Difference between revisions of "Attributable risk"

Latest revision as of 11:45, 18 September 2016

Attributable risk is a fraction of total risk that can be attributed to a particular cause. There are a few different ways to calculate it. Population attributable fraction of an exposure agent is the fraction of disease that would disappear if the exposure to that agent would disappear in a population. Etiologic fraction is the fraction of cases that have occurred earlier than they would have occurred (if at all) without exposure. Etiologic fracion cannot typically be calculated based on risk ratio (RR) alone, but it requires knowledge about biological mechanisms.

Question

How to calculate attributable risk? What different approaches are there, and what are their differences in interpretation and use?

Answer

Risk ratio (RR)

risk among the exposed divided by the risk among the unexposed

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): RR = \frac{R_1}{R_0}.

Excess fraction

(sometimes called attributable fraction) the fraction of cases among the exposed that would not have occurred if the exposure would not have taken place:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): XF = \frac{RR - 1}{RR}

Population attributable fraction

the fraction of cases among the total population that would not have occurred if the exposure would not have taken place. The most useful formulas are

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = 1 - \frac{1}{\sum_{i=0}^k p_i (RR_i)}

for use with several population subgroups (typically with different exposure levels). Not valid when confounding exists. Subscript i refers to the i^th subgroup. p_i = proportion of total population in i^th subgroup.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = 1- \sum_{i=0}^k \frac{p_{di}}{RR_i} = \sum_i p_{di} \frac{p_{ie}(RR_i - 1)}{p_{ie}(RR_i - 1) + 1}

which produces valid estimates when confounding exists but with a problem that parameters are often not known. p_di is the proportion of cases falling in subgroup i (so that Σ_ip_di = 1), p_ie is the proportion of exposed people within subgroup i (and 1-p_ie is the fraction of unexposed)

Etiologic fraction

Fraction of cases among the exposed that would have occurred later (if at all) if the exposure had not taken place. It cannot be calculated without understanding of the biological mechanism, but there are equations for several specific cases. If survival functions are known, the lower limit of EF can be calculated:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \int_G [f_1(u) - f_0(u)]\mathrm{d}u / [1 - S_1(t)],

where 1 means the exposed group, 0 means the unexposed group, f is the proportion of population dying at particular time points, S is the survival function (and thus f(u) = -dS(u)/du), t is the length of the observation time, u the observation time and G is the set of all u < t such that f₁(u) > f₀(u).

In a specific case where the survival distribution is exponential, the following formula can be used for the lowest possible EF. However, the exponential survival model says nothing about which individuals are affected and lose how much life years, and therefore in this model the actual EF may be between the lower bound and 1.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): EF_l = \frac{RR - 1}{RR^{RR/(RR-1)}}.

Finally, it should be remembered that if the rank preserving assumption holds (i.e. the rank of individual deaths is not affected by exposure: everyone dies in the same order as without exposure, just sooner), the EF can be as high as 1.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): EF_u = 1

With this code, you can compare excess fraction and lower (assuming exponential survival distribution) and upper bounds of etiological fraction.

library(OpasnetUtils)
library(psych)
AF <- function(x) {return(data.frame(RR = x, XF = (x-1)/x, EF_exp_lower = (x-1)/x^(x/(x-1)), EF_upper = 1))}

oprint(AF(RR))

This code creates a simulated population of 200 individuals that are now 60 years of age. It calculates their survival and excess and etiologic fractions in different mechanistic settings. Relative risk of 1.2 and a constant hazard rate will be applied in all scenarios.

