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3_simulation.R
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3_simulation.R
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#---------------------------------------------------------------------------------------------------------
#This code entails the simulation study in 'Sensitivity analysis for bias due to a misclassfied confoundin
#g variable in marginal structural models'------------------------------------------------------------------------
#Author: Linda Nab, l.nab@lumc.nl-------------------------------------------------------------------------
#Date: 05122019-------------------------------------------------------------------------------------------
#---------------------------------------------------------------------------------------------------------
require(ipw)
require(survey)
require(rsimsum)
require(ggplot2)
require("xtable")
#---------------------------------------------------------------------------------------------------------
# 0. Bias formulae----------------------------------------------------------------------------------------
#---------------------------------------------------------------------------------------------------------
#read for the derivations of these bias expressions the appendix of the original paper--------------------
#---------------------------------------------------------------------------------------------------------
#bias in conditional model
bias_cm <- function(par){
alpha <- 1
beta <- par[1]; gamma <- par[2]; lambda <- par[3]
p_0 <- par[4]; p_1 <- par[5]; pi_0 <- par[6]; pi_1 <- par[7]
#omega = P(A)
omega <- pi_0 * (1-lambda) + pi_1 * lambda
#ell = P(L^*)
ell <- p_0 * (1-lambda) + p_1 * lambda
#pi_star = P(A|L_star=l_star)
pi_star <- function(l_star){
term0 <- pi_0 * ((1-p_0)^(1-l_star)*p_0^l_star*(1-lambda)) / ((1-ell)^(1-l_star)*ell^l_star)
term1 <- pi_1 * ((1-p_1)^(1-l_star)*p_1^l_star*lambda) / ((1-ell)^(1-l_star)*ell^l_star)
return(term0 + term1)
}
#phi = P(L|A=a,L_star=l_star)
phi <- function(a, l_star){
out <- (lambda*(1-pi_1)^(1-a)*pi_1^a*(1-p_1)^(1-l_star)*p_1^l_star) /
((1-pi_star(l_star))^(1-a)*pi_star(l_star)^a*(1-ell)^(1-l_star)*ell^l_star)
return(out)
}
#bias
temp <- (pi_star(1)*ell*(1-omega)-pi_star(1)*(pi_star(1) - pi_star(0))*ell*(1-ell)) /
(omega*(1-omega)-(pi_star(1)-pi_star(0))^2*ell*(1-ell))
bias_cm <- gamma * (phi(1,0) - phi(0,0)) * (1 - temp) +
gamma * (phi(1,1) - phi(0,1)) * temp
return(bias_cm)
}
#bias in msm estimated using ipw
bias_msm <- function(par){
alpha <- 1
beta <- par[1]; gamma <- par[2]; lambda <- par[3]
p_0 <- par[4]; p_1 <- par[5]; pi_0 <- par[6]; pi_1 <- par[7]
#omega = P(A)
omega <- pi_0 * (1-lambda) + pi_1 * lambda
#ell = P(L^*)
ell <- p_0 * (1-lambda) + p_1 * lambda
#pi_star = P(A|L_star=l_star)
pi_star <- function(l_star){
term0 <- pi_0 * ((1-p_0)^(1-l_star)*p_0^l_star*(1-lambda)) / ((1-ell)^(1-l_star)*ell^l_star)
term1 <- pi_1 * ((1-p_1)^(1-l_star)*p_1^l_star*lambda) / ((1-ell)^(1-l_star)*ell^l_star)
return(term0 + term1)
}
#phi = P(L|A=a,L_star=l_star)
phi <- function(a, l_star){
out <- (lambda*(1-pi_1)^(1-a)*pi_1^a*(1-p_1)^(1-l_star)*p_1^l_star) /
((1-pi_star(l_star))^(1-a)*pi_star(l_star)^a*(1-ell)^(1-l_star)*ell^l_star)
return(out)
}
#bias
bias_msm <- gamma * (phi(1,0) - phi(0,0)) * (1 - ell) +
gamma * (phi(1,1) - phi(0,1)) * ell
return(bias_msm)
}
