Chapter 5 Traditional regression models
Traditional regression models can be applied in the absence of an intermediate confounder \(L(1)\) of the \(M-Y\) relationship affected by the exposure \(A\) (Causal model 1). They can be used for two-way, three-way and four-way decomposition of the average total effect.
In the following examples, we use the df1_int.csv data set with a \(A \star M\) interaction effect on the outcome.
If we assumed that there was no \(A \star M\) interaction, then the A0_PM2.5:M_diabetes interaction terms should be removed from the models below (applicable if we use the df1.csv data set).
5.1 Estimation of the Average Total Effect (ATE)
The average total effect is the difference between the mean outcome had the whole population been exposed to high levels of \(\text{PM}_{2.5}\), compared to the mean outcome had the whole population been unexposed: \(\text{ATE} = \mathbb{E}(Y_{A=1}) - \mathbb{E}(Y_{A=0})\).
For the quantitative outcome, the ATE of the exposure to high levels of \(\text{PM}_{2.5}\) \((A =1 \text{ versus } A=0)\) on the quality of life score \(Y\) can be estimated using a traditional linear regression of Y_qol on A0_PM2.5, adjusted for the baseline confounders L0_male and L0_parent_low_educ_lv.
For the binary outcome (death), we can estimate a risk difference applying a Generalized Linear Model with a Gaussian distribution and identity link, as suggested by Naimi et al (Naimi and Whitcomb 2020).
The regression coefficient of the exposure variable \(A\) is used to estimate the risk difference or the average difference. \[\begin{equation} \mathbb{E}(Y \mid A, L(0)) = \alpha_0 + \alpha_A A + \alpha_{L(0)} L(0) \tag{5.1} \end{equation}\]
\[ \hat{\Psi}_{\text{trad}}^{\text{ATE}} = \hat{\alpha}_A\]
## Import data
rm(list=ls())
df1_int <- read.csv(file = "./data/df1_int.csv")
## For quantitative outcomes, apply a linear regression of Y on A (A0_PM2.5),
## adjusted for the baseline confounders L(0):
trad_ATE_qol <- lm(Y_qol ~ A0_PM2.5 + L0_male + L0_soc_env,
data = df1_int)
## For binary outcomes, apply a GLM of Y on A with a Gaussian distribution and
## identity link, adjusted for the baseline confounders:
trad_ATE_death <- glm(Y_death ~ A0_PM2.5 + L0_male + L0_soc_env,
family = gaussian("identity"),
data = df1_int)
## Use the regression coefficient of the exposure (A0_PM2.5) to estimate the ATE
ATE_trad_qol <- coefficients(trad_ATE_qol)["A0_PM2.5"]
# -7.210089
ATE_trad_death <- coefficients(trad_ATE_death)["A0_PM2.5"]
# 0.07720726The estimation of 95% confidence intervals could be obtained directly from the linear regression with quantitative outcomes (equation (5.1)). However, using a robust (sandwich) variance estimator or applying a bootstrap procedure is recommended (Naimi and Whitcomb 2020).
library(sandwich)
?sandwich
## the sandwich() function returns a sandwich covariance matrix estimate
## for the Quality of Life outcome
ATE_trad_qol <- list(ATE = coef(trad_ATE_qol)["A0_PM2.5"],
lo = coef(trad_ATE_qol)["A0_PM2.5"] - qnorm(0.975) *
sqrt(sandwich(trad_ATE_qol)["A0_PM2.5","A0_PM2.5"]),
hi = coef(trad_ATE_qol)["A0_PM2.5"] + qnorm(0.975) *
sqrt(sandwich(trad_ATE_qol)["A0_PM2.5","A0_PM2.5"]))
ATE_trad_qol
# ATE = -7.210089 , IC95% = [-7.978234 ; -6.441944]
## for death outcome
ATE_trad_death <- list(ATE = coef(trad_ATE_death)["A0_PM2.5"],
lo = coef(trad_ATE_death)["A0_PM2.5"] - qnorm(0.975) *
sqrt(sandwich(trad_ATE_death)["A0_PM2.5","A0_PM2.5"]),
hi = coef(trad_ATE_death)["A0_PM2.5"] + qnorm(0.975) *
sqrt(sandwich(trad_ATE_death)["A0_PM2.5","A0_PM2.5"]))
ATE_trad_death
# ATE = 0.07720726 , IC95% = [0.04945859 ; 0.1049559]
## 95% CI calculation applying a bootstrap procedure
library(boot)
bootfunc <- function(data,index){
boot_dat <- data[index,]
mod.qol <- lm(Y_qol ~ A0_PM2.5 + L0_male + L0_soc_env,
data = boot_dat)
mod.death <- glm(Y_death ~ A0_PM2.5 + L0_male + L0_soc_env,
family = gaussian("identity"),
data = boot_dat)
est <- c(coef(mod.qol)["A0_PM2.5"],
coef(mod.death)["A0_PM2.5"])
return(est)
}
set.seed(1234)
boot_est <- boot(df1_int, bootfunc, R = 1000)
## the 95% CI for the estimation of the ATE of ACE on QoL is:
boot.ci(boot_est, index = 1, type = "norm")
# (-7.955, -6.445 )
## the 95% CI for the estimation of the ATE of ACE on death is:
boot.ci(boot_est, index = 2, type = "norm")
# ( 0.0501, 0.1046 ) Alternatively for binary outcomes, the total effect conditional on baseline confounders can be expressed on an Odds Ratio scale \(\text{OR}^{\text{TE}}\), using the logistic regression (5.2).
