# Validation of extended formuals with effect modification using bootstrap

#### 2024-01-06

Since there is no gold standard to verify our extended formulas incorporating effect modifications, we use bootstrap to check the point estimates and standard errors given in one single run of regmedint(). The original estimation for standard errors uses delta method, which agrees with bootstrap asymptotically.

We use simulated data, and present the estimates from 5000 times of boostrap. For the purpose of demonstration, we include all three effect modification terms (i.e. $$A\times C$$ in mediator model, $$A\times C$$ in outcome model, and $$M\times C$$ in outcome model). Due to the long computational time, the code chunks are commented out and only the summary tables are shown. Readers are free to run the code on their side to replicate the results.

library(regmedint)
library(tidyverse)

expit <- function(x){exp(x)/(1 + exp(x))}

# Parallel computing setup

library(parallel)
library(MASS)
numCores <- detectCores()
numCores
## [1] 12
# Number of bootstrap
trials <- 1:5000
seed <-  3104

# Simulated data generating process

# Model 1: M linear, Y linear
datamaker.s4.m1 <- function(n, k){
C <- matrix(rnorm(n*1, 0, 2), ncol = 1)
A <- rbinom(n, 1, expit(C + C^2))
M <- 0.2 + 0.4*A + 0.5*C + 0.2*A*C + rnorm(n, 0, 0.5)
Y <- 0.5 + 0.3*A + 0.2*M + 0.5*A*M + 0.2*A*C + k*M*C + rnorm(n, 0, 0.5)
list(C = C, A = A, M = M, Y = Y)
}

# Model 2: M logistic, Y linear
datamaker.s4.m2 <- function(n, k){
C <- matrix(rnorm(n*1, 0, 2), ncol = 1)
A <- rbinom(n, 1, expit(C + C^2))
M <- rbinom(n, 1, expit(0.2 + 0.4*A + 0.5*C + 0.2*A*C))
Y <- 0.5 + 0.3*A + 0.2*M + 0.5*A*M + 0.2*A*C + k*M*C + rnorm(n, 0, 0.5)
list(C = C, A = A, M = M, Y = Y)
}

# Model 3: M linear, Y logistic
datamaker.s4.m3 <- function(n, k){
C <- matrix(rnorm(n*1, 0, 2), ncol = 1)
A <- rbinom(n, 1, expit(C + C^2))
M <- (0.2 + 0.4*A + 0.5*C + 0.2*A*C + rnorm(n, 0, 0.5)/5)
Y <- rbinom(n, 1, expit((0.5 + 0.3*A + 0.6*M + 0.4*C + 0.5*A*M + 0.2*A*C + k*M*C)))
list(C = C, A = A, M = M, Y = Y)
}

# Model 4: M logistic, Y logistic
datamaker.s4.m4 <- function(n, k){
C <- matrix(rnorm(n*1, 0, 2), ncol = 1)
A <- rbinom(n, 1, expit(C + C^2))
M <- rbinom(n, 1, expit(0.2 + 0.4*A + 0.5*C + 0.2*A*C))
Y <- rbinom(n, 1, expit(0.5 + 0.3*A + 0.2*M + 0.1*C + 0.5*A*M + 0.2*A*C + k*M*C))
list(C = C, A = A, M = M, Y = Y)
}

## Generate datasets

set.seed(seed)
dat_linear_M_linear_Y     <- as.data.frame(datamaker.s4.m1(n = 5000, k = 0.3))
dat_logistic_M_linear_Y   <- as.data.frame(datamaker.s4.m2(n = 5000, k = 0.3))
dat_linear_M_logistic_Y   <- as.data.frame(datamaker.s4.m3(n = 5000, k = 0.7))
dat_logistic_M_logistic_Y <- as.data.frame(datamaker.s4.m4(n = 5000, k = 0.3))

