# Simulation Model Notes

This simulation is the same as ''Marshall_notes_recentered_dists'' but now instead of the marker having a mixture of normals with zero mean, we set the mean as the crossing point of the two risk curves (ie where the trt effect given Y is zero).

link to the notes for the previous model

## plot of interest

Recall that:

• the old model: marker distributions are centered at zero
• the new model: marker distributions are centered at $$y_0$$ s.t. $$\Delta(y = y_0) = 0$$. (See below for more details regarding the simulation model)

This figure shows $$p_0 = Pr(D=1|T=0)$$ and $$p_1 = Pr(D=1|T=1)$$ for the new model (red dot) compared to the $$p_0$$ and $$p_1$$ for the old model (black vertical and horizontal lines). We see that shifting the marker mean causes the largest change from old to new when the difference between $$q_0$$ and $$q_1$$ is large. The shift changes the values of $$p_0$$ and $$p_1$$ to be more similar to each other, which defeats the purpose of trying to set the marginals and look at the coverage.

• pros to new method: changing $$p$$ now has the desired effect of varying the proportion of observations with small treatment effects. see here
• cons: we can't set the marginals.

For the figure below $$k = 4$$ and $$p = 0.1$$, but this is largely irrelevant because these plots look very similar for all values of $$k$$ and $$p$$.

## Modeling assumptions

• Y: mixture of normals with: $$f(y) = p\phi(-\beta_1/\psi_2, c_1) + (1-p)\phi(-\beta_1/\psi_2, c_2)$$

• $$p\in(0,1)$$
• $$k = c_2/c_1$$
• $$var(Y) = 1$$.
• $$logit(Pr(D = 1 | T,Y)) = \beta_0 + \beta_1T +\psi_1Y + \psi_2YT$$

• $$\psi_1 = \psi_2=1$$ and $$\beta_0, \beta_1$$ chosen to fix:
• $$q_0 = expit(\beta_0)$$
• $$q_1 = expit(\beta_0 + \beta_1)$$

and

• $$p0 = Pr(D=1|T=0)$$
• $$p1 = Pr(D=1|T=1)$$

## Risk Curves(top)

Risk by F(y) given trt by q0 and q1. True value of Theta in grey.

## Treatment Effect Distribution(top)

Distribution of $$\Delta(y) = Pr(D=1|T=0, Y) - Pr(D-1|T=1, Y)$$ by $$F(y)$$.

## Marker Distribution(top)

pdf of y colored by p. The vertical black line indicates where $$\Delta(y)=0$$.

## Proportion of markers near the boundary(top)

This is a plot of p (x-axis) vs $$F(y_{\Delta=0} + 0.01) - F(y_{\Delta=0} - 0.01)$$ where $$\Delta(y_{\Delta=0}) = 0$$. This is a very arbitrary measure of proportion of observations near the boundary of interest.