\[\begin{equation} \text{logit} P(Y = 1 \mid A, L(0)) = \alpha_0 + \alpha_A A + \alpha_{L(0)}^\prime L(0) \tag{5.2} \end{equation}\]
\[ \text{OR}^{\text{TE}} \mid L(0) = \exp \hat{\alpha}_A \]
TE_death_model <- glm(Y_death ~ A0_PM2.5 + L0_male + L0_soc_env,
family = "binomial",
data = df1_int)
res_TE_death <- summary(TE_death_model)
tot.effect.death.OR <- list(OR = exp(coef(res_TE_death)["A0_PM2.5","Estimate"]),
lo = exp(coef(res_TE_death)["A0_PM2.5","Estimate"] -
qnorm(0.975) *
coef(res_TE_death)["A0_PM2.5","Std. Error"]),
hi = exp(coef(res_TE_death)["A0_PM2.5","Estimate"] +
qnorm(0.975) *
coef(res_TE_death)["A0_PM2.5","Std. Error"]))
tot.effect.death.OR
# OR = 1.523254 , 95% CI = [ 1.323317 ; 1.753398]5.2 Two-way decomposition
In order to carry-out two-way decomposition mediation analyses, with a binary mediator and a continuous outcome, Valeri and VanderWeele suggest using the following linear regression of the outcome and logistic regression of the mediator:(Valeri and VanderWeele 2013) \[\begin{equation} \mathbb{E}(Y \mid A, M,L(0),L(1)) = \gamma_0 + \gamma_A A + \gamma_M M + \gamma_{A \ast M} (A \ast M) + \gamma_{L(0)}^\prime L(0) + \gamma_{L(1)}^\prime L(1) \tag{5.3} \end{equation}\]
\[\begin{equation} \text{logit}P(M=1 \mid A,L(0),L(1)) = \beta_0 + \beta_A A + \beta_{L(0)}^\prime L(0) + \beta_{L(1)}^\prime L(1) \tag{5.4} \end{equation}\]
If the outcome is binary, they suggest using the following logistic regression of the outcome instead of the previous linear regression: \[\begin{equation} \small \text{logit}P(Y \mid A, M,L(0),L(1)) = \gamma_0 + \gamma_A A + \gamma_M M + \gamma_{A \ast M} (A \ast M) + \gamma_{L(0)}^\prime L(0) + \gamma_{L(1)}^\prime L(1) \tag{5.5} \end{equation}\]
trad_qol_am <- lm(Y_qol ~ A0_PM2.5 + M_diabetes + A0_PM2.5:M_diabetes +
L0_male + L0_soc_env + L1,
data = df1_int)
gamma.A.q <- coef(trad_qol_am)["A0_PM2.5"]
gamma.M.q <- coef(trad_qol_am)["M_diabetes"]
gamma.AM.q <- coef(trad_qol_am)["A0_PM2.5:M_diabetes"]
trad_m <- glm(M_diabetes ~ A0_PM2.5 + L0_male + L0_soc_env + L1,
family = "binomial",
data = df1_int)
beta.0 <- coef(trad_m)["(Intercept)"]
beta.A <- coef(trad_m)["A0_PM2.5"]
trad_death_am <- glm(Y_death ~ A0_PM2.5 + M_diabetes + A0_PM2.5:M_diabetes +
L0_male + L0_soc_env + L1,
family = "binomial",
data = df1_int)
gamma.A.d <- coef(trad_death_am)["A0_PM2.5"]
gamma.M.d <- coef(trad_death_am)["M_diabetes"]
gamma.AM.d <- coef(trad_death_am)["A0_PM2.5:M_diabetes"]5.2.1 Controlled Direct Effect
The Controlled Direct Effect is defined as \(\text{CDE}_m = \mathbb{E}(Y_{A=1,M=m}) - \mathbb{E}(Y_{A=0,M=m})\):
For continuous outcome, using parameters from equation (5.3), it can be estimated by: \[\text{CDE}_m = \hat{\gamma}_A + \hat{\gamma}_{A \ast M} \times m\]
### For a continuous outcome
# setting the mediator to M=0
trad_CDE_qol_m0 <- gamma.A.q + gamma.AM.q * 0
trad_CDE_qol_m0
# -3.715265
# setting the mediator to M=1
trad_CDE_qol_m1 <- gamma.A.q + gamma.AM.q * 1
trad_CDE_qol_m1
# -9.330657For binary outcomes, using parameters from equation (5.5), it can be estimated on the OR scale by: \[OR^{\text{CDE}_m} = \text{exp}\left(\hat{\gamma}_A + \hat{\gamma}_{A \ast M} \times m\right)\]
5.2.2 Natural Direct and Indirect effects
The Pure Natural Direct Effect (PNDE) and the Total Natural Indirect Effect (TNIE) are defined as:
- \(\text{PNDE} = \mathbb{E}\left(Y_{A=1,M_{A=0}}) - \mathbb{E}(Y_{A=0,M_{A=0}}\right)\),
- \(\text{TNIE} = \mathbb{E}\left(Y_{A=1,M_{A=1}}) - \mathbb{E}(Y_{A=1,M_{A=0}}\right)\).
Alternatively, one can use the Total Natural Direct Effect (TNDE) and the Pure Natural Indirect Effect (PNIE):
- \(\text{TNDE} = \mathbb{E}\left(Y_{A=1,M_{A=1}}) - \mathbb{E}(Y_{A=0,M_{A=1}}\right)\),
- \(\text{PNIE} = \mathbb{E}\left(Y_{A=0,M_{A=1}}) - \mathbb{E}(Y_{A=0,M_{A=0}}\right)\).
With a continuous outcome and a binary mediator, the PNDE and TNDE can be estimated using the linear regression of the outcome (equation (5.3)) and the logistic regression of the mediator (equation (5.4)): \[\text{PNDE} \mid L(0), L(1) = \hat{\gamma}_A + \hat{\gamma}_{A \ast M} \frac{exp(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}\]
\[\scriptsize \text{TNIE} \mid L(0), L(1) = \left( \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1 \right) \left[ \frac{exp(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))} - \frac{exp(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}\right]\] The alternative TNDE ad PNIE can be estimated by: \[\text{TNDE} \mid L(0), L(1) = \hat{\gamma}_A + \hat{\gamma}_{A \ast M}\frac{exp(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}\] \[\scriptsize \text{PNIE} \mid L(0), L(1) = \left( \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right) \left[ \frac{exp(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))} - \frac{exp(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}\right]\] Conditional on participants for which \(L(0)=0\) and \(L(1)=0\), these expressions are simplified: \[\text{PNDE} \Big| (L(0)=0, L(1)=0) = \hat{\gamma}_A + \hat{\gamma}_{A \ast M}\frac{exp(\hat{\beta}_0)}{1 + exp(\hat{\beta}_0)}\] \[\text{TNIE} \Big| (L(0)=0, L(1)=0) = \left( \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \right) \left[ \frac{exp(\hat{\beta}_0 + \hat{\beta}_A )}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A )} - \frac{exp(\hat{\beta}_0 )}{1 + exp(\hat{\beta}_0)}\right]\] and \[\text{TNDE} \Big| (L(0)=0, L(1)=0) = \hat{\gamma}_A + \hat{\gamma}_{A \ast M}\frac{exp(\hat{\beta}_0 + \hat{\beta}_A)}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A)}\] \[\text{PNIE} \Big| (L(0)=0, L(1)=0) = \hat{\gamma}_M \left[ \frac{exp(\hat{\beta}_0 + \hat{\beta}_A )}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A )} - \frac{exp(\hat{\beta}_0 )}{1 + exp(\hat{\beta}_0)}\right]\]
### For a continuous outcome, in the subgroup with L(0)=0 and L(1)=0
## The PNDE and TNIE are:
trad_PNDE_qol <- gamma.A.q + gamma.AM.q * (exp(beta.0)) / (1 + exp(beta.0))
trad_PNDE_qol
# -4.845089
trad_TNIE_qol <- (gamma.M.q + gamma.AM.q) *
(exp(beta.0 + beta.A) / (1 + exp(beta.0 + beta.A)) -
exp(beta.0) / (1 + exp(beta.0)))
trad_TNIE_qol
# -1.50119
## The TNDE and PNIE are:
trad_TNDE_qol <- gamma.A.q +
gamma.AM.q * exp(beta.0 + beta.A) / (1 + exp(beta.0 + beta.A))
trad_TNDE_qol
# -5.436773
trad_PNIE_qol <- gamma.M.q *
(exp(beta.0 + beta.A) /
(1 + exp(beta.0 + beta.A)) - exp(beta.0) / (1 + exp(beta.0)))
trad_PNIE_qol
# -0.9095061For binary outcomes, total, direct and indirect effects can be expressed on relative risk or odds ratio scales:
The total effect risk ratio is equal to : \[\text{RR}^\text{TE}=\frac{\mathbb{E}(Y_1)}{\mathbb{E}(Y_0)}=\frac{\mathbb{E}(Y_{1,M_1})}{\mathbb{E}(Y_{0,M_0})}\]
