This example demonstrates how to conduct sensitivity analysis for measurement error by using cmsens. For this purpose, we simulate some data containing a continuous baseline confounder \(C_1\), a binary baseline confounder \(C_2\), a binary exposure \(A\), a binary mediator \(M\) and a binary outcome \(Y\). The true regression models for \(A\), \(M\) and \(Y\) are: \[logit(E(A|C_1,C_2))=0.2+0.5C_1+0.1C_2\] \[logit(E(M|A,C_1,C_2))=1+2A+1.5C_1+0.8C_2\] \[logit(E(Y|A,M,C_1,C_2)))=-3-0.4A-1.2M+0.5AM+0.3C_1-0.6C_2\]

Then, we generate some measurement errors for \(C_1\) and \(A\).

set.seed(1)
expit <- function(x) exp(x)/(1+exp(x))
n <- 10000
C1 <- rnorm(n, mean = 1, sd = 0.5)
C1_error <- C1 + rnorm(n, 0, 0.05)
C2 <- rbinom(n, 1, 0.6)
A <- rbinom(n, 1, expit(0.2 + 0.5*C1 + 0.1*C2))
mc <- matrix(c(0.9,0.1,0.1,0.9), nrow = 2)
A_error <- A
for (j in 1:2) {
  A_error[which(A_error == c(0,1)[j])] <-
    sample(x = c(0,1), size = length(which(A_error == c(0,1)[j])),
           prob = mc[, j], replace = TRUE)
}
M <- rbinom(n, 1, expit(1 + 2*A + 1.5*C1 + 0.8*C2))
Y <- rbinom(n, 1, expit(-3 - 0.4*A - 1.2*M + 0.5*A*M + 0.3*C1 - 0.6*C2))
data <- data.frame(A, A_error, M, Y, C1, C1_error, C2)

The DAG for this scientific setting is:

library(CMAverse)
cmdag(outcome = "Y", exposure = "A", mediator = "M",
      basec = c("C1", "C2"), postc = NULL, node = TRUE, text_col = "white")

A Continuous Variable Measured with Error

Firstly, we assume \(C1\) was measured with error. \(C_1\) is continuous, so the measurement error can be corrected by regression calibration or SIMEX. We use the regression-based approach for illustration. The naive results obtained by fitting data with measurement error:

res_naive_cont <- cmest(data = data, model = "rb", outcome = "Y", exposure = "A",
                        mediator = "M", basec = c("C1_error", "C2"), EMint = TRUE,
                        mreg = list("logistic"), yreg = "logistic",
                        astar = 0, a = 1, mval = list(1), 
                        estimation = "paramfunc", inference = "delta")
summary(res_naive_cont)
## Causal Mediation Analysis
## 
## # Outcome regression:
## 
## Call:
## glm(formula = Y ~ A + M + A * M + C1_error + C2, family = binomial(), 
##     data = getCall(x$reg.output$yreg)$data, weights = getCall(x$reg.output$yreg)$weights)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -0.5658  -0.2097  -0.1763  -0.1527   3.1992  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -2.5688     0.2608  -9.852  < 2e-16 ***
## A            -0.8882     0.7581  -1.172  0.24139    
## M            -1.8387     0.2858  -6.434 1.24e-10 ***
## C1_error      0.4856     0.1481   3.280  0.00104 ** 
## C2           -0.6142     0.1500  -4.094 4.24e-05 ***
## A:M           1.1063     0.7790   1.420  0.15555    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1858.7  on 9999  degrees of freedom
## Residual deviance: 1799.6  on 9994  degrees of freedom
## AIC: 1811.6
## 
## Number of Fisher Scoring iterations: 7
## 
## 
## # Mediator regressions: 
## 
## Call:
## glm(formula = M ~ A + C1_error + C2, family = binomial(), data = getCall(x$reg.output$mreg[[1L]])$data, 