# Model fit

## 1. Linear mediator model, linear outcome model

regmedint1 <- regmedint(data = dat_linear_M_linear_Y,
yvar = "Y",
avar = "A",
mvar = "M",
cvar = c("C"),
emm_ac_mreg = c("C"),
emm_ac_yreg = c("C"),
emm_mc_yreg = c("C"),
eventvar = NULL,
a0 = 0,
a1 = 1,
m_cde = 0.5012509,
c_cond = -0.0434094,
mreg = "linear",
yreg = "linear",
interaction = TRUE,
casecontrol = FALSE,
na_omit = FALSE)
summary(regmedint1)
data1 <- dat_linear_M_linear_Y
boot1 <- function(trials){
ind <- sample(5000, 5000, replace = TRUE)
dat <- data1[ind,]

regmedint1 <- regmedint(data = dat,
yvar = "Y",
avar = "A",
mvar = "M",
cvar = c("C"),
emm_ac_mreg = c("C"),
emm_ac_yreg = c("C"),
emm_mc_yreg = c("C"),
eventvar = NULL,
a0 = 0,
a1 = 1,
m_cde = 0.5012509,
c_cond = -0.0434094,
mreg = "linear",
yreg = "linear",
interaction = TRUE,
casecontrol = FALSE,
na_omit = FALSE)

out <- summary(regmedint1)
cde.est.boot <- out$summary_myreg[1,1] pnde.est.boot <- out$summary_myreg[2,1]
tnie.est.boot <- out$summary_myreg[3,1] tnde.est.boot <- out$summary_myreg[4,1]
pnie.est.boot <- out$summary_myreg[5,1] te.est.boot <- out$summary_myreg[6,1]
pm.est.boot <- out$summary_myreg[7,1] return(c(cde.est.boot, pnde.est.boot, tnie.est.boot, tnde.est.boot, pnie.est.boot, te.est.boot, pm.est.boot)) } set.seed(seed) system.time({ results1 <- mclapply(trials, boot1, mc.cores = numCores) }) results1.df <- as.data.frame(do.call(rbind, results1)) apply(results1.df, 2, mean) apply(results1.df, 2, sd) ## 2. Logistic mediator model, linear outcome model regmedint2 <- regmedint(data = dat_logistic_M_linear_Y, yvar = "Y", avar = "A", mvar = "M", cvar = c("C"), emm_ac_mreg = c("C"), emm_ac_yreg = c("C"), emm_mc_yreg = c("C"), eventvar = NULL, a0 = 0, a1 = 1, m_cde = 0, c_cond = -0.0434094, mreg = "logistic", yreg = "linear", interaction = TRUE, casecontrol = FALSE, na_omit = FALSE) summary(regmedint2) data2 <- dat_logistic_M_linear_Y boot2 <- function(trials){ ind <- sample(5000, 5000, replace = TRUE) dat <- data2[ind,] regmedint2 <- regmedint(data = dat, yvar = "Y", avar = "A", mvar = "M", cvar = c("C"), emm_ac_mreg = c("C"), emm_ac_yreg = c("C"), emm_mc_yreg = c("C"), eventvar = NULL, a0 = 0, a1 = 1, m_cde = 0, c_cond = -0.0434094, mreg = "logistic", yreg = "linear", interaction = TRUE, casecontrol = FALSE, na_omit = FALSE) out <- summary(regmedint2) cde.est.boot <- out$summary_myreg[1,1]
pnde.est.boot <- out$summary_myreg[2,1] tnie.est.boot <- out$summary_myreg[3,1]
tnde.est.boot <- out$summary_myreg[4,1] pnie.est.boot <- out$summary_myreg[5,1]
te.est.boot <- out$summary_myreg[6,1] pm.est.boot <- out$summary_myreg[7,1]
return(c(cde.est.boot,
pnde.est.boot, tnie.est.boot,
tnde.est.boot, pnie.est.boot,
te.est.boot, pm.est.boot))
}

set.seed(seed)
system.time({
results2 <- mclapply(1:100, boot2, mc.cores = numCores)
})

results2.df <- as.data.frame(do.call(rbind, results2))
apply(results2.df, 2, mean)
apply(results2.df, 2, sd)

## 3. Linear mediator model, logistic outcome model

regmedint3 <- regmedint(data = dat_linear_M_logistic_Y,
yvar = "Y",
avar = "A",
mvar = "M",
cvar = c("C"),
emm_ac_mreg = c("C"),
emm_ac_yreg = c("C"),
emm_mc_yreg = c("C"),
eventvar = NULL,
a0 = 0,
a1 = 1,
m_cde = 0.5012509,
c_cond = 0.5,
mreg = "linear",
yreg = "logistic",
interaction = TRUE,
casecontrol = FALSE,
na_omit = FALSE)
summary(regmedint3)
data3 <- dat_linear_M_logistic_Y
boot3 <- function(trials){
ind <- sample(5000, 5000, replace = TRUE)
dat <- data3[ind,]

