MR.reg {MultiRobust}

Multiply Robust Estimation for (Mean) Regression

Description

MR.reg() is used for (mean) regression under generalized linear models with missing responses and/or missing covariates. Multiple missingness probability models and imputation models are allowed.

Usage

MR.reg(formula, family = gaussian, imp.model = NULL,
  mis.model = NULL, moment = NULL, order = 1, L = 30, data,
  bootstrap = FALSE, bootstrap.size = 300, alpha = 0.05, ...)

Arguments

formula

The formula of the regression model of interest.

family

A description of the error distribution and link function to be used for the GLM of interest.

imp.model

A list of possibly multiple lists of the form list(list.1, list.2, ..., list.K), where K is the total number of different imputation models. For the k-th imputation model, list.k is a list of possibly multiple models, each of which is defined by glm.work and imputes one single missing variable marginally. See details.

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.

...

Addition arguments for the function glm.

Details

The function MR.reg() currently deals with data with one missingness pattern. When multiple variables are subject to missingness, their values are missing simultaneously. The methods in Han (2016) and Zhang and Han (2019) specify an imputation model by modeling the joint distribution of the missing variables conditional on the fully observed variables. In contrast, the function MR.reg() specifies an imputation model by separately modeling the marginal distribution of each missing variable conditional on the fully observed variables. These marginal distribution models for different missing variables constitute one joint imputation model. Different imputation models do not need to model the marginal distribution of each missing variable differently.

Value

coefficients

The estimated regression coefficients.

SE

The bootstrap standard error of coefficients when bootstrap = TRUE.

CI

A Wald-type confidence interval based on coefficients and SE when bootstrap = TRUE.

References

Han, P. (2014). Multiply robust estimation in regression analysis with missing data. Journal of the American Statistical Association, 109(507), 1159–1173.

Han, P. (2016). Combining inverse probability weighting and multiple imputation to improve robustness of estimation. Scandinavian Journal of Statistics, 43, 246–260.

Zhang, S. and Han, P. (2019). A simple implementation of multiply robust estimation for GLMs with missing data. Unpublished manuscript.

See Also

glm.

Examples

# Simulated data set
set.seed(123)
n <- 400
gamma0 <- c(1, 2, 3)
alpha0 <- c(-0.8, -0.5, 0.3)
S <- runif(n, min = -2.5, max = 2.5) # auxiliary variables
X1 <- rbinom(n, size = 1, prob = 0.5) # covariate X1
X2 <- rexp(n) # covariate X2
p.obs <- 1 / (1 + exp(alpha0[1] + alpha0[2] * S + alpha0[3] * S ^ 2)) # non-missingness probability
R <- rbinom(n, size = 1, prob = p.obs)
a.x <- gamma0[1] + gamma0[2] * X1 + gamma0[3] * X2
Y <- rnorm(n, a.x)
dat <- data.frame(S, X1, X2, Y)
dat[R == 0, c(2, 4)] <- NA # X1 and Y may be missing

# marginal imputation models for X1
impX1.1 <- glm.work(formula = X1 ~ S, family = binomial(link = logit))
impX1.2 <- glm.work(formula = X1 ~ S + X2, family = binomial(link = cloglog))
# marginal imputation models for Y
impY.1 <- glm.work(formula = Y ~ S, family = gaussian)
impY.2 <- glm.work(formula = Y ~ S + X2, family = gaussian)
# missingness probability models
mis1 <- glm.work(formula = R ~ S + I(S ^ 2), family = binomial(link = logit))
mis2 <- glm.work(formula = R ~ I(S ^ 2), family = binomial(link = cloglog))
# this example considers the following K = 3 imputation models for imputing the missing (X1, Y)
imp1 <- list(impX1.1, impY.1)
imp2 <- list(impX1.1, impY.2)
imp3 <- list(impX1.2, impY.1)

results <- MR.reg(formula = Y ~ X1 + X2, family = gaussian, imp.model = list(imp1, imp2, imp3),
                  mis.model = list(mis1, mis2), L = 10, data = dat)
results$coefficients
MR.reg(formula = Y ~ X1 + X2, family = gaussian, 
       moment = c(S, X2), order = 2, data = dat)$coefficients