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

library(dplyr)
load("../simulation/simu_qte_v2.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

• Total sample size: n = 400
• Analysis time $L = 1$
• Covariates $X$ is generated from $\mathcal{N}(0.5, 1)$
• Treatment group $A$ is generated from $Bernoulli(p)$, where $p=logit^{-1}(-0.25 + 0.5 * x)$ such that the randomization ratio is close to 1:1.
• Yilong: in clinical trial, we typically use block randomization, so it shall have lower variability than this setup.
• Outcome $Y$ is generated from $\mathcal{N}(\eta, 1)$, where $\eta = 0.2 + 0.8 * A$.
• Death time: $T \sim \exp(\lambda)$, where $\lambda = exp(-2 + 0.5 * X)$ (for a simple Cox PH model)
• Censoring time: $C \sim \exp(0.2)$
• Observed time: $U = T \wedge C$

• Event indicator: $\delta = T < C$

• Missing data: y_obs in simulated data

• $Y$ is missing if $U < L$
• we summarize the missing data proportion based on the event indicator.
• Actual data: the underline true value of y_actual in simulated data
• y_actual = $\min(y) - 1$ if $T < L$

Example of simulated data

head(db) %>% select(a, x, y, y_obs, y_actual, 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 $\pi$: Coxph model: oxph(Surv(obs_time, 1 - status) ~ x, 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

• n: sample size in each group
• 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_y: quantile ($\xi=0.5$) based on simulated data $Y$.
• quantile_actual: quantile ($\xi=0.5$) based on actual data y_actual (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.