regmedint3 <- regmedint(data = dat,
yvar = "Y",
avar = "A",
mvar = "M",
cvar = c("C"),
emm_ac_mreg = c("C"),
emm_ac_yreg = c("C"),
emm_mc_yreg = c("C"),
eventvar = NULL,
a0 = 0,
a1 = 1,
m_cde = 0.5012509,
c_cond = 0.5,
mreg = "linear",
yreg = "logistic",
interaction = TRUE,
casecontrol = FALSE,
na_omit = FALSE)

out <- summary(regmedint3)
cde.est.boot <- out$summary_myreg[1,1] pnde.est.boot <- out$summary_myreg[2,1]
tnie.est.boot <- out$summary_myreg[3,1] tnde.est.boot <- out$summary_myreg[4,1]
pnie.est.boot <- out$summary_myreg[5,1] te.est.boot <- out$summary_myreg[6,1]
pm.est.boot <- out$summary_myreg[7,1] return(c(cde.est.boot, pnde.est.boot, tnie.est.boot, tnde.est.boot, pnie.est.boot, te.est.boot, pm.est.boot)) } set.seed(seed) system.time({ results3 <- mclapply(trials, boot3, mc.cores = numCores) }) results3.df <- as.data.frame(do.call(rbind, results3)) apply(results3.df, 2, mean) apply(results3.df, 2, sd) ## 4. Logistic mediator model, logistic outcome model regmedint4 <- regmedint(data = dat_logistic_M_logistic_Y, yvar = "Y", avar = "A", mvar = "M", cvar = c("C"), emm_ac_mreg = c("C"), emm_ac_yreg = c("C"), emm_mc_yreg = c("C"), eventvar = NULL, a0 = 0, a1 = 1, m_cde = 0, c_cond = -0.0434094, mreg = "logistic", yreg = "logistic", interaction = TRUE, casecontrol = FALSE, na_omit = FALSE) summary(regmedint4) data4 <- dat_logistic_M_logistic_Y boot4 <- function(trials){ ind <- sample(5000, 5000, replace = TRUE) dat <- data4[ind,] regmedint4 <- regmedint(data = dat, yvar = "Y", avar = "A", mvar = "M", cvar = c("C"), emm_ac_mreg = c("C"), emm_ac_yreg = c("C"), emm_mc_yreg = c("C"), eventvar = NULL, a0 = 0, a1 = 1, m_cde = 0, c_cond = -0.0434094, mreg = "logistic", yreg = "logistic", interaction = TRUE, casecontrol = FALSE, na_omit = FALSE) out <- summary(regmedint4) cde.est.boot <- out$summary_myreg[1,1]
pnde.est.boot <- out$summary_myreg[2,1] tnie.est.boot <- out$summary_myreg[3,1]
tnde.est.boot <- out$summary_myreg[4,1] pnie.est.boot <- out$summary_myreg[5,1]
te.est.boot <- out$summary_myreg[6,1] pm.est.boot <- out$summary_myreg[7,1]
return(c(cde.est.boot,
pnde.est.boot, tnie.est.boot,
tnde.est.boot, pnie.est.boot,
te.est.boot, pm.est.boot))
}

set.seed(seed)
system.time({
results4 <- mclapply(trials, boot4, mc.cores = numCores)
})

results4.df <- as.data.frame(do.call(rbind, results4))
apply(results4.df, 2, mean)
apply(results4.df, 2, sd)

# Results comparison

The following tables shows the point estimates and standard errors from one single run of regmedint() and bootstrap.

## Error in library(kableExtra): there is no package called 'kableExtra'
## Error in library(formattable): there is no package called 'formattable'

## 1. Linear mediator model, linear outcome model

## Error in add_header_above(., c(  = 1, Non-bootstrap = 2, Bootstrap = 2)): could not find function "add_header_above"

## 2. Logistic mediator model, linear outcome model

## Error in add_header_above(., c(  = 1, Non-bootstrap = 2, Bootstrap = 2)): could not find function "add_header_above"

## 3. Linear mediator model, logistic outcome model

## Error in add_header_above(., c(  = 1, Non-bootstrap = 2, Bootstrap = 2)): could not find function "add_header_above"

## 4. Logistic mediator model, logistic outcome model

## Error in add_header_above(., c(  = 1, Non-bootstrap = 2, Bootstrap = 2)): could not find function "add_header_above"