#---------------------------------------------------------------------------------------------------------
# 1. Five scenarios 's'# ---------------------------------------------------------------------------------
# s is a matrix with entries: 1) beta = 1, 2) gamma = 2, 3) lambda, 4) p_0, 5) p_1, 6) pi_0, 7) pi_1------
#---------------------------------------------------------------------------------------------------------
# Scenario 0 ('s0'): no measurement error-----------------------------------------------------------------
# lambda = 0.5; p_0 = 0; p_1 = 1, pi_0 = 0.5; pi_1 = 0.75
s0 <- matrix(nrow = 1, ncol = 7)
s0[,1] <- 1; s0[,2] <- 2; s0[,3] <- 0.5; s0[,4] <- 0; s0[,5] <- 1; s0[,6] <- 0.5; s0[,7] <- 0.75
bias_cm(s0)
bias_msm(s0)
# Scenario 1 ('s1')---------------------------------------------------------------------------------------
# lambda = 0.5; p_0 = 0.05; p_1 = 0.9; pi_0 = 0.9; pi_1 = 0.45, (pi_1/pi_0 = 0.5)
s1 <- matrix(nrow = 1, ncol = 7)
s1[,1] <- 1; s1[,2] <- 2; s1[,3] <- 0.5; s1[,4] <- 0.05; s1[,5] <- 0.9; s1[,6] <- 0.9; s1[,7] <- 0.45
bias_cm(s1)
bias_msm(s1)
# Scenario 2 ('s2')---------------------------------------------------------------------------------------
# lambda = 0.8, p_0 = 0.05, p_1 = 0.9, pi_0 = 0.5, pi_1 = 0.75, (pi_1/pi_0 = 3)
s2 <- matrix(nrow = 1, ncol = 7)
s2[,1] <- 1; s2[,2] <- 2; s2[,3] <- 0.8; s2[,4] <- 0.05; s2[,5] <- 0.9; s2[,6] <- 0.5; s2[,7] <- 0.75
bias_cm(s2)
bias_msm(s2)
# Scenario 3 ('s3')---------------------------------------------------------------------------------------
# lambda = 0.8, p_0 = 0.05, p_1 = 0.9, pi_0 = 0.25, pi_1 = 0.75, (pi_1/pi_0 = 3)
s3 <- matrix(nrow = 1, ncol = 7)
s3[,1] <- 1; s3[,2] <- 2; s3[,3] <- 0.8; s3[,4] <- 0.05; s3[,5] <- 0.9; s3[,6] <- 0.25; s3[,7] <- 0.75
bias_cm(s3)
bias_msm(s3)
# Scenario 4 ('s4'): equal bias---------------------------------------------------------------------------
# lambda = 0.45, p_0 = 0.05, p_1 = 0.9, pi_0 = 0.5, pi_1 = 0.75
# (pi_1/pi_0 = 3)
s4 <- matrix(nrow = 1, ncol = 7)
s4[,1] <- 1; s4[,2] <- 2; s4[,3] <- 0.45; s4[,4] <- 0.05; s4[,5] <- 0.9; s4[,6] <- 0.5; s4[,7] <- 0.75
bias_cm(s4)
bias_msm(s4)
# Function that generates data----------------------------------------------------------------------------
data_gen <- function(m, nobs, seed){
set.seed(seed)
alpha <- 1
beta <- m[1]
gamma <- m[2]
lambda <- m[3]
p_0 <- m[4]
p_1 <- m[5]
pi_0 <- m[6]
pi_1 <- m[7]
L <- rbinom(nobs, 1, lambda)
A <- ifelse(L==1, rbinom(nobs, 1, pi_1), rbinom(nobs, 1, pi_0))
Y <- alpha + beta * A + gamma * L + rnorm(nobs, 0, 1)
Ls <- ifelse(L==1, rbinom(nobs, 1, p_1), rbinom(nobs, 1, p_0))
data <- data.frame(cbind(L, A, Y, Ls))
return(data)
}
# Function that performs simulation-----------------------------------------------------------------------
sim <- function(m, S, nobs, seeds){
out <- data.frame(dataset = numeric(2*S), method = numeric(2*S),
b = numeric(2*S), se = numeric(2*S))
for(i in 1:S){
d <- data_gen(m, nobs, seeds[i])
#sim msm
out[(2*i - 1), 1] <- i; out[i + (i - 1), 2] <- "msm"
temp <- ipwpoint(exposure = A,
family = "binomial",
link = "logit",
denominator = ~ Ls,
numerator = ~ 1,
data = d)
msm <- svyglm(Y ~ A,
design = svydesign(~ 1,
weights = ~ temp$ipw.weights,
data = d))
out[i + (i - 1), 3] <- msm$coefficients[2]
out[i + (i - 1), 4] <- summary(msm)$coefficients[2,2]
#sim cond m
fit <- lm(Y ~ A + Ls, d)
out[(2*i), 1] <- i; out[(2*i), 2] <- "cm"