The total effect risk ratio can be decomposed as the product of the PNDE risk ratio and the TNIE risk ratio: \[\text{RR}^\text{TE}=\frac{\mathbb{E}(Y_{1,M_1})}{\mathbb{E}(Y_{0,M_0})}=\frac{\mathbb{E}(Y_{1,M_0})}{\mathbb{E}(Y_{0,M_0})} \times \frac{\mathbb{E}(Y_{1,M_1})}{\mathbb{E}(Y_{1,M_0})}=\text{RR}^\text{PNDE} \times \text{RR}^\text{TNIE}\]
Similarly, the total effect risk ratio can be decomposed as the product of the TNDE risk ratio and the PNIE risk ratio: \[\text{RR}^\text{TE}=\frac{\mathbb{E}(Y_{1,M_1})}{\mathbb{E}(Y_{0,M_0})}=\frac{\mathbb{E}(Y_{1,M_1})}{\mathbb{E}(Y_{0,M_1})} \times \frac{\mathbb{E}(Y_{0,M_1})}{\mathbb{E}(Y_{0,M_0})}=\text{RR}^\text{TNDE} \times \text{RR}^\text{PNIE}\]
PNDE, TNIE, TNDE and PNIE can also be given on the OR scale, \[\text{OR}^\text{PNDE} = \frac{\frac{P\left(Y_{A=1,M_{A=0}}=1\right)}{1 - P\left(Y_{A=1,M_{A=0}}=1\right)}}{\frac{P\left(Y_{A=0,M_{A=0}}=1\right)}{1 - P\left(Y_{A=0,M_{A=0}}=1\right)}} \quad \text{,} \quad \text{OR}^\text{TNIE} = \frac{\frac{P\left(Y_{A=1,M_{A=1}}=1\right)}{1 - P\left(Y_{A=1,M_{A=1}}=1\right)}}{\frac{P\left(Y_{A=1,M_{A=0}}=1\right)}{1 - P\left(Y_{A=1,M_{A=0}}=1\right)}}\] and \[\text{OR}^\text{TNDE} = \frac{\frac{P\left(Y_{A=1,M_{A=0}}=1\right)}{1 - P\left(Y_{A=1,M_{A=0}}=1\right)}}{\frac{P\left(Y_{A=0,M_{A=0}}=1\right)}{1 - P\left(Y_{A=0,M_{A=0}}=1\right)}} \quad \text{and} \quad \text{OR}^\text{PNIE} = \frac{\frac{P\left(Y_{A=1,M_{A=1}}=1\right)}{1 - P\left(Y_{A=1,M_{A=1}}=1\right)}}{\frac{P\left(Y_{A=1,M_{A=0}}=1\right)}{1 - P\left(Y_{A=1,M_{A=0}}=1\right)}}.\] If the outcome is rare, we have \(P(Y=1) \approx \frac{P(Y=1)}{1-P(Y=1)}\) so that, \(\text{OR}^{\text{PNDE}}\) AND \(\text{OR}^{\text{TNIE}}\) can be estimated using the logistic model of the outcome (equation (5.5)) and the logistic model of the mediator (equation (5.4)): \[\small \text{OR}^\text{PNDE} \mid L(0), L(1) \approx \frac{\text{exp}(\hat{\gamma}_A \times 1) \left[ 1 + \text{exp}(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1 + \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))\right]}{ \text{exp}(\hat{\gamma}_A \times 0) \left[ 1 + \text{exp}(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 + \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))\right]} \]
and
\[ \scriptsize \text{OR}^\text{TNIE} \mid L(0), L(1) \approx \frac{\left[1 + \text{exp}(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1)) \right] \left[ 1 + \text{exp}(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1 + \hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))\right]}{ \left[1 + \text{exp}(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1)) \right] \left[ 1 + \text{exp}(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1 + \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))\right]} \] Similarly, if the outcome is rare, \(\text{OR}^{\text{TNDE}}\) AND \(\text{OR}^{\text{PNIE}}\) can be estimated using the logistic regression models for the outcome and the mediator (equations (5.5) and (5.4)): \[\small \text{OR}^\text{TNDE} \mid L(0), L(1) \approx \frac{\text{exp}\left(\hat{\gamma}_A \times 1 \right) \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1 + \hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1)\right)\right]}{ \text{exp}\left(\hat{\gamma}_A \times 0 \right) \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 + \hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1)\right)\right]}\]
and \[\scriptsize \text{OR}^\text{PNIE} \mid L(0), L(1) \approx \frac{\left[1 + \text{exp}\left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1)\right) \right] \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 + \hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1)\right)\right]}{ \left[1 + \text{exp}\left(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1)\right) \right] \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 + \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1)\right)\right]} \] Conditional on participants for which \(L(0)=0\) and \(L(1)=0\), these expressions are simplified: \[\text{OR}^\text{PNDE} \Bigg| \left(L(0) = 0, L(1) = 0\right) \approx \frac{\text{exp}\left(\hat{\gamma}_A\right) \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} + \hat{\beta}_0\right)\right]}{ \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\beta}_0 \right)\right]}\] and \[\text{OR}^\text{TNIE} \Bigg| \left(L(0)=0, L(1)=0\right) \approx \frac{\left[1 + \text{exp}\left(\hat{\beta}_0\right) \right] \times \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} + \hat{\beta}_0 + \hat{\beta}_A\right)\right]}{ \left[1 + \text{exp}\left(\hat{\beta}_0 + \hat{\beta}_A \right) \right] \times \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} + \hat{\beta}_0 \right)\right]} \] Similarly, conditional on \(L(0)=0\) and \(L(1)=0\), \[\text{OR}^\text{TNDE} \Bigg| \left(L(0)=0, L(1)=0\right) \approx \frac{\text{exp}\left(\hat{\gamma}_A\right) \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\gamma}_{A \ast M} + \hat{\beta}_0 + \hat{\beta}_A \right)\right]}{ \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\beta}_0 + \hat{\beta}_A \right)\right]}\] \[\text{OR}^\text{PNIE} \Bigg| \left(L(0)=0, L(1)=0\right) \approx \frac{\left[1 + \text{exp}\left(\hat{\beta}_0 \right) \right] \times \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\beta}_0 + \hat{\beta}_A \right)\right]}{ \left[1 + \text{exp}\left(\hat{\beta}_0 + \hat{\beta}_A \right) \right] \times \left[ 1 + \text{exp}\left(\hat{\gamma}_M + \hat{\beta}_0 \right)\right]} \]
### For a binary outcome, in the subgroup with L(0)=0 and L(1)=0
## The PNDE and TNIE are (on the OR scale):
trad_OR_PNDE_death <- exp(gamma.A.d) *
(1 + exp(gamma.M.d + gamma.AM.d + beta.0 )) /
(1 + exp(gamma.M.d + beta.0))
trad_OR_PNDE_death
# 1.448035
trad_OR_TNIE_death <- (1 + exp(beta.0)) *
(1 + exp(gamma.M.d + gamma.AM.d + beta.0 + beta.A)) /
((1 + exp(beta.0 + beta.A)) * (1 + exp(gamma.M.d + gamma.AM.d + beta.0)))
trad_OR_TNIE_death
# 1.050029
## The TNDE and PNIE are (on the OR scale):
trad_OR_TNDE_death <- exp(gamma.A.d) *
(1 + exp(gamma.M.d + gamma.AM.d + beta.0 + beta.A)) /
(1 + exp(gamma.M.d + beta.0 + beta.A))
trad_OR_TNDE_death
# 1.450344
trad_OR_PNIE_death <- (1 + exp(beta.0)) *
(1 + exp(gamma.M.d + beta.0 + beta.A)) /
((1 + exp(beta.0 + beta.A)) * (1 + exp(gamma.M.d + beta.0)))
trad_OR_PNIE_death
# 1.048358The regmedint package (Regression-Based Causal Mediation Analysis with Interaction and Effect Modification Terms) can be used for two-way decomposition. Estimations of the CDE, PNDE, TNIE, TNDE and PNIE presented above can be obtained as we show in the following example.
