# Is augmentation of treatment effect caused significantly by mediator?

I want to find out if the influence of potential mediator P on treatment effect D is significant. In the two models below $\beta_1$ and $\lambda_1$ are different and I don't know how to test if the difference is significant. Consider D as a perfect random assignment.

$$\hat{y} = \beta_0 + \beta_1D + \epsilon$$

and

$$\hat{y} = \lambda_0 + \lambda_1D + \lambda_2P + \mu$$

I could find a lot of answers comparing differences between two groups. Since I'm considering only one group I think this problem has not been answered before.

I give following example.

# Data example

df1 <- structure(list(Y = c(5, 8, 6, 7, 8, 8, 10, 6, 7, 7, 5, 7, 8,
7, 7, 5, 8, 6, 7, 5, 10, 10, 6, 7, 4, 7, 9, 8, 2, 6, 6, 5, 6,
6, 6, 7, 6, 4, 7, 2, 6, 5, 8, 7, 5, 6, 6, 8, 6, 6), P = c(5,
5, 4, 5, 5, 4, 5, 5, 4, 5, 4, 5, 5, 4, 3, 4, 5, 3, 5, 5, 5, 5,
4, 4, 2, 6, 6, 5, 4, 5, 5, 4, 5, 6, 5, 5, 5, 4, 5, 6, 5, 2, 5,
4, 4, 5, 5, 6, 4, 5), D = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)), .Names = c("Y",
"P", "D"), row.names = c("159.31", "174.31", "106.31", "264.31",
"424.31", "371.31", "379.31", "399.31", "177.31", "269.31", "133.31",
"89.31", "78.31", "385.31", "130.31", "330.31", "260.31", "406.31",
"278.31", "455.31", "245.31", "225.31", "285.31", "173.31", "447.31",
"310.3", "251.3", "337.3", "122.3", "229.3", "362.3", "241.3",
"151.3", "140.3", "398.3", "402.3", "169.3", "152.3", "219.3",
"283.3", "110.3", "359.3", "442.3", "354.3", "415.3", "105.3",
"246.3", "459.3", "113.3", "161.3"), class = "data.frame")


# Models

s0 <- summary(f0 <- lm(Y ~ D , df1))
s1 <- summary(f1 <- lm(Y ~ D + P , df1))
> texreg::screenreg(list(f0, f1))

================================
Model 1    Model 2
--------------------------------
(Intercept)   6.96 ***   3.55 **
(0.32)     (1.14)
D            -0.96 *    -1.30 **
(0.46)     (0.43)
P                        0.77 **
(0.25)
--------------------------------
R^2           0.08       0.24
Num. obs.    50         50
RMSE          1.61       1.49
================================
*** p < 0.001, ** p < 0.01, * p < 0.05


Taking P into the model P is significant and D is augmented. Very well. But how can I test if this augmentation is significant?

When I plot the coefficients the confidence intervals do overlap.

# Plot

m <- rbind(data.frame(var="D", coef=coef(s0)[2, 1], se=coef(s0)[2, 2], mn="w/o P"),
data.frame(var="D", coef=coef(s1)[2, 1], se=coef(s1)[2, 2], mn="w/ P"))

library(ggplot2)
library(ggstance)
ggplot(m, aes(color=mn)) +
geom_pointrangeh(aes(y=mn, x=coef, xmin=coef - se*1.96, xmax=coef + se*1.96)) +
theme_bw() + scale_color_manual(values=c("black", "blue")) +
guides(color=FALSE)


But I suppose the overlap is not an ultimate proof that there's no significant difference between the two models, though.

An anova() test between both models only would show me if P is significant. But I need to test if the shift of D is significant. How would I do that?

With my real data the AIC difference between both models is $39>2$. According to a "rule of thumb" mentioned in an answer this could be evidence of significant difference. Such a statement would be very thin, though, and I'm not sure.

How can I test if the augmented treatment effect is significant?

• I think the delta method/generalized estimating equations might be helpful here, but I don't know enough to give a good answer. With GEE, you can set up a system of equations corresponding to the two models, and create a new parameter which is the difference between the two coefficients. – Noah May 19 '18 at 2:54
• @Noah Thanks for your comment. Hm since I've heard the first time of GEE I'd need more information of how to setup these equations. Perhaps you could cite a straightforward source related to my issue? I've heard about a Blinder-Oaxaca decomposition approach as well. Could eventually anybody else formulate an answer? – jay.sf May 19 '18 at 10:05

The effect of D on Y may be confounded by P. Your second model can be thought of as adjusting for confounding by P. Alternatively, P may be an effect modifier of D on Y. To test for this, test for the interaction of P on D by entering Y~D+P+D*P. In either case, use of the AIC is reasonable to detect the information gain or loss between models. However, you will need to understand the model you built to gain any information from the AIC or other significance tests.
• Thanks for your answer. Since D is randomly assigned (I've updated that information to my question), and there is no causal path from P to D we can exclude the assumption the treatment effect might be confounded by P. Taking the interaction D:P into model 2, AIC rises slightly (with real data) and $\chi^2$ is not significant. But this only tells us that the effect is not distinguishable between treatment groups, doesn't it? And we still don't know if the shift in coefficient D between models 1 and 2 is significant. Or am I wrong? – jay.sf May 19 '18 at 16:52