# ANCOVA when the covariate is a mediator affected by treatment

This is fake data that simulates an experiment to measure effect of treatment on fat weight in mice. The treatment is “diet” with two levels: “control” (blue dots) and “treated” (gold dots). Diet has a large effect on total body weight. The simulated data are in the plot above - these look very much like the real data. The question is, what are problems with using an “ancova” linear model to estimate the direct effect of treatment on fat weight?

# Using Wright's rules and a DAG to compute the bias of an effect when we measure proxies for X and Y

This is a skeletal post to work up an answer to a twitter question using Wright’s rules of path models. Using this figure from Panel A of a figure from Hernan and Cole. The scribbled red path coefficients are added the question is I want to know about A->Y but I measure A* and Y*. So in figure A, is the bias the backdoor path from A* to Y* through A and Y?

# What is the bias in the estimation of an effect given an omitted interaction term?

Some background (due to Sewall Wright’s method of path analysis) Given a generating model: $$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3$$$ where $$x_3 = x_1 x_2$$; that is, it is an interaction variable. The total effect of $$x_1$$ on $$y$$ is $$\beta_1 + \frac{\mathrm{COV}(x_1, x_2)}{\mathrm{VAR}(x_1)} \beta_2 + \frac{\mathrm{COV}(x_1, x_3)}{\mathrm{VAR}(x_1)} \beta_3$$. If $$x_3$$ (the interaction) is missing, its component on the total efffect is added to the coefficient of $$x_1$$.

# Expected covariances in a causal network

This is a skeleton post Standardized variables (Wright’s rules) n <- 10^5 # z is the common cause of g1 and g2 z <- rnorm(n) # effects of z on g1 and g2 b1 <- 0.7 b2 <- 0.7 r12 <- b1*b2 g1 <- b1*z + sqrt(1-b1^2)*rnorm(n) g2 <- b2*z + sqrt(1-b2^2)*rnorm(n) var(g1) # E(VAR(g1)) = 1 ## [1] 1.001849 var(g2) # E(VAR(g2)) = 1 ## [1] 1.006102 cor(g1, g2) # E(COR(g1,g2)) = b1*b2 ## [1] 0.

#### R doodles. Some ecology. Some physiology. Much fake data.

Thoughts on R, statistical best practices, and teaching applied statistics to Biology majors.

Jeff Walker, Professor of Biological Sciences

University of Southern Maine, Portland, Maine, United States