Fit an Age-Period-Cohort model to tabular data.

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Description

Fits the classical five models to tabulated rate data (cases, person-years) classified by two of age, period, cohort: Age, Age-drift, Age-Period, Age-Cohort and Age-Period-Cohort. There are no assumptions about the age, period or cohort classes being of the same length, or that tabulation should be only by two of the variables. Only requires that mean age and period for each tabulation unit is given.

Usage

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apc.fit( data,
            A,
            P,
            D,
            Y,
        ref.c,
        ref.p,
          dist = c("poisson","binomial"),
         model = c("ns","bs","ls","factor"),
       dr.extr = "weighted",
          parm = c("ACP","APC","AdCP","AdPC","Ad-P-C","Ad-C-P","AC-P","AP-C"),
          npar = c( A=5, P=5, C=5 ),
         scale = 1,
         alpha = 0.05,
     print.AOV = TRUE )
  

Arguments

data

Data frame with (at least) variables, A (age), P (period), D (cases, deaths) and Y (person-years). Cohort (date of birth) is computed as P-A. If this argument is given the arguments A, P, D and Y are ignored.

A

Age; numerical vector with mean age at diagnosis for each unit.

P

Period; numerical vector with mean date of diagnosis for each unit.

D

Cases, deaths; numerical vector.

Y

Person-years; numerical vector. Also used as denominator for binomial data, see the dist argument.

ref.c

Reference cohort, numerical. Defaults to median date of birth among cases. If used with parm="AdCP" or parm="AdPC", the residual cohort effects will be 1 at ref.c

ref.p

Reference period, numerical. Defaults to median date of diagnosis among cases.

dist

Distribution (or more precisely: Likelihood) used for modeling. if a binomial model us used, Y is assumed to be the denominator; "binomial" gives a binomial model with logit link.

model

Type of model fitted:

  • ns fits a model with natural splines for each of the terms, with npar parameters for the terms.

  • bs fits a model with B-splines for each of the terms, with npar parameters for the terms.

  • ls fits a model with linear splines.

  • factor fits a factor model with one parameter per value of A, P and C. npar is ignored in this case.

dr.extr

Character or numeric. How the drift parameter should be extracted from the age-period-cohort model. Specifies the inner product used for definition of orthogonality of the period / cohort effects to the linear effects — in terms of a diagonal matrix.

"weighted" (or "t") (default) uses the no. cases, D, corresponding to the observed information about the log-rate (usually termed "theta", hence the "t"). "r" or "l" uses Y*Y/D corresponding to the observed information about the rate (usually termed "lambda", hence the "l"). "y" uses the person-years as the weight in the inner product. If given "n" (Naive) (well, in fact any other character value) will induce the use of the standard inner product putting equal weight on all units in the dataset.

If given as a numeric vector this is used as the diagonal of the matrix inducing the inner product.

The setting of this parameter has no effect on the fit of the model, only on the parametrization.

parm

Character. Indicates the parametrization of the effects. The first four refer to the ML-fit of the Age-Period-Cohort model, the last four give Age-effects from a smaller model and residuals relative to this. If one of the latter is chosen, the argument dr.extr is ignored. Possible values for parm are:

  • "ACP": ML-estimates. Age-effects as rates for the reference cohort. Cohort effects as RR relative to the reference cohort. Period effects constrained to be 0 on average with 0 slope.

  • "APC": ML-estimates. Age-effects as rates for the reference period. Period effects as RR relative to the reference period. Cohort effects constrained to be 0 on average with 0 slope.

  • "AdCP": ML-estimates. Age-effects as rates for the reference cohort. Cohort and period effects constrained to be 0 on average with 0 slope. In this case returned effects do not multiply to the fitted rates, the drift is missing and needs to be included to produce the fitted values.

  • "AdPC": ML-estimates. Age-effects as rates for the reference period. Cohort and period effects constrained to be 0 on average with 0 slope. In this case returned effects do not multiply to the fitted rates, the drift is missing and needs to be included to produce the fitted values.

  • "Ad-C-P": Age effects are rates for the reference cohort in the Age-drift model (cohort drift). Cohort effects are from the model with cohort alone, using log(fitted values) from the Age-drift model as offset. Period effects are from the model with period alone using log(fitted values) from the cohort model as offset.

  • "Ad-P-C": Age effects are rates for the reference period in the Age-drift model (period drift). Period effects are from the model with period alone, using log(fitted values) from the Age-drift model as offset. Cohort effects are from the model with cohort alone using log(fitted values) from the period model as offset.