##     weights = getCall(x$reg.output$mreg[[1L]])$weights)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -3.4202   0.0948   0.1395   0.2539   1.0725  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   0.9916     0.1160   8.552  < 2e-16 ***
## A             2.1481     0.1473  14.586  < 2e-16 ***
## C1_error      1.4448     0.1193  12.109  < 2e-16 ***
## C2            0.6769     0.1193   5.673  1.4e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2825.7  on 9999  degrees of freedom
## Residual deviance: 2306.7  on 9996  degrees of freedom
## AIC: 2314.7
## 
## Number of Fisher Scoring iterations: 7
## 
## 
## # Effect decomposition on the risk ratio scale via the regression-based approach
##  
## Closed-form parameter function estimation with 
##  delta method standard errors, confidence intervals and p-values 
##  
##                 Estimate Std.error   95% CIL 95% CIU    P.val    
## Rcde             1.24383   0.22304   0.87524   1.768 0.223674    
## Rpnde            1.01905   0.17645   0.72578   1.431 0.913235    
## Rtnde            1.20938   0.21034   0.86004   1.701 0.274370    
## Rpnie            0.80077   0.05414   0.70138   0.914 0.001017 ** 
## Rtnie            0.95034   0.06599   0.82942   1.089 0.463197    
## Rte              0.96844   0.15560   0.70683   1.327 0.841789    
## ERcde            0.18846   0.16640  -0.13769   0.515 0.257410    
## ERintref        -0.16941   0.09996  -0.36533   0.027 0.090128 .  
## ERintmed         0.14862   0.08786  -0.02358   0.321 0.090726 .  
## ERpnie          -0.19923   0.05414  -0.30535  -0.093 0.000234 ***
## ERcde(prop)     -5.97114  34.38901 -73.37236  61.430 0.862152    
## ERintref(prop)   5.36768  26.42572 -46.42577  57.161 0.839039    
## ERintmed(prop)  -4.70891  23.17884 -50.13861  40.721 0.839013    
## ERpnie(prop)     6.31237  31.15632 -54.75290  67.378 0.839445    
## pm               1.60346   8.40147 -14.86311  18.070 0.848639    
## int              0.65876   3.24847  -5.70811   7.026 0.839298    
## pe               6.97114  34.38901 -60.43008  74.372 0.839359    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Rcde: controlled direct effect risk ratio; Rpnde: pure natural direct effect risk ratio; Rtnde: total natural direct effect risk ratio; Rpnie: pure natural indirect effect risk ratio; Rtnie: total natural indirect effect risk ratio; Rte: total effect risk ratio; ERcde: excess relative risk due to controlled direct effect; ERintref: excess relative risk due to reference interaction; ERintmed: excess relative risk due to mediated interaction; ERpnie: excess relative risk due to pure natural indirect effect; ERcde(prop): proportion ERcde; ERintref(prop): proportion ERintref; ERintmed(prop): proportion ERintmed; ERpnie(prop): proportion ERpnie; pm: overall proportion mediated; int: overall proportion attributable to interaction; pe: overall proportion eliminated)
## 
## Relevant variable values: 
## $a
## [1] 1
## 
## $astar
## [1] 0
## 
## $yval
## [1] "1"
## 
## $mval
## $mval[[1]]
## [1] 1
## 
## 
## $basecval
## $basecval[[1]]
## [1] 0.996522
## 
## $basecval[[2]]
## [1] 0.5936

