AF.cc {AF} | R Documentation |

`AFglm`

(for unmatched case-control studies) or `AFclogit`

(for matched case-control studies).`AF.cc`

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

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

`formula` |
an 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 (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 ( |

`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.

`AF.est` |
estimated attributable fraction. |

`AF.var` |
estimated variance of |

`log.or` |
a vector of the estimated log odds ratio for every individual.
then
then |

`object` |
the fitted model. Fitted using logistic regression, |

Elisabeth Dahlqwist, Arvid Sjölander

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.

The new and more general version of the function: `AFglm`

for non-matched and `AFclogit`

for matched case-control sampling designs. `glm`

and `gee`

used for fitting the logistic regression model (for non-matched case-control) and the conditional logistic regression model (for matched case-control).

```
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)
```

[Package *AF* version 0.1.5 Index]