AFivglm function

Attributable fraction function based on Instrumental Variables (IV) regression as an ivglm object in the ivtools package.

AFivglm estimates the model-based adjusted attributable fraction from a Instrumental Variable regression from a ivglm object. The IV regression can be estimated by either G-estimation or Two Stage estimation for a binary exposure and outcome.

AFivglm(object, data)

Arguments

• object: a fitted Instrumental Variable regression of class "ivglm".
• data: an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment (formula), typically the environment from which the function is called.

Returns

• AF.est: estimated attributable fraction.

• AF.var: estimated variance of AF.est. The variance is obtained by combining the delta methods with the sandwich formula.

Details

AFivglm estimates the attributable fraction for an IV regression under the hypothetical scenario where a binary exposure X is eliminated from the population. The estimate can be adjusted for IV-outcome confounders L in the ivglm function. Let the AF function be defined as

where $Pr(Y0=1)$ denotes the counterfactual outcome prevalence had everyone been unexposed and $Pr(Y=1)$ denotes the factual outcome prevalence. If the instrument Z is valid, conditional on covariates L, i.e. fulfills the IV assumptions 1) the IV should have a (preferably strong) association with the exposure, 2) the effect of the IV on the outcome should only go through the exposure and 3) the IV-outcome association should be unconfounded (Imbens and Angrist, 1994) then $Pr(Y0=1)$ can be estimated.

Examples

# Example 1
set.seed(2)
n <- 5000
## parameter a0 determines the outcome prevalence
a0 <- -4
psi.true <- 1
l <- rbinom(n, 1, 0.5)
u <- rbinom(n, 1, 0.5)
z <- rbinom(n, 1, plogis(a0))
x <- rbinom(n, 1, plogis(a0+3*z+ u))
y <- rbinom(n, 1, exp(a0+psi.true*x+u))
d <- data.frame(z,u,x,y,l)
## Outcome prevalence 
mean(d$y)

####### G-estimation
## log CRR
fitz.l <- glm(z~1, family=binomial, data=d)
gest_log <- ivglm(estmethod="g", X="x", Y="y",
                  fitZ.L=fitz.l, data=d, link="log")
AFgestlog <- AFivglm(gest_log, data=d)
summary(AFgestlog)

## log COR
## Associational model, saturated
fit_y <- glm(y~x+z+x*z, family="binomial", data=d)
## Estimations of COR and AF
gest_logit <- ivglm(estmethod="g", X="x", Y="y",
                    fitZ.L=fitz.l, fitY.LZX=fit_y,
                    data=d, link="logit")
AFgestlogit <- AFivglm(gest_logit, data = d)
summary(AFgestlogit)

####### TS estimation
## log CRR
# First stage
fitx <- glm(x ~ z, family=binomial, data=d)
# Second stage
fity <- glm(y ~ x, family=poisson, data=d)
## Estimations of CRR and AF
TSlog <- ivglm(estmethod="ts", X="x", Y="y",
               fitY.LX=fity, fitX.LZ=fitx, data=d, link="log")
AFtslog <- AFivglm(TSlog, data=d)
summary(AFtslog)

## log COR
# First stage
fitx_logit <- glm(x ~ z, family=binomial, data=d)
# Second stage
fity_logit <- glm(y ~ x, family=binomial, data=d)
## Estimations of COR and AF
TSlogit <- ivglm(estmethod="ts", X="x", Y="y",
                 fitY.LX=fity_logit, fitX.LZ=fitx_logit,
                  data=d, link="logit")
AFtslogit <- AFivglm(TSlogit, data=d)
summary(AFtslogit)

## Example 2: IV-outcome confounding by L
####### G-estimation
## log CRR
fitz.l <- glm(z~l, family=binomial, data=d)
gest_log <- ivglm(estmethod="g", X="x", Y="y",
                  fitZ.L=fitz.l, data=d, link="log")
AFgestlog <- AFivglm(gest_log, data=d)
summary(AFgestlog)

## log COR
## Associational model
fit_y <- glm(y~x+z+l+x*z+x*l+z*l, family="binomial", data=d)
## Estimations of COR and AF
gest_logit <- ivglm(estmethod="g", X="x", Y="y",
                    fitZ.L=fitz.l, fitY.LZX=fit_y,
                    data=d, link="logit")
AFgestlogit <- AFivglm(gest_logit, data = d)
summary(AFgestlogit)

####### TS estimation
## log CRR
# First stage
fitx <- glm(x ~ z+l, family=binomial, data=d)
# Second stage
fity <- glm(y ~ x+l, family=poisson, data=d)
## Estimations of CRR and AF
TSlog <- ivglm(estmethod="ts", X="x", Y="y",
               fitY.LX=fity, fitX.LZ=fitx, data=d,
               link="log")
AFtslog <- AFivglm(TSlog, data=d)
summary(AFtslog)

## log COR
# First stage
fitx_logit <- glm(x ~ z+l, family=binomial, data=d)
# Second stage
fity_logit <- glm(y ~ x+l, family=binomial, data=d)
## Estimations of COR and AF
TSlogit <- ivglm(estmethod="ts", X="x", Y="y",
                 fitY.LX=fity_logit, fitX.LZ=fitx_logit,
                 data=d, link="logit")
AFtslogit <- AFivglm(TSlogit, data=d)
summary(AFtslogit)

References

Dahlqwist E., Kutalik Z., , A. (2019). Using Instrumental Variables to estimate the attributable fraction. Manuscript.

See Also

ivglm used for fitting the causal risk ratio or odds ratio using the G-estimator or Two stage estimator.

Author(s)

Elisabeth Dahlqwist, Arvid