The results corrected by regression calibration:

res_rc_cont <- cmsens(object = res_naive_cont, sens = "me", MEmethod = "rc", 
                      MEvariable = "C1_error", MEvartype = "con", MEerror = 0.05)
summary(res_rc_cont)
## Sensitivity Analysis For Measurement Error 
##  
## The variable measured with error: C1_error
## Type of the variable measured with error: continuous
## 
## # Measurement error 1: 
## [1] 0.05
## 
## ## Error-corrected regressions for measurement error 1: 
## 
## ### Outcome regression:
## Call:
## rcreg(reg = getCall(x$sens[[1L]]$reg.output$yreg)$reg, formula = Y ~ 
##     A + M + A * M + C1_error + C2, data = getCall(x$sens[[1L]]$reg.output$yreg)$data, 
##     MEvariable = "C1_error", MEerror = 0.05, variance = TRUE, 
##     nboot = 400, weights = getCall(x$sens[[1L]]$reg.output$yreg)$weights)
## 
## Naive coefficient estimates: 
## (Intercept)           A           M    C1_error          C2         A:M 
##  -2.5688363  -0.8881518  -1.8387439   0.4856104  -0.6142071   1.1063464 
## 
## Naive var-cov estimates: 
##              (Intercept)             A            M      C1_error            C2
## (Intercept)  0.067990581 -0.0545603114 -0.048204212 -0.0154891419 -0.0071194416
## A           -0.054560311  0.5747507824  0.054868704 -0.0004875592  0.0004879114
## M           -0.048204212  0.0548687045  0.081668670 -0.0080216586 -0.0027107101
## C1_error    -0.015489142 -0.0004875592 -0.008021659  0.0219233192 -0.0001013951
## C2          -0.007119442  0.0004879114 -0.002710710 -0.0001013951  0.0225065272
## A:M          0.055848645 -0.5747208888 -0.077864595 -0.0011835634 -0.0008171310
##                      A:M
## (Intercept)  0.055848645
## A           -0.574720889
## M           -0.077864595
## C1_error    -0.001183563
## C2          -0.000817131
## A:M          0.606845225
## 
## Variable measured with error:
## C1_error
## Measurement error:
## 0.05
## Error-corrected results:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -2.5717     0.2542 -10.116  < 2e-16 ***
## A            -0.8886     4.4880  -0.198  0.84306    
## M            -1.8405     0.2816  -6.535 6.65e-11 ***
## C1_error      0.4905     0.1456   3.368  0.00076 ***
## C2           -0.6142     0.1495  -4.108 4.03e-05 ***
## A:M           1.1063     4.5006   0.246  0.80583    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## ### Mediator regressions: 
## Call:
## rcreg(reg = getCall(x$sens[[1L]]$reg.output$mreg[[1L]])$reg, 
##     formula = M ~ A + C1_error + C2, data = getCall(x$sens[[1L]]$reg.output$mreg[[1L]])$data, 
##     MEvariable = "C1_error", MEerror = 0.05, variance = TRUE, 
##     nboot = 400, weights = getCall(x$sens[[1L]]$reg.output$mreg[[1L]])$weights)
## 
## Naive coefficient estimates: 
## (Intercept)           A    C1_error          C2 
##   0.9915629   2.1481051   1.4448263   0.6769050 
## 
## Naive var-cov estimates: 
##              (Intercept)             A      C1_error            C2
## (Intercept)  0.013444822 -0.0040002083 -0.0092978270 -0.0070380421
## A           -0.004000208  0.0216896845 -0.0008364503  0.0002395075
## C1_error    -0.009297827 -0.0008364503  0.0142375505  0.0008566472
## C2          -0.007038042  0.0002395075  0.0008566472  0.0142371683
## 
## Variable measured with error:
## C1_error
## Measurement error:
## 0.05
## Error-corrected results:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.9784     0.1208   8.101 6.10e-16 ***
## A             2.1465     0.1509  14.229  < 2e-16 ***
## C1_error      1.4591     0.1274  11.454  < 2e-16 ***
## C2            0.6769     0.1208   5.602 2.18e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## ## Error-corrected causal effects on the risk ratio scale for measurement error 1: 
##                 Estimate Std.error   95% CIL 95% CIU    P.val    
## Rcde             1.24332   0.21790   0.88187   1.753 0.214000    
## Rpnde            1.01851   0.51560   0.37762   2.747 0.971104    
## Rtnde            1.20880   0.21487   0.85320   1.713 0.286051    
## Rpnie            0.80064   0.05377   0.70189   0.913 0.000931 ***
## Rtnie            0.95024   0.40376   0.41320   2.185 0.904375    
## Rte              0.96782   0.15935   0.70088   1.336 0.842532    
## ERcde            0.18801   0.16191  -0.12932   0.505 0.245553    
## ERintref        -0.16951   0.50414  -1.15760   0.819 0.736699    
## ERintmed         0.14867   0.44221  -0.71805   1.015 0.736722    
## ERpnie          -0.19936   0.05377  -0.30475  -0.094 0.000209 ***
## ERcde(prop)     -5.84268  33.30469 -71.11868  59.433 0.860741    
## ERintref(prop)   5.26756  25.15247 -44.03038  54.566 0.834116    
## ERintmed(prop)  -4.62011  22.05610 -47.84926  38.609 0.834081    
## ERpnie(prop)     6.19523  30.72226 -54.01929  66.410 0.840188    
## pm               1.57512  17.92074 -33.54889  36.699 0.929961    
## int              0.64746   3.09806  -5.42463   6.720 0.834458    
## pe               6.84268  33.30469 -58.43331  72.119 0.837215    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## ----------------------------------------------------------------
## 
## (Rcde: controlled direct effect risk ratio; Rpnde: pure natural direct effect risk ratio; Rtnde: total natural direct effect risk ratio; Rpnie: pure natural indirect effect risk ratio; Rtnie: total natural indirect effect risk ratio; Rte: total effect risk ratio; ERcde: excess relative risk due to controlled direct effect; ERintref: excess relative risk due to reference interaction; ERintmed: excess relative risk due to mediated interaction; ERpnie: excess relative risk due to pure natural indirect effect; ERcde(prop): proportion ERcde; ERintref(prop): proportion ERintref; ERintmed(prop): proportion ERintmed; ERpnie(prop): proportion ERpnie; pm: overall proportion mediated; int: overall proportion attributable to interaction; pe: overall proportion eliminated)
## 
## Relevant variable values: 
## $a
## [1] 1
## 
## $astar
## [1] 0
## 
## $yval
## [1] "1"
## 
## $mval
## $mval[[1]]
## [1] 1
## 
## 
## $basecval
## $basecval[[1]]
## [1] 0.996522
## 
## $basecval[[2]]
## [1] 0.5936

