Package 'AF'

Title:	Model-Based Estimation of Confounder-Adjusted Attributable Fractions
Description:	Estimates the attributable fraction in different sampling designs adjusted for measured confounders using logistic regression (cross-sectional and case-control designs), conditional logistic regression (matched case-control design), Cox proportional hazard regression (cohort design with time-to- event outcome), gamma-frailty model with a Weibull baseline hazard and instrumental variables analysis. An exploration of the AF with a genetic exposure can be found in the package 'AFheritability' Dahlqwist E et al. (2019) <doi:10.1007/s00439-019-02006-8>.
Authors:	Elisabeth Dahlqwist and Arvid Sjolander
Maintainer:	Elisabeth Dahlqwist <[email protected]>
License:	GPL-2 \| GPL-3
Version:	0.1.5
Built:	2024-11-27 06:28:14 UTC
Source:	CRAN

Help Index

Attributable fraction for mached and non-matched case-control sampling designs. NOTE! Deprecated function. Use AFglm (for unmatched case-control studies) or AFclogit (for matched case-control studies).
Attributable fraction function for cohort sampling designs with time-to-event outcomes. NOTE! Deprecated function. Use AFcoxph.
Attributable fraction for cross-sectional sampling designs. NOTE! Deprecated function. Use AFglm.
Attributable fraction estimation based on a conditional logistic regression model as a clogit object (commonly used for matched case-control sampling designs).
Attributable fraction function based on a Cox Proportional Hazard regression model as a coxph object (commonly used for cohort sampling designs with time-to-event outcomes).
Attributable fraction estimation based on a logistic regression model from a glm object (commonly used for cross-sectional or case-control sampling designs).
Attributable fraction function based on Instrumental Variables (IV) regression as an ivglm object in the ivtools package.
Attributable fraction function based on a Weibull gamma-frailty model as a parfrailty object (commonly used for cohort sampling family designs with time-to-event outcomes).
Birthweight data clustered on the mother.
Plot function for objects of class "AF" from the function AFcoxph or AFparfrailty.
Cohort study on breast cancer patients from the Netherlands.
Case-control study on oesophageal cancer in Chinese Singapore men.
Summary function for objects of class "AF".

Attributable fraction for mached and non-matched case-control sampling designs. NOTE! Deprecated function. Use `AFglm` (for unmatched case-control studies) or `AFclogit` (for matched case-control studies).

Description

AF.cc estimates the model-based adjusted attributable fraction for data from matched and non-matched case-control sampling designs.

Usage

AF.cc(formula, data, exposure, clusterid, matched = FALSE)
AF.cc(formula, data, exposure, clusterid, matched = FALSE)

Arguments

`formula`	an object of class "`formula`" (or one that can be coerced to that class): a symbolic description of the model used for confounder adjustment. The exposure and confounders should be specified as independent (right-hand side) variables. The outcome should be specified as dependent (left-hand side) variable. The formula is used to object a logistic regression by `glm` for non-matched case-control and conditional logistic regression by `gee` (in package `drgee`) for matched case-control.
`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.
`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 (e.g. matched).
`matched`	a logical that specifies if the sampling design is matched (TRUE) or non-matched (FALSE) case-control. Default setting is non-matched (`matched = FALSE`).

Details

Af.cc 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 for unmatched case-control (glm) and conditional logistic regression for matched case-control (gee). The estimation assumes that the outcome is rare so that the risk ratio can be approximated by the odds ratio, for details see Bruzzi et. al. Let the AF be defined as

$AF = 1 - \frac{Pr(Y_0=1)}{Pr(Y = 1)}$

where $Pr(Y_0=1)$ 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 $Pr(Y_0=1)$ can be expressed as

$Pr(Y_0=1)=E_Z\{Pr(Y=1\mid{X}=0,Z)\}.$

Using Bayes' theorem this implies that the AF can be expressed as

$AF = 1-\frac{E_Z\{Pr(Y=1\mid X=0,Z)\}}{Pr(Y=1)}=1-E_Z\{RR^{-X}(Z)\mid{Y = 1}\}$

where $RR(Z)$ is the risk ratio

$\frac{Pr(Y=1\mid{X=1,Z})}{Pr(Y=1\mid{X=0,Z})}.$

Moreover, the risk ratio can be approximated by the odds ratio if the outcome is rare. Thus,

$AF \approx 1 - E_Z\{OR^{-X}(Z)\mid{Y = 1}\}.$

The odds ratio is estimated by logistic regression or conditional logistic regression. 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 `AF.est`. The variance is obtained by combining the delta methods with the sandwich formula.
`log.or`	a vector of the estimated log odds ratio for every individual. `log.or` contains the estimated coefficient for the exposure variable `X` for every level of the confounder `Z` as specified by the user in the formula. If the model to be estimated is $logit\{Pr(Y=1\|X,Z)\} = \alpha+\beta{X}+\gamma{Z}$ then `log.or` is the estimate of $\beta$ . If the model to be estimated is $logit\{Pr(Y=1\|X,Z)\}=\alpha+\beta{X}+\gamma{Z}+\psi{XZ}$ then `log.odds` is the estimate of $\beta + \psi{Z}$ .
`object`	the fitted model. Fitted using logistic regression, `glm`, for non-matched case-control and conditional logistic regression, `gee`, for matched case-control.

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.