For continuous outcomes:
library(regmedint)
regmedint_cont <- regmedint(data = df1_int,
## Variables
yvar = "Y_qol", # outcome variable
avar = "A0_PM2.5", # exposure
mvar = "M_diabetes", # mediator
cvar = c("L0_male", # confounders
"L0_soc_env",
"L1"),
#eventvar = "event", # only for survival outcome
## Values at which effects are evaluated
a0 = 0,
a1 = 1,
m_cde = 0,
c_cond = c(0,0,0), # covariate level
## Model types
mreg = "logistic",
yreg = "linear",
## Additional specification
interaction = TRUE, # presence of A:M interaction term
# in the outcome model
casecontrol = FALSE)
summary(regmedint_cont)
#### Mediation analysis
# est se Z p lower upper
# cde -3.7152652 0.41600219 -8.930879 0.000000e+00 -4.5306145 -2.8999159
# pnde -4.8450888 0.35052810 -13.822255 0.000000e+00 -5.5321113 -4.1580663
# tnie -1.5011902 0.20821830 -7.209694 5.608847e-13 -1.9092905 -1.0930898
# tnde -5.4367728 0.34049175 -15.967414 0.000000e+00 -6.1041244 -4.7694213
# pnie -0.9095061 0.12266064 -7.414817 1.219025e-13 -1.1499166 -0.6690957
# te -6.3462790 0.38788368 -16.361294 0.000000e+00 -7.1065170 -5.5860409
# pm 0.2365465 0.02947624 8.024991 1.110223e-15 0.1787742 0.2943189
# note: te = total effect = (pnde + tnie) = (tnde + pnie)
# pm = proportion mediated = tnie / teFor binary outcomes:
regmedint_bin <- regmedint(data = df1_int,
## Variables
yvar = "Y_death", # outcome variable
avar = "A0_PM2.5", # exposure
mvar = "M_diabetes", # mediator
cvar = c("L0_male", # confounders
"L0_soc_env",
"L1"),
#eventvar = "event", # only for survival outcome
## Values at which effects are evaluated
a0 = 0,
a1 = 1,
m_cde = 0,
c_cond = c(0,0,0), # covariate level
## Model types
mreg = "logistic",
yreg = "logistic",
## Additional specification
interaction = TRUE,
casecontrol = FALSE)
results.binary <- summary(regmedint_bin)
# taking the exponential of the estimations
exp(results.binary$summary_myreg[c("cde","pnde","tnie","tnde","pnie","te"),
c("est","lower","upper")])
#### Mediation analysis
# est lower upper
# cde 1.442942 1.191195 1.747893
# pnde 1.448035 1.245470 1.683545
# tnie 1.050029 1.013842 1.087509
# tnde 1.450344 1.257042 1.673371
# pnie 1.048358 1.029285 1.067783
# te 1.520479 1.316954 1.755457The regmedint package gives the same results as those calculated manually using the regression coefficients.
5.3 Three-way decomposition
In order to carry-out a three-way decomposition with standard regressions, we will use the same models as for the two-way decomposition (equations (5.3), (5.5) and (5.4)).
(VanderWeele 2013) defines:
- the \(\text{PNDE} = \mathbb{E}\left(Y_{A=1,M_{A=0}}) - \mathbb{E}(Y_{A=0,M_{A=0}}\right)\),
- the \(\text{PNIE} = \mathbb{E}\left(Y_{A=0,M_{A=1}}) - \mathbb{E}(Y_{A=0,M_{A=0}}\right)\),
- and the mediated interactive effect \(\text{MIE} = \mathbb{E}\left( \left[ Y_{1,1} - Y_{1,0} - Y_{0,1} - Y_{0,0}\right] \times \left[M_1 - M_0 \right]\right)\).
The sum of these 3 components is equal to the Average total effect (ATE).
With a continuous outcome and a binary mediator, the PNDE and PNIE can be estimated as for the two-way decomposition (section 5.2.2) using the linear regression of the outcome (equation (5.3)) and the logistic regression of the mediator (equation (5.4)).
The mediated interactive effect can be estimated using the same equations (5.3) and (5.4), by: \[ \scriptsize \text{MIE} \mid (L(0), L(1)) = \hat{\gamma}_{A \ast M} \left[ \frac{\exp (\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + \exp (\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))} - \frac{\exp (\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + \exp (\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}\right]\]
Conditional on participants for which \(L(0)=0\) and \(L(1)=0\), the expression is simplified: \[\text{MIE} \Big| (L(0)=0, L(1)=0) = \hat{\gamma}_{A \ast M} \left[ \frac{\exp (\hat{\beta}_0 + \hat{\beta}_A )}{1 + \exp (\hat{\beta}_0 + \hat{\beta}_A )} - \frac{\exp (\hat{\beta}_0)}{1 + \exp (\hat{\beta}_0)}\right]\]
### For a continuous outcome, in the subgroup with L(0)=0 and L(1)=0
## The PNDE is:
trad_PNDE_qol <- gamma.A.q + gamma.AM.q * (exp(beta.0)) / (1 + exp(beta.0))
trad_PNDE_qol
# -4.845089
## The PNIE is:
trad_PNIE_qol <- gamma.M.q *
(exp(beta.0 + beta.A) /
(1 + exp(beta.0 + beta.A)) - exp(beta.0) / (1 + exp(beta.0)))
trad_PNIE_qol
# -0.9095061
## The MIE is:
trad_MIE_qol <- gamma.AM.q *
(exp(beta.0 + beta.A) / (1 + exp(beta.0 + beta.A)) -
exp(beta.0) / (1 + exp(beta.0)))
trad_MIE_qol
# -0.591684With a binary outcome, the total effect, the direct and indirect effects can be expressed using risk ratios or odds ratios. In order to express the mediated interactive effect, (VanderWeele 2013) suggested decomposing the excess relative risk of the total effect \((\text{RR}^{\text{TE}}-1)\), which enables the expression of the mediated interactive effect on an additive scale.