The results corrected by SIMEX:

res_simex_cont <- cmsens(object = res_naive_cont, sens = "me", MEmethod = "simex", 
                         MEvariable = "C1_error", MEvartype = "con", MEerror = 0.05)
summary(res_simex_cont)
## Sensitivity Analysis For Measurement Error 
##  
## The variable measured with error: C1_error
## Type of the variable measured with error: continuous
## 
## # Measurement error 1: 
## [1] 0.05
## 
## ## Error-corrected regressions for measurement error 1: 
## 
## ### Outcome regression:
## Call:
## simexreg(reg = getCall(x$sens[[1L]]$reg.output$yreg)$reg, formula = Y ~ 
##     A + M + A * M + C1_error + C2, data = getCall(x$sens[[1L]]$reg.output$yreg)$data, 
##     MEvariable = "C1_error", MEvartype = "continuous", MEerror = 0.05, 
##     variance = TRUE, lambda = c(0.5, 1, 1.5, 2), B = 200, weights = getCall(x$sens[[1L]]$reg.output$yreg)$weights)
## 
## Naive coefficient estimates: 
## (Intercept)           A           M    C1_error          C2         A:M 
##  -2.5688363  -0.8881518  -1.8387439   0.4856104  -0.6142071   1.1063464 
## 
## Naive var-cov estimates: 
##              (Intercept)             A            M      C1_error            C2
## (Intercept)  0.067990581 -0.0545603114 -0.048204212 -0.0154891419 -0.0071194416
## A           -0.054560311  0.5747507824  0.054868704 -0.0004875592  0.0004879114
## M           -0.048204212  0.0548687045  0.081668670 -0.0080216586 -0.0027107101
## C1_error    -0.015489142 -0.0004875592 -0.008021659  0.0219233192 -0.0001013951
## C2          -0.007119442  0.0004879114 -0.002710710 -0.0001013951  0.0225065272
## A:M          0.055848645 -0.5747208888 -0.077864595 -0.0011835634 -0.0008171310
##                      A:M
## (Intercept)  0.055848645
## A           -0.574720889
## M           -0.077864595
## C1_error    -0.001183563
## C2          -0.000817131
## A:M          0.606845225
## 
## Variable measured with error:
## C1_error
## Measurement error:
## [1] 0.05
## 
## Error-corrected results:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -2.5697     0.2612  -9.839  < 2e-16 ***
## A            -0.8885     0.7581  -1.172  0.24125    
## M            -1.8396     0.2859  -6.435 1.30e-10 ***
## C1_error      0.4875     0.1497   3.255  0.00114 ** 
## C2           -0.6141     0.1500  -4.093 4.29e-05 ***
## A:M           1.1065     0.7790   1.420  0.15554    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## ### Mediator regressions: 
## Call:
## simexreg(reg = getCall(x$sens[[1L]]$reg.output$mreg[[1L]])$reg, 
##     formula = M ~ A + C1_error + C2, data = getCall(x$sens[[1L]]$reg.output$mreg[[1L]])$data, 
##     MEvariable = "C1_error", MEvartype = "continuous", MEerror = 0.05, 
##     variance = TRUE, lambda = c(0.5, 1, 1.5, 2), B = 200, weights = getCall(x$sens[[1L]]$reg.output$mreg[[1L]])$weights)
## 
## Naive coefficient estimates: 
## (Intercept)           A    C1_error          C2 
##   0.9915629   2.1481051   1.4448263   0.6769050 
## 
## Naive var-cov estimates: 
##              (Intercept)             A      C1_error            C2
## (Intercept)  0.013444822 -0.0040002083 -0.0092978270 -0.0070380421
## A           -0.004000208  0.0216896845 -0.0008364503  0.0002395075
## C1_error    -0.009297827 -0.0008364503  0.0142375505  0.0008566472
## C2          -0.007038042  0.0002395075  0.0008566472  0.0142371683
## 
## Variable measured with error:
## C1_error
## Measurement error:
## [1] 0.05
## 
## Error-corrected results:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.9816     0.1165   8.426  < 2e-16 ***
## A             2.1466     0.1473  14.572  < 2e-16 ***
## C1_error      1.4580     0.1205  12.102  < 2e-16 ***
## C2            0.6768     0.1194   5.670 1.47e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## ## Error-corrected causal effects on the risk ratio scale for measurement error 1: 
##                 Estimate Std.error   95% CIL 95% CIU    P.val    
## Rcde             1.24357   0.22301   0.87503   1.767 0.224154    
## Rpnde            1.01920   0.17645   0.72592   1.431 0.912545    
## Rtnde            1.20915   0.21032   0.85985   1.700 0.274902    
## Rpnie            0.80111   0.05410   0.70180   0.914 0.001024 ** 
## Rtnie            0.95042   0.06585   0.82975   1.089 0.462970    
## Rte              0.96867   0.15569   0.70691   1.327 0.842992    
## ERcde            0.18834   0.16647  -0.13794   0.515 0.257913    
## ERintref        -0.16914   0.09979  -0.36473   0.026 0.090093 .  
## ERintmed         0.14836   0.08769  -0.02352   0.320 0.090694 .  
## ERpnie          -0.19889   0.05410  -0.30492  -0.093 0.000237 ***
## ERcde(prop)     -6.01072  34.85624 -74.32769  62.306 0.863089    
## ERintref(prop)   5.39808  26.78430 -47.09817  57.894 0.840277    
## ERintmed(prop)  -4.73472  23.48910 -50.77251  41.303 0.840252    
## ERpnie(prop)     6.34736  31.57520 -55.53889  68.234 0.840680    
## pm               1.61264   8.50887 -15.06443  18.290 0.849682    
## int              0.66336   3.29679  -5.79824   7.125 0.840532    
## pe               7.01072  34.85624 -61.30625  75.328 0.840595    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## ----------------------------------------------------------------
## 
## (Rcde: controlled direct effect risk ratio; Rpnde: pure natural direct effect risk ratio; Rtnde: total natural direct effect risk ratio; Rpnie: pure natural indirect effect risk ratio; Rtnie: total natural indirect effect risk ratio; Rte: total effect risk ratio; ERcde: excess relative risk due to controlled direct effect; ERintref: excess relative risk due to reference interaction; ERintmed: excess relative risk due to mediated interaction; ERpnie: excess relative risk due to pure natural indirect effect; ERcde(prop): proportion ERcde; ERintref(prop): proportion ERintref; ERintmed(prop): proportion ERintmed; ERpnie(prop): proportion ERpnie; pm: overall proportion mediated; int: overall proportion attributable to interaction; pe: overall proportion eliminated)
## 
## Relevant variable values: 
## $a
## [1] 1
## 
## $astar
## [1] 0
## 
## $yval
## [1] "1"
## 
## $mval
## $mval[[1]]
## [1] 1
## 
## 
## $basecval
## $basecval[[1]]
## [1] 0.996522
## 
## $basecval[[2]]
## [1] 0.5936