Examples

expit <- function(x) 1 / (1 + exp( - x))
NN <- 1000000
n <- 500

# Example 1: 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
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), ]

# Estimation of the attributable fraction
AF.cc_est <- AF.cc(formula = Y ~ X + Z + X * Z, data = data, exposure = "X")
summary(AF.cc_est)

# Example 2: matched case-control
# Duplicate observations in order to create a matched data sample
# Create an unobserved confounder U common for each pair of individuals
U  <- rnorm(n = NN)
Z1 <- rnorm(n = NN)
Z2 <- rnorm(n = NN)
X1 <- rbinom(n = NN, size = 1, prob = expit(U + Z1))
X2 <- rbinom(n = NN, size = 1, prob = expit(U + Z2))
Y1 <- rbinom(n = NN, size = 1, prob = expit(intercept + U + Z1 + X1))
Y2 <- rbinom(n = NN, size = 1, prob = expit(intercept + U + Z2 + X2))
# Select discordant pairs
discordant <- which(Y1!=Y2)
id <- rep(1:n, 2)
# Sample from discordant pairs
incl <- sample(x = discordant, size = n, replace = TRUE)
data <- data.frame(id = id, Y = c(Y1[incl], Y2[incl]), X = c(X1[incl], X2[incl]),
                   Z = c(Z1[incl], Z2[incl]))

# Estimation of the attributable fraction
AF.cc_match <- AF.cc(formula = Y ~ X + Z + X * Z, data = data,
                         exposure = "X", clusterid = "id", matched = TRUE)
summary(AF.cc_match)
expit <- function(x) 1 / (1 + exp( - x))
NN <- 1000000
n <- 500

# Example 1: 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
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), ]

# Estimation of the attributable fraction
AF.cc_est <- AF.cc(formula = Y ~ X + Z + X * Z, data = data, exposure = "X")
summary(AF.cc_est)

# Example 2: matched case-control
# Duplicate observations in order to create a matched data sample
# Create an unobserved confounder U common for each pair of individuals
U  <- rnorm(n = NN)
Z1 <- rnorm(n = NN)
Z2 <- rnorm(n = NN)
X1 <- rbinom(n = NN, size = 1, prob = expit(U + Z1))
X2 <- rbinom(n = NN, size = 1, prob = expit(U + Z2))
Y1 <- rbinom(n = NN, size = 1, prob = expit(intercept + U + Z1 + X1))
Y2 <- rbinom(n = NN, size = 1, prob = expit(intercept + U + Z2 + X2))
# Select discordant pairs
discordant <- which(Y1!=Y2)
id <- rep(1:n, 2)
# Sample from discordant pairs
incl <- sample(x = discordant, size = n, replace = TRUE)
data <- data.frame(id = id, Y = c(Y1[incl], Y2[incl]), X = c(X1[incl], X2[incl]),
                   Z = c(Z1[incl], Z2[incl]))

# Estimation of the attributable fraction
AF.cc_match <- AF.cc(formula = Y ~ X + Z + X * Z, data = data,
                         exposure = "X", clusterid = "id", matched = TRUE)
summary(AF.cc_match)

Attributable fraction function for cohort sampling designs with time-to-event outcomes. NOTE! Deprecated function. Use `AFcoxph`.

Description

AF.ch estimates the model-based adjusted attributable fraction function for data from cohort sampling designs with time-to-event outcomes.

Usage

AF.ch(formula, data, exposure, ties = "breslow", times, clusterid)
AF.ch(formula, data, exposure, ties = "breslow", times, clusterid)

Arguments

`formula`	a formula object, with the response on the left of a ~ operator, and the terms on the right. The response must be a survival object as returned by the `Surv` function (`Surv`). The exposure and confounders should be specified as independent (right-hand side) variables. The time-to-event outcome should be specified by the survival object. The formula is used to fit a Cox proportional hazards model.
`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.
`exposure`	the name of the exposure variable as a string. The exposure must be binary (0/1) where unexposed is coded as 0.
`ties`	a character string specifying the method for tie handling. If there are no tied death times all the methods are equivalent. Uses the Breslow method by default.
`times`	a scalar or vector of time points specified by the user for which the attributable fraction function is estimated. If not specified the observed death times will be used.
`clusterid`	the name of the cluster identifier variable as a string, if data are clustered.

Details

Af.ch estimates the attributable fraction for a time-to-event outcome under the hypothetical scenario where a binary exposure X is eliminated from the population. The estimate is adjusted for confounders Z by the Cox proportional hazards model (coxph). Let the AF function be defined as

$AF=1-\frac{\{1-S_0(t)\}}{\{1-S(t)\}}$

where $S_0(t)$ denotes the counterfactual survival function for the event if the exposure would have been eliminated from the population at baseline and $S(t)$ denotes the factual survival function. If Z is sufficient for confounding control, then $S_0(t)$ can be expressed as $E_Z\{S(t\mid{X=0,Z })\}$ . The function uses Cox proportional hazards regression to estimate $S(t\mid{X=0,Z})$ , and the marginal sample distribution of Z to approximate the outer expectation (Sjölander and Vansteelandt, 2014). If clusterid is supplied, then a clustered sandwich formula is used in all variance calculations.

Value

`AF.est`	estimated attributable fraction function for every time point specified by `times`.
`AF.var`	estimated variance of `AF.est`. The variance is obtained by combining the delta methods with the sandwich formula.
`S.est`	estimated factual survival function; $S(t)$ .
`S.var`	estimated variance of `S.est`. The variance is obtained by the sandwich formula.
`S0.est`	estimated counterfactual survival function if exposure would be eliminated; $S_0(t)$ .
`S0.var`	estimated variance of `S0.est`. The variance is obtained by the sandwich formula.
`object`	the fitted model. Fitted using Cox proportional hazard, `coxph`.

Author(s)

Elisabeth Dahlqwist, Arvid Sjölander

References

Chen, L., Lin, D. Y., and Zeng, D. (2010). Attributable fraction functions for censored event times. Biometrika 97, 713-726.

Sjölander, A. and Vansteelandt, S. (2014). Doubly robust estimation of attributable fractions in survival analysis. Statistical Methods in Medical Research. doi: 10.1177/0962280214564003.