On the difference scale, the total effect can be decomposed as the sum of the PNDE, the PNIE and the MIE: \[\begin{array}{rl} \mathbb{E}(Y_1) - \mathbb{E}(Y_0) = & \left[ \mathbb{E}\left(Y_{1M_0}\right) - \mathbb{E}\left(Y_{0M_0}\right) \right] + \left[ \mathbb{E}\left(Y_{0M_1}\right) - \mathbb{E}\left(Y_{0M_0}\right) \right] \\ & + \left[ \left[ \mathbb{E}\left(Y_{1M_1}\right) - \mathbb{E}\left(Y_{1M_0}\right) \right] - \left[ \mathbb{E}\left(Y_{0M_1}\right) - \mathbb{E}\left(Y_{0M_0}\right) \right] \right] \\ = & \text{PNDE + PNIE} \\ & \text{ + MIE} \end{array}\]
Dividing by \(\mathbb{E}(Y_0) = \mathbb{E}\left(Y_{0M_0}\right)\), we obtain the excess relative risk of the total effect decomposition: \[\begin{array}{rl} \frac{\mathbb{E}(Y_1)}{\mathbb{E}(Y_0)}-1 = & \left[ \frac{\mathbb{E}\left( Y_{1M_0}\right)}{\mathbb{E}\left( Y_{0M_0}\right)} - 1 \right] + \left[ \frac{\mathbb{E}\left( Y_{0M_1}\right)}{\mathbb{E}\left( Y_{0M_0}\right)} - 1 \right] \\ & + \left[ \frac{\mathbb{E}\left( Y_{1M_1}\right)}{\mathbb{E}\left( Y_{0M_0}\right)} - \frac{\mathbb{E}\left( Y_{1M_0}\right)}{\mathbb{E}\left( Y_{0M_0}\right)} - \frac{\mathbb{E}\left( Y_{0M_1}\right)}{\mathbb{E}\left( Y_{0M_0}\right)} + 1 \right] \end{array}\] where the first component is the excess relative risk due to the PNDE, the second component is the excess relative risk due to the PNIE and the third component is the mediated excess relative risk due to interaction.
If the outcome is rare, relative risks are approximately equal to odds ratios, and the 3 components of the excess relative risk can be estimated using the logistic regression of the outcome (equation (5.5)) and the logistic regression of the mediator (equation (5.4)).
The component of the excess relative risk due to the PNDE is approximately equal to: \[\scriptsize \text{RR}^\text{PNDE} - 1 \approx \frac{ \exp \left[ \hat{\gamma}_A (1 - 0) \right] \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1 \right) \right]}{ \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right) \right] } - 1 \]
The component of the excess relative risk due to the PNIE is approximately equal to: \[\scriptsize \text{RR}^\text{PNIE} - 1 \approx \frac{\left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right] \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right) \right]}{\left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right] \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right) \right] } - 1 \]
The component of the excess relative risk due to the mediated interactive effect is approximately equal to: \[\small \begin{array}{rl} \text{RERI}_{mediated} \approx & \frac{\exp \left[ \hat{\gamma}_A (1 - 0)\right] \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1\right) \right] \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1)\right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0\right)\right] \left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right]} \\ & - \frac{\left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0\right) \right] \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right)\right] \left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right]} \\ & - \frac{\exp \left[ \hat{\gamma}_A (1 - 0)\right] \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1 \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right)\right]} + 1 \end{array}\]
Conditional on participants for which \(L(0)=0\) and \(L(1)=0\), these expressions are simplified: \[\text{RR}^\text{PNDE} - 1 \approx \frac{ \exp \left( \hat{\gamma}_A \right) \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \right) \right]}{ \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\gamma}_M \right) \right] } - 1 \]
\[\text{RR}^\text{PNIE} - 1 \approx \frac{\left[ 1 + \exp \left( \hat{\beta}_0 \right) \right] \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A + \hat{\gamma}_M \right) \right]}{\left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \right) \right] \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\gamma}_M \right) \right] } - 1 \]
and \[\begin{array}{rl} \text{RERI}_\text{mediated} \approx & \frac{ \exp \left( \hat{\gamma}_A \right) \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \right) \right] \left[1 + \exp \left(\hat{\beta}_0 \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M \right)\right] \left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \right) \right]} - \frac{\left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A + \hat{\gamma}_M \right) \right] \left[1 + \exp \left(\hat{\beta}_0 \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M \right)\right] \left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \right) \right]} \\ & - \frac{ \exp \left( \hat{\gamma}_A \right) \left[1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M \right)\right]} + 1 \end{array}\]
### For a binary outcome, in the subgroup with L(0)=0 and L(1)=0
## The excess relative risk is 52.3% (calculated from the OR of the total effect)
1.523254 - 1
# 0.523254
## The excess relative risk is decomposed into 3 components:
## The component of the excess relative risk due to PNDE is:
comp_PNDE_death <- exp(gamma.A.d) * (1 + exp(beta.0 + gamma.M.d + gamma.AM.d)) /
(1 + exp(beta.0 + gamma.M.d)) - 1
comp_PNDE_death
# 0.4480347
## The component of the excess relative risk due to PNIE is:
comp_PNIE_death <- (1 + exp(beta.0)) * (1 + exp(beta.0 + beta.A + gamma.M.d)) /
((1 + exp(beta.0 + beta.A)) * (1 + exp(beta.0 + gamma.M.d))) - 1
comp_PNIE_death
# 0.04835753
## The component of the excess relative risk due to the mediated interactive
## effect is:
comp_MIE_qol <- exp(gamma.A.d) *
(1 + exp(beta.0 + beta.A + gamma.M.d + gamma.AM.d)) * (1 + exp(beta.0)) /
((1 + exp(beta.0 + gamma.M.d)) * (1 + exp(beta.0 + beta.A))) -
(1 + exp(beta.0 + beta.A + gamma.M.d)) * (1 + exp(beta.0)) /
((1 + exp(beta.0 + gamma.M.d)) * (1 + exp(beta.0 + beta.A))) -
exp(gamma.A.d) * (1 + exp(beta.0 + gamma.M.d + gamma.AM.d)) /
(1 + exp(beta.0 + gamma.M.d)) + 1
comp_MIE_qol
# 0.02408674In this example, the excess relative risk of the exposure to high levels of \(\text{PM}_{2.5}\) is \(\approx 52.3\%\), and of this excess relative risk…:
- \(\approx 44.8\%\) is attributable to the PNDE of \(\text{PM}_{2.5}\),
- \(\approx 4.8\%\) is attributable to the PNIE of \(\text{PM}_{2.5}\) through type-2 diabetes,
- \(\approx 2.4\%\) is attributable to the mediated interactive effect between \(\text{PM}_{2.5}\) and type-2 diabetes
Note: in this simulated data, the probability of death is around \(20\%\), so that the requirement of a rare outcome is not really fulfilled (usually, we would consider \(< 10\%\) to be acceptable).