A Categorical Variable Measured with Error

Then, we assume \(A\) was measured with error. \(A\) is categorical, so only SIMEX can be used. The naive results obtained by fitting data with measurement error:

res_naive_cat <- cmest(data = data, model = "rb", outcome = "Y", exposure = "A_error",
                       mediator = "M", basec = c("C1", "C2"), EMint = TRUE,
                       mreg = list("logistic"), yreg = "logistic",
                       astar = 0, a = 1, mval = list(1), 
                       estimation = "paramfunc", inference = "delta")
summary(res_naive_cat)
## Causal Mediation Analysis
## 
## # Outcome regression:
## 
## Call:
## glm(formula = Y ~ A_error + M + A_error * M + C1 + C2, family = binomial(), 
##     data = getCall(x$reg.output$yreg)$data, weights = getCall(x$reg.output$yreg)$weights)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -0.5652  -0.2098  -0.1765  -0.1529   3.1640  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -2.5907     0.2727  -9.500  < 2e-16 ***
## A_error      -0.5077     0.5703  -0.890 0.373318    
## M            -1.7375     0.2882  -6.029 1.65e-09 ***
## C1            0.5037     0.1487   3.386 0.000709 ***
## C2           -0.6133     0.1500  -4.089 4.34e-05 ***
## A_error:M     0.5931     0.5945   0.998 0.318446    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1858.7  on 9999  degrees of freedom
## Residual deviance: 1801.3  on 9994  degrees of freedom
## AIC: 1813.3
## 
## Number of Fisher Scoring iterations: 7
## 
## 
## # Mediator regressions: 
## 
## Call:
## glm(formula = M ~ A_error + C1 + C2, family = binomial(), data = getCall(x$reg.output$mreg[[1L]])$data, 
##     weights = getCall(x$reg.output$mreg[[1L]])$weights)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -3.3245   0.1157   0.1712   0.2666   1.0700  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   1.1369     0.1164   9.769  < 2e-16 ***
## A_error       1.5434     0.1310  11.783  < 2e-16 ***
## C1            1.5322     0.1190  12.879  < 2e-16 ***
## C2            0.6753     0.1181   5.718 1.08e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2825.7  on 9999  degrees of freedom
## Residual deviance: 2428.4  on 9996  degrees of freedom
## AIC: 2436.4
## 
## Number of Fisher Scoring iterations: 7
## 
## 
## # Effect decomposition on the risk ratio scale via the regression-based approach
##  
## Closed-form parameter function estimation with 
##  delta method standard errors, confidence intervals and p-values 
##  
##                Estimate Std.error  95% CIL 95% CIU    P.val    
## Rcde            1.08912   0.18279  0.78381   1.513 0.611003    
## Rpnde           0.98700   0.15917  0.71952   1.354 0.935309    
## Rtnde           1.06299   0.17173  0.77449   1.459 0.705322    
## Rpnie           0.86553   0.04070  0.78932   0.949 0.002133 ** 
## Rtnie           0.93218   0.04836  0.84205   1.032 0.175816    
## Rte             0.92005   0.14238  0.67934   1.246 0.590294    
## ERcde           0.07373   0.14960 -0.21949   0.367 0.622139    
## ERintref       -0.08673   0.08188 -0.24722   0.074 0.289503    
## ERintmed        0.06753   0.06388 -0.05767   0.193 0.290433    
## ERpnie         -0.13447   0.04070 -0.21424  -0.055 0.000954 ***
## ERcde(prop)    -0.92222   3.45691 -7.69764   5.853 0.789642    
## ERintref(prop)  1.08489   2.06437 -2.96120   5.131 0.599215    
## ERintmed(prop) -0.84468   1.60658 -3.99352   2.304 0.599050    
## ERpnie(prop)    1.68201   3.03652 -4.26946   7.633 0.579628    
## pm              0.83733   1.71924 -2.53231   4.207 0.626232    
## int             0.24020   0.45912 -0.65966   1.140 0.600849    
## pe              1.92222   3.45691 -4.85321   8.698 0.578176    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Rcde: controlled direct effect risk ratio; Rpnde: pure natural direct effect risk ratio; Rtnde: total natural direct effect risk ratio; Rpnie: pure natural indirect effect risk ratio; Rtnie: total natural indirect effect risk ratio; Rte: total effect risk ratio; ERcde: excess relative risk due to controlled direct effect; ERintref: excess relative risk due to reference interaction; ERintmed: excess relative risk due to mediated interaction; ERpnie: excess relative risk due to pure natural indirect effect; ERcde(prop): proportion ERcde; ERintref(prop): proportion ERintref; ERintmed(prop): proportion ERintmed; ERpnie(prop): proportion ERpnie; pm: overall proportion mediated; int: overall proportion attributable to interaction; pe: overall proportion eliminated)
## 
## Relevant variable values: 
## $a
## [1] 1
## 
## $astar
## [1] 0
## 
## $yval
## [1] "1"
## 
## $mval
## $mval[[1]]
## [1] 1
## 
## 
## $basecval
## $basecval[[1]]
## [1] 0.9967315
## 
## $basecval[[2]]
## [1] 0.5936