#This is code 6211/ on page [[Attributable risk]]
library(OpasnetUtils)
library(reshape2)
library(ggplot2)

cat("Analysis of variation in etiologic fraction. Parameters:\n")
if(linear) cat("Uniform survival distribution (people die between 60 and 80 years)\n") else
  cat("Exponential survival distribution (remaining life expectancy at 60 year is 10 years\n")
if(shuffle == 1) {
  cat("Preserve rank order of individual lifetimes\n")
} else {
  if(shuffle == 2) {
    cat("Minimize EF by accumulating life loss to the hardy\n")
  } else {
    cat("Shuffle lifetimes with approximate rank correlation\n")
  }
}

#linear <- TRUE
#scenario <- c(1, 2, 3)
#crr <- NULL
RR. <- 1.2

objects.latest("Op_en6007", code_name = "answer") # Fetch correlvar

lifetime <- data.frame(
  Unexposed = if(linear) seq(0, 20, 0.1) else qexp((1:200)/201, 1/10)
)

yll <- mean(lifetime$Unexposed) * (RR. - 1) / RR.

if(1 %in% scenario) {
  lifetime$ConstantSurvShift <- lifetime$Unexposed - yll
}
#if(2 %in% scenario) {
#  sequ <- RR. / (RR. - 1)
#  temp <- round(1:(nrow(lifetime) / sequ) * sequ)
#  lifetime$AFdistribution <- lifetime$Unexposed
#  lifetime$AFdistribution[temp] <- lifetime$Unexposed[temp] - yll * sequ
#}

if(2 %in% scenario) {
  lifetime$CompetingCauses <- if(linear) {
    seq(0, by = 0.1/RR., length.out = nrow(lifetime))
  } else {
    qexp((1:200)/201, 1/10*RR.)
  }
}

cat("Individual lifetimes in the population when order is preserved.\n")
oprint(lifetime)

# Minimize EF by sorting

if(shuffle == 2) {
  for(j in colnames(lifetime)[!colnames(lifetime) %in% c("Id", "Unexposed")]) {
    for(i in 1:nrow(lifetime)) {
      pos <- match(TRUE, lifetime$Unexposed[i] <= lifetime[i:nrow(lifetime) , j]) + i - 1
      if(pos > i & !is.na(pos)) {
        block1 <- if(i < 2) numeric() else 1:(i - 1)
        block2 <- if(pos == nrow(lifetime)) numeric() else (pos+1):nrow(lifetime)
        temp <- c(block1, pos, i:(pos-1), block2)
        if(length(temp) == nrow(lifetime)) {
          lifetime[[j]] <- lifetime[temp , j]
        } else {
          warning("Vectors do not match: i ", i, ", pos ", pos, ", temp ", temp)
        }
      }
    }
  }
  cat("Individual lifetimes in the population when life loss is accumulated to the hardy.\n")
  oprint(lifetime)
}

# Shuffle individuals in different scenarios

if(shuffle == 3) {
  Sigma <- matrix(crr, nrow = ncol(lifetime), ncol = ncol(lifetime)) + diag(ncol(lifetime))*(1 - crr)
  lifetime <- correlvar(lifetime, Sigma)
  lifetime <- lifetime[order(lifetime$Unexposed) , ]

  for(j in colnames(lifetime)[colnames(lifetime) != "Unexposed"]) {
    for(i in order(lifetime[[j]], decreasing = TRUE)) {
      pos <- match(TRUE, lifetime[i,j] <= lifetime$Unexposed)
      if(pos > i & !is.na(pos)) {
        block1 <- if(i < 2) numeric() else 1:(i - 1)
        block2 <- if(pos == nrow(lifetime)) numeric() else (pos+1):nrow(lifetime)
        temp <- c(block1, (i+1):pos, i, block2)
        if(length(temp) == nrow(lifetime)) {
          lifetime[[j]] <- lifetime[temp , j]
        } else {
          warning("Vectors do not match: i ", i, ", pos ", pos, ", temp ", temp)
        }
      }
    }
  }
}

cat("Rank correlation coefficients.\n")
oprint(cor(lifetime, method = "spearman"))

plot(lifetime)

lifetime$Id <- 1:nrow(lifetime)
objects.latest("Op_en6211", code_name = "EF")

