# Multiply Robust Estimation of the Marginal Quantile `MR.quantile()` is used to estimate the marginal quantile of a variable which is subject to missingness. Multiple missingness probability models and imputation models are allowed. ```r MR.quantile(response, tau = 0.5, imp.model = NULL, mis.model = NULL, moment = NULL, order = 1, L = 30, data, bootstrap = FALSE, bootstrap.size = 300, alpha = 0.05) ``` ## Arguments - `response`: The response variable of interest whose marginal quantile is to be estimated. - `tau`: A numeric value in (0,1). The quantile to be estimated. - `imp.model`: A list of imputation 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. - `L`: Number of imputations. - `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 - **`q`**: The estimated value of the marginal quantile. - **`SE`**: The bootstrap standard error of `q` when `bootstrap = TRUE`. - **`CI`**: A Wald-type confidence interval based on `q` and `SE` when `bootstrap = TRUE`. - **`weights`**: The calibration weights if any `imp.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 imp1 <- glm.work(formula = Y ~ X + exp(X), family = gaussian) imp2 <- 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.quantile(response = Y, tau = 0.25, imp.model = list(imp1, imp2), mis.model = list(mis1, mis2), L = 10, data = dat) MR.quantile(response = Y, tau = 0.25, moment = c(X), order = 2, data = dat) ``` ## References Han, P., Kong, L., Zhao, J. and Zhou, X. (2019). A general framework for quantile estimation with incomplete data. **Journal of the Royal Statistical Society: Series B (Statistical Methodology)**. 81 (2), 305--333.

MR.quantile function