The results corrected by SIMEX:

res_simex_cat <- cmsens(object = res_naive_cat, sens = "me", MEmethod = "simex", 
                         MEvariable = "A_error", MEvartype = "cat", MEerror = list(mc))
summary(res_simex_cat)
## Sensitivity Analysis For Measurement Error 
##  
## The variable measured with error: A_error
## Type of the variable measured with error: categorical
## 
## # Measurement error 1: 
##      [,1] [,2]
## [1,]  0.9  0.1
## [2,]  0.1  0.9
## 
## ## Error-corrected regressions for measurement error 1: 
## 
## ### Outcome regression:
## Call:
## simexreg(reg = getCall(x$sens[[1L]]$reg.output$yreg)$reg, formula = Y ~ 
##     A_error + M + A_error * M + C1 + C2, data = getCall(x$sens[[1L]]$reg.output$yreg)$data, 
##     MEvariable = "A_error", MEvartype = "categorical", MEerror = c(0.9, 
##     0.1, 0.1, 0.9), variance = TRUE, lambda = c(0.5, 1, 1.5, 
##     2), B = 200, weights = getCall(x$sens[[1L]]$reg.output$yreg)$weights)
## 
## Naive coefficient estimates: 
## (Intercept)     A_error           M          C1          C2   A_error:M 
##  -2.5906738  -0.5077290  -1.7374904   0.5036807  -0.6133152   0.5930976 
## 
## Naive var-cov estimates: 
##              (Intercept)       A_error            M            C1            C2
## (Intercept)  0.074373396 -0.0608915941 -0.054071017 -1.576936e-02 -7.164751e-03
## A_error     -0.060891594  0.3252487905  0.060788376 -1.130950e-04  5.120972e-04
## M           -0.054071017  0.0607883762  0.083061875 -8.233291e-03 -2.664520e-03
## C1          -0.015769362 -0.0001130950 -0.008233291  2.212526e-02 -9.887795e-05
## C2          -0.007164751  0.0005120972 -0.002664520 -9.887795e-05  2.250262e-02
## A_error:M    0.062066763 -0.3252496367 -0.079058624 -1.368648e-03 -8.758372e-04
##                 A_error:M
## (Intercept)  0.0620667626
## A_error     -0.3252496367
## M           -0.0790586237
## C1          -0.0013686479
## C2          -0.0008758372
## A_error:M    0.3534194114
## 
## Variable measured with error:
## A_error
## Measurement error:
##     0   1
## 0 0.9 0.1
## 1 0.1 0.9
## 
## Error-corrected results:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -2.5706     0.2726  -9.429  < 2e-16 ***
## A_error1     -0.6697     0.7186  -0.932 0.351439    
## M            -1.7728     0.2976  -5.957 2.66e-09 ***
## C1            0.5014     0.1490   3.366 0.000764 ***
## C2           -0.6139     0.1500  -4.092 4.31e-05 ***
## A_error1:M    0.7803     0.7398   1.055 0.291608    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## ### Mediator regressions: 
## Call:
## simexreg(reg = getCall(x$sens[[1L]]$reg.output$mreg[[1L]])$reg, 
##     formula = M ~ A_error + C1 + C2, data = getCall(x$sens[[1L]]$reg.output$mreg[[1L]])$data, 
##     MEvariable = "A_error", MEvartype = "categorical", MEerror = c(0.9, 
##     0.1, 0.1, 0.9), variance = TRUE, lambda = c(0.5, 1, 1.5, 
##     2), B = 200, weights = getCall(x$sens[[1L]]$reg.output$mreg[[1L]])$weights)
## 
## Naive coefficient estimates: 
## (Intercept)     A_error          C1          C2 
##   1.1368565   1.5434255   1.5321997   0.6752983 
## 
## Naive var-cov estimates: 
##              (Intercept)       