# pif: Potential Impact Fraction In pifpaf: Potential Impact Fraction and Population Attributable Fraction for Cross-Sectional Data

## Description

Function for estimating the Potential Impact Fraction `pif` from a cross-sectional sample of the exposure `X` with known Relative Risk function `rr` with parameter `theta`, where the Potential Impact Fraction is given by:

PIF = (mean(rr(X; theta)) - mean(rr(cft(X);theta)))/mean(rr(X; theta)) .

## Usage

 ```1 2 3 4 5``` ```pif(X, thetahat, rr, cft = NA, method = "empirical", weights = rep(1/nrow(as.matrix(X)), nrow(as.matrix(X))), Xvar = var(X), deriv.method.args = list(), deriv.method = "Richardson", adjust = 1, n = 512, ktype = "gaussian", bw = "SJ", check_exposure = TRUE, check_integrals = TRUE, check_rr = TRUE, is_paf = FALSE) ```

## Arguments

 `X` Random sample (`data.frame`) which includes exposure and covariates or sample `mean` if `"approximate"` method is selected. `thetahat` Asymptotically consistent or Fisher consistent estimator (`vector`) of `theta` for the Relative Risk function. `rr` `function` for Relative Risk which uses parameter `theta`. The order of the parameters should be `rr(X, theta)`. **Optional** `cft` Function `cft(X)` for counterfactual. Leave empty for the Population Attributable Fraction `paf` where counterfactual is that of a theoretical minimum risk exposure `X0` such that `rr(X0,theta) = 1`. `method` Either `"empirical"` (default), `"kernel"` or `"approximate"`. `weights` Normalized survey `weights` for the sample `X`. `Xvar` Variance of exposure levels (for `"approximate"` method). `deriv.method.args` `method.args` for `hessian` (for `"approximate"` method). `deriv.method` `method` for `hessian`. Don't change this unless you know what you are doing (for `"approximate"` method). `adjust` Adjust bandwith parameter (for `"kernel"` method) from `density`. `n` Number of equally spaced points at which the density (for `"kernel"` method) is to be estimated (see `density`). `ktype` `kernel` type: `"gaussian"`, `"epanechnikov"`, `"rectangular"`, `"triangular"`, `"biweight"`, `"cosine"`, `"optcosine"` (for `"kernel"` method). Additional information on kernels in `density`. `bw` Smoothing bandwith parameter (for `"kernel"` method) from `density`. Default `"SJ"`. `check_exposure` Check exposure `X` is positive and numeric. `check_integrals` Check that counterfactual of theoretical minimum risk exposure and relative risk's expected values are well defined for this scenario. `check_rr` Check that Relative Risk function `rr` equals `1` when evaluated at `0`. `is_paf` Boolean forcing evaluation of `paf`. This forces the `pif` function ignore the inputed counterfactual and set the relative risk to the theoretical minimum risk value of `1`.

## Details

The `"empirical"` method estimates the `pif` by

PIF = 1 - weighted.mean(rr(cft(X), theta), weights)/ weighted.mean(rr(cft(X), theta), weights).

The `"kernel"` method approximates the `density` of the exposure `X` and estimates its expected value from that approximation:

PIF = 1 - integrate(rr(cft(X), theta)*f(x))/integrate(rr(X, theta)*f(x)).

The `"approximate"` method conducts a Laplace approximation of the `pif`. Additional information on the methods is dicussed in the package's vignette: `browseVignettes("pifpaf")`.

In practice `"approximate"` method should be the last choice. Simulations have shown that `"empirical"`'s convergence is faster than `"kernel"` for most functions. In addition, the scope of `"kernel"` is limited as it does not work with multivariate exposure data `X`.

## Value

pif Estimate of Potential Impact Fraction.

## Note

For more information on kernels see `density`.

Do not use the `\$` operator when using `"approximate"` `method`.

## Author(s)

Rodrigo Zepeda-Tello [email protected]

## References

Vander Hoorn, S., Ezzati, M., Rodgers, A., Lopez, A. D., & Murray, C. J. (2004). Estimating attributable burden of disease from exposure and hazard data. Comparative quantification of health risks: global and regional burden of disease attributable to selected major risk factors. Geneva: World Health Organization, 2129-40.