Examples

# Simulate a sample from a cohort sampling design with time-to-event outcome
expit <- function(x) 1 / (1 + exp( - x))
n <- 500
time <- c(seq(from = 0.2, to = 1, by = 0.2))
Z <- rnorm(n = n)
X <- rbinom(n = n, size = 1, prob = expit(Z))
Tim <- rexp(n = n, rate = exp(X + Z))
C <- rexp(n = n, rate = exp(X + Z))
Tobs <- pmin(Tim, C)
D <- as.numeric(Tobs < C)
#Ties created by rounding
Tobs <- round(Tobs, digits = 2)

# Example 1: non clustered data from a cohort sampling design with time-to-event outcomes
data <- data.frame(Tobs, D, X,  Z)

# Estimation of the attributable fraction
AF.ch_est <- AF.ch(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data,
                   exposure = "X", times = time)
summary(AF.ch_est)

# Example 2: clustered data from a cohort sampling design with time-to-event outcomes
# Duplicate observations in order to create clustered data
id <- rep(1:n, 2)
data <- data.frame(Tobs = c(Tobs, Tobs), D = c(D, D), X = c(X, X), Z = c(Z, Z), id = id)

# Estimation of the attributable fraction
AF.ch_clust <- AF.ch(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data,
                         exposure = "X", times = time, clusterid = "id")
summary(AF.ch_clust)
plot(AF.ch_clust, CI = TRUE)
# Simulate a sample from a cohort sampling design with time-to-event outcome
expit <- function(x) 1 / (1 + exp( - x))
n <- 500
time <- c(seq(from = 0.2, to = 1, by = 0.2))
Z <- rnorm(n = n)
X <- rbinom(n = n, size = 1, prob = expit(Z))
Tim <- rexp(n = n, rate = exp(X + Z))
C <- rexp(n = n, rate = exp(X + Z))
Tobs <- pmin(Tim, C)
D <- as.numeric(Tobs < C)
#Ties created by rounding
Tobs <- round(Tobs, digits = 2)

# Example 1: non clustered data from a cohort sampling design with time-to-event outcomes
data <- data.frame(Tobs, D, X,  Z)

# Estimation of the attributable fraction
AF.ch_est <- AF.ch(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data,
                   exposure = "X", times = time)
summary(AF.ch_est)

# Example 2: clustered data from a cohort sampling design with time-to-event outcomes
# Duplicate observations in order to create clustered data
id <- rep(1:n, 2)
data <- data.frame(Tobs = c(Tobs, Tobs), D = c(D, D), X = c(X, X), Z = c(Z, Z), id = id)

# Estimation of the attributable fraction
AF.ch_clust <- AF.ch(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data,
                         exposure = "X", times = time, clusterid = "id")
summary(AF.ch_clust)
plot(AF.ch_clust, CI = TRUE)

Attributable fraction for cross-sectional sampling designs. NOTE! Deprecated function. Use `AFglm`.

Description

AF.cs estimates the model-based adjusted attributable fraction for data from cross-sectional sampling designs.

Usage

AF.cs(formula, data, exposure, clusterid)
AF.cs(formula, data, exposure, clusterid)

Arguments

`formula`	an object of class "`formula`" (or one that can be coerced to that class): a symbolic description of the model used for adjusting for confounders. The exposure and confounders should be specified as independent (right-hand side) variables. The outcome should be specified as dependent (left-hand side) variable. The formula is used to object a logistic regression by `glm`.
`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.
`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.

Details

Af.cs 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 (glm). Let the AF be defined as

$AF=1-\frac{Pr(Y_0=1)}{Pr(Y=1)}$

Value

`AF.est`	estimated attributable fraction.
`AF.var`	estimated variance of `AF.est`. The variance is obtained by combining the delta method with the sandwich formula.
`P.est`	estimated factual proportion of cases; $Pr(Y=1)$ .
`P.var`	estimated variance of `P.est`. The variance is obtained by the sandwich formula.
`P0.est`	estimated counterfactual proportion of cases if exposure would be eliminated; $Pr(Y_0=1)$ .
`P0.var`	estimated variance of `P0.est`. The variance is obtained by the sandwich formula.
`object`	the fitted model. Fitted using logistic regression, `glm`.

Author(s)

Elisabeth Dahlqwist, Arvid Sjölander

References

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.

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)

# Estimation of the attributable fraction
AF.cs_est <- AF.cs(formula = Y ~ X + Z + X * Z, data = data, exposure = "X")
summary(AF.cs_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))

# Estimation of the attributable fraction
AF.cs_clust <- AF.cs(formula = Y ~ X + Z + X * Z, data = data,
                         exposure = "X", clusterid = "id")
summary(AF.cs_clust)
# 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)

# Estimation of the attributable fraction
AF.cs_est <- AF.cs(formula = Y ~ X + Z + X * Z, data = data, exposure = "X")
summary(AF.cs_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))

# Estimation of the attributable fraction
AF.cs_clust <- AF.cs(formula = Y ~ X + Z + X * Z, data = data,
                         exposure = "X", clusterid = "id")
summary(AF.cs_clust)

Attributable fraction estimation based on a conditional logistic regression model as a `clogit` object (commonly used for matched case-control sampling designs).

Description

AFclogit estimates the model-based adjusted attributable fraction from a conditional logistic regression model in form of a clogit object. This model is model is commonly used for data from matched case-control sampling designs.

Usage

AFclogit(object, data, exposure, clusterid)
AFclogit(object, data, exposure, clusterid)

Arguments

`object`	a fitted conditional logistic regression model object of class "`clogit`".
`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.
`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. Because conditional logistic regression is only used for clustered data, this argument must be supplied.

Details

AFclogit 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 conditional logistic regression. The estimation assumes that the outcome is rare so that the risk ratio can be approximated by the odds ratio, for details see Bruzzi et. al. Let the AF be defined as

$AF = 1 - \frac{Pr(Y_0=1)}{Pr(Y = 1)}$

$Pr(Y_0=1)=E_Z\{Pr(Y=1\mid{X}=0,Z)\}.$

Using Bayes' theorem this implies that the AF can be expressed as

$AF = 1-\frac{E_Z\{Pr(Y=1\mid X=0,Z)\}}{Pr(Y=1)}=1-E_Z\{RR^{-X}(Z)\mid{Y = 1}\}$

where $RR(Z)$ is the risk ratio

$\frac{Pr(Y=1\mid{X=1,Z})}{Pr(Y=1\mid{X=0,Z})}.$

Moreover, the risk ratio can be approximated by the odds ratio if the outcome is rare. Thus,

$AF \approx 1 - E_Z\{OR^{-X}(Z)\mid{Y = 1}\}.$

The odds ratio is estimated by conditional logistic regression. The function gee in the drgee package is used to get the score contributions for each cluster and the hessian. A clustered sandwich formula is used in the variance calculation.