A_error            C1            C2
## (Intercept)  0.013541986 -0.0044464941 -0.0091359634 -0.0068278415
## A_error     -0.004446494  0.0171570798 -0.0006625250  0.0001919242
## C1          -0.009135963 -0.0006625250  0.0141546164  0.0007819793
## C2          -0.006827841  0.0001919242  0.0007819793  0.0139496499
## 
## Variable measured with error:
## A_error
## Measurement error:
##     0   1
## 0 0.9 0.1
## 1 0.1 0.9
## 
## Error-corrected results:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   1.0152     0.1193   8.512  < 2e-16 ***
## A_error1      1.9701     0.1676  11.757  < 2e-16 ***
## C1            1.4739     0.1203  12.255  < 2e-16 ***
## C2            0.6761     0.1197   5.649 1.66e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## ## Error-corrected causal effects on the risk ratio scale for measurement error 1: 
##                Estimate Std.error  95% CIL 95% CIU    P.val    
## Rcde            1.11696   0.23958  0.73360   1.701 0.606087    
## Rpnde           0.96730   0.20112  0.64354   1.454 0.872960    
## Rtnde           1.09045   0.22744  0.72455   1.641 0.678042    
## Rpnie           0.82467   0.05202  0.72876   0.933 0.002245 ** 
## Rtnie           0.92966   0.06727  0.80674   1.071 0.313472    
## Rte             0.89926   0.17449  0.61478   1.315 0.584228    
## ERcde           0.09294   0.18747 -0.27449   0.460 0.620044    
## ERintref       -0.12564   0.10510 -0.33164   0.080 0.231910    
## ERintmed        0.10729   0.08991 -0.06894   0.284 0.232769    
## ERpnie         -0.17533   0.05202 -0.27729  -0.073 0.000751 ***
## ERcde(prop)    -0.92262   3.41032 -7.60672   5.761 0.786746    
## ERintref(prop)  1.24722   2.29278 -3.24654   5.741 0.586456    
## ERintmed(prop) -1.06501   1.95713 -4.90092   2.771 0.586326    
## ERpnie(prop)    1.74041   3.09718 -4.32996   7.811 0.574162    
## pm              0.67540   1.48233 -2.22991   3.581 0.648653    
## int             0.18221   0.33721 -0.47870   0.843 0.588946    
## pe              1.92262   3.41032 -4.76148   8.607 0.572913    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## ----------------------------------------------------------------
## 
## (Rcde: controlled direct effect risk ratio; Rpnde: pure natural direct effect risk ratio; Rtnde: total natural direct effect risk ratio; Rpnie: pure natural indirect effect risk ratio; Rtnie: total natural indirect effect risk ratio; Rte: total effect risk ratio; ERcde: excess relative risk due to controlled direct effect; ERintref: excess relative risk due to reference interaction; ERintmed: excess relative risk due to mediated interaction; ERpnie: excess relative risk due to pure natural indirect effect; ERcde(prop): proportion ERcde; ERintref(prop): proportion ERintref; ERintmed(prop): proportion ERintmed; ERpnie(prop): proportion ERpnie; pm: overall proportion mediated; int: overall proportion attributable to interaction; pe: overall proportion eliminated)
## 
## Relevant variable values: 
## $a
## [1] 1
## 
## $astar
## [1] 0
## 
## $yval
## [1] "1"
## 
## $mval
## $mval[[1]]
## [1] 1
## 
## 
## $basecval
## $basecval[[1]]
## [1] 0.9967315
## 
## $basecval[[2]]
## [1] 0.5936