RR <- EvalOutput(RR)

cat("Relative risks observed in the model.\n")
oprint(RR@output)

lif <- lif + 60 
# Only after the RR and le have been calculated, we can start talking about the 
# total life expectancy rather than the remaining life expectancy at 60 a.

metrices <- EvalOutput(metrices)

cat("Different etiologic and attributable fractions.\n")
oprint(unkeep(metrices, sources = TRUE))

oline <- data.frame(A = c(
  min(result(lif)[lif$Scenario == "Unexposed"]),
  max(result(lif)[!lif$Scenario %in% c("Id", "Unexposed")])
))

plotting <- lif[lif$Scenario == "Unexposed" , colnames(lif@output) != "Scenario"]
plotting <- plotting + lif - lif

BS <- 24

ggplot()+geom_point(data = plotting@output, aes(x = lifResult, y = Result, colour = Scenario))+
  geom_line(data = oline, aes(x = A, y = A)) + theme_gray(base_size = BS)+
  labs(
    title = "Scatter plot of individual lifetimes",
    x = "Unexposed (years)",
    y = "Exposed (years)"
  )

ggplot(lif@output, aes(x = Id, y = lifResult, colour = Scenario))+geom_point()+
 theme_gray(base_size = BS) + labs(title = "Life expectancies of 200 individuals", y = "Age at death", x = "Individual")

ggplot(fr@output, aes(x = Time, y = frResult, colour = Scenario, group = Scenario))+
  geom_line() + 
  theme_gray(base_size = BS)+
  theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
  labs(title = "fraction of people dying at different time groups")
  
ggplot(surv@output, aes(x = Time, y = survResult, colour = Scenario, group = Scenario))+geom_line() + 
 theme_gray(base_size = BS) + labs(title = "Survival curves in different scenarios")+
  theme(axis.text.x = element_text(angle = 90, hjust = 1))
  
ggplot(EF_eq9@output, aes(x = Time, y = EF_eq9Result, colour = Scenario, group = Scenario))+geom_line()+
 theme_gray(base_size = BS) + labs(title = "Development of etiologic fraction in time")+
  theme(axis.text.x = element_text(angle = 90, hjust = 1))

Rationale

Definitions of terms

There are several different kinds of proportions that sound alike but are not. Therefore, we explain the specific meaning of several terms.

Number of people (N)

The number of people in the total population considered, including cases, non-cases, exposed and unexposed. N₁ and N_o are the numbers of exposed and unexposed people in the population, respectively.

Classifications

There are three classifications, and every person in the total population belongs to exactly one group in each classification.

• Disease (D): classes case (c) and non-case (nc)
• Exposure (E): classes exposed (1) and unexposed (0)
• Population subgroup (S): classes i = 0, 1, 2, ..., k (typically based on different exposure levels)
• Confounders (C): other factors correlating with exposure and disease and thus potentially causing bias in estimates unless measured and adjusted for.

Excess fraction (XF)

The proportion of exposed cases that would not have occurred without exposure on population level.

Etiologic fraction (EF)

The proportion of exposed cases that would have occurred later (if at all) without exposure on individual level.

Hazard fraction (HF)

The proportion of hazard rate that would not be there without exposure, HF = [h₁(t) - h₀(t)]/h₁(t) = [R(t) - 1]/R(t), where h(t) is hazard rate at time t and R(t) = h₁(t)/h₀(t).

Attributable fraction (AF)

An ambiguous term that has been used for excess fraction, etiologic fraction and hazard fraction without being specific. Therefore, its use is not recommended.

Population attributable fraction (PAF)

The proportion of all cases (exposed and unexposed) that would not have occurred without exposure on population level. PAF_i is PAF of subgroup i.

