R: Fit Lee-Carter-type models for rates to arbitrarily shaped...

LCa.fit {Epi} | R Documentation

## Fit Lee-Carter-type models for rates to arbitrarily shaped observations of rates in a Lexis diagram.

### Description

The Lee-Carter model is originally defined as a model for rates observed in A-sets (age by period) of a Lexis diagram, as log(rate(x,t)) = a(x) + b(x)k(t), using one parameter per age(x) and period(t). This function uses natural splines for a(), b() and k(), placing knots for each effect such that the number of events is the same between knots. Also fits the continuous time counterparts of all models supported by the `lca.rh` function from the `ilc` package (see details).

### Usage

    LCa.fit( data, A, P, D, Y,
             model = "APa",    # or one of "ACa", "APaC", "APCa" or "APaCa"
             a.ref,            # age reference for the interactions
            pi.ref = a.ref,    # age reference for the period interaction
            ci.ref = a.ref,    # age reference for the cohort interaction
             p.ref,            # period reference for the interaction
             c.ref,            # cohort reference for the interactions
              npar = c(a = 6,  # no. knots for main age-effect
                       p = 6,  # no. knots for period-effect
                       c = 6,  # no. knots for cohort-effect
                      pi = 6,  # no. knots for age in the period interaction
                      ci = 6), # no. knots for age in the cohort interaction
                VC = TRUE,     # numerical calculation of the Hessian?
             alpha = 0.05,     # 1 minus confidence level
               eps = 1e-6,     # convergence criterion
             maxit = 100,      # max. no iterations
             quiet = TRUE )    # cut the crap
    ## S3 method for class 'LCa'
    print( x, ... )
    ## S3 method for class 'LCa'
    summary( object, show.est=FALSE, ... )
    ## S3 method for class 'LCa'
    plot( x, ... )
    ## S3 method for class 'LCa'
    predict( object, newdata,
                            alpha = 0.05,
                            level = 1-alpha,
                              sim = ( "vcov" %in% names(object) ),
                              ... )

### Arguments

`data` | A data frame. Must have columns `A`(age), `P`(period, that is calendar time), `D`(no. of events) and `Y`(person-time, exposure). Alternatively these four quantities can be given as separate vectors:
`A` | Vector of ages (midpoint of observation).
`P` | Vector of period (midpoint of observation).
`D` | Vector of no. of events.
`Y` | Vector of person-time. Demographers would say "exposure", bewildering epidemiologists.
`a.ref` | Reference age for the age-interaction term(s) `pi(x)` and/or `pi(x)`, where `pi(a.ref)=1` and `ci(a.ref)=1`.
`pi.ref` | Same, but specifically for the interaction with period.
`ci.ref` | Same, but specifically for the interaction with cohort.
`p.ref` | Reference period for the time-interaction term `kp(t)` where `kp(p.ref)=0`.
`c.ref` | Reference period for the time-interaction term `kp(t)` where `kc(c.ref)=0`.
`model` | Character, either `"APa"` which is the classical Lee-Carter model for log-rates, other possibilities are `"ACa"`, `"APCa"`, `"APaC"` or `"APaCa"`, see details.
`npar` | A (possibly named) vector or list, with either the number of knots or the actual vectors of knots for each term. If unnamed, components are taken to be in the order (a,b,t), if the model is "APaCa" in the order (a,p,c,pi,ci). If a vector, the three integers indicate the number of knots for each term; these will be placed so that there is an equal number of events (`D`) between each, and half as many below the first and above the last knot. If `npar` is a list of scalars the behavior is the same. If `npar` is a list of vectors, these are taken as the knots for the natural splines. See details for naming convention.
`VC` | Logical. Should the variance-covariance matrix of the parameters be computed by numerical differentiation? See details.
`alpha` | 1 minus the confidence level used when calculating confidence intervals for estimates in `LCa.fit` and for predictions by `predict.LCa`.
`eps` | Convergence criterion for the deviance, we use the the relative difference between deviance from the two models fitted.
`maxit` | Maximal number of iterations.
`quiet` | Shall I shut up or talk extensively to you about iteration progression etc.?
`object` | An `LCa` object, see under "Value".
`show.est` | Logical. Should the estimates be printed?
`x` | An `LCa` object, see under "Value".
`newdata` | Prediction data frame, must have columns `A` and `P`. Any `Y` column is ignored, predictions are given in units of the `Y` supplied for the call that generated the `LCa` object.
`level` | Confidence level.
`sim` | Logical or numeric. If `TRUE`, prediction c.i.s will be based on 1000 simulations from the posterior parameters. If numeric, it will be based on that number of simulations.
`...` | Additional parameters passed on to the method.

