# The Hirano and Imbens estimator This function estimates the GPS function and estimates the ADRF. The GPS score is based on different treatment models. The treatment is linearly related to Xs. ```r hi_est(Y, treat, treat_formula, outcome_formula, data, grid_val, treat_mod, link_function, ...) ``` ## Arguments - `Y`: is the the name of the outcome variable contained in `data`. - `treat`: is the name of the treatment variable contained in `data`. - `treat_formula`: an object of class "formula" (or one that can be coerced to that class) that regresses `treat` on a linear combination of `X`: a symbolic description of the model to be fitted. - `outcome_formula`: is the formula used for fitting the outcome surface. gps is one of the independent variables to use in the outcome_formula. ie. `Y ~ treat+ I(treat^2) + gps + I(gps^2) + treat * gps` or a variation of this. Use `gps` as the name of the variable representing the gps in `outcome_formula`. - `data`: is a dataframe containing `Y`, `treat`, and `X`. - `grid_val`: contains the treatment values to be evaluated. - `treat_mod`: a description of the error distribution to be used in the model for treatment. Options include: `"Normal"` for normal model, `"LogNormal"` for lognormal model, `"Sqrt"` for square-root transformation to a normal treatment, `"Poisson"` for Poisson model, `"NegBinom"` for negative binomial model, `"Gamma"` for gamma model, `"Binomial"` for binomial model. - `link_function`: For `treat_mod = "Gamma"` (fitted using glm) alternatives are "log" or "inverse". For `treat_mod = "Binomial"` (fitted using glm) alternatives are "logit", "probit", "cauchit", "log" and "cloglog". - `...`: additional arguments to be passed to the outcome lm() function. ## Returns `hi_est` returns an object of class "causaldrf", a list that contains the following components: - **param**: parameter estimates for a hi fit. - **t_mod**: the result of the treatment model fit. - **out_mod**: the result of the outcome model fit. - **call**: the matched call. ## Details Hirano (2004) (HI) introduced this imputation-type method that includes a GPS component. The idea is to fit a parametric observable (outcome) model, which includes the estimated GPS as a covariate, to impute missing potential outcomes. The method requires several steps. First, a model is used to relate treatment to the recorded covariates. For example, $T_i|\textbf{X}_i \sim \mathcal{N}(\textbf{X}_i^T \boldsymbol{\beta}, \sigma^2)$ and then estimate the $\boldsymbol{\beta}$ parameters. Next, the GPS for each unit is estimated $$ \hat{R}_i(t) = \frac{1}{\sqrt{2 \pi \hat{\sigma}^2} } e^{-\frac{(t - \textbf{X}_i^T \boldsymbol{\hat{\beta}})^2}{2 \hat{\sigma}^2}} $$ These GPS estimates are used in the outcome or observable model. The outcome is modeled as a function of $T_i$ and $\hat{R}_i$ parametrically. For example, $$ E[Y_i | T_i, R_i] = \alpha_0 + \alpha_1 T_i + \alpha_2 T_i^2 + \alpha_3 \hat{R}_i + \alpha_4 \hat{R}_i^2 + \alpha_5 \hat{R}_i\cdot T_i $$ After collecting the estimated parameters in the outcome and treatment models, plug-in the treatment values into the model to estimate the missing potential outcomes of each individual at that treatment level. For example, if we plug in $T_i = t$ into the estimated models, then each unit will have a potential outcome estimated at treatment level $T_i= t$. $$ \hat{Y}_i(t) = \hat{\alpha}_0 + \hat{\alpha}_1 t + \hat{\alpha}_2 t^2 + \hat{\alpha}_3 \hat{R}_i(t) + \hat{\alpha}_4 \hat{R}_i^2(t) + \hat{\alpha}_5 \hat{R}_i(t) \cdot t $$ The next step is to aggregate these estimated potential outcomes to get an average treatment effect at dose level $T_i = t$. The mean outcome at dose-level $T_i = t$ is given by: $$ \hat{\mu}(t) = \frac{1}{N}\sum_i^N \hat{\alpha}_0 + \hat{\alpha}_1 t + \hat{\alpha}_2 t^2 + \hat{\alpha}_3 \hat{R}_i(t) + \hat{\alpha}_4 \hat{R^2}_i(t) + \hat{\alpha}_5 \hat{R}_i(t) \cdot t $$ Different treatment levels are plugged into the previous equation to estimate the missing potential outcomes. If many $t$ values are evaluated, then it is possible to trace out an ADRF. ## Examples ```r ## Example from Schafer (2015). example_data <- sim_data hi_list <- hi_est(Y = Y, treat = T, treat_formula = T ~ B.1 + B.2 + B.3 + B.4 + B.5 + B.6 + B.7 + B.8, outcome_formula = Y ~ T + I(T^2) + gps + I(gps^2) + T * gps, data = example_data, grid_val = seq(8, 16, by = 1), treat_mod = "Normal") sample_index <- sample(1:1000, 100) plot(example_data$T[sample_index], example_data$Y[sample_index], xlab = "T", ylab = "Y", main = "hi estimate") lines(seq(8, 16, by = 1), hi_list$param, lty = 2, lwd = 2, col = "blue") legend('bottomright', "hi estimate", lty=2, lwd = 2, col = "blue", bty='Y', cex=1) rm(example_data, hi_list, sample_index) ## Example from van der Wal, Willem M., and Ronald B. Geskus. (2011) #Simulate data with continuous confounder and outcome, binomial exposure. #Marginal causal effect of exposure on outcome: 10. n <- 1000 simdat <- data.frame(l = rnorm(n, 10, 5)) a.lin <- simdat$l - 10 pa <- exp(a.lin)/(1 + exp(a.lin)) simdat$a <- rbinom(n, 1, prob = pa) simdat$y <- 10*simdat$a + 0.5*simdat$l + rnorm(n, -10, 5) simdat[1:5,] temp_hi <- hi_est(Y = y, treat = a, treat_formula = a ~ l, outcome_formula = y ~ gps, data = simdat, grid_val = c(0, 1), treat_mod = "Binomial", link_function = "logit") temp_hi[[1]] # estimated coefficients ``` ## References Schafer, J.L., Galagate, D.L. (2015). Causal inference with a continuous treatment and outcome: alternative estimators for parametric dose-response models. **Manuscript in preparation**. Hirano, Keisuke, Imbens, Guido W (2004). The propensity score with continuous treatments. **Applied Bayesian modeling and causal inference from incomplete-data perspectives.** ## See Also `nw_est`, `iw_est`, `hi_est`, `gam_est`, `add_spl_est`, `bart_est`, etc. for other estimates. `t_mod`, `overlap_fun` to prepare the `data` for use in the different estimates.

hi_est function