Value

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.

log.or

a vector of the estimated log odds ratio for every individual. log.or contains the estimated coefficient for the exposure variable X for every level of the confounder Z as specified by the user in the formula. If the model to be estimated is

$logit\{Pr(Y=1|X,Z)\} = \alpha+\beta{X}+\gamma{Z}$

then log.or is the estimate of $\beta$ . If the model to be estimated is

$logit\{Pr(Y=1|X,Z)\}=\alpha+\beta{X}+\gamma{Z}+\psi{XZ}$

then log.odds is the estimate of $\beta + \psi{Z}$ .

Author(s)

Elisabeth Dahlqwist, Arvid Sjölander

References

Examples

expit <- function(x) 1 / (1 + exp( - x))
NN <- 1000000
n <- 500

# Example 1: matched case-control
# Duplicate observations in order to create a matched data sample
# Create an unobserved confounder U common for each pair of individuals
intercept <- -6
U  <- rnorm(n = NN)
Z1 <- rnorm(n = NN)
Z2 <- rnorm(n = NN)
X1 <- rbinom(n = NN, size = 1, prob = expit(U + Z1))
X2 <- rbinom(n = NN, size = 1, prob = expit(U + Z2))
Y1 <- rbinom(n = NN, size = 1, prob = expit(intercept + U + Z1 + X1))
Y2 <- rbinom(n = NN, size = 1, prob = expit(intercept + U + Z2 + X2))
# Select discordant pairs
discordant <- which(Y1!=Y2)
id <- rep(1:n, 2)
# Sample from discordant pairs
incl <- sample(x = discordant, size = n, replace = TRUE)
data <- data.frame(id = id, Y = c(Y1[incl], Y2[incl]), X = c(X1[incl], X2[incl]),
                   Z = c(Z1[incl], Z2[incl]))

# Fit a clogit object
fit <- clogit(formula = Y ~ X + Z + X * Z + strata(id), data = data)

# Estimate the attributable fraction from the fitted conditional logistic regression
AFclogit_est <- AFclogit(fit, data, exposure = "X", clusterid="id")
summary(AFclogit_est)
expit <- function(x) 1 / (1 + exp( - x))
NN <- 1000000
n <- 500

# Example 1: matched case-control
# Duplicate observations in order to create a matched data sample
# Create an unobserved confounder U common for each pair of individuals
intercept <- -6
U  <- rnorm(n = NN)
Z1 <- rnorm(n = NN)
Z2 <- rnorm(n = NN)
X1 <- rbinom(n = NN, size = 1, prob = expit(U + Z1))
X2 <- rbinom(n = NN, size = 1, prob = expit(U + Z2))
Y1 <- rbinom(n = NN, size = 1, prob = expit(intercept + U + Z1 + X1))
Y2 <- rbinom(n = NN, size = 1, prob = expit(intercept + U + Z2 + X2))
# Select discordant pairs
discordant <- which(Y1!=Y2)
id <- rep(1:n, 2)
# Sample from discordant pairs
incl <- sample(x = discordant, size = n, replace = TRUE)
data <- data.frame(id = id, Y = c(Y1[incl], Y2[incl]), X = c(X1[incl], X2[incl]),
                   Z = c(Z1[incl], Z2[incl]))

# Fit a clogit object
fit <- clogit(formula = Y ~ X + Z + X * Z + strata(id), data = data)

# Estimate the attributable fraction from the fitted conditional logistic regression
AFclogit_est <- AFclogit(fit, data, exposure = "X", clusterid="id")
summary(AFclogit_est)

Attributable fraction function based on a Cox Proportional Hazard regression model as a `coxph` object (commonly used for cohort sampling designs with time-to-event outcomes).

Description

AFcoxph estimates the model-based adjusted attributable fraction function from a Cox Proportional Hazard regression model in form of a coxph object. This model is commonly used for data from cohort sampling designs with time-to-event outcomes.

Usage

AFcoxph(object, data, exposure, times, clusterid)
AFcoxph(object, data, exposure, times, clusterid)

Arguments

`object`	a fitted Cox Proportional Hazard regression model object of class "`coxph`". Method for handling ties must be breslow since this is assumed in the calculation of the standard errors. No special terms such as `cluster`, `strata` and `tt` is allowed in the formula for the fitted object.
`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.
`exposure`	the name of the exposure variable as a string. The exposure must be binary (0/1) where unexposed is coded as 0.
`times`	a scalar or vector of time points specified by the user for which the attributable fraction function is estimated. If not specified the observed event times will be used.
`clusterid`	the name of the cluster identifier variable as a string, if data are clustered. Cluster robust standard errors will be calculated.

Details

AFcoxph estimates the attributable fraction for a time-to-event outcome under the hypothetical scenario where a binary exposure X is eliminated from the population. The estimate is adjusted for confounders Z by the Cox proportional hazards model (coxph). Let the AF function be defined as

$AF=1-\frac{\{1-S_0(t)\}}{\{1-S(t)\}}$

where $S_0(t)$ denotes the counterfactual survival function for the event if the exposure would have been eliminated from the population at baseline and $S(t)$ denotes the factual survival function. If Z is sufficient for confounding control, then $S_0(t)$ can be expressed as $E_Z\{S(t\mid{X=0,Z })\}$ . The function uses a fitted Cox proportional hazards regression to estimate $S(t\mid{X=0,Z})$ , and the marginal sample distribution of Z to approximate the outer expectation (Sjölander and Vansteelandt, 2014). If clusterid is supplied, then a clustered sandwich formula is used in all variance calculations.

