This is an extension of the regression-based causal mediation analysis first proposed by Valeri and VanderWeele (2013) and Valeri and VanderWeele (2015). The current version supports including effect measure modification by covariates (treatment-covariate and mediator-covariate product terms in mediator and outcome regression models). It also accommodates the original ‘SAS’ macro (can be found at Dr. VanderWeele’s Tools and Tutorials) and PROC CAUSALMED procedure in ‘SAS’ when there is no effect measure modification. Linear and logistic models are supported for the mediator model. Linear, logistic, loglinear, Poisson, negative binomial, Cox, and accelerated failure time (exponential and Weibull) models are supported for the outcome model.
To cite this software, please use: regmedint (v1.0.0; Yoshida, Li, & Mathur, 2021)
The following grid of models are implemented. yreg refers to the
outcome model and mreg refers to the mediator model.
| yreg \\ mreg | linear | logistic |
|---|---|---|
| linear | ✔️ | ✔️ |
| logistic1 | ✔️ | ✔️ |
| loglinear | ✔️2 | ✔️2 |
| poisson | ✔️ | ✔️ |
| negbin | ✔️ | ✔️ |
| survCox1 | ✔️ | ✔️ |
| survAFT exp | ✔️ | ✔️ |
| survAFT weibull | ✔️ | ✔️ |
1 Approximation depends on the rare event assumptions.
2 Implemented as a modified Poisson model (log link with robust variance) as in Z2004.
See the corresponding vignettes (Articles on the package website) for how to perform bootstrapping and multiple imputation.
For the developmental version on Github, use the following commands to install the package.
# install.packages("devtools") # If you do not have devtools already.
devtools::install_github("kaz-yos/regmedint")
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The CRAN version can be installed as follows.
install.packages("regmedint")
We use VV2015 dataset for demonstration.
library(regmedint)
data(vv2015)
The regmedint function is the user interface for constructing a result
object of class regmedint. The interface is similar to the original
SAS macro. For survival outcomes, the indicator variable is an event
indicator (1 for event, 0 for censoring). c_cond vector is required be
specified. This vector is the vector of covariate values at which the
conditional effects are evaluated at.
- When there is no effect measure modification by covariates,
emm_ac_mreg = NULL,emm_ac_yreg = NULL,emm_mc_yreg = NULL.
regmedint_obj1 <- regmedint(data = vv2015,
## Variables
yvar = "y",
avar = "x",
mvar = "m",
cvar = c("c"),
eventvar = "event",
## Values at which effects are evaluated
a0 = 0,
a1 = 1,
m_cde = 1,
c_cond = 3,
## Model types
mreg = "logistic",
yreg = "survAFT_weibull",
## Additional specification
interaction = TRUE,
casecontrol = FALSE)
summary(regmedint_obj1)
## ### Mediator model
##
## Call:
## glm(formula = m ~ x + c, family = binomial(link = "logit"), data = data)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5143 -1.1765 0.9177 1.1133 1.4602
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.3545 0.3252 -1.090 0.276