Risk of disease (hazard rates)

R₁ and R₀ are the risks of disease in the exposed and unexposed group, respectively, and RR = R₁ / R₀. RR_i = relative risk comparing i^th exposure level with unexposed group (i = 0). Note that often texts are not clear when they talk about risk proportion = number of cases / number of population and thus risk ratio; and when about hazard rates = number of cases / observation time and thus rate ratio. RR may mean either one. If occurrence of cases is small, risk ratio and rate ratio approach each other, because then cases hardly shorten the observation time in the population.

Proportion exposed (p_e, p_ie, p_ed)

proportion of exposed among the total population or within subgroup i or within cases (we use subscript d as diseased rather than c as cases to distinguish it from subscript e): p_e = N(E=1)/N, p_ie = N(E=1,S=i)/N(S=i), p_ed = N(E=1,D=c)/N(D=c)

Proportion of population (p_i)

proportion of population in subgroups i among the total population: N(S=i)/N. p'_i is the fraction of population in a counterfactual ideal situation (where the exposure is typically lower).

Proportion of cases of the disease (p_di)

proportion of cases in subgroups i among the total cases: N(D=c,S=i)/N(D=c) (so that Σ_ip_di = 1).

Excess fraction

Rockhill et al.^[1] give an extensive description about different ways to calculate excess fraction (XF) and population attributable fraction (PAF) and assumptions needed in each approach. Modern Epidemiology ^[2] is the authoritative source of epidemiology. They first define excess fraction XF for a cohort of people (pages 295-297). It is the fraction of cases among the exposed that would not have occurred if the exposure would not have taken place.R↻ However, both sources use the term attributable fraction rather than excess fraction.

Impact of confounders

The problem with the two PAF equations (see Answer) is that the former has easier-to-collect input, but it is not valid if there is confounding. It is still often mistakenly used. The latter equation would produce an unbiased estimate, but the data needed is harder to collect. Darrow and Steenland^[3] have studied the impact of confounding on the bias in attributable fraction. This is their summary:

Population attributable fraction

The population attributable fraction PAF is the fraction of all cases (exposed and unexposed) that would not have occurred if the exposure had been absent.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{N_1 (R_1 - R_0)}{N_1 R_1 + N_0 R_0} = \frac{N_1 (R_1 - R_0)/R_0}{N_1 R_1/R_0 + N_0 R_0/R_0} = \frac{N_1 (RR - 1)}{N_1 RR + N_0}

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): = \frac{ \frac{N_1 (RR - 1)}{N_1 + N_0} }{ \frac{N_1 RR + N_0}{N_1 + N_0}} = \frac{ p_e (RR - 1) }{ \frac{N_1 RR - N_1 + (N_1 + N_0)}{N_1 + N_0}} = \frac{p_e (RR - 1)}{p_e RR - p_e + 1} = \frac{p_e (RR - 1)}{p_e (RR - 1) + 1}.

Note that there is a typo in the Modern Epidemiology book: the denominator should be p(RR-1)+1, not p(RR-1)-1.

Population attributable fraction can be calculated as a weighted average based on subgroup data:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \Sigma_i p_{di} PAF_{i}.

Specifically, we can divide the cohort into subgroups based on exposure (in the simplest case exposed and unexposed), so we get

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = p_{ed} \frac{1(RR - 1)}{1(RR - 1) + 1} + (1 - p_{ed}) \frac{0(RR - 1)}{0(RR - 1) +1} = p_{ed} \frac{RR - 1}{RR},

where p_c is the proportion of cases in the exposed group among all cases; this is the same as exposure prevalence among cases.

WHO approach

According to WHO, PAF is ^[9]

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{\sum_{i=0}^k p_i RR_i - \Sigma_{i=0}^k p'_i RR_i}{\Sigma_{i=0}^k p_i RR_i}.