5.4 Four-way decomposition
The same models as for the two-way and three-way decomposition (equations (5.3), (5.5) and (5.4)) will be used in order to apply the four-way decomposition.
(VanderWeele 2014) defines:
- the \(\text{CDE}_{M=0} = \mathbb{E}\left( Y_{1,0}\right) - \mathbb{E}\left(Y_{0,0} \right)\),
- the mediated interaction effect \(\text{MIE} = \mathbb{E}\left( \left[ Y_{1,1} - Y_{1,0} - Y_{0,1} - Y_{0,0}\right] \times \left[M_1 - M_0 \right]\right)\),
- the reference interaction effect \(\text{RIE} = \mathbb{E}\left( \left[Y_{1,1} - Y_{1,0} - Y_{0,1} - Y_{0,0}\right] \times M_0 \right)\),
- and the \(\text{PNIE} = \mathbb{E}\left(Y_{A=0,M_{A=1}}) - \mathbb{E}(Y_{A=0,M_{A=0}} \right)\).
The sum of these 4 components is equal to the Average total effect (ATE), and if the exposure affects the outcome, then at least one of these 4 components should be non-null.
With a continuous outcome and a binary mediator, the \(\text{CDE}_{M=0}\) and \(\text{PNIE}\) can be estimated as for the two-way decomposition (sections 5.2.1 and 5.2.2), and the \(\text{MIE}\) can be estimated as for the three-way decomposition (section 5.3), using the linear regression of the outcome (equation (5.3)) and the logistic regression of the mediator (equation (5.4)).
\[\text{CDE}_{M=0} \mid \left(L(0),L(1) \right) = \hat{\gamma}_A + \hat{\gamma}_{A \ast M} \times 0\]
\[\scriptsize \text{PNIE} \mid L(0), L(1) = \left( \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right) \left[ \frac{exp(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))} - \frac{exp(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + exp(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}\right]\]
and
\[ \scriptsize \text{MIE} \mid (L(0), L(1)) = \hat{\gamma}_{A \ast M} \left[ \frac{\exp (\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + \exp (\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))} - \frac{\exp (\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + \exp (\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}\right]\] The RIE can be estimated by: \[\scriptsize \text{RIE} \mid (L(0), L(1)) = \hat{\gamma}_{A \ast M} \left[ \frac{\exp (\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))}{1 + \exp (\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1))} - 0 \right]\]
### For a continuous outcome, in the subgroup with L(0)=0 and L(1)=0
## The CDE_(M=0) is:
trad_CDE_qol_m0 <- gamma.A.q + gamma.AM.q * 0
trad_CDE_qol_m0
# -3.715265
## The PNIE is:
trad_PNIE_qol <- gamma.M.q *
(exp(beta.0 + beta.A) /
(1 + exp(beta.0 + beta.A)) - exp(beta.0) / (1 + exp(beta.0)))
trad_PNIE_qol
# -0.9095061
## The MIE is:
trad_MIE_qol <- gamma.AM.q *
(exp(beta.0 + beta.A) / (1 + exp(beta.0 + beta.A)) -
exp(beta.0) / (1 + exp(beta.0)))
trad_MIE_qol
# -0.591684
## The RIE is:
trad_RIE_qol <- gamma.AM.q * (exp(beta.0)) / (1 + exp(beta.0))
trad_RIE_qol
# -1.129824With a binary outcome and a binary mediator, (VanderWeele 2014) suggested decomposing the excess relative risk of the total effect \((\text{RR}^{\text{TE}}-1)\) (as for the 3-way decomposition), which enables the expression of the MIE and the RIE on an additive scale.
If the outcome is rare,
The component of the excess relative risk due to the CDE is approximately equal to:
\[\small \begin{array}{rl} \frac{\mathbb{E}\left( Y_{0,0} \mid L(0), L(1) \right)}{\mathbb{E}\left( Y_0 \mid L(0), L(1) \right)} \left(\frac{\mathbb{E}\left(Y_{1,0} \mid L(0),L(1) \right)}{\mathbb{E}\left(Y_{0,0} \mid L(0),L(1)\right)} - 1\right) \approx & \frac{\exp \left( \hat{\gamma}_A(1-0) + \hat{\gamma}_M \times 0 + \hat{\gamma}_{A \ast M} \times 1 \times 0 \right) \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(0)}^\prime L(1) \right) \right]}{1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(0)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0\right)} \\ & - \frac{\exp \left( \hat{\gamma}_M \times 0 + \hat{\gamma}_{A \ast M} \times 0 \times 0\right) \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(0)}^\prime L(1) \right) \right]}{1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(0)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right)} \end{array}\]
The component of the excess relative risk due to the PNIE is approximately equal to: \[\scriptsize \text{RR}^\text{PNIE} - 1 \approx \frac{\left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right] \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right) \right]}{\left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right] \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right) \right] } - 1 \]
The component of the excess relative risk due to the mediated interactive effect is approximately equal to: \[\small \begin{array}{rl} \text{RERI}_{mediated} \approx & \frac{\exp \left[ \hat{\gamma}_A (1 - 0)\right] \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1\right) \right] \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1)\right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0\right)\right] \left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right]} \\ & - \frac{\left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0\right) \right] \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right)\right] \left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 1 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right]} \\ & - \frac{\exp \left[ \hat{\gamma}_A (1 - 0)\right] \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1 \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right)\right]} + 1 \end{array}\]