## x 0.3842 0.4165 0.922 0.356
## c 0.2694 0.2058 1.309 0.191
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 138.59 on 99 degrees of freedom
## Residual deviance: 136.08 on 97 degrees of freedom
## AIC: 142.08
##
## Number of Fisher Scoring iterations: 4
##
## ### Outcome model
##
## Call:
## survival::survreg(formula = Surv(y, event) ~ x + m + x:m + c,
## data = data, dist = "weibull")
## Value Std. Error z p
## (Intercept) -1.0424 0.1903 -5.48 0.000000043
## x 0.4408 0.3008 1.47 0.14
## m 0.0905 0.2683 0.34 0.74
## c -0.0669 0.0915 -0.73 0.46
## x:m 0.1003 0.4207 0.24 0.81
## Log(scale) -0.0347 0.0810 -0.43 0.67
##
## Scale= 0.966
##
## Weibull distribution
## Loglik(model)= -11.4 Loglik(intercept only)= -14.5
## Chisq= 6.31 on 4 degrees of freedom, p= 0.18
## Number of Newton-Raphson Iterations: 5
## n= 100
##
## ### Mediation analysis
## est se Z p lower upper
## cde 0.541070807 0.29422958 1.8389409 0.06592388 -0.03560858 1.11775019
## pnde 0.505391952 0.21797147 2.3186151 0.02041591 0.07817572 0.93260819
## tnie 0.015988820 0.03171597 0.5041252 0.61417338 -0.04617334 0.07815098
## tnde 0.513662425 0.22946248 2.2385465 0.02518544 0.06392423 0.96340062
## pnie 0.007718348 0.02398457 0.3218047 0.74760066 -0.03929055 0.05472725
## te 0.521380773 0.22427066 2.3247837 0.02008353 0.08181835 0.96094319
## pm 0.039039346 0.07444080 0.5244348 0.59997616 -0.10686194 0.18494063
##
## Evaluated at:
## avar: x
## a1 (intervened value of avar) = 1
## a0 (reference value of avar) = 0
## mvar: m
## m_cde (intervend value of mvar for cde) = 1
## cvar: c
## c_cond (covariate vector value) = 3
##
## Note that effect estimates can vary over m_cde and c_cond values when interaction = TRUE.
- When there is effect measure modification by covariates,
emm_ac_mreg,emm_ac_yregandemm_mc_yregcan take a sub-vector of covariates incvar.
regmedint_obj2 <- regmedint(data = vv2015,
## Variables
yvar = "y",
avar = "x",
mvar = "m",
cvar = c("c"),
emm_ac_mreg = c("c"),
emm_ac_yreg = c("c"),
emm_mc_yreg = c("c"),
eventvar = "event",
## Values at which effects are evaluated
a0 = 0,
a1 = 1,
m_cde = 1,
c_cond = 3,
## Model types
mreg = "logistic",
yreg = "survAFT_weibull",
## Additional specification
interaction = TRUE,
casecontrol = FALSE)
summary(regmedint_obj2)
## ### Mediator model
##
## Call:
## glm(formula = m ~ x + c + x:c, family = binomial(link = "logit"),
## data = data)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5689 -1.1585 0.8925 1.1242 1.4342
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.32727 0.34979 -0.936 0.349
## x 0.30431 0.56789 0.536 0.592
## c 0.24085 0.24688 0.976 0.329
## x:c 0.09216 0.44624 0.207 0.836
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 138.59 on 99 degrees of freedom