We can see that this reduces to PAF equation 2 when we limit our examination to a situation where there are only two population groups, one exposed to background level (with relative risk 1) and the other exposed to a higher level (with relative risk RR). In the counterfactual situation nobody is exposed. in this specific case, p_i = p_e. Thus, we get

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{(p_e RR + (1-p_e)*1) - (0*RR + 1*1)}{p_e RR + (1-p_e)*1}

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{p_e RR - p_e}{p_i RR + 1 - p_e}

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{p_e(RR - 1)}{p_e(RR -1) + 1}

--#: Constant background assumption section was archived because it was only relevant for a previous HIA model version. --Jouni (talk) 13:17, 25 April 2016 (UTC)

Etiologic fraction

Etiologic fraction (EF) is defined as the fraction of cases that are advanced in time because of exposure.^[10]R↻ In other words, those cases would have occurred later (if at all), if there had not been exposure. EF can also be called probability of causation, which has importance in court. It can also be used to calculate premature cases, but that term is ambiguous and sometimes it is used to mean cases that have been substantially advanced in time, in contrast to the harvesting effect where an exposure kills people that would have died anyway within a few days. There has been a heated discussion about harvesting effect related to fine particles. Therefore, sometimes excess fraction is used instead to calculate what they call premature mortality, but unfortunately that practice causes even more confusion.R↻ Therefore, it is important to explicitly explain what is meant by the word premature.

Robins and Greenland^[10] studied the estimability of etiologic fraction. They concluded that observations are not enough to conclude about the precise value of EF, because irrespective of observation, the same amount of observed life years lost may be due to many people losing a short time each, or due to a few losing a long time each. The upper limit in theory is always 1, and the lower bound they estimated by this equation (equation 9 in the article):

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \int_G [f_1(u) - f_0(u)]\mathrm{d}u / [1 - S_1(t)],

where 1 means the exposed group, 0 means the unexposed group, f is the proportion of population dying at particular time points, S is the survival function (and thus f(u) = -dS(u)/du), t is the length of the observation time, u the observation time and G is the set of all u < t such that f₁(u) > f₀(u).

Although the exact value of etiologic fraction cannot be estimated directly from risk ratio (RR), different models offer equations to estimate EF. It is just important to understand, discuss, and communicate, which of the models most closely represents the actual situation observed. Three models are explained here.R↻

Rank-preserving model says that everyone dies at the same rank order as without exposure, but that the deaths occur earlier. If the exposed population loses life years compared with unexposed population, it is in theory always possible that everyone dies a bit earlier and thus

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): EF_u = 1.

Competing causes model is the most commonly assumed model, but often people do not realise that they make such an assumption. The model says that the exposure of interest and other causes of death are constantly competing, and that the impact of the exposure is relative to the other competing causes. In other words, the hazard rate in the exposed population is h₁(t) = RR h₀(t). Hazard rates are functions of time, and may become very high in very old populations. In any case, the proportional impact of the exposure stays constant.

In the case where competing causes model and independence assumtption applies, lower end of EF range is often close to the excess fraction XF. (But it can be lower, as the next example with a skewed exponential distribution demonstrates.)

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): EF_l = XF = \frac{RR - 1}{RR}.

Exponential survival model assumes that the hazard rate is constant and the deaths occur following the exponential distribution. Although this model has very elegant formulas, it is typically far from plausible, as the differences in survival may be very large. E.g. with average life expectancy of 70 years, 10 % of the population would die before 8 years of age, while 10 % would live beyond 160 years. In situations where exponential survival model can be used, the lower bound of EF (equation 9^[10]) is as low as

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): EF_l = \frac{RR - 1}{RR^{RR/(RR-1)}}.

For an illustration of the behaviour of EF, see the code "Test different etiologic fractions" in the Answer. Also the true etiologic fraction is calculated for this simulated population, because in the simulation we assume that we know exactly what happens to each individual in each scenario and how much their lengths of lives change. By testing with several inputs, we can see the following pattern (table).

As we can see from the table, true etiologic fraction can vary substantially - in theory. High values assume that most people are affected by a small life loss. This might be true with causes that worsen general health, thus killing the person a bit earlier than what would have happened if the person had been in a hardier state.

