---
title: "Cox Regression with Dependent Error in Covariates"
author: "Yijian Huang (yhuang5@emory.edu)"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Cox Regression with dependent error in covariates}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

Package `coxerr` performs the functional modeling methods of Huang and
Wang (2018) to accommodate dependent error in covariates of the proportional
hazards model. The adopted measurement error model has minimal
assumptions on the dependence structure, and an instrumental variable is
supposed to be available.

## Installation
`coxerr` is available on CRAN:
```{r install, eval=FALSE, message=FALSE, warning=FALSE}
install.packages("coxerr")
```

## Cox regression with dependent error in covariates
Simulate a dataset for the purpose of illustration, following Scenario 1 of
Table 1 in Huang and Wang (2018):
```{r simulation, eval=TRUE, message=FALSE, warning=FALSE}
size <- 300
bt0 <- 1
## true covariate
x <- rnorm(size)
## survival time, censoring time, follow-up time, censoring indicator
s <- rexp(size) * exp(-bt0 * x)
c <- runif(size) * ifelse(x <= 0, 4.3, 8.6)
t <- pmin(s, c)
dlt <- as.numeric(s <= c)
## mismeasured covariate with heterogeneous error, IV
w <- x + rnorm(size) * sqrt(pnorm(x) * 2) * 0.5 + 1
u <- x * 0.8 + rnorm(size) * 0.6
wuz <- cbind(w, u)
```
Run the two proposed methods:
```{r coxerr, eval=TRUE, message=FALSE, warning=FALSE}
library(coxerr)
## estimation using PROP1
fit1 <- coxerr(t, dlt, wuz, 1)
fit1
## estimation using PROP2
fit2 <- coxerr(t, dlt, wuz, 2)
fit2
```

## References
Huang, Y. and Wang, C. Y. (2018) Cox Regression with dependent
error in covariates, _Biometrics_ 74, 118--126.