## Residual deviance: 136.04 on 96 degrees of freedom
## AIC: 144.04
##
## Number of Fisher Scoring iterations: 4
##
## ### Outcome model
##
## Call:
## survival::survreg(formula = Surv(y, event) ~ x + m + x:m + c +
## x:c + m:c, data = data, dist = "weibull")
## Value Std. Error z p
## (Intercept) -0.9959 0.2071 -4.81 0.0000015
## x 0.4185 0.3354 1.25 0.21
## m -0.0216 0.3112 -0.07 0.94
## c -0.1339 0.1405 -0.95 0.34
## x:m 0.0905 0.4265 0.21 0.83
## x:c 0.0327 0.2242 0.15 0.88
## m:c 0.1275 0.1861 0.69 0.49
## Log(scale) -0.0406 0.0814 -0.50 0.62
##
## Scale= 0.96
##
## Weibull distribution
## Loglik(model)= -11.1 Loglik(intercept only)= -14.5
## Chisq= 6.78 on 6 degrees of freedom, p= 0.34
## Number of Newton-Raphson Iterations: 5
## n= 100
##
## ### Mediation analysis
## est se Z p lower upper
## cde 0.60705735 0.52594922 1.1542128 0.2484129 -0.4237842 1.6378989
## pnde 0.57902523 0.51447701 1.1254638 0.2603926 -0.4293312 1.5873816
## tnie 0.05333600 0.10591830 0.5035579 0.6145721 -0.1542601 0.2609321
## tnde 0.58889505 0.51488644 1.1437377 0.2527324 -0.4202638 1.5980539
## pnie 0.04346618 0.09107534 0.4772552 0.6331804 -0.1350382 0.2219706
## te 0.63236123 0.52776615 1.1981845 0.2308452 -0.4020414 1.6667639
## pm 0.11082259 0.20960355 0.5287248 0.5969964 -0.2999928 0.5216380
##
## Evaluated at:
## avar: x
## a1 (intervened value of avar) = 1
## a0 (reference value of avar) = 0
## mvar: m
## m_cde (intervend value of mvar for cde) = 1
## cvar: c
## c_cond (covariate vector value) = 3
##
## Note that effect estimates can vary over m_cde and c_cond values when interaction = TRUE.
The summary method gives the summary for mreg, yreg, and mediation
analysis results. The exponentiate option will add the exponentiated
estimate and confidence limits if the outcome model is not a linear
model. The pure natural direct effect (pnde) is what is typically
called the natural direct effect (NDE). The total natural indirect
effect (tnie) is the corresponding natural indirect effect (NIE).
summary(regmedint_obj2, exponentiate = TRUE)
## ### Mediator model
##
## Call:
## glm(formula = m ~ x + c + x:c, family = binomial(link = "logit"),
## data = data)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5689 -1.1585 0.8925 1.1242 1.4342
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.32727 0.34979 -0.936 0.349
## x 0.30431 0.56789 0.536 0.592
## c 0.24085 0.24688 0.976 0.329
## x:c 0.09216 0.44624 0.207 0.836
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 138.59 on 99 degrees of freedom
## Residual deviance: 136.04 on 96 degrees of freedom
## AIC: 144.04
##
## Number of Fisher Scoring iterations: 4
##
## ### Outcome model
##
## Call:
## survival::survreg(formula = Surv(y, event) ~ x + m + x:m + c +
## x:c + m:c, data = data, dist = "weibull")
## Value Std. Error z p
## (Intercept) -0.9959 0.2071 -4.81 0.0000015
## x 0.4185 0.3354 1.25 0.21
## m -0.0216 0.3112 -0.07 0.94
## c -0.1339 0.1405 -0.95 0.34
## x:m 0.0905 0.4265 0.21 0.83
## x:c 0.0327 0.2242 0.15 0.88
## m:c 0.1275 0.1861 0.69 0.49
## Log(scale) -0.0406 0.0814 -0.50 0.62
##
## Scale= 0.96
##
## Weibull distribution
## Loglik(model)= -11.1 Loglik(intercept only)= -14.5
## Chisq= 6.78 on 6 degrees of freedom, p= 0.34
## Number of Newton-Raphson Iterations: 5
## n= 100
##
## ### Mediation analysis
## est se Z p lower upper exp(est) exp(lower) exp(upper)
## cde 0.60705735 0.52594922 1.1542128 0.2484129 -0.4237842 1.6378989 1.835024 0.6545651 5.144349
## pnde 0.57902523 0.51447701 1.1254638 0.2603926 -0.4293312 1.5873816 1.784298 0.6509443 4.890926
## tnie 0.05333600 0.10591830 0.5035579 0.6145721 -0.1542601 0.2609321 1.054784 0.8570491 1.298139
## tnde 0.58889505 0.51488644 1.1437377 0.2527324 -0.4202638 1.5980539 1.801996 0.6568735 4.943403
## pnie 0.04346618 0.09107534 0.4772552 0.6331804 -0.1350382 0.2219706 1.044425 0.8736825 1.248535
## te 0.63236123 0.52776615 1.1981845 0.2308452 -0.4020414 1.6667639 1.882049 0.6689530 5.295005
## pm 0.11082259 0.20960355 0.5287248 0.5969964 -0.2999928 0.5216380 NA NA NA
##
## Evaluated at:
## avar: x
## a1 (intervened value of avar) = 1
## a0 (reference value of avar) = 0
## mvar: m
## m_cde (intervend value of mvar for cde) = 1
## cvar: c
## c_cond (covariate vector value) = 3
##
## Note that effect estimates can vary over m_cde and c_cond values when interaction = TRUE.
Use coef to extract the mediation analysis results only.