When we compare equations 9 and 11, we can see that the former never performs worse than the latter. This is simply because equation 11 was derived from equation 9 by making an additional assmuption that the survival distribution is exponential. Indeed, in such a case they produce identical values but in other cases equation 11 underestimates EF compared with equation 9. A practical conclusion is that if survival curves for exposed and unexposed groups are available, equation 9 rather than equation 11 should always be used. Even excess fraction is usually a better estimate than an estimate from equation 11, with the exception of exponential survival distribution.

Calculations

⇤# : UPDATE AF TO REFLECT THE CURRENT IMPLEMENTATION OF ERF Exposure-response function --Jouni (talk) 05:20, 13 June 2015 (UTC)

# This is code Op_en6211/AF on page [[Attributable risk]]
# Parameters: none

library(OpasnetUtils)

# AF = attributable fraction
# EF = etiologic fraction
# PAF = population attributable fraction using 
EF <- Ovariable("EF", 
	dependencies = data.frame(Name = c(
		"RR" # Risk ratio
	)),
	
	formula = function(...) {

		R <- unkeep(RR, sources = TRUE, prevresults = TRUE)
		EF <- (RR - 1) / R^(R/(R-1))
		EF <- EF * Ovariable("temp", data = data.frame(
			EFestimate = c("Low", "High"),
			Result = 1
		))
		result(EF)[EF$EFestimate == "High"] <- 1

		return(EF)
	}
)

AF <- Ovariable("AF", 
	dependencies = data.frame(Name = c(
		"RR" # Risk ratio
	)),
	
	formula = function(...) {

		AF <- (RR - 1) / unkeep(RR, sources = TRUE, prevresults = TRUE)

		return(AF)
	}
)

PAF <- Ovariable("PAF", 
	dependencies = data.frame(Name = c(
		"RR", # Risk ratio
		"pci", # proportion of cases falling subgroup i among all cases
		"pei" # proportion of exposed people within subgroup i
	)),
	
	formula = function(...) {

		peirri <- pei * (RR - 1)
		peirri <- unkeep(peirri, sources = TRUE, prevresults = TRUE)

		PAF <- pci * peirri / (peirri + 1) # The population subgroup could be summed up.

		return(PAF)
	}
)

objects.store(EF, AF, PAF)
cat("Ovariables EF, AF, PAF stored.\n")

A previous version of code looked at RRs of all exposure agents and summed PAFs up.

Some interesting model runs:

• Population with exponentially distributed lifetimes
• Life loss to a fraction of people across the whole population
• Give all life loss to the people living the longest

#This is code Op_en6211/EF on page [[Attributable risk]]

library(OpasnetUtils)

lif <- Ovariable(
  "lif",
  dependencies = data.frame(Name = "lifetime"),
  formula = function(...) {
    out <- melt(
       lifetime, 
       id.vars = "Id", 
       value.name = "Result", 
       variable.name = "Scenario"
    )
    out <- Ovariable(
      output = out, 
      marginal = c(TRUE, TRUE, FALSE)
    )
    return(out)
  }
)

le <- Ovariable(
  "le",
  dependencies = data.frame(Name = "lif"),
  formula = function(...) {
    le <- oapply(lif, INDEX = "Scenario", FUN = sum) / 
    oapply(lif, INDEX = "Scenario", FUN = length)
    return(le)
  }
)