Compare the error-corrected results with the true results:

res_true <- cmest(data = data, model = "rb", outcome = "Y", exposure = "A",
                       mediator = "M", basec = c("C1", "C2"), EMint = TRUE,
                       mreg = list("logistic"), yreg = "logistic",
                       astar = 0, a = 1, mval = list(1), 
                       estimation = "paramfunc", inference = "delta")
summary(res_true)
## Causal Mediation Analysis
## 
## # Outcome regression:
## 
## Call:
## glm(formula = Y ~ A + M + A * M + C1 + C2, family = binomial(), 
##     data = getCall(x$reg.output$yreg)$data, weights = getCall(x$reg.output$yreg)$weights)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -0.5659  -0.2096  -0.1762  -0.1524   3.1928  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -2.5768     0.2611  -9.869  < 2e-16 ***
## A            -0.8888     0.7581  -1.172 0.241026    
## M            -1.8429     0.2859  -6.447 1.14e-10 ***
## C1            0.4970     0.1489   3.337 0.000845 ***
## C2           -0.6144     0.1500  -4.095 4.22e-05 ***
## A:M           1.1064     0.7790   1.420 0.155512    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1858.7  on 9999  degrees of freedom
## Residual deviance: 1799.2  on 9994  degrees of freedom
## AIC: 1811.2
## 
## Number of Fisher Scoring iterations: 7
## 
## 
## # Mediator regressions: 
## 
## Call:
## glm(formula = M ~ A + C1 + C2, family = binomial(), data = getCall(x$reg.output$mreg[[1L]])$data, 
##     weights = getCall(x$reg.output$mreg[[1L]])$weights)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -3.4257   0.0947   0.1396   0.2558   1.0542  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   0.9817     0.1163   8.442  < 2e-16 ***
## A             2.1466     0.1473  14.574  < 2e-16 ***
## C1            1.4572     0.1200  12.145  < 2e-16 ***
## C2            0.6764     0.1193   5.668 1.44e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2825.7  on 9999  degrees of freedom
## Residual deviance: 2305.6  on 9996  degrees of freedom
## AIC: 2313.6
## 
## Number of Fisher Scoring iterations: 7
## 
## 
## # Effect decomposition on the risk ratio scale via the regression-based approach
##  
## Closed-form parameter function estimation with 
##  delta method standard errors, confidence intervals and p-values 
##  
##                 Estimate Std.error   95% CIL 95% CIU    P.val    
## Rcde             1.24309   0.22291   0.87472   1.767 0.224946    
## Rpnde            1.01814   0.17635   0.72506   1.430 0.917328    
## Rtnde            1.20855   0.21018   0.85947   1.699 0.276073    
## Rpnie            0.80040   0.05419   0.70093   0.914 0.001009 ** 
## Rtnie            0.95009   0.06605   0.82906   1.089 0.461459    
## Rte              0.96733   0.15543   0.70599   1.325 0.836210    
## ERcde            0.18777   0.16621  -0.13800   0.514 0.258600    
## ERintref        -0.16963   0.10003  -0.36569   0.026 0.089947 .  
## ERintmed         0.14878   0.08790  -0.02351   0.321 0.090547 .  
## ERpnie          -0.19960   0.05419  -0.30582  -0.093 0.000231 ***
## ERcde(prop)     -5.74662  32.11420 -68.68930  57.196 0.857982    
## ERintref(prop)   5.19141  24.66335 -43.14787  53.531 0.833285    
## ERintmed(prop)  -4.55337  21.62860 -46.94465  37.838 0.833257    
## ERpnie(prop)     6.10859  29.09367 -50.91396  63.131 0.833697    
## pm               1.55521   7.88784 -13.90468  17.015 0.843698    
## int              0.63803   3.03635  -5.31311   6.589 0.833565    
## pe               6.74662  32.11420 -56.19606  69.689 0.833604    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Rcde: controlled direct effect risk ratio; Rpnde: pure natural direct effect risk ratio; Rtnde: total natural direct effect risk ratio; Rpnie: pure natural indirect effect risk ratio; Rtnie: total natural indirect effect risk ratio; Rte: total effect risk ratio; ERcde: excess relative risk due to controlled direct effect; ERintref: excess relative risk due to reference interaction; ERintmed: excess relative risk due to mediated interaction; ERpnie: excess relative risk due to pure natural indirect effect; ERcde(prop): proportion ERcde; ERintref(prop): proportion ERintref; ERintmed(prop): proportion ERintmed; ERpnie(prop): proportion ERpnie; pm: overall proportion mediated; int: overall proportion attributable to interaction; pe: overall proportion eliminated)
## 
## Relevant variable values: 
## $a
## [1] 1
## 
## $astar
## [1] 0
## 
## $yval
## [1] "1"
## 
## $mval
## $mval[[1]]
## [1] 1
## 
## 
## $basecval
## $basecval[[1]]
## [1] 0.9967315
## 
## $basecval[[2]]
## [1] 0.5936