See `pif.confidence` for confidence interval estimation, and `paf` for Population Attributable Fraction estimation.
Sensitivity analysis plots can be done with `pif.plot`, `pif.sensitivity`, and `pif.heatmap`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236``` ```#Example 1: Exponential Relative Risk #-------------------------------------------- set.seed(18427) X <- data.frame(Exposure = rnorm(100,3,1)) thetahat <- 0.12 rr <- function(X, theta){exp(theta*X)} #Without specifying counterfactual pif matches paf pif(X, thetahat, rr) paf(X, thetahat, rr) #Same example with kernel method pif(X, thetahat, rr, method = "kernel") #Same example with approximate method Xmean <- data.frame(Exposure = mean(X[,"Exposure"])) Xvar <- var(X[,"Exposure"]) pif(Xmean, thetahat, rr, method = "approximate", Xvar = Xvar) #Same example considering counterfactual of halving exposure cft <- function(X){ 0.5*X } pif(X, thetahat, rr, cft, method = "empirical") #Example 2: Linear Relative Risk #-------------------------------------------- set.seed(18427) X <- data.frame(Exposure = rbeta(100,3,1)) thetahat <- 0.12 rr <- function(X, theta){theta*X + 1} cft <- function(X){ 0.5*X } weights <- runif(100) normalized_weights <- weights/sum(weights) pif(X, thetahat, rr, cft, weights = normalized_weights) #Same example with more complex counterfactual that reduces #only the values > 0.75 are halved cft <- function(X){ #Indentify the ones with "a lot" of exposure: where_excess_exposure <- which(X[,"Exposure"] > 0.75) #Halve their exposure X[where_excess_exposure, "Exposure"] <- X[where_excess_exposure, "Exposure"]/2 return(X) } pif(X, thetahat, rr, cft, weights = normalized_weights) #Example 3: Multivariate Linear Relative Risk #-------------------------------------------- set.seed(18427) X1 <- rnorm(100,4,1) X2 <- rnorm(100,2,0.4) X <- data.frame(Exposure = X1, Covariate = X2) thetahat <- c(0.12, 0.03) #When creating relative risks and counterfactuals avoid using \$ operator #as it doesn't work under approximate method rr_not <- function(X, theta){ exp(theta[1]*X\$Exposure + theta[2]*X\$Covariate) } rr_better <- function(X, theta){ exp(theta[1]*X[,"Exposure"] + theta[2]*X[,"Covariate"]) } #Creating a counterfactual. cft <- function(X){ Y <- X Y[,"Exposure"] <- 0.5*X[,"Exposure"] Y[,"Covariate"] <- 1.1*X[,"Covariate"] + 1 return(Y) } pif(X, thetahat, rr_better, cft) #Same multivariate example for approximate method calculating #mean and variance Xmean <- data.frame(Exposure = mean(X\$Exposure), Covariate = mean(X\$Covariate)) Xvar <- var(X) pif(Xmean, thetahat, rr_better, method = "approximate", Xvar = Xvar) ## Not run: #The one with \$ operators doesn't work: pif(Xmean, thetahat, rr_not, method = "approximate", Xvar = Xvar) ## End(Not run) ## Not run: #Warning: Multivariate cases cannot be evaluated with kernel method pif(X, thetahat, rr_better, method = "kernel") ## End(Not run) #Example 4: Categorical Relative Risk & Exposure #-------------------------------------------- set.seed(18427) mysample <- sample(c("Normal","Overweight","Obese"), 100, replace = TRUE, prob = c(0.4, 0.1, 0.5)) X <- data.frame(Exposure = mysample) thetahat <- c(1, 1.2, 1.5) #Categorical relative risk function rr <- function(X, theta){ #Create return vector with default risk of 1 r_risk <- rep(1, nrow(X)) #Assign