Value

`AF.est`	estimated attributable fraction function for every time point specified by `times`.
`AF.var`	estimated variance of `AF.est`. The variance is obtained by combining the delta methods with the sandwich formula.
`S.est`	estimated factual survival function; $S(t)$ .
`S.var`	estimated variance of `S.est`. The variance is obtained by the sandwich formula.
`S0.est`	estimated counterfactual survival function if exposure would be eliminated; $S_0(t)$ .
`S0.var`	estimated variance of `S0.est`. The variance is obtained by the sandwich formula.

Author(s)

Elisabeth Dahlqwist, Arvid Sjölander

References

Chen, L., Lin, D. Y., and Zeng, D. (2010). Attributable fraction functions for censored event times. Biometrika 97, 713-726.

Sjölander, A. and Vansteelandt, S. (2014). Doubly robust estimation of attributable fractions in survival analysis. Statistical Methods in Medical Research. doi: 10.1177/0962280214564003.

Examples

# Simulate a sample from a cohort sampling design with time-to-event outcome
expit <- function(x) 1 / (1 + exp( - x))
n <- 500
time <- c(seq(from = 0.2, to = 1, by = 0.2))
Z <- rnorm(n = n)
X <- rbinom(n = n, size = 1, prob = expit(Z))
Tim <- rexp(n = n, rate = exp(X + Z))
C <- rexp(n = n, rate = exp(X + Z))
Tobs <- pmin(Tim, C)
D <- as.numeric(Tobs < C)
#Ties created by rounding
Tobs <- round(Tobs, digits = 2)

# Example 1: non clustered data from a cohort sampling design with time-to-event outcomes
data <- data.frame(Tobs, D, X,  Z)

# Fit a Cox PH regression model
fit <- coxph(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data, ties="breslow")

# Estimate the attributable fraction from the fitted Cox PH regression model
AFcoxph_est <- AFcoxph(fit, data=data, exposure ="X", times = time)
summary(AFcoxph_est)

# Example 2: clustered data from a cohort sampling design with time-to-event outcomes
# Duplicate observations in order to create clustered data
id <- rep(1:n, 2)
data <- data.frame(Tobs = c(Tobs, Tobs), D = c(D, D), X = c(X, X), Z = c(Z, Z), id = id)

# Fit a Cox PH regression model
fit <- coxph(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data, ties="breslow")

# Estimate the attributable fraction from the fitted Cox PH regression model
AFcoxph_clust <- AFcoxph(object = fit, data = data,
                         exposure = "X", times = time, clusterid = "id")
summary(AFcoxph_clust)
plot(AFcoxph_clust, CI = TRUE)

# Estimate the attributable fraction from the fitted Cox PH regression model, time unspecified
AFcoxph_clust_no_time <- AFcoxph(object = fit, data = data,
                         exposure = "X", clusterid = "id")
summary(AFcoxph_clust_no_time)
plot(AFcoxph_clust, CI = TRUE)
# Simulate a sample from a cohort sampling design with time-to-event outcome
expit <- function(x) 1 / (1 + exp( - x))
n <- 500
time <- c(seq(from = 0.2, to = 1, by = 0.2))
Z <- rnorm(n = n)
X <- rbinom(n = n, size = 1, prob = expit(Z))
Tim <- rexp(n = n, rate = exp(X + Z))
C <- rexp(n = n, rate = exp(X + Z))
Tobs <- pmin(Tim, C)
D <- as.numeric(Tobs < C)
#Ties created by rounding
Tobs <- round(Tobs, digits = 2)

# Example 1: non clustered data from a cohort sampling design with time-to-event outcomes
data <- data.frame(Tobs, D, X,  Z)

# Fit a Cox PH regression model
fit <- coxph(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data, ties="breslow")

# Estimate the attributable fraction from the fitted Cox PH regression model
AFcoxph_est <- AFcoxph(fit, data=data, exposure ="X", times = time)
summary(AFcoxph_est)

# Example 2: clustered data from a cohort sampling design with time-to-event outcomes
# Duplicate observations in order to create clustered data
id <- rep(1:n, 2)
data <- data.frame(Tobs = c(Tobs, Tobs), D = c(D, D), X = c(X, X), Z = c(Z, Z), id = id)

# Fit a Cox PH regression model
fit <- coxph(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data, ties="breslow")

# Estimate the attributable fraction from the fitted Cox PH regression model
AFcoxph_clust <- AFcoxph(object = fit, data = data,
                         exposure = "X", times = time, clusterid = "id")
summary(AFcoxph_clust)
plot(AFcoxph_clust, CI = TRUE)

# Estimate the attributable fraction from the fitted Cox PH regression model, time unspecified
AFcoxph_clust_no_time <- AFcoxph(object = fit, data = data,
                         exposure = "X", clusterid = "id")
summary(AFcoxph_clust_no_time)
plot(AFcoxph_clust, CI = TRUE)

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)
AFglm(object, data, exposure, clusterid, case.control = FALSE)

Arguments

`object`	a fitted logistic regression model object of class "`glm`".
`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.
`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 `TRUE` if the data is from a non-matched case control study. By default `case.control` is set to `FALSE` which is used for cross-sectional sampling designs.