  • "AC-P": Age effects are rates for the reference cohort in the Age-Cohort model, cohort effects are RR relative to the reference cohort. Period effects are from the model with period alone, using log(fitted values) from the Age-Cohort model as offset.

  • "AP-C": Age effects are rates for the reference period in the Age-Period model, period effects are RR relative to the reference period. Cohort effects are from the model with cohort alone, using log(fitted values) from the Age-Period model as offset.

npar

The number of parameters/knots to use for each of the terms in the model. If it is vector of length 3, the numbers are taken as the no. of knots for Age, Period and Cohort, respectively. Unless it has a names attribute with vales "A", "P" and "C" in which case these will be used. The knots chosen are the quantiles (1:nk-0.5)/nk of the events (i.e. of rep(A,D))

npar may also be a named list of three numerical vectors with names "A", "P" and "C", in which case these taken as the knots for the age, period and cohort effect, the smallest and largest element in each vector are used as the boundary knots.

alpha

The significance level. Estimates are given with (1-alpha) confidence limits.

scale

numeric(1), factor multiplied to the rate estimates before output.

print.AOV

Should the analysis of deviance table for the models be printed?

Details

Each record in the input data frame represents a subset of a Lexis diagram. The subsets need not be of equal length on the age and period axes, in fact there are no restrictions on the shape of these; they could be Lexis triangels for example. The requirement is that A and P are coded with the mean age and calendar time of observation in the subset. This is essential since A and P are used as quantitative variables in the models.

This is a different approach relative to the vast majority of the uses of APC-models in the literature where a factor model is used for age, perido and cohort effects. The latter can be obtained by using model="factor".

Value

An object of class "apc" (recognized by apc.plot and apc.lines) — a list with components:

Type

Text describing the model and parametrization returned

Model

The model object(s) on which the parametrization is based.

Age

Matrix with 4 columns: A.pt with the ages (equals unique(A)) and three columns giving the estimated rates with c.i.s.

Per

Matrix with 4 columns: P.pt with the dates of diagnosis (equals unique(P)) and three columns giving the estimated RRs with c.i.s.

Coh

Matrix with 4 columns: C.pt with the dates of birth (equals unique(P-A)) and three columns giving the estimated RRs with c.i.s.

Drift

A 3 column matrix with drift-estimates and c.i.s: The first row is the ML-estimate of the drift (as defined by drift), the second row is the estimate from the Age-drift model. The first row name indicates which type of inner product were used for projections. For the sequential parametrizations, only the latter is given.

Ref

Numerical vector of length 2 with reference period and cohort. If ref.p or ref.c was not supplied the corresponding element is NA.

AOV

Analysis of deviance table comparing the five classical models.

Type

Character string explaining the model and the parametrization.

Knots

If model is one of "ns" or "bs", a list with three components: Age, Per, Coh, each one a vector of knots. The max and the min are the boundary knots.

Author(s)

Bendix Carstensen, http://BendixCarstensen.com

References

The considerations behind the parametrizations used in this function are given in details in: B. Carstensen: Age-Period-Cohort models for the Lexis diagram. Statistics in Medicine, 10; 26(15):3018-45, 2007.

Various links to course material etc. is available through http://BendixCarstensen.com/APC

See Also

LCa.fit, apc.frame, apc.lines, apc.plot.

Examples

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library( Epi )
data(lungDK)

# Taylor a dataframe that meets the requirements
exd <- lungDK[,c("Ax","Px","D","Y")]
names(exd)[1:2] <- c("A","P")

# Two different ways of parametrizing the APC-model, ML
ex.H <- apc.fit( exd, npar=7, model="ns", dr.extr="Holford",  parm="ACP", scale=10^5 )
ex.W <- apc.fit( exd, npar=7, model="ns", dr.extr="weighted", parm="ACP", scale=10^5 )

# Sequential fit, first AC, then P given AC.
ex.S <- apc.fit( exd, npar=7, model="ns", parm="AC-P", scale=10^5 )

# Show the estimated drifts
ex.H[["Drift"]]
ex.W[["Drift"]]
ex.S[["Drift"]]

# Plot the effects
fp <- apc.plot( ex.H )
apc.lines( ex.W, frame.par=fp, col="red" )
apc.lines( ex.S, frame.par=fp, col="blue" )

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