coef(summary(regmedint_obj2, exponentiate = TRUE))
## est se Z p lower upper exp(est) exp(lower) exp(upper)
## cde 0.60705735 0.52594922 1.1542128 0.2484129 -0.4237842 1.6378989 1.835024 0.6545651 5.144349
## pnde 0.57902523 0.51447701 1.1254638 0.2603926 -0.4293312 1.5873816 1.784298 0.6509443 4.890926
## tnie 0.05333600 0.10591830 0.5035579 0.6145721 -0.1542601 0.2609321 1.054784 0.8570491 1.298139
## tnde 0.58889505 0.51488644 1.1437377 0.2527324 -0.4202638 1.5980539 1.801996 0.6568735 4.943403
## pnie 0.04346618 0.09107534 0.4772552 0.6331804 -0.1350382 0.2219706 1.044425 0.8736825 1.248535
## te 0.63236123 0.52776615 1.1981845 0.2308452 -0.4020414 1.6667639 1.882049 0.6689530 5.295005
## pm 0.11082259 0.20960355 0.5287248 0.5969964 -0.2999928 0.5216380 NA NA NA
Note that the estimates can be re-evaluated at different m_cde and
c_cond without re-fitting the underlyng models.
coef(summary(regmedint_obj2, exponentiate = TRUE, m_cde = 0, c_cond = 5))
## est se Z p lower upper exp(est) exp(lower) exp(upper)
## cde 0.58192722 1.0143233 0.5737098 0.5661642 -1.4061100 2.5699644 1.789484 0.2450949 13.065360
## pnde 0.65642157 0.9349234 0.7021127 0.4826089 -1.1759946 2.4888377 1.927881 0.3085120 12.047265
## tnie 0.07541287 0.1873908 0.4024363 0.6873630 -0.2918664 0.4426921 1.078329 0.7468683 1.556893
## tnde 0.66420100 0.9330958 0.7118251 0.4765731 -1.1646332 2.4930352 1.942937 0.3120371 12.097940
## pnie 0.06763343 0.1720653 0.3930683 0.6942690 -0.2696084 0.4048753 1.069973 0.7636785 1.499116
## te 0.73183444 0.9597352 0.7625379 0.4457390 -1.1492119 2.6128808 2.078891 0.3168864 13.638283
## pm 0.13996739 0.3295286 0.4247503 0.6710187 -0.5058969 0.7858316 NA NA NA
See here for the following formulas.
| yreg \\ mreg | linear | logistic |
|---|---|---|
| linear | Formulas (1) - (5) | Formulas (11) - (15) |
| logistic | Formulas (21) - (25) | Formulas (31) - (35) |
| loglinear | Formulas (21) - (25) | Formulas (31) - (35) |
| poisson | Formulas (21) - (25) | Formulas (31) - (35) |
| negbin | Formulas (21) - (25) | Formulas (31) - (35) |
| survCox | Formulas (21) - (25) | Formulas (31) - (35) |
| survAFT exp | Formulas (21) - (25) | Formulas (31) - (35) |
| survAFT weibull | Formulas (21) - (25) | Formulas (31) - (35) |
| yreg \\ mreg | linear | logistic |
|---|---|---|
| linear | Formulas (6) - (10) | Formulas (16) - (20) |
| logistic | Formulas (26) - (30) | Formulas (36) - (40) |
| loglinear | Formulas (26) - (30) | Formulas (36) - (40) |
| poisson | Formulas (26) - (30) | Formulas (36) - (40) |
| negbin | Formulas (26) - (30) | Formulas (36) - (40) |
| survCox | Formulas (26) - (30) | Formulas (36) - (40) |
| survAFT exp | Formulas (26) - (30) | Formulas (36) - (40) |
| survAFT weibull | Formulas (26) - (30) | Formulas (36) - (40) |
Note: The point estimate and standard error formulas (multivariate delta method) were derived based on the following references.
