AFglm {AF} | R Documentation |
Attributable fraction estimation based on a logistic regression model from a glm
object (commonly used for cross-sectional or case-control sampling designs).
Description
AFglm
estimates the model-based adjusted attributable fraction for data from a logistic regression model in the form of a glm
object. This model is commonly used for data from a cross-sectional or non-matched case-control sampling design.
Usage
AFglm(object, data, exposure, clusterid, case.control = FALSE)
Arguments
object |
a fitted logistic regression model object of class " |
data |
an optional data frame, list or environment (or object coercible by |
exposure |
the name of the exposure variable as a string. The exposure must be binary (0/1) where unexposed is coded as 0. |
clusterid |
the name of the cluster identifier variable as a string, if data are clustered. Cluster robust standard errors will be calculated. |
case.control |
can be set to |
Details
AFglm
estimates the attributable fraction for a binary outcome Y
under the hypothetical scenario where a binary exposure X
is eliminated from the population.
The estimate is adjusted for confounders Z
by logistic regression using the (glm
) function.
The estimation strategy is different for cross-sectional and case-control sampling designs even if the underlying logististic regression model is the same.
For cross-sectional sampling designs the AF can be defined as
where denotes the counterfactual probability of the outcome if
the exposure would have been eliminated from the population and
denotes the factual probability of the outcome.
If
Z
is sufficient for confounding control, then can be expressed as
The function uses logistic regression to estimate
, and the marginal sample distribution of
Z
to approximate the outer expectation (Sjölander and Vansteelandt, 2012).
For case-control sampling designs the outcome prevalence is fixed by sampling design and absolute probabilities (P.est
and P0.est
) can not be estimated.
Instead adjusted log odds ratios (log.or
) are estimated for each individual.
This is done by setting case.control
to TRUE
. It is then assumed that the outcome is rare so that the risk ratio can be approximated by the odds ratio.
For case-control sampling designs the AF be defined as (Bruzzi et. al)
where denotes the counterfactual probability of the outcome if
the exposure would have been eliminated from the population. If
Z
is sufficient for confounding control then the probability can be expressed as
Using Bayes' theorem this implies that the AF can be expressed as
where is the risk ratio
Moreover, the risk ratio can be approximated by the odds ratio if the outcome is rare. Thus,
If clusterid
is supplied, then a clustered sandwich formula is used in all variance calculations.
Value
AF.est |
estimated attributable fraction. |
AF.var |
estimated variance of |
P.est |
estimated factual proportion of cases; |
P.var |
estimated variance of |
P0.est |
estimated counterfactual proportion of cases if exposure would be eliminated; |
P0.var |
estimated variance of |
log.or |
a vector of the estimated log odds ratio for every individual.
then
then |
Author(s)
Elisabeth Dahlqwist, Arvid Sjölander
References
Bruzzi, P., Green, S. B., Byar, D., Brinton, L. A., and Schairer, C. (1985). Estimating the population attributable risk for multiple risk factors using case-control data. American Journal of Epidemiology 122, 904-914.
Greenland, S. and Drescher, K. (1993). Maximum Likelihood Estimation of the Attributable Fraction from logistic Models. Biometrics 49, 865-872.
Sjölander, A. and Vansteelandt, S. (2011). Doubly robust estimation of attributable fractions. Biostatistics 12, 112-121.
See Also
glm
used for fitting the logistic regression model. For conditional logistic regression (commonly for data from a matched case-control sampling design) see AFclogit
.
Examples
# Simulate a cross-sectional sample
expit <- function(x) 1 / (1 + exp( - x))
n <- 1000
Z <- rnorm(n = n)
X <- rbinom(n = n, size = 1, prob = expit(Z))
Y <- rbinom(n = n, size = 1, prob = expit(Z + X))
# Example 1: non clustered data from a cross-sectional sampling design
data <- data.frame(Y, X, Z)
# Fit a glm object
fit <- glm(formula = Y ~ X + Z + X * Z, family = binomial, data = data)
# Estimate the attributable fraction from the fitted logistic regression
AFglm_est <- AFglm(object = fit, data = data, exposure = "X")
summary(AFglm_est)
# Example 2: clustered data from a cross-sectional sampling design
# Duplicate observations in order to create clustered data
id <- rep(1:n, 2)
data <- data.frame(id = id, Y = c(Y, Y), X = c(X, X), Z = c(Z, Z))
# Fit a glm object
fit <- glm(formula = Y ~ X + Z + X * Z, family = binomial, data = data)
# Estimate the attributable fraction from the fitted logistic regression
AFglm_clust <- AFglm(object = fit, data = data,
exposure = "X", clusterid = "id")
summary(AFglm_clust)
# Example 3: non matched case-control
# Simulate a sample from a non matched case-control sampling design
# Make the outcome a rare event by setting the intercept to -6
expit <- function(x) 1 / (1 + exp( - x))
NN <- 1000000
n <- 500
intercept <- -6
Z <- rnorm(n = NN)
X <- rbinom(n = NN, size = 1, prob = expit(Z))
Y <- rbinom(n = NN, size = 1, prob = expit(intercept + X + Z))
population <- data.frame(Z, X, Y)
Case <- which(population$Y == 1)
Control <- which(population$Y == 0)
# Sample cases and controls from the population
case <- sample(Case, n)
control <- sample(Control, n)
data <- population[c(case, control), ]
# Fit a glm object
fit <- glm(formula = Y ~ X + Z + X * Z, family = binomial, data = data)
# Estimate the attributable fraction from the fitted logistic regression
AFglm_est_cc <- AFglm(object = fit, data = data, exposure = "X", case.control = TRUE)
summary(AFglm_est_cc)