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

$AF=1-\frac{Pr(Y_0=1)}{Pr(Y=1)}$

where $Pr(Y_0=1)$ denotes the counterfactual probability of the outcome if the exposure would have been eliminated from the population and $Pr(Y = 1)$ denotes the factual probability of the outcome. If Z is sufficient for confounding control, then $Pr(Y_0=1)$ can be expressed as $E_Z\{Pr(Y=1\mid{X=0,Z})\}.$ The function uses logistic regression to estimate $Pr(Y=1\mid{X=0,Z})$ , 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)

$AF = 1 - \frac{Pr(Y_0=1)}{Pr(Y = 1)}$

$Pr(Y_0=1)=E_Z\{Pr(Y=1\mid{X}=0,Z)\}.$

Using Bayes' theorem this implies that the AF can be expressed as

$AF = 1-\frac{E_Z\{Pr(Y=1\mid X=0,Z)\}}{Pr(Y=1)}=1-E_Z\{RR^{-X}(Z)\mid{Y = 1}\}$

where $RR(Z)$ is the risk ratio

$\frac{Pr(Y=1\mid{X=1,Z})}{Pr(Y=1\mid{X=0,Z})}.$

Moreover, the risk ratio can be approximated by the odds ratio if the outcome is rare. Thus,

$AF \approx 1 - E_Z\{OR^{-X}(Z)\mid{Y = 1}\}.$

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 `AF.est`. The variance is obtained by combining the delta method with the sandwich formula.
`P.est`	estimated factual proportion of cases; $Pr(Y=1)$ . Returned by default when `case.control = FALSE`.
`P.var`	estimated variance of `P.est`. The variance is obtained by the sandwich formula. Returned by default when `case.control = FALSE`.
`P0.est`	estimated counterfactual proportion of cases if exposure would be eliminated; $Pr(Y_0=1)$ . Returned by default when `case.control = FALSE`.
`P0.var`	estimated variance of `P0.est`. The variance is obtained by the sandwich formula. Returned by default when `case.control = FALSE`.
`log.or`	a vector of the estimated log odds ratio for every individual. `log.or` contains the estimated coefficient for the exposure variable `X` for every level of the confounder `Z` as specified by the user in the formula. If the model to be estimated is $logit\{Pr(Y=1\|X,Z)\} = \alpha+\beta{X}+\gamma{Z}$ then `log.or` is the estimate of $\beta$ . If the model to be estimated is $logit\{Pr(Y=1\|X,Z)\}=\alpha+\beta{X}+\gamma{Z}+\psi{XZ}$ then `log.odds` is the estimate of $\beta + \psi{Z}$ . Only returned if argument `case.control` is set to `TRUE`.

Author(s)

Elisabeth Dahlqwist, Arvid Sjölander

References

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.

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)
# 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)

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

Description

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.

Usage

AFivglm(object, data)
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.

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

$AF=1-\frac{Pr(Y_0=1)}{Pr(Y=1)}$

where $Pr(Y_0=1)$ 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 $Pr(Y_0=1)$ can be estimated.

Value

`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.

Author(s)

Elisabeth Dahlqwist, Arvid Sjölander

References

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

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)
# 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)

Attributable fraction function based on a Weibull gamma-frailty model as a `parfrailty` object (commonly used for cohort sampling family designs with time-to-event outcomes).

Description

AFparfrailty estimates the model-based adjusted attributable fraction function from a shared Weibull gamma-frailty model in form of a parfrailty object. This model is commonly used for data from cohort sampling familty designs with time-to-event outcomes.

Usage

AFparfrailty(object, data, exposure, times, clusterid)
AFparfrailty(object, data, exposure, times, clusterid)

Arguments

`object`	a fitted Weibull gamma-parfrailty object of class "`parfrailty`".
`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.
`exposure`	the name of the exposure variable as a string. The exposure must be binary (0/1) where unexposed is coded as 0.
`times`	a scalar or vector of time points specified by the user for which the attributable fraction function is estimated. If not specified the observed death times will be used.
`clusterid`	the name of the cluster identifier variable as a string, if data are clustered.

Details

AFparfrailty estimates the attributable fraction for a time-to-event outcome under the hypothetical scenario where a binary exposure X is eliminated from the population. The estimate is adjusted for confounders Z by the shared frailty model (parfrailty). The baseline hazard is assumed to follow a Weibull distribution and the unobserved shared frailty effects U are assumed to be gamma distributed. Let the AF function be defined as

$AF=1-\frac{\{1-S_0(t)\}}{\{1-S(t)\}}$

where $S_0(t)$ denotes the counterfactual survival function for the event if the exposure would have been eliminated from the population at baseline and $S(t)$ denotes the factual survival function. If Z and U are sufficient for confounding control, then $S_0(t)$ can be expressed as $E_Z\{S(t\mid{X=0,Z })\}$ . The function uses a fitted Weibull gamma-frailty model to estimate $S(t\mid{X=0,Z})$ , and the marginal sample distribution of Z to approximate the outer expectation. A clustered sandwich formula is used in all variance calculations.

Value

`AF.est`	estimated attributable fraction function for every time point specified by `times`.
`AF.var`	estimated variance of `AF.est`. The variance is obtained by combining the delta methods with the sandwich formula.
`S.est`	estimated factual survival function; $S(t)$ .
`S.var`	estimated variance of `S.est`. The variance is obtained by the sandwich formula.
`S0.est`	estimated counterfactual survival function if exposure would be eliminated; $S_0(t)$ .
`S0.var`	estimated variance of `S0.est`. The variance is obtained by the sandwich formula.

Author(s)

Elisabeth Dahlqwist, Arvid Sjölander

Examples

# Example 1: clustered data with frailty U
expit <- function(x) 1 / (1 + exp( - x))
n <- 100
m <- 2
alpha <- 1.5
eta <- 1
phi <- 0.5
beta <- 1
id <- rep(1:n,each=m)
U <- rep(rgamma(n, shape = 1 / phi, scale = phi), each = m)
Z <- rnorm(n * m)
X <- rbinom(n * m, size = 1, prob = expit(Z))
# Reparametrize scale as in rweibull function
weibull.scale <- alpha / (U * exp(beta * X)) ^ (1 / eta)
t <- rweibull(n * m, shape = eta, scale = weibull.scale)

# Right censoring
c <- runif(n * m, 0, 10)
delta <- as.numeric(t < c)
t <- pmin(t, c)

data <- data.frame(t, delta, X, Z, id)

# Fit a parfrailty object
library(stdReg)
fit <- parfrailty(formula = Surv(t, delta) ~ X + Z + X * Z, data = data, clusterid = "id")
summary(fit)

# Estimate the attributable fraction from the fitted frailty model

time <- c(seq(from = 0.2, to = 1, by = 0.2))