out[(2*i), 3] <- fit$coef[2]
out[(2*i), 4] <- summary(fit)$coefficients[2,2]}
return(out)
}
# Simulate data ------------------------------------------------------------------------------------------
# generate seeds
# seeds <- sample(1:1e8, 10000, replace = F)
simsum_dir <- paste0(getwd(), "/sims_simsum") #directory where folder sims_simsum is located
# saveRDS(seeds, paste0(simsum_dir, "/seeds.rds"))
seeds <- readRDS(paste0(simsum_dir, "/seeds.rds"))
# sample size 1000
sim_s0_1000 <- sim(s0, 5000, 1e3, seeds[1:5000])
sim_s1_1000 <- sim(s1, 5000, 1e3, seeds[1:5000])
sim_s2_1000 <- sim(s2, 5000, 1e3, seeds[1:5000])
sim_s3_1000 <- sim(s3, 5000, 1e3, seeds[1:5000])
sim_s4_1000 <- sim(s4, 5000, 1e3, seeds[1:5000])
# sample size 100
sim_s0_100 <- sim(s0, 5000, 1e2, seeds[5001:1e4])
sim_s1_100 <- sim(s1, 5000, 1e2, seeds[5001:1e4])
sim_s2_100 <- sim(s2, 5000, 1e2, seeds[5001:1e4])
sim_s3_100 <- sim(s3, 5000, 1e2, seeds[5001:1e4])
sim_s4_100 <- sim(s4, 5000, 1e2, seeds[5001:1e4])
# save simulated sets------------------------------------------------------------------------------------
saveRDS(sim_s0_1000, paste0(simsum_dir, "/sim_s0_1000.rds"))
saveRDS(sim_s1_1000, paste0(simsum_dir, "/sim_s1_1000.rds"))
saveRDS(sim_s2_1000, paste0(simsum_dir, "/sim_s2_1000.rds"))
saveRDS(sim_s3_1000, paste0(simsum_dir, "/sim_s3_1000.rds"))
saveRDS(sim_s4_1000, paste0(simsum_dir, "/sim_s4_1000.rds"))
saveRDS(sim_s0_100, paste0(simsum_dir, "/sim_s0_100.rds"))
saveRDS(sim_s1_100, paste0(simsum_dir, "/sim_s1_1000.rds"))
saveRDS(sim_s2_100, paste0(simsum_dir, "/sim_s2_1000.rds"))
saveRDS(sim_s3_100, paste0(simsum_dir, "/sim_s3_1000.rds"))
saveRDS(sim_s4_100, paste0(simsum_dir, "/sim_s4_1000.rds"))
# read simulated sets-------------------------------------------------------------------------------------
sim_s0_1000 <- readRDS(file = paste0(simsum_dir, "/sim_s0_1000.rds"))
sim_s1_1000 <- readRDS(file = paste0(simsum_dir, "/sim_s1_1000.rds"))
sim_s2_1000 <- readRDS(file = paste0(simsum_dir, "/sim_s2_1000.rds"))
sim_s3_1000 <- readRDS(file = paste0(simsum_dir, "/sim_s3_1000.rds"))
sim_s4_1000 <- readRDS(file = paste0(simsum_dir, "/sim_s4_1000.rds"))
sim_s0_100 <- readRDS(file = paste0(simsum_dir, "/sim_s0_100.rds"))
sim_s1_100 <- readRDS(file = paste0(simsum_dir, "/sim_s1_1000.rds"))
sim_s2_100 <- readRDS(file = paste0(simsum_dir, "/sim_s2_1000.rds"))
sim_s3_100 <- readRDS(file = paste0(simsum_dir, "/sim_s3_1000.rds"))
sim_s4_100 <- readRDS(file = paste0(simsum_dir, "/sim_s4_1000.rds"))
# create simsum objects of simulated sets ---------------------------------------------------------------
simsum.sim_s0_1000 <- simsum(sim_s0_1000, estvarname = "b", true = s0[1],
se = "se", methodvar = "method", x = TRUE)
simsum.sim_s1_1000 <- simsum(sim_s1_1000, estvarname = "b", true = s1[1],
se = "se", methodvar = "method", x = TRUE)
simsum.sim_s2_1000 <- simsum(sim_s2_1000, estvarname = "b", true = s2[1],
se = "se", methodvar = "method", x = TRUE)
simsum.sim_s3_1000 <- simsum(sim_s3_1000, estvarname = "b", true = s3[1],
se = "se", methodvar = "method", x = TRUE)
simsum.sim_s4_1000 <- simsum(sim_s4_1000, estvarname = "b", true = s4[1],
se = "se", methodvar = "method", x = TRUE)
simsum.sim_s0_100 <- simsum(sim_s0_100, estvarname = "b", true = s0[1],
se = "se", methodvar = "method", x = TRUE)
simsum.sim_s1_100 <- simsum(sim_s1_100, estvarname = "b", true = s1[1],