RR <- Ovariable(
  "RR",
  dependencies = data.frame(Name = "le"),
  formula = function(...) {
    RR <- le[le$Scenario == "Unexposed" , ]
    RR <- unkeep(RR, cols = c("Scenario", "lifResult"))
    RR <- RR / le
    RR <- unkeep(RR, prevresults = TRUE)
    return(RR)
  }
)

fr <- Ovariable("fr", 
  dependencies = data.frame(
    Name = "lif"
  ),
  formula = function(...) {
    out <- lif
    temp2 <- cut(result(out), breaks = 12)
    out$Time <- temp2
    out <- out * 0 + 1/oapply(out, cols = c("Id", "Time"), FUN = length)
    temp <- Ovariable(
      "temp", 
      data = data.frame(
        Time = levels(temp2),
        Result = 0
      )
    )
    out <- combine(EvalOutput(temp), out)
    out <- oapply(out, cols = "Id", FUN = sum) # Automatic fillna is OK.
    return(out)
  }
)

surv <- Ovariable(
  "surv",
  dependencies = data.frame(Name = "fr"),
  formula = function(...) {
    out <- fr[order(fr$Time) , ]
    temp <- data.frame()
    for(i in unique(out$Scenario)) {
      temp2 <- out[out$Scenario == i , ]
      result(temp2) <- 1 - cumsum(result(temp2))
      temp <- rbind(temp, temp2@output)
    }
    out@output <- temp
    return(out)
  }
)

EF_eq9 <- Ovariable(
  "EF_eq9",
  dependencies = data.frame(Name = c("fr", "surv")),
  formula = function(...) {
    BAU <- fr[fr$Scenario == "Unexposed" , ]
    BAU <- unkeep(BAU, prevresults = TRUE, sources = TRUE, cols = "Scenario")
    
    out <- fr
    result(out) <- pmax(0, result(out - BAU))
    
    out <- out[order(out$Time) , ]
    temp <- data.frame()
    for(i in unique(out$Scenario)) {
      temp2 <- out[out$Scenario == i , ]
      result(temp2) <- cumsum(result(temp2))
      temp <- rbind(temp, temp2@output)
    }
    out@output <- temp
    out <- out / (1 - surv)
    
    return(out)
  }
)

EF_true <- Ovariable(
  "EF_true", 
  dependencies = data.frame(Name = "lif"),
  formula = function(...) {
    BAU <- lif[lif$Scenario == "Unexposed" , ]
    BAU <- unkeep(BAU, cols = "Scenario", prevresults = TRUE, sources = TRUE)
    out <- lif < BAU
    out <- oapply(out, cols = "Id", FUN = sum) / 
      oapply(out, cols = "Id", FUN = length)
    
    return(out)
  }
)

metrices <- Ovariable(
  "metrices", 
  dependencies = data.frame(Name = c("RR", "lif", "EF_true", "EF_eq9")),
  formula = function(...) {
    out <- (RR - 1) / RR
    out$Metric <- "Attributable fraction"
    temp <- (RR - 1)/(RR^(RR/(RR-1)))
    temp$Metric <- "EF_low from eq 11"
    out <- combine(out, temp)
#    result(temp) <- 1
#    temp$Metric <- "EF_up theoretical"
#    out <- combine(out, temp)
    temp <- unkeep(EF_true, sources = TRUE, prevresults = TRUE)
    temp$Metric <- "EF_true"
    out <- combine(out, temp)
    temp <- unkeep(EF_eq9[EF_eq9$Time == levels(EF_eq9$Time)[length(levels(EF_eq9$Time))] , ],
           cols = "Time", sources = TRUE, prevresults = TRUE
    )
    temp$Metric <- "EF_low from Eq 9"
    out <- combine(out, temp)
    

    return(out)
  }
)

objects.store(lif, le, RR, fr, surv, EF_eq9, EF_true, metrices)
cat("Ovariables lif, le, RR, fr, surv, EF_eq9, EF_true, metrices stored.\n")

See also

• Health impact assessment
• Attributable risk in Wikipedia.
• Jacqueline C. M. Witteman, Ralph B. D'Agostino, Theo Stijnen, William B. Kannel, Janet C. Cobb, Maria A. J. de Ridder, Albert Hofman and James M. Robins. G-estimation of Causal Effects: Isolated Systolic Hypertension and Cardiovascular Death in the Framingham Heart Study. Am. J. Epidemiol. (1998) 148 (4): 390-401. [8]

References