AFparfrailty_est <- AFparfrailty(object = fit, data = data, exposure = "X",
                                  times = time, clusterid = "id")
summary(AFparfrailty_est)
plot(AFparfrailty_est, CI = TRUE, ylim=c(0.1,0.7))
# Example 1: clustered data with frailty U
expit <- function(x) 1 / (1 + exp( - x))
n <- 100
m <- 2
alpha <- 1.5
eta <- 1
phi <- 0.5
beta <- 1
id <- rep(1:n,each=m)
U <- rep(rgamma(n, shape = 1 / phi, scale = phi), each = m)
Z <- rnorm(n * m)
X <- rbinom(n * m, size = 1, prob = expit(Z))
# Reparametrize scale as in rweibull function
weibull.scale <- alpha / (U * exp(beta * X)) ^ (1 / eta)
t <- rweibull(n * m, shape = eta, scale = weibull.scale)

# Right censoring
c <- runif(n * m, 0, 10)
delta <- as.numeric(t < c)
t <- pmin(t, c)

data <- data.frame(t, delta, X, Z, id)

# Fit a parfrailty object
library(stdReg)
fit <- parfrailty(formula = Surv(t, delta) ~ X + Z + X * Z, data = data, clusterid = "id")
summary(fit)

# Estimate the attributable fraction from the fitted frailty model

time <- c(seq(from = 0.2, to = 1, by = 0.2))

AFparfrailty_est <- AFparfrailty(object = fit, data = data, exposure = "X",
                                  times = time, clusterid = "id")
summary(AFparfrailty_est)
plot(AFparfrailty_est, CI = TRUE, ylim=c(0.1,0.7))

Birthweight data clustered on the mother.

Description

This dataset is borrowed from "An introduction to Stata for health reserachers" (Juul and Frydenberg, 2010). The dataset contains data on 189 mothers who have given birth to one or several children. In total, the dataset contains data on 487 births.

Usage

data(clslowbwt)
data(clslowbwt)

Format

The dataset is structured so that each row corresponds to one birth/child. It contains the following variables:

id: the identification number of the mother.
birth: the number of the birth, i.e. "1" for the mother's first birth, "2" for the mother's second birth etc.
smoke: a categorical variable indicating if the mother is a smoker or not with levels "0. No" and "1. Yes".
race: the race of the mother with levels "1. White", "2. Black" or "3. Other".
age: the age of the mother at childbirth.
lwt: weight of the mother at last menstruational period (in pounds).
bwt: birthweight of the newborn.
low: a categorical variable indicating if the newborn is categorized as a low birthweight baby (<2500 grams) or not with levels "0. No" and "1. Yes".
smoker: a numeric indicator if the mother is a smoker or not. Recoded version of the variable "smoke" where "0.No" is recoded as "0" and "1.Yes" is recoded as "1".
lbw: a numeric indicator of whether the newborn is categorized as a low birthweight baby (<2500 grams) or not. Recoded version of the variable "low" where "0.No" is recoded as "0" and "1.Yes" is recoded as "1".

The following changes have been made to the original data in Juul & Frydenberg (2010):

- The variable "low" is recoded into the numeric indicator variable "lbw":

clslowbwt$lbw <- as.numeric(clslowbwt$low == "1. Yes")

- The variable "smoke" is recoded into the numeric indicator variable "smoker":

clslowbwt$smoker <- as.numeric(clslowbwt$smoke == "1. Yes")

References

Juul, Svend & Frydenberg, Morten (2010). An introduction to Stata for health researchers, Texas, Stata press, 2010 (Third edition).

http://www.stata-press.com/data/ishr3.html

Plot function for objects of class "`AF`" from the function `AFcoxph` or `AFparfrailty`.

Description

Creates a simple scatterplot for the AF function with time sequence (specified by the user as times in the AFcoxph function) on the x-axis and the AF function estimate on the y-axis.

Usage

## S3 method for class 'AF'
plot(x, CI = TRUE, confidence.level, CI.transform, xlab,
  main, ylim, ...)
## S3 method for class 'AF'
plot(x, CI = TRUE, confidence.level, CI.transform, xlab,
  main, ylim, ...)

Arguments

`x`	an object of class `AF` from the `AFcoxph` or `AFparfrailty` function.
`CI`	if TRUE confidence intervals are estimated and ploted in the graph.
`confidence.level`	user-specified confidence level for the confidence intervals. If not specified it defaults to 95 percent. Should be specified in decimals such as 0.95 for 95 percent.
`CI.transform`	user-specified transformation of the Wald confidence interval(s). Options are `untransformed`, `log` and `logit`. If not specified untransformed will be calculated.
`xlab`	label on the x-axis. If not specified the label "Time" will be displayed.
`main`	main title of the plot. If not specified the lable "Estimate of the attributable fraction function" will be displayed.
`ylim`	limits on the y-axis of the plot. If not specified the minimum value of the lower bound of the confidence interval will be used as the minimal value and the maximum value of the upper bound of the confidence interval will be used as the maximum of y-axis of the plot.
`...`	further arguments to be passed to the plot function. See `plot`.

Author(s)

Elisabeth Dahlqwist, Arvid Sjölander

Cohort study on breast cancer patients from the Netherlands.

Description

This dataset is borrowed from "Flexible parametric survival analysis using Stata: beyond the Cox model" (Roystone and Lambert, 2011). It contains follow-up data on 2982 woman with breast cancer who have gone through breast surgery. The women are followed from the time of surgery until death, relapse or censoring.

Usage

data(rott2)
data(rott2)

Format

The dataset rott2 contains the following variables:

pid: patient ID number.
year: year of breast surgery (i.e. year of enrollment into the study), between the years 1978-1993.
rf: relapse free interval measured in months.
rfi: relapse indicator.
m: metastasis free.
mfi: metastasis status.
os: overall survival
osi: overall survival indicator
age: age at surgery measured in years.
meno: menopausal status with levels "pre" and "post".
size: tumor size in three classes: <=20mm, >20-50mmm and >50mm.
grade: differentiation grade with levels 2 or 3.
pr: progesterone receptors, fmol/l.
er: oestrogen receptors, fmol/l.
nodes: the number of positive lymph nodes.
hormon: hormonal therapy with levels "no" and "yes".
chemo: categorical variable indicating whether the patient recieved chemotheraphy or not, with levels "no" and "yes".
recent: a numeric indicator of whether the tumor was discovered recently with levels "1978-87" and "1988-93".
no.chemo: a numerical indicator of whether the patient did not recieved chemotherapy. Recoded version of "chemo" where "yes" is recoded as 0 and "no" is recoded as 1.