se = "se", methodvar = "method", x = TRUE)
simsum.sim_s2_100 <- simsum(sim_s2_100, estvarname = "b", true = s2[1],
se = "se", methodvar = "method", x = TRUE)
simsum.sim_s3_100 <- simsum(sim_s3_100, estvarname = "b", true = s3[1],
se = "se", methodvar = "method", x = TRUE)
simsum.sim_s4_100 <- simsum(sim_s4_100, estvarname = "b", true = s4[1],
se = "se", methodvar = "method", x = TRUE)
# Create tables for in paper -----------------------------------------------------------------------------
summ_s0_1000 <- summary(simsum.sim_s0_1000)$summ
summ_s1_1000 <- summary(simsum.sim_s1_1000)$summ
summ_s2_1000 <- summary(simsum.sim_s2_1000)$summ
summ_s3_1000 <- summary(simsum.sim_s3_1000)$summ
summ_s4_1000 <- summary(simsum.sim_s4_1000)$summ
summ_s0_100 <- summary(simsum.sim_s0_100)$summ
summ_s1_100 <- summary(simsum.sim_s1_100)$summ
summ_s2_100 <- summary(simsum.sim_s2_100)$summ
summ_s3_100 <- summary(simsum.sim_s3_100)$summ
summ_s4_100 <- summary(simsum.sim_s4_100)$summ
#
summ_list <- list(summ_s0_1000, summ_s1_1000, summ_s2_1000, summ_s3_1000, summ_s4_1000,
summ_s0_100, summ_s1_100, summ_s2_100, summ_s3_100, summ_s4_100)
#
stat_from_summ <- function(summ, stat, method, digitsb, digitse){
stats_from_summ <- summ[summ$stat == stat,]
stat_from_summ <- stats_from_summ[stats_from_summ$method == method,]
out <- paste0(round(stat_from_summ$est, digitsb),
" (", round(stat_from_summ$mcse, digitse), ")")
return(out)
}
results <- data.frame(method = numeric(20), sample_size = numeric(20), scenario = numeric(20),
bias_form = numeric(20), bias = numeric(20),
mse = numeric(20), coverage = numeric(20), empse = numeric(20),
modelse = numeric(20))
results$method <- c(rep("msm", 10), rep("cm", 10))
results$sample_size <- c(rep("1000", 5), rep("100", 5), rep("1000", 5), rep("100", 5))
results$scenario <- rep(c("0", "1", "2", "3", "4"), 4)
#results$scenario <- c(rep("s0", 4), rep("s1", 4), rep("s2", 4), rep("s3", 4), rep("s4", 4))
#results$sample_size <- rep(c("1000", "1000", "100", "100"), 5)
#results$method <- rep(c("msm", "cm"), 10)
results$bias_form[1] <- round(bias_msm(s0), 2)
results$bias_form[2] <- round(bias_msm(s1), 2)
results$bias_form[3] <- round(bias_msm(s2), 2)
results$bias_form[4] <- round(bias_msm(s3), 2)
results$bias_form[5] <- round(bias_msm(s4), 2)
results$bias_form[6:10] <- results$bias_form[1:5]
#
results$bias_form[11] <- round(bias_cm(s0), 2)
results$bias_form[12] <- round(bias_cm(s1), 2)
results$bias_form[13] <- round(bias_cm(s2), 2)
results$bias_form[14] <- round(bias_cm(s3), 2)
results$bias_form[15] <- round(bias_cm(s4), 2)
results$bias_form[16:20] <- results$bias_form[11:15]
#bias
for(j in 1:2){
for(i in 1:10){
results$bias[10*(j-1)+i] <- stat_from_summ(summ_list[[i]], "bias", results$method[10*(j-1)+i], 2, 3)
}
}
#mse
for(j in 1:2){
for(i in 1:10){
results$mse[10*(j-1)+i] <- stat_from_summ(summ_list[[i]], "mse", results$method[10*(j-1)+i], 2, 3)
}
}
#coverage
for(j in 1:2){
for(i in 1:10){
results$coverage[10*(j-1)+i] <- stat_from_summ(summ_list[[i]], "cover", results$method[10*(j-1)+i], 2, 3)
}
}
#empse
for(j in 1:2){
for(i in 1:10){
results$empse[10*(j-1)+i] <- stat_from_summ(summ_list[[i]], "empse", results$method[10*(j-1)+i], 2, 3)
}
}
#modelse
for(j in 1:2){
for(i in 1:10){
results$modelse[10*(j-1)+i] <- stat_from_summ(summ_list[[i]], "modelse", results$method[10*(j-1)+i], 2, 3)
}
}
xtable(results, include.rownames = F)