and the component of the excess relative risk due to the reference interaction effect is approximately equal to: \[ \begin{array}{rl} \text{RERI}_\text{ref} \approx & \frac{\exp \left[ \hat{\gamma}_A (1 - 0)\right] \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 1\right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0\right)\right] } - 1 \\ & - \frac{\exp \left[ \hat{\gamma}_A (1 - 0) + \hat{\gamma}_M \times 0 + \hat{\gamma}_{A \ast M} 1 \times 0 \right] \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right]}{ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right)} \\ & + \frac{\exp \left( \hat{\gamma}_M \times 0 + \hat{\gamma}_{A \ast M} \times 0 \times 0 \right) \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) \right) \right]}{ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \times 0 + \hat{\beta}_{L(0)}^\prime L(0) + \hat{\beta}_{L(1)}^\prime L(1) + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \times 0 \right) } \end{array}\]
Conditional on participants for which \(L(0)=0\) and \(L(1)=0\), these expressions are simplified:
The component of the excess relative risk due to the CDE is approximately equal to: \[ \approx \frac{\exp \left( \hat{\gamma}_A \right) \left[ 1 + \exp \left( \hat{\beta}_0 \right) \right]}{1 + \exp \left( \hat{\beta}_0 + \hat{\gamma}_M \right)} - \frac{ \left[ 1 + \exp \left( \hat{\beta}_0 \right) \right]}{1 + \exp \left( \hat{\beta}_0 + \hat{\gamma}_M \right)}\]
The component of the excess relative risk due to the PNIE is approximately equal to: \[\text{RR}^\text{PNIE} - 1 \approx \frac{\left[ 1 + \exp \left( \hat{\beta}_0 \right) \right] \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A + \hat{\gamma}_M \right) \right]}{\left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\beta}_A \right) \right] \left[ 1 + \exp \left( \hat{\beta}_0 + \hat{\gamma}_M \right) \right] } - 1 \]
The component of the excess relative risk due to the mediated interactive effect is approximately equal to: \[ \begin{array}{rl} \text{RERI}_{mediated} \approx & \frac{\exp \left( \hat{\gamma}_A \right) \left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \right) \right] \left[1 + \exp \left(\hat{\beta}_0 \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M \right)\right] \left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \right) \right]} - \frac{\left[1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A + \hat{\gamma}_M \right) \right] \left[1 + \exp \left(\hat{\beta}_0 \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M \right)\right] \left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\beta}_A \right) \right]} \\ & - \frac{\exp \left( \hat{\gamma}_A \right) \left[1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M \right)\right]} + 1 \end{array} \]
and the component of the excess relative risk due to the reference interaction effect is approximately equal to: \[\small \text{RERI}_\text{ref} \approx \frac{\exp \left( \hat{\gamma}_A\right) \left[1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M + \hat{\gamma}_{A \ast M} \right) \right]}{\left[ 1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M \right)\right] } - 1 - \frac{\exp \left( \hat{\gamma}_A \right) \left[1 + \exp \left(\hat{\beta}_0 \right) \right]}{ 1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M \right)} + \frac{ \left[1 + \exp \left(\hat{\beta}_0 \right) \right]}{ 1 + \exp \left(\hat{\beta}_0 + \hat{\gamma}_M \right) } \]
### For a binary outcome, in the subgroup with L(0)=0 and L(1)=0
## The excess relative risk is 52.3% (calculated from the OR of the total effect)
1.523254 - 1
# 0.523254
## The excess relative risk is decomposed into 4 components:
## The component of the excess relative risk due to the CDE(M=0) is:
comp_CDE_death_m0 <- exp(gamma.A.d) * (1 + exp(beta.0)) /
(1 + exp(beta.0 + gamma.M.d)) -
(1 + exp(beta.0)) / (1 + exp(beta.0 + gamma.M.d))
comp_CDE_death_m0
# 0.402041
## The component of the excess relative risk due to the PNIE is:
comp_PNIE_death <- (1 + exp(beta.0)) * (1 + exp(beta.0 + beta.A + gamma.M.d)) /
((1 + exp(beta.0 + beta.A)) * (1 + exp(beta.0 + gamma.M.d))) - 1
comp_PNIE_death
# 0.04835753
## The component of the excess relative risk due to the MIE is:
comp_MIE_death <- exp(gamma.A.d) * (1 + exp(beta.0 + beta.A + gamma.M.d +
gamma.AM.d)) * (1 + exp(beta.0)) /
((1 + exp(beta.0 + gamma.M.d)) * (1 + exp(beta.0 + beta.A))) -
(1 + exp(beta.0 + beta.A + gamma.M.d)) * (1 + exp(beta.0)) /
((1 + exp(beta.0 + gamma.M.d)) * (1 + exp(beta.0 + beta.A))) -
exp(gamma.A.d) * (1 + exp(beta.0 + gamma.M.d + gamma.AM.d)) /
(1 + exp(beta.0 + gamma.M.d)) + 1
comp_MIE_death
# 0.02408674
## The component of the excess relative risk due to the RIE is:
comp_RIE_death <- exp(gamma.A.d) * (1 + exp(beta.0 + gamma.M.d + gamma.AM.d)) /
(1 + exp(beta.0 + gamma.M.d)) - 1 -
exp(gamma.A.d) * (1 + exp(beta.0)) / (1 + exp(beta.0 + gamma.M.d)) +
(1 + exp(beta.0)) / (1 + exp(beta.0 + gamma.M.d))
comp_RIE_death
# 0.04599376In this example, the excess relative risk of the exposure to high levels of \(\text{PM}_{2.5}\) is \(\approx 52.3\%\), and of this excess relative risk…:
- \(\approx 40.2\%\) is attributable to the CDE of \(\text{PM}_{2.5}\),
- \(\approx 4.8\%\) is attributable to the PNIE of \(\text{PM}_{2.5}\) through type-2 diabetes,
- \(\approx 2.4\%\) is attributable to the mediated interactive effect between \(\text{PM}_{2.5}\) and type-2 diabetes
- and \(\approx 4.6\%\) is attributable to the (\(\text{PM}_{2.5}\) \(\ast\) type-2 diabetes) reference interactive effect.
Note: in this simulated data, the probability of death is around \(20\%\), so that the requirement of a rare outcome is not really fulfilled (usually, we would consider \(< 10\%\) to be acceptable).
R package for 3-way and 4-way decomposition
The CMAverse R package (a suite of functions for causal mediation analysis) can be used for 2-way, 3-way and 4-way decompositions. Estimations of the CDE(M=0), PNIE, MIE and INTref presented above can be obtained as we show in the following example.
Standard errors and confidence intervals can be computed by delta-method or by bootstrap.
Note that the results from regression based methods (applying the estimation = "paramfunc" argument) are not marginal expectations, but conditional expectations of direct, indirect, total, and interaction effects.
