Validation of extended formuals with effect modification using bootstrap

About this document

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] 8

# 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.

1. Linear mediator model, linear outcome model

	Non-bootstrap		Bootstrap
	Point Estimate	Standard Error	Point Estimate	Standard Error
CDE	0.54832158	0.02201021	0.5476528	0.02170323
PNDE	0.37202753	0.02273628	0.3719104	0.02241993
TNIE	0.28120386	0.01480052	0.2807417	0.01490996
TNDE	0.58513575	0.02334807	0.5843535	0.02312960
PNIE	0.06809564	0.01196713	0.0682987	0.01233101
TE	0.65323139	0.02075177	0.6526522	0.02123155
PM	0.43048124	0.02380022	0.4304126	0.02344534

2. Logistic mediator model, linear outcome model

	Non-bootstrap		Bootstrap
	Point Estimate	Standard Error	Point Estimate	Standard Error
CDE	0.2674768	0.02753958	0.27144584	0.02890304
PNDE	0.5532311	0.02144037	0.55465628	0.02129513
TNIE	0.0869584	0.01342216	0.08809824	0.01480665
TNDE	0.6227790	0.02032353	0.62500131	0.01874403
PNIE	0.0174105	0.00459631	0.01775321	0.00467464
TE	0.6401895	0.02035298	0.64275452	0.01879988
PM	0.1358323	0.02033647	0.13703290	0.02272391

3. Linear mediator model, logistic outcome model

	Non-bootstrap		Bootstrap
	Point Estimate	Standard Error	Point Estimate	Standard Error
CDE	0.9731831	0.2696701	0.9779244	0.2712117
PNDE	1.0001259	0.2838602	1.0044763	0.2857491
TNIE	0.6608177	0.2557117	0.6606327	0.2528156
TNDE	0.6089993	0.3558141	0.6078226	0.3542694
PNIE	1.0519444	0.3044409	1.0572864	0.3059525
TE	1.6609436	0.1614044	1.6651091	0.1599455
PM	0.5969717	0.1616112	0.5776469	0.1685690

4. Logistic mediator model, logistic outcome model

	Non-bootstrap		Bootstrap
	Point Estimate	Standard Error	Point Estimate	Standard Error
CDE	0.20341749	0.11831758	0.20089667	0.12168979
PNDE	0.76137841	0.08613598	0.76132524	0.08675318
TNIE	0.06535229	0.01561403	0.06497380	0.01562340
TNDE	0.82960632	0.08759429	0.82951442	0.08783860
PNIE	-0.00287562	0.01138042	-0.00321537	0.01164672
TE	0.82673070	0.08688896	0.82629904	0.08757983
PM	0.11246227	0.02606395	0.11228721	0.02634418

Yi Li, Kazuki Yoshida

2022-04-06

About this document

Parallel computing setup

Simulated data generating process

Generate datasets

Model fit

1. Linear mediator model, linear outcome model

2. Logistic mediator model, linear outcome model

3. Linear mediator model, logistic outcome model

4. Logistic mediator model, logistic outcome model

Results comparison

1. Linear mediator model, linear outcome model

2. Logistic mediator model, linear outcome model

3. Linear mediator model, logistic outcome model

4. Logistic mediator model, logistic outcome model