categorical relative risk r_risk[which(X[,"Exposure"] == "Normal")] <- thetahat[1] r_risk[which(X[,"Exposure"] == "Overweight")] <- thetahat[2] r_risk[which(X[,"Exposure"] == "Obese")] <- thetahat[3] return(r_risk) } pif(X, thetahat, rr, check_rr = FALSE) #Counterfactual of reducing all obesity to normality cft <- function(X){ X[which(X[,"Exposure"] == "Obese"),] <- "Normal" return(X) } pif(X, thetahat, rr, cft, check_rr = FALSE) #Example 5: Categorical Relative Risk & continuous exposure #---------------------------------------------------------- set.seed(18427) BMI <- data.frame(Exposure = rlnorm(100, 3.1, sdlog = 0.1)) #Theoretical minimum risk exposure is at 20kg/m^2 in borderline "Normal" category BMI_adjusted <- BMI - 20 thetahat <- c(Malnourished = 2.2, Normal = 1, Overweight = 1.8, Obese = 2.5) rr <- function(X, theta){ #Create return vector with default risk of 1 r_risk <- rep(1, nrow(X)) #Assign categorical relative risk r_risk[which(X[,"Exposure"] < 0)] <- theta[1] #Malnourished r_risk[intersect(which(X[,"Exposure"] >= 0), which(X[,"Exposure"] < 5))] <- theta[2] #Normal r_risk[intersect(which(X[,"Exposure"] >= 5), which(X[,"Exposure"] < 10))] <- theta[3] #Overweight r_risk[which(X[,"Exposure"] >= 10)] <- theta[4] #Obese return(r_risk) } #Counterfactual of everyone in normal range cft <- function(bmi){ bmi <- data.frame(rep(2.5, nrow(bmi)), ncol = 1) colnames(bmi) <- c("Exposure") return(bmi) } pif(BMI_adjusted, thetahat, rr, cft, check_exposure = FALSE, method = "empirical") #Example 6: Bivariate exposure and rr ("classical PAF") #------------------------------------------------------------------ set.seed(18427) mysample <- sample(c("Exposed","Unexposed"), 1000, replace = TRUE, prob = c(0.1, 0.9)) X <- data.frame(Exposure = mysample) theta <- c("Exposed" = 2.5, "Unexposed" = 1.2) rr <- function(X, theta){ #Create relative risk function r_risk <- rep(1, nrow(X)) #Assign values of relative risk r_risk[which(X[,"Exposure"] == "Unexposed")] <- theta["Unexposed"] r_risk[which(X[,"Exposure"] == "Exposed")] <- theta["Exposed"] return(r_risk) } #Counterfactual of reducing the exposure in half of the individuals cft <- function(X){ #Find out which ones are exposed Xexp <- which(X[,"Exposure"] == "Exposed") #Use only half of the exposed randomly reduc <- sample(Xexp, length(Xexp)/2) #Unexpose those individuals X[reduc, "Exposure"] <- "Unexposed" return(X) } pif(X, theta, rr, cft) #Example 7: Continuous exposure, several covariates #------------------------------------------------------------------ X <- data.frame(Exposure = rbeta(100, 2, 3), Age = runif(100, 20, 100), Sex = sample(c("M","F"), 100, replace = TRUE), BMI = rlnorm(100, 3.2, 0.2)) thetahat <- c(-0.1, 0.05, 0.2, -0.4, 0.3, 0.1) rr <- function(X, theta){ #Create risk vector Risk <- rep(1, nrow(X)) #Identify subpopulations males <- which(X[,"Sex"] == "M") females <- which(X[,"Sex"] == "F") #Calculate population specific rr Risk[males] <- theta[1]*X[males,"Exposure"] + theta[2]*X[males,"Age"]^2 + theta[3]*X[males,"BMI"]/2 Risk[females] <- theta[4]*X[females,"Exposure"] + theta[5]*X[females,"Age"]^2 + theta[6]*X[females,"BMI"]/2 return(Risk) } #Counterfactual of reducing BMI cft <- function(X){ excess_bmi <- which(X[,"BMI"] > 25) X[excess_bmi,"BMI"] <- 25 return(X) } pif(X, thetahat, rr, cft) ```