For continuous outcomes:
library(CMAverse)
?CMAverse::cmest
### For the continuous outcome
## Closed-form parameter function estimation and delta method inferece
res_rb_param_delta <- cmest(data = df1_int,
model = "rb", # for "regression based" (rb) approach
outcome = "Y_qol", # outcome variable
exposure = "A0_PM2.5", # exposure variable
mediator = "M_diabetes", # mediator
basec = c("L0_male", # confounders
"L0_soc_env",
"L1"),
EMint = TRUE, # exposures*mediator interaction
mreg = list("logistic"), # model of the mediator
yreg = "linear", # model of the outcome
astar = 0,
a = 1,
mval = list(0),
basecval = list(0,0,0), # covariate level
estimation = "paramfunc", # closed-form parameter
# function estimation
inference = "delta") # IC95% : "delta" or "bootstrap"
summary(res_rb_param_delta)
# Closed-form parameter function estimation with
# delta method standard errors, confidence intervals and p-values
#
# Estimate Std.error 95% CIL 95% CIU P.val
# cde -3.71527 0.41600 -4.53061 -2.900 < 2e-16 *** CDE(M=0)
# pnde -4.84509 0.35053 -5.53211 -4.158 < 2e-16 ***
# tnde -5.43677 0.34049 -6.10412 -4.769 < 2e-16 ***
# pnie -0.90951 0.12266 -1.14992 -0.669 1.22e-13 *** PNIE
# tnie -1.50119 0.20822 -1.90929 -1.093 5.61e-13 ***
# te -6.34628 0.38788 -7.10652 -5.586 < 2e-16 ***
# intref -1.12982 0.13824 -1.40076 -0.859 2.22e-16 *** INTref
# intmed -0.59168 0.10398 -0.79547 -0.388 1.27e-08 *** MIE
# cde(prop) 0.58542 0.04505 0.49712 0.674 < 2e-16 ***
# intref(prop) 0.17803 0.02560 0.12786 0.228 3.51e-12 ***
# intmed(prop) 0.09323 0.01586 0.06216 0.124 4.11e-09 ***
# pnie(prop) 0.14331 0.01655 0.11087 0.176 < 2e-16 ***
# pm 0.23655 0.02948 0.17877 0.294 1.11e-15 ***
# int 0.27126 0.03780 0.19717 0.345 7.20e-13 ***
# pe 0.41458 0.04505 0.32627 0.503 < 2e-16 ***
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## for the 3-way decomposition, the PNDE, PNIE, and MIE are given by:
data.frame("Estimate" = res_rb_param_delta$effect.pe,
"lower95CI" = res_rb_param_delta$effect.ci.low,
"upper95CI" = res_rb_param_delta$effect.ci.high,
"P.value" = res_rb_param_delta$effect.pval)[c("pnde","pnie", "intmed"),]
# Estimate lower95CI upper95CI P.value
# pnde -4.8450888 -5.5321113 -4.1580663 0.000000e+00
# pnie -0.9095061 -1.1499166 -0.6690957 1.219025e-13
# intmed -0.5916840 -0.7954739 -0.3878941 1.266206e-08
## for the 4-way decomposition, the CDE(M=0), Intref, MIE and PNIE are given by:
data.frame("Estimate" = res_rb_param_delta$effect.pe,
"lower95CI" = res_rb_param_delta$effect.ci.low,
"upper95CI" = res_rb_param_delta$effect.ci.high,
"P.value" = res_rb_param_delta$effect.pval)[c("cde","intref",
"intmed","pnie"),]
# Estimate lower95CI upper95CI P.value
# cde -3.7152652 -4.5306145 -2.8999159 0.000000e+00
# intref -1.1298236 -1.4007600 -0.8588871 2.220446e-16
# intmed -0.5916840 -0.7954739 -0.3878941 1.266206e-08
# pnie -0.9095061 -1.1499166 -0.6690957 1.219025e-13For binary outcomes:
### For the binary outcome
## Closed-form parameter function estimation and delta method inferece
res_rb_param_delta <- cmest(data = df1_int,
model = "rb", # for "regression based" (rb) approach
outcome = "Y_death", # outcome variable
exposure = "A0_PM2.5", # exposure variable
mediator = "M_diabetes", # mediator
basec = c("L0_male", # confounders
"L0_soc_env",
"L1"),
EMint = TRUE, # exposures*mediator interaction
mreg = list("logistic"), # model of the mediator
yreg = "logistic", # model of the outcome
astar = 0,
a = 1,
mval = list(0),
basecval = list(0,0,0), # covariate level
estimation = "paramfunc", # closed-form parameter
# function estimation
inference = "delta") # IC95% : delta method
summary(res_rb_param_delta)
# 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 1.44294 0.14115 1.19120 1.748 0.000178 ***
# Rpnde 1.44803 0.11133 1.24547 1.684 1.47e-06 ***
# Rtnde 1.45034 0.10585 1.25704 1.673 3.50e-07 ***
# Rpnie 1.04836 0.00982 1.02929 1.068 4.62e-07 ***
# Rtnie 1.05003 0.01879 1.01384 1.088 0.006368 **
# Rte 1.52048 0.11148 1.31695 1.755 1.10e-08 ***
# ERcde 0.40204 0.12699 0.15315 0.651 0.001546 ** CDE(M=0)
# ERintref 0.04599 0.04821 -0.04850 0.140 0.340093 INTref
# ERintmed 0.02409 0.02543 -0.02576 0.074 0.343607 MIE
# ERpnie 0.04836 0.00982 0.02911 0.068 8.47e-07 *** PNIE
# ERcde(prop) 0.77244 0.14282 0.49252 1.052 6.36e-08 ***
# ERintref(prop) 0.08837 0.09311 -0.09413 0.271 0.342600
# ERintmed(prop) 0.04628 0.04901 -0.04978 0.142 0.345058
# ERpnie(prop) 0.09291 0.02602 0.04192 0.144 0.000356 ***
# pm 0.13919 0.05498 0.03142 0.247 0.011359 *
# int 0.13465 0.14186 -0.14340 0.413 0.342562
# pe 0.22756 0.14282 -0.05237 0.507 0.111092
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## for the 3-way decomposition, the excess relative risk due to
## the PNDE, PNIE and MIE are given by:
data.frame("Estimate" = res_rb_param_delta$effect.pe - 1,
"lower95CI" = res_rb_param_delta$effect.ci.low - 1,
"upper95CI" = res_rb_param_delta$effect.ci.high - 1)[c("Rpnde","Rpnie"),]
# Estimate lower95CI upper95CI
# Rpnde 0.44803473 0.24546995 0.68354491
# Rpnie 0.04835753 0.02928532 0.06778313
# and the excess relative risk due to MIE is given by:
data.frame("Estimate" = res_rb_param_delta$effect.pe,
"lower95CI" = res_rb_param_delta$effect.ci.low,
"upper95CI" = res_rb_param_delta$effect.ci.high,
"P.value" = res_rb_param_delta$effect.pval)[c("ERintmed"),]
# Estimate lower95CI upper95CI P.value
# ERintmed 0.02408674 -0.02576124 0.07393472 3.436070e-01
## for the 4-way decomposition, the CDE(M=0), the excess relative risk due to
## CDE(M=0), Intref, MIE and PNIE are given by
data.frame("Estimate" = res_rb_param_delta$effect.pe,
"lower95CI" = res_rb_param_delta$effect.ci.low,
"upper95CI" = res_rb_param_delta$effect.ci.high,
"P.value" = res_rb_param_delta$effect.pval)[c("ERcde","ERintref",
"ERintmed","ERpnie"),]
# Estimate lower95CI upper95CI P.value
# ERcde 0.40204098 0.15315072 0.65093124 1.545523e-03
# ERintref 0.04599376 -0.04850084 0.14048835 3.400930e-01
# ERintmed 0.02408674 -0.02576124 0.07393472 3.436070e-01
# ERpnie 0.04835753 0.02910970 0.06760535 8.473169e-07