# erl: Compute survival functions from rates and expected residual... In Epi: Statistical Analysis in Epidemiology

## Description

These functions compute survival functions from a set of mortality and disease incidence rates in an illness-death model. Expected residual life time can be computed under various scenarios by the `erl` function, and areas between survival functions can be computed under various scenarios by the `yll` function. Rates are assumed supplied for equidistant intervals of length `int`.

## Usage

 ```1 2 3 4 5 6 7 8``` ``` surv1( int, mu , age.in = 0, A = NULL ) erl1( int, mu , age.in = 0 ) surv2( int, muW, muD, lam, age.in = 0, A = NULL ) erl( int, muW, muD, lam=NULL, age.in = 0, A = NULL, immune = is.null(lam), yll=TRUE, note=TRUE ) yll( int, muW, muD, lam=NULL, age.in = 0, A = NULL, immune = is.null(lam), note=TRUE ) ```

## Arguments

 `int` Scalar. Length of intervals that rates refer to. `mu` Numeric vector of mortality rates at midpoints of intervals of length `int` `muW` Numeric vector of mortality rates among persons in the "Well" state at midpoints of intervals of length `int`. Left endpoint of first interval is `age.in`. `muD` Numeric vector of mortality rates among persons in the "Diseased" state at midpoints of intervals of length `int`. Left endpoint of first interval is `age.in`. `lam` Numeric vector of disease incidence rates among persons in the "Well" state at midpoints of intervals of length `int`. Left endpoint of first interval is `age.in`. `age.in` Scalar indicating the age at the left endpoint of the first interval. `A` Numeric vector of conditioning ages for calculation of survival functions. `immune` Logical. Should the years of life lost to the disease be computed using assumptions that non-diseased individuals are immune to the disease (`lam`=0) and that their mortality is yet still `muW`. `note` Logical. Should a warning of silly assumptions be printed? `yll` Logical. Should years of life lost be included in the result?

## Details

The mortality rates given are supposed to refer to the ages `age.in+(i-1/2)*int`, `i=1,2,3,...`.

The units in which `int` is given must correspond to the units in which the rates `mu`, `muW`, `muD` and `lam` are given. Thus if `int` is given in years, the rates must be given in the unit of events per year.

The ages in which the survival curves are computed are from `age.in` and then at the end of `length(muW)` (`length(mu)`) intervals each of length `int`.

The `age.in` argument is merely a device to account for rates only available from a given age. It has two effects, one is that labeling of the interval endpoint is offset by this quantity, thus starting at `age.in`, and the other that the conditioning ages given in the argument `A` will refer to the ages defined by this.

The `immune` argument is `FALSE` whenever the disease incidence rates are supplied. If set to `TRUE`, the years of life lost is computed under the assumption that individuals without the disease at a given age are immune to the disease in the sense that the disease incidence rate is 0, so transitions to the diseased state (with presumably higher mortality rates) are assumed not to occur. This is a slightly peculiar assumption (but presumably the most used in the epidemiological literature) and the resulting object is therefore given an attribute, `NOTE`, that point this out.

If however `muW` is the total mortality in the population (including the diseased) the result is a good approximation to the correct YLL.

The default of the `surv2` function is to take the possibility of disease into account.

## Value

`surv1` and `surv2` return a matrix whose first column is the ages at the ends of the intervals, thus with `length(mu)+1` rows. The following columns are the survival functions (since `age.in`), and conditional on survival till ages as indicated in `A`, thus a matrix with `length(A)+2` columns. Columns are labeled with the actual conditioning ages; if `A` contains values that are not among the endpoints of the intervals used, the nearest smaller interval border is used as conditioning age, and columns are named accordingly.

`surv1` returns the survival function for a simple model with one type of death, occurring at intensity `mu`.

`surv2` returns the survival function for a person in the "Well" state of an illness-death model, taking into account that the person may move to the "Diseased" state, thus requiring all three transition rates to be specified. The conditional survival functions are conditional on being in the "Well" state at ages given in `A`.

`erl1` returns a three column matrix with columns `age`, `surv` (survival function) and `erl` (expected residual life time) with `length(mu)+1` rows.

`erl` returns a two column matrix, columns labeled "Well" and "Dis", and with row-labels `A`. The entries are the expected residual life times given survival to `A`. If `yll=TRUE` the difference between the columns is added as a third column, labeled "YLL".

## Author(s)

Bendix Carstensen, b@bxc.dk

`ci.cum`
 ``` 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``` ```library( Epi ) data( DMlate ) # Naive Lexis object Lx <- Lexis( entry = list( age = dodm-dobth ), exit = list( age = dox -dobth ), exit.status = factor( !is.na(dodth), labels=c("DM","Dead") ), data = DMlate ) # Cut follow-up at insulin inception Lc <- cutLexis( Lx, cut = Lx\$doins-Lx\$dob, new.state = "DM/ins", precursor.states = "DM" ) summary( Lc ) # Split in small age intervals Sc <- splitLexis( Lc, breaks=seq(0,120,2) ) summary( Sc ) # Overview of object boxes( Sc, boxpos=TRUE, show.BE=TRUE, scale.R=100 ) # Knots for splines a.kn <- 2:9*10 # Mortality among DM mW <- glm( lex.Xst=="Dead" ~ Ns( age, knots=a.kn ), offset = log(lex.dur), family = poisson, data = subset(Sc,lex.Cst=="DM") ) # Mortality among insulin treated mI <- update( mW, data = subset(Sc,lex.Cst=="DM/ins") ) # Total motality mT <- update( mW, data = Sc ) # Incidence of insulin inception lI <- update( mW, lex.Xst=="DM/ins" ~ . ) # From these we can now derive the fitted rates in intervals of 1 year's # length. In real applications you would use much smaller interval like # 1 month: # int <- 1/12 int <- 1 # Prediction frame to return rates in units of cases per 1 year # - we start at age 40 since rates of insulin inception are largely # indeterminate before age 40 nd <- data.frame( age = seq( 40+int, 110, int ) - int/2, lex.dur = 1 ) muW <- predict( mW, newdata = nd, type = "response" ) muD <- predict( mI, newdata = nd, type = "response" ) lam <- predict( lI, newdata = nd, type = "response" ) # Compute the survival function, and the conditional from ages 50 resp. 70 s1 <- surv1( int, muD, age.in=40, A=c(50,70) ) round( s1, 3 ) s2 <- surv2( int, muW, muD, lam, age.in=40, A=c(50,70) ) round( s2, 3 ) # How much is YLL overrated by ignoring insulin incidence? round( YLL <- cbind( yll( int, muW, muD, lam, A = 41:90, age.in = 40 ), yll( int, muW, muD, lam, A = 41:90, age.in = 40, immune=TRUE ) ), 2 )[seq(1,51,10),] par( mar=c(3,3,1,1), mgp=c(3,1,0)/1.6, bty="n", las=1 ) matplot( 40:90, YLL, type="l", lty=1, lwd=3, ylim=c(0,10), yaxs="i", xlab="Age" ) ```