In trbrick/MICr: Matrices of Implied Causation in R

1. Introduction
2. Mediation model MICs
3. Potential Intervention Evaluation
4. Multiple Objective Optimization

Introduction

In this vignette, we provide an introduction to using MICs to select among a set of interventions. We assume the mediation model described in Brick & Bailey (submitted), in which Z partially mediates the effect of X on Y, and in which there are several approaches for

library(MICr)
library(umx)
library(knitr)
drawPlots <- library(semPlot, logical.return = TRUE)  # Plot only if there's a plotting function.
knitr::opts_chunk$set(fig.align="center")

Mediation Model MICs

We can construct a multiple mediator data set with a few quick lines of R code:

set.seed(1234)
# Predictors
Predictor1 <- rnorm(100)
Predictor2 <- rnorm(100)
# Mediator
Mediator  <- .3*Predictor1 + .7*Predictor2 + rnorm(100)
#Outcome
Outcome  <-0*Predictor1 + 0.4*Predictor2 + .7*Mediator + rnorm(100) 
# Combined Data
theData <- data.frame(Predictor1, Predictor2, Mediator, Outcome)

We'll start out by just building the mediation model. We'll use umx here for simplicity and semPaths for plotting.

model <- umxRAM("MediationModel", data=theData,
                # Predictor Effects
                umxPath(fromEach=c("Predictor1", "Predictor2"), to=c("Mediator", "Outcome")),

                # Mediator Effect
                umxPath(from="Mediator", to="Outcome"),
                # Variances, means, and intercepts
                umxPath(v.m. = c("Predictor1", "Predictor2", "Mediator", "Outcome"))
)
if(drawPlots) semPaths(model)

We can generate the causal implications of this model by computing the MIC. This table shows the model-implied causal effects of each variable on each other variable. These are loading weights for cases where there is only a direct effect, and total effects when there are both direct and indirect effects.

MIC(model)

We can format them more precisely (and with standard errors) using the MICTable() function:

MICTable(model, se=TRUE)

Evaluating potential interventions

In this case, we will examine the effects of three different potential interventions on the outcome: one intervention (ModifyP1) that improves Predictor1 by two points, another (ModifyP2) that increases Predictor2 by one point, and a joint intervention ModifyBoth that increases both by a half point.

ixnModel <- umxRAM("MediationModel", data=theData,
                # Predictor Effects
                umxPath(fromEach=c("Predictor1", "Predictor2"), to=c("Mediator", "Outcome")),

                # (Latent) Intervention Effects
                umxPath(from="ModifyP1", to="Predictor1", fixedAt=2.0),
                umxPath(from="ModifyP2", to="Predictor2", fixedAt=1.0),
                umxPath(from="ModifyBoth", to=c("Predictor1", "Predictor2"), fixedAt=c(.5, .5)),

                # Mediator Effect
                umxPath(from="Mediator", to="Outcome"),
                # Variances, means, and intercepts
                umxPath(v.m. = c("Predictor1", "Predictor2", "Mediator", "Outcome"))
)
if(drawPlots) semPaths(ixnModel)

The intervention model's MIC looks like this:

MIC(ixnModel)

The MIC is somewhat intricate, so we'll again use the MICTable format:

MICTable(ixnModel, caption="Intervention Results")

Even here, there's complexity. It can be helpful to look directly at the effects of the interventions on the outcomes:

MICTable(ixnModel, caption="Implied Intervention Effects On Output", from=c("ModifyP1", "ModifyP2", "ModifyBoth"), to="Outcome")

ModifyP2 has the strongest predicted influence on the outcome, partially because the direct effects of P2 are so strong. In case the mediator is also of interest, we can check the influence on that as well.

MICTable(ixnModel, caption="Implied Intervention Effects On Output", from=c("ModifyP1", "ModifyP2", "ModifyBoth"), to="Mediator")

If we're interested in modifying both the mediator and the outcome, ModifyP1 may be a better choice. If we are interested in modifying only the outcome, ModifyP2 is better.

Multiobjective Optimization

In cases with many possible outcomes, it may be helpful to generate a pareto plot that balances the cost against the influence of the outcome. A Pareto plot provides a tradeoff surface showing the maximum influence of outcome at a given level of cost. We can do this by assigning costs to each of the interactions in the MIC Table for the outcome.

The cost function here is fairly simple:

costFrame <- data.frame(name=c("ModifyP1", "ModifyP2", "ModifyBoth"), cost=c(1, 2, 1.5))
costFrame

And then we can compute the pareto front and plot it directly from the MIC package.

paretoFront(mic=ixnModel, outcome="Outcome", costs=costFrame)

To find the optimal effect at a given level of cost, trace upward to the line at the selected cost level, and then follow the line left to the first observed value. That represents the intervention with the largest effect on the outcome that can be achieved for that cost. If several have a common effect size, the optimal is the one with the lowest cost.

For better comparisons, we can add a scale column, for cases where the intervention may have a quantitative rather than discrete effect, e.g. where more intervention (e.g. more clinical sessions) is expected to have a linear causal effect on the outcome.

costFrame <- data.frame(name=c("ModifyP1", "ModifyP2", "ModifyBoth", "ModifyP1", "ModifyP2", "ModifyBoth"), cost=c(1, 2, 1.5, 2, 4, 3), scale=c(1,1,1,2,2,2))
costFrame

Here, the pareto front may be more informative:

paretoFront(mic=ixnModel, outcome="Outcome", costs=costFrame)

Here, it is clear that the ModifyBoth intervention is sub-optimal. Indeed, ModifyP1 is so efficient that it is better to apply it with double effect than to apply ModifyP2 at scale 1. The cost for these two interventions is the same, but the effect is stronger for ModifyP1.

Note that the scale here is a multiplier for the effect, not for the intervention. That is, it may take 4 training sessions to have twice the effect of a single session. These nonlinear effects should be reflected in the cost--they are not included in the model, and thus are not accounted for in the MIC.

trbrick/MICr documentation built on March 7, 2020, 3:30 p.m.