### Details

The Lee-Carter model is non-linear in age and time so does not fit in the classical glm-Poisson framework. But for fixed `b(x)` it is a glm, and also for fixed `a(x)`, `k(t)`. The function alternately fits the two versions until the same fit is produced (same deviance).

The multiplicative age by period term could equally well have been a multiplicative age by cohort or even both. Thus the most extensive model has 5 continuous functions:

log( lambda(a,p)) = f(a) + b_p(a)k_p(p) + b_c(a)k_c(p-a)

Each of these is fitted by a natural spline, with knots placed at the quantiles of the events along the age (a), calendar time (p) respective cohort (p-a) scales. Alternatively the knots can be specified explicitly in the argument `npar` as a named list, where `a` refers to f(a), `p` refers to k_p(p), `c` refers to k_c(p-a), `pi` (`p`eriod codeinteraction) refers to b_p(a) and `ci` (`c`ohort `i`nteraction) refers to b_c(p-a).

The naming convention for the models is a capital `P` and/or `C` if the effect is in the model followed by a lower case `a` if there is an interaction with age. Thus there are 5 different models that can be fitted: `APa`, `ACa`, `APaC` `APCa` and `APaCa`.

The standard errors of the parameters from the two separate model fits in the iterations are however wrong; they are conditional on a subset of the parameters having a fixed value. However, analytic calculation of the Hessian is a bit of a nightmare, so this is done numerically using the `hessian` function from the `numDeriv` package if `VC=TRUE`.

The coefficients and the variance-covariance matrix of these are used in `predict.LCa` for a parametric bootstrap (that is, a simulation from a multivariate normal with mean equal to the parameter estimates and variance as the estimated variance-covariance) to get confidence intervals for the predictions if `sim` is `TRUE` — which it is by default if they are part of the object.

The `plot` for `LCa` objects merely produces between 3 and 5 panels showing each of the terms in the model. These are mainly for preliminary inspection; real reporting of the effects should use proper relative scaling of the effects.

### Value

`LCa.fit` returns an object of class `LCa` (smooth effects `L`ee-`Ca`rter model); it is a list with the following components:

`model` | Character, either `APa`, `ACa`, `APaC`, `APCa` or `APaCa`, indicating the variable(s) interacting with age.
`ax` | 3-column matrix of age-effects, c.i. from the age-time model. Row names are the unique occurring ages in the dataset. Estimates are rates.
`pi` | 3-column matrix of age-period interaction effects, c.i. from the age model. Row names are the actually occurring ages in the dataset. Estimates are multipliers of the log-RRs in `kp`, centered at 1 at `pi.ref`.
`kp` | 3-column matrix of period-effects, with c.i.s from the age-time model. Row names are the actually occurring times in the dataset. Estimates are rate-ratios centered at 1 at `p.ref`.
`ci` | 3-column matrix of age-cohort interaction effects, c.i. from the age model. Row names are the actually occurring ages in the dataset. Estimates are multipliers of the log-RRs in `kc`, centered at 1 at `ci.ref`.
`kc` | 3-column matrix of cohort-effects, with c.i.s from the age-time model. Row names are the actually occurring times in the dataset. Estimates are rate-ratios centered at 1 at `c.ref`.
`mod.at` | `glm` object with the final age-time model — estimates the terms `ax`, `kp`, `kc`. Gives the same fit as the `mod.b` model after convergence.
`mod.b` | `glm` object with the final age model — estimates the terms `pi`, `ci`. Gives the same fit as the `mod.at` model after convergence.
`coef` | All coefficients from both models, in the order `ax`, `kp`, `kc`, `pi`, `ci`. Only present if `LCa.fit` were called with `VC=TRUE` (the default).
`vcov` | Variance-covariance matrix of coefficients from both models, in the same order as in the `coef`. Only present if `LCa.fit` were called with `VC=TRUE`.
`knots` | List of vectors of knots used in for the age, period and cohort effects.
`refs` | List of reference points used for the age, period and cohort terms in the interactions.
`deviance` | Deviance of the model
`df.residual` | Residual degrees of freedom
`iter` | Number of iterations used to reach convergence.

