# Proper Average Treatment Effect estimators (and standard errors) for Generalized Linear Models with log link?

What is the proper formula for estimating the Average Treatment Effect with a simple main effects generalized linear model?

My first pass at defining the Average Treatment Effect for a GLM with a log link:

$$ATE = \biggl[\frac{\mathbb{E}(Y|A=1,\mathbf{X})}{\mathbb{E}(Y|A=0,\mathbf{X})}\biggr]\\ = \biggl[\frac{e^{\beta_0+\beta_A\cdot 1+\mathbf{\beta}\mathbf{X}}}{e^{\beta_0+\beta_A\cdot0+\mathbf{\beta}\mathbf{X}}}\biggr] \\ =e^{\beta_A}$$

where the 95% confidence interval for the ATE estimate would be $$e^{\beta_A}\pm e^{1.96 \times SE(\beta_{A})}$$.

For a project at work I'm doing a difference-in-differences case-control healthcare analysis to estimate the % change in episode-level medical costs from period=0 to period=1 for treatment=1 relative to treatment=0, using pooled cross-sectional observational data. (i.e., not longitudinal records but independent observations in each period $$\times$$ treatment group). Expanding the above ATE formula:

$$ATE = \biggl[\frac{\mathbb{E}(Y|P=1,A=1,\mathbf{X})/\mathbb{E}(Y|P=0,A=1,\mathbf{X})}{\mathbb{E}(Y|P=1,A=0,\mathbf{X})/\mathbb{E}(Y|P=0,A=0,\mathbf{X})}\biggr]\\ = \biggl[\frac{e^{\beta_0+\beta_P\cdot1+\beta_A\cdot1+\beta_{PA}\cdot1+\mathbf{\beta}\mathbf{X}}/e^{\beta_0+\beta_P\cdot0+\beta_A\cdot1+\beta_{PA}\cdot0+\mathbf{\beta}\mathbf{X}}}{e^{\beta_0+\beta_P\cdot1+\beta_A\cdot0+\beta_{PA}\cdot0+\mathbf{\beta}\mathbf{X}}/e^{\beta_0+\beta_P\cdot0+\beta_A\cdot0+\beta_{PA}\cdot0+\mathbf{\beta}\mathbf{X}}}\biggr] \\ =(e^{\beta_P +\beta_{PA}}/e^{\beta_P})\\ =e^{\beta_{PA}}$$

where the 95% confidence interval for the ATE estimate would be $$e^{\beta_{PA}}\pm e^{1.96 \times SE(\beta_{PA})}$$.

If I want to use inverse propensity score weighting, doubly robust methods, etc. for estimating the ATE, observed records would fall in the "treatment" group for the above difference-in-differences study if P=1 and A=1, with the "control" group containing records where P=0 and A=1, P=1 and A=0, or P=0 and A=0.

Is my reasoning on-track here?

I believe there is a parameteric solution to estimating the Average Treatment Effect for a difference-in-differences study using the $$g$$-computation method:

1) Fit a GLM to the data, including terms for the time period and treatment group plus interactions:

$$\mathbb{E}[Y|P,T,\mathbf{X}]=e^{\beta_0+\beta_P\cdot P+\beta_T\cdot T+\beta_{PT}\cdot P\cdot T+. . .}$$

2) For each record in the data compute the three counterfactual expected values of $$Y$$ (in addition to the existing expected value in the original data) by substituting alternative values for the the time period and treatment variables into the regression formula: $$\mathbb{E}[Y_i|P_i=0, T_i=0, \mathbf{X}_i]$$$$\mathbb{E}[Y_i|P_i=0, T_i=1,\mathbf{X}_i]$$$$\mathbb{E}[Y_i|P_i=1, T_i=0,\mathbf{X}_i]$$$$\mathbb{E}[Y_i|P_i=1, T_i=1,\mathbf{X}_i]$$

3) Compute the Average Treatment Effect by taking the geometric mean of the individual treatment effects:

$$ATE = \biggl[\prod_{i=1}^n\frac{\mathbb{E}(Y_i|P_i=1,A_i=1,\mathbf{X}_i)/\mathbb{E}(Y_i|P_i=0,A_i=1,\mathbf{X}_i)}{\mathbb{E}(Y_i|P_i=1,A_i=0,\mathbf{X}_i)/\mathbb{E}(Y_i|P_i=0,A_i=0,\mathbf{X}_i)}\biggr]^{\frac{1}{n}}$$

The 95% confidence interval for the $$ATE$$ estimate can be estimated with bootstrapping, computer resources permitting.