \ldots, S^{(B)}$` that are larger/smaller than zero.

If `$S^{(M)} < 0 \leq S^{(M+1)}$` for `$1 \leq M \leq B$`.

`$$p = 2\min\{M/B, 1-M/B\}$$`

and zero otherwise
]

???
Note: many bootstraps! parametric, wild, sieve, block, etc. and many methods (basic, studentized, bias corrected and accelerated

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# Example from Preacher and Hayes (2004)

> Suppose an investigator is interested in the effects of a new cognitive therapy on life satisfaction after retirement.

>Residents of a retirement home diagnosed as clinically
depressed are randomly assigned to receive 10 sessions of a
new cognitive therapy `$(X = 1)$` or 10 sessions of an alternative
(standard) therapeutic method `$(X = 0)$`.

> After Session 8, the positivity of the attributions the residents
make for a recent failure experience is assessed `$(M)$`.

> Finally, at the end of Session 10, the residents are given a
measure of life satisfaction `$(Y)$`.
The question is whether the cognitive therapy’s effect on life
satisfaction is mediated by the positivity of their causal
attributions of negative experiences. ”

]
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# Defaults of linear SEM

- Definitions contingent on model 
   - (causal quantities have a meaning regardless of estimation method)
- Linearity assumption not generalizable.
   - effect constant over individuals/levels

Additional untestable assumption of uncorrelated disturbances (no unmeasured confounders).
]
.pull-right-narrow[
![](img/12/spherical_cow.png)
.small[Keenan Crane]
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# Assumptions of causal mediation

Need assumptions to hold (and correct model!) to derive causal statements

- Potential confounding can be accounted for with explanatories.
- Careful with what is included (colliders)! 
 - *as-if* randomization assumption
 
- Generalizations to interactions, multiple mediators, etc. should require careful acknowledgement of confounding.