Multiply Robust Estimation of the Marginal Mean

Description

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.

Usage

MR.mean(response, reg.model = NULL, mis.model = NULL, moment = NULL,
  order = 1, data, bootstrap = FALSE, bootstrap.size = 300,
  alpha = 0.05)

Arguments

Value

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.

Examples

# 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)

Multiply Robust Estimation of the Marginal Quantile

Description

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.

Usage

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

Value

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.

Examples

# 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)

Multiply Robust Estimation for Quantile Regression

Description

MR.quantreg() is used for quantile regression with missing responses and/or missing covariates. Multiple missingness probability models and imputation models are allowed.

Usage

MR.quantreg(formula, 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

Details

The function MR.quantreg() currently deals with data with one missingness pattern. When multiple variables are subject to missingness, their values are missing simultaneously. The method in Han et al. (2019) specifies an imputation model by modeling the joint distribution of the missing variables conditional on the fully observed variables. In contrast, the function MR.quantreg() 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

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.

See Also

rq.

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.quantreg(formula = Y ~ X1 + X2, tau = 0.75, imp.model = list(imp1, imp2, imp3),
                       mis.model = list(mis1, mis2), L = 10, data = dat)
results$coefficients
MR.quantreg(formula = Y ~ X1 + X2, tau = 0.75, 
            moment = c(S, X2), order = 2, data = dat)$coefficients

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

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

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

Define a Generalized Linear Model

Description

Define a generalized linear model. All the arguments in glm are allowed except for data. Supported types of family include gaussian, binomial, poisson, Gamma and inverse.gaussian.

Usage

glm.work(formula, family = gaussian, weights = NULL, ...)

Arguments

See Also

glm.

Examples

# A logistic regression with response R and covariates X1 and X2
mis1 <- glm.work(formula = R ~ X1 + X2, family = binomial(link = logit))