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R0DecompKitagawa {Lotka} R Documentation

Decompose differences in R0 into fertility, sex ratio and survival components.


The function uses a 2-step Kitagawa decomposition technique to decompose the difference between two R0 estimates into 3 age-specific contribution vectors: fertility, sex ratio and survival. This is a non-symmetrical decomposition technique specific for the case of R0.


R0DecompKitagawa(fx1, px1, Lx1, fx2, px2, Lx2)


6 vectors of age-specific values, all of equal length:
fx1, fx2 vector of age-specific all-birth fertility by age of mother.
px1, px2 vector of proportion female of births by age of mother.
Lx1, Lx2 vector of lifetable person-years lived between the interval x, x+1 (livings) calculated using a radix of 1. Any Lx vector can be converted to radix 1 by dividing the entire vector by the value used for l0 in the survival function column of the lifetable.


All vectors must refer to the same ages. These may include pre- and post-reproductive ages. In the case of px and fx, where non-reproductive ages are used, the vectors must contain no NAs- instead impute zeros in these ages.


A list with 8 components. The list is given the class "R0DecompKitagawa" in order to define plot and summary methods.

Epsilon A vector of the age-specific net contributions to differences in R0.
Fert A vector of the age-specific gross fertility contributions to changes in R0.
SRB A vector of the age-specific contributions to changes in R0 due to differences in the sex ratio at birth (given as the proportion female in the arguments.
Surv A vector of the age-specific survival contributions to changes in R0.
R01 The R0 estimate corresponding with the first set of observations, (population or observation 1).
R02 The R0 estimate corresponding with the second set of observations, (population or observation 2).
R0diff The difference in the two R0 estimates, R01-R02.


This function has not undergone substantial testing, and the formulas used in it have not yet been peer reviewed.


Timothy Riffe


Kitagawa, E.M. (1955) Components of a difference between two rates. Journal of the American Statistical Association. Vol. 50. No. 272. pp. 1168-1194.

Riffe, Timothy (2011) unpublished paper. Decomposing Net Reproduction (R0). Available at https://sites.google.com/site/timriffepersonal/documents/R0DecompMath.pdf

See Also

See Also plot.R0DecompKitagawa, summary.R0DecompKitagawa, plot.R0Decomp


# Fertility
fx2 <- LotkaData[,"Bx2"]/LotkaData[,"Nx2"]
fx1 <- LotkaData[,"Bx1"]/LotkaData[,"Nx1"]
# Proportion births female
px2 <- LotkaData[,"px2"]
px1 <- LotkaData[,"px1"]
# Proportion births female
Lx2 <- LotkaData[,"Lx2"]
Lx1 <- LotkaData[,"Lx1"]

(DecompK <- R0DecompKitagawa(fx1,px1,Lx1,fx2,px2,Lx2))

## The function is currently defined as
SurvComponent <- FertComponent <- AgeComponent <- rep(0,length(fx1))
N <- length(fx1)
R01 <- sum(fx1*px1*Lx1)
R02 <- sum(fx2*px2*Lx2)
R0diff <- R01-R02
AgeComponent <- (fx1*Lx1)-(fx2*Lx2)
FertComponent <- (fx1-fx2)*(Lx1+Lx2)*(px1+px2)*.25
SexRatComponent <- (fx1+fx2)*(Lx1+Lx2)*(px1-px2)*.25
SurvComponent <- (Lx1-Lx2)*(fx1*px1+fx2*px2)*.5
output <- list(Epsilon=AgeComponent,Fert=FertComponent,SRB=SexRatComponent,Surv=SurvComponent,R01=R01,R02=R02,R0diff=R0diff)
class(output) <- "R0DecompKitagawa"

[Package Lotka version 1.1 Index]