# Multiply Robust Estimation of the Marginal Mean `MR.mean()` is used to estimate the marginal mean of a variable which is subject to missingness. Multiple missingness probability models and outcome regression models can be accommodated. ```r MR.mean(response, reg.model = NULL, mis.model = NULL, moment = NULL, order = 1, data, bootstrap = FALSE, bootstrap.size = 300, alpha = 0.05) ``` ## Arguments - `response`: The response variable of interest whose marginal mean is to be estimated. - `reg.model`: A list of outcome regression models defined by `glm.work`. - `mis.model`: A list of missingness probability models defined by `glm.work`. The dependent variable is always specified as `R`. - `moment`: A vector of auxiliary variables whose moments are to be calibrated. - `order`: A numeric value. The order of moments up to which to be calibrated. - `data`: A data frame with missing data encoded as `NA`. - `bootstrap`: Logical. If `bootstrap = TRUE`, the bootstrap will be applied to calculate the standard error and construct a Wald confidence interval. - `bootstrap.size`: A numeric value. Number of bootstrap resamples generated if `bootstrap = TRUE`. - `alpha`: Significance level used to construct the 100(1 - `alpha`)% Wald confidence interval. ## Returns - **`mu`**: The estimated value of the marginal mean. - **`SE`**: The bootstrap standard error of `mu` when `bootstrap = TRUE`. - **`CI`**: A Wald-type confidence interval based on `mu` and `SE` when `bootstrap = TRUE`. - **`weights`**: The calibration weights if any `reg.model`, `mis.model` or `moment` is specified. ## Examples ```r # Simulated data set set.seed(123) n <- 400 gamma0 <- c(1, 2, 3) alpha0 <- c(-0.8, -0.5, 0.3) X <- runif(n, min = -2.5, max = 2.5) p.mis <- 1 / (1 + exp(alpha0[1] + alpha0[2] * X + alpha0[3] * X ^ 2)) R <- rbinom(n, size = 1, prob = 1 - p.mis) a.x <- gamma0[1] + gamma0[2] * X + gamma0[3] * exp(X) Y <- rnorm(n, a.x, sd = sqrt(4 * X ^ 2 + 2)) dat <- data.frame(X, Y) dat[R == 0, 2] <- NA # Define the outcome regression models and missingness probability models reg1 <- glm.work(formula = Y ~ X + exp(X), family = gaussian) reg2 <- glm.work(formula = Y ~ X + I(X ^ 2), family = gaussian) mis1 <- glm.work(formula = R ~ X + I(X ^ 2), family = binomial(link = logit)) mis2 <- glm.work(formula = R ~ X + exp(X), family = binomial(link = cloglog)) MR.mean(response = Y, reg.model = list(reg1, reg2), mis.model = list(mis1, mis2), data = dat) MR.mean(response = Y, moment = c(X), order = 2, data = dat) ``` ## References Han, P. and Wang, L. (2013). Estimation with missing data: beyond double robustness. **Biometrika**, 100 (2), 417--430. Han, P. (2014). A further study of the multiply robust estimator in missing data analysis. **Journal of Statistical Planning and Inference**, 148 , 101--110.

MR.mean function