- V2015: VanderWeele (2015) Explanation in Causal Inference.
- VV2013A: Valeri & VanderWeele (2013) Appendix
- VV2015A: Valeri & VanderWeele (2015) Appendix
| yreg \\ mreg | linear | logistic |
|---|---|---|
| linear | V2015 p466 Proposition 2.3 | V2015 p471 Proposition 2.5 |
| logistic | V2015 p468 Proposition 2.4 | V2015 p473 Proposition 2.6 |
| loglinear | VV2013A p8 Use Proposition 2.4 | VV2013A p8 Use Proposition 2.6 |
| poisson | VV2013A p8 Use Proposition 2.4 | VV2013A p8 Use Proposition 2.6 |
| negbin | VV2013A p8 Use Proposition 2.4 | VV2013A p8 Use Proposition 2.6 |
| survCox | V2015 p496 Proposition 4.4 (Use 2.4) | V2015 p499 Proposition 4.6 (Use 2.6) |
| survAFT exp | V2015 p494 Proposition 4.1 (Use 2.4) | V2015 p495 Proposition 4.3 (Use 2.6) |
| survAFT weibull | V2015 p494 Proposition 4.1 (Use 2.4) | V2015 p495 Proposition 4.3 (Use 2.6) |
| yreg \\ mreg | linear | logistic |
|---|---|---|
| linear | V2015 p466 Proposition 2.3 | V2015 p471 Proposition 2.5 |
| logistic | V2015 p468 Proposition 2.4 | V2015 p473 Proposition 2.6 |
| loglinear | VV2013A p8 Use Proposition 2.4 | VV2013A p8 Use Proposition 2.6 |
| poisson | VV2013A p8 Use Proposition 2.4 | VV2013A p8 Use Proposition 2.6 |
| negbin | VV2013A p8 Use Proposition 2.4 | VV2013A p8 Use Proposition 2.6 |
| survCox | V2015 p496 Use Proposition 2.4 | V2015 p499 Use Proposition 2.6 |
| survAFT exp | V2015 p494 Use Proposition 2.4 | V2015 p495 Use Proposition 2.6 |
| survAFT weibull | V2015 p494 Use Proposition 2.4 | V2015 p495 Use Proposition 2.6 |
- mediation (simulation-based): https://CRAN.R-project.org/package=mediation
- medflex (natural effect model): https://CRAN.R-project.org/package=medflex
- intmed (interventional analogue): https://CRAN.R-project.org/package=intmed
- CMAverse (regression-based approach, weighting-based approach, inverse odd-ratio weighting, natural effect model, marginal structural model, g-formula approach): https://bs1125.github.io/CMAverse/
- mediator (regression-based): https://github.com/GerkeLab/mediator
- causalMediation (regression-based): https://github.com/harvard-P01/causalMediation
- SAS macro (original regression-based) https://www.hsph.harvard.edu/tyler-vanderweele/tools-and-tutorials/
- SAS PROC CAUSALMED (regression-based) https://support.sas.com/rnd/app/stat/procedures/causalmed.html
- V2015: VanderWeele (2015) Explanation in Causal Inference.
- VV2013: Valeri & VanderWeele (2013) Psych Method. 18:137.
- VV2015: Valeri & VanderWeele (2015) Epidemiology. 26:e23.
- Z2004: Zou (2004) Am J Epidemiol 159:702.