The following changes have been made to the original data in Roystone and Lambert (2011):

- The variable "chemo" is recoded into the numeric indicator variable "no.chemo":

rott22$no.chemo <- as.numeric(rott2$chemo == "no")

The follwing variables have been removed from the original dataset: enodes, pr_1, enodes_1, _st, _d, _t, _t0 since they are recodings of some existing variables which are not used in this analysis.

References

Royston, Patrick & Lambert, Paul. C (2011). Flexible parametric survival analysis using Stata: beyond the Cox model. College Station, Texas, U.S, Stata press.

http://www.stata-press.com/data/fpsaus.html

Case-control study on oesophageal cancer in Chinese Singapore men.

Description

This dataset is borrowed from "Aetiological factors in oesophageal cancer in Singapore Chinese" by De Jong UW, Breslow N, Hong JG, Sridharan M, Shanmugaratnam K (1974).

Usage

data(singapore)
data(singapore)

Format

The dataset contains the following variables:

Age: age of the patient.
Dial: dialect group where 1 represent "Hokhien/Teochew" and 0 represent "Cantonese/Other".
Samsu: a numeric indicator of whether the patient consumes Samsu wine or not.
Cigs: number of cigarettes smoked per day.
Bev: number of beverage at "burning hot" temperatures ranging between 0 to 3 different drinks per day.
Everhotbev: a numeric indicator of whether the patients ever drinks "burning hot beverage" or not. Recoded from the variable "Bev".
Set: matched set identification number.
CC: a numeric variable where 1 represent if the patient is a case, 2 represent if the patient is a control from the same ward as the case and 3 represent if the patient is control from orthopedic hospital.
Oesophagealcancer: a numeric indicator variable of whether the patient is a case of oesophageal cancer or not.

The following changes have been made to the data from the original data in De Jong UW (1974):

- The variable "Bev" is recoded into the numeric indicator variable "Everhotbev":

singapore$Everhotbev <- ifelse(singapore$Bev >= 1, 1, 0)

References

De Jong UW, Breslow N, Hong JG, Sridharan M, Shanmugaratnam K. (1974). Aetiological factors in oesophageal cancer in Singapore Chinese. Int J Cancer Mar 15;13(3), 291-303.

http://faculty.washington.edu/heagerty/Courses/b513/WEB2002/datasets.html

Summary function for objects of class "`AF`".

Description

Gives a summary of the AF estimate(s) including z-value, p-value and confidence interval(s).

Usage

## S3 method for class 'AF'
summary(object, digits = max(3L, getOption("digits") - 3L),
  confidence.level, CI.transform, ...)
## S3 method for class 'AF'
summary(object, digits = max(3L, getOption("digits") - 3L),
  confidence.level, CI.transform, ...)

Arguments

`object`	an object of class `AF` from `AFglm`, `AFcoxph`, `AFclogit`, `AFparfrailty` or `AFivglm` functions.
`digits`	maximum number of digits.
`confidence.level`	user-specified confidence level for the confidence intervals. If not specified it defaults to 95 percent. Should be specified in decimals such as 0.95 for 95 percent.
`CI.transform`	user-specified transformation of the Wald confidence interval(s). Options are `untransformed`, `log` and `logit`. If not specified untransformed will be calculated.
`...`	further arguments to be passed to the summary function. See `summary`.

Author(s)

Elisabeth Dahlqwist, Arvid Sjölander

Package 'AF'

Help Index

Attributable fraction for mached and non-matched case-control sampling designs. NOTE! Deprecated function. Use AFglm (for unmatched case-control studies) or AFclogit (for matched case-control studies).

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Attributable fraction function for cohort sampling designs with time-to-event outcomes. NOTE! Deprecated function. Use AFcoxph.

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Attributable fraction for cross-sectional sampling designs. NOTE! Deprecated function. Use AFglm.

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Attributable fraction estimation based on a conditional logistic regression model as a clogit object (commonly used for matched case-control sampling designs).

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Attributable fraction function based on a Cox Proportional Hazard regression model as a coxph object (commonly used for cohort sampling designs with time-to-event outcomes).

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Attributable fraction estimation based on a logistic regression model from a glm object (commonly used for cross-sectional or case-control sampling designs).

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

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

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Attributable fraction function based on a Weibull gamma-frailty model as a parfrailty object (commonly used for cohort sampling family designs with time-to-event outcomes).

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Attributable fraction for mached and non-matched case-control sampling designs. NOTE! Deprecated function. Use `AFglm` (for unmatched case-control studies) or `AFclogit` (for matched case-control studies).

Attributable fraction function for cohort sampling designs with time-to-event outcomes. NOTE! Deprecated function. Use `AFcoxph`.

Attributable fraction for cross-sectional sampling designs. NOTE! Deprecated function. Use `AFglm`.

Attributable fraction estimation based on a conditional logistic regression model as a `clogit` object (commonly used for matched case-control sampling designs).

Attributable fraction function based on a Cox Proportional Hazard regression model as a `coxph` object (commonly used for cohort sampling designs with time-to-event outcomes).

Attributable fraction estimation based on a logistic regression model from a `glm` object (commonly used for cross-sectional or case-control sampling designs).

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

Attributable fraction function based on a Weibull gamma-frailty model as a `parfrailty` object (commonly used for cohort sampling family designs with time-to-event outcomes).

Plot function for objects of class "`AF`" from the function `AFcoxph` or `AFparfrailty`.

Summary function for objects of class "`AF`".