In elong0527/qte: Quantile Treatment Effects under Censoring and Truncation by Death

library(dplyr)
load("../simulation/simu_qte.Rdata")

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

Simulation Setup

We conducted simulation studies to verify the validity of the proposed IPW and AIPW estimators.

Simulation set up

• Sample size per group: n = 200
• Analysis time $L = 1$
• Covariates $X$ is generated from $\mathcal{N}(\mu_a, 1)$, where $\mu_0 = 0$ and $\mu_1 = 0.5$.
• Outcome $Y$ is generated from $\mathcal{N}(\eta_a, 1)$, where $\eta_0 = 0.6$ and $\eta_1 = 0.1$
• Death indicator $D$ is derived by outcome $Y$ such that $D = I(Y < y_0)$, we select $y_0 = - 0.5$ that result in approximately 10% of death in the simulated data.
• In reality, we will not observe those outcome $Y$ less than $y_0$
• This is to mimic the assumption that those missing data due to death are worst outcome.
• Death time:
• if $Y < y_0$, the event time $T$ is generated from a uniform distribution $Unif[0,L]$
• if $Y \ge y_0$, the event time $T$ is generated from a scaled exponential distribution $L + exp(5)$ when $a=0$ and $L+exp(3)$ when $a=1$
• Censoring time:
• censoring time $C$ is generated from an exponential distribution $L + exp(5)$
• Observed time: $U = T \wedge C$
• Event indicator: $\delta = T < C$
• Observed Outcome $Y$ is observed if
• subject is alive at the analysis time $L$ and not censored.

Example of simulated data

head(db) %>% select(a, x, y, y_obs, event_time, censor_time, obs_time, status, analysis_time) %>% 
             knitr::kable(digits = 3)

Model Estimation

The calculation is within each treatment group

1. Propensity score: Fit a logistic model: glm(a ~ x, family = "binomial", data = db)
2. IPW weight $\pi$: KM estimator for censoring time: survfit(Surv(obs_time, 1 - status) ~ 1, data = db) and calculate the probability at the minimal of obs_time and analysis_time
3. $F(y\mid x; \theta)$
4. Fit a linear model based on observed outcome lm(y_obs ~ x, data = db)
5. Calculate the scaled CDF for f_q

db$y_mean  <- predict(fit_lm_obs, newdata = db)
db$y_sigma <- summary(fit_lm_obs)$sigma
y_std <- (q -  db$y_mean)/ db$y_sigma
f_q <- pmax(db$rho, pnorm(y_std))

1. $R$: (obs_time > analysis_time) | (obs_time <= analysis_time & status = TRUE)

Simulation Results

• rho: Probability subject is dead
• pct_missing_death: Percent of subject dead on orbefore analysis time $L$
• pct_missing_censor: Percent of subject censored on or before analysis time $L$
• quantile_all: quantile ($\xi=0.5$) based on all simulated data (underline truth)
• quantile_obs: quantile ($\xi=0.5$) based on all observed data
• it is expected to over estimate the quantile
• ipw: results from formula (5)
• aipw: results from formula (6)

t1 %>% select(- naive) %>% knitr::kable(digits = 3)

elong0527/qte documentation built on Jan. 19, 2021, 10:24 p.m.