`plot.LCa` plots the estimated effects in separate panels, using a log-scale for the baseline rates (`ax`) and the time-RR (`kt`). For the `APaCa` model 5 panels are plotted.

`summary.LCa` returns (invisibly) a matrix with the parameters from the models and a column of the conditional se.s and additionally of the se.s derived from the numerically computed Hessian (if `LCa.fit` were called with `VC=TRUE`.)

`predict.LCa` returns a matrix with one row per row in `newdata`. If `LCa.fit` were called with `VC=TRUE` there will be 3 columns, namely prediction (1st column) and c.i.s based on a simulation of parameters from a multivariate normal with mean `coef` and variance `vcov` using the median and `alpha`/2 quantiles from the `sim` simulations. If `LCa.fit` were called with `VC=FALSE` there will be 6 columns, namely estimates and c.i.s from age-time model (`mod.at`), and from the age-interaction model (`mod.b`), both using conditional variances, and therefore likely with too narrow confidence limits.

### Author(s)

Bendix Carstensen, http://BendixCarstensen.com

This function was conceived while teaching a course on APC models at the Max Planck Institute of Demographic Research (MPIDR, https://www.demogr.mpg.de/en/) in Rostock in May 2016 (http://bendixcarstensen.com/APC/MPIDR-2016/), and finished during a week long research stay there, kindly sponsored by the MPIDR.

### See Also

`apc.fit`, `apc.LCa`, `lca.rh`, `lca`

### Examples

    library( Epi )
    # Load the testis cancer data by Lexis triangles
    data( testisDK )
    tc <- subset( testisDK, A>14 & A<60 )
    head( tc )

    # We want to see rates per 100,000 PY
    tc$Y <- tc$Y / 10^5

    # Fit the Lee-Carter model with age-period interaction (default)
    LCa.tc <- LCa.fit( tc, model="ACa", a.ref=30, p.ref=1980, quiet=FALSE, eps=10e-4, maxit=50 )

    LCa.tc
    summary( LCa.tc )

    # Inspect what we got
    names( LCa.tc )

    # show the estimated effects
    par( mfrow=c(1,3) )
    plot( LCa.tc )

    # Prediction data frame for ages 15 to 60 for two time points:
    nd <- data.frame( A=15:60 )
    # LCa predictions
    p70 <- predict.LCa( LCa.tc, newdata=cbind(nd,P=1970), sim=1000 )
    p90 <- predict.LCa( LCa.tc, newdata=cbind(nd,P=1990), sim=1000 )

    # Inspect the curves from the parametric bootstrap (simulation):
    par( mfrow=c(1,1) )
    head( cbind(p70,p90) )
    matplot( nd$A, cbind(p70,p90), type="l", lwd=c(6,3,3), lty=c(1,3,3),
             col=rep( 2:3, each=3 ), log="y",
             ylab="Testis cancer incidence per 100,000 PY in 1970 resp. 1990", xlab="Age" )

[Package Epi version 2.40 Index]