lca.rh: A class of generalised Lee-Carter models
In ilc: Lee-Carter Mortality Models using Iterative Fitting Algorithms
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
A purpose-built regression routine to fit any of the six variants of the class of Lee-Carter model structures using an iterative Newton-Raphson fitting method.
Usage
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Arguments
dat
source data object of demogdata class
year
vector of years to be included in the regression (all available years by default)
age
vector of ages to be included in the regression (all available ages by default)
series
numerical index corresponding to the target series to be used from the source data
max.age
highest age to be used in the regression
dec.conv
number of decimal places used to achieve convergence. The lower the value the faster the convergence of the fitting algorithm.
clip
number of marginal birth cohorts to exclude from the regression (i.e., give 0 weights). It is only applicable to the first 5 models (see below)
error
type of error structure of the model choice (Poisson distribution of the errors by default)
model
a character (see usage) or a numeric value (1-6) to specify the model choice
restype
types of residuals, which also controls the type of the fitted value.
Thus, in the cases of logrates and rates the function returns as fitted values the log and untransformed mortality rates, respectively. Likewise, the choices of deaths and deviance correspond to the fitted number of deaths
scale
logical, if TRUE, re-scale the interaction parameters so that the k_t has drift parameter equal to 1 (see also lca)
interpolate
logical, if TRUE, replace before regression all zero or missing values in the mortality rates of dat argument by interpolation across calendar years (see also smooth.demogdata)
verbose
logical, if TRUE, the program prints out the updated deviance values along with the starting and final parameter estimates
spar
numerical smoothing spline parameter in the interval (0,1] (with a recommended value of 0.6). If it is not NULL, the interaction effects (i.e. β_x^{(0,1)}) are smoothed out after the initial regression. Consequently, the period and/or cohort effects are adjusted (smoothed out) accordingly.
Details
Implements the modelling approach proposed in Renshaw and Haberman (2006), which extends the basic Lee-Carter model within the GLM framework. The function makes use of tailored iterative Newton-Raphson fitting algorithms to estimate the graduation parameters of the six variants within this class of extended Lee-Carter models.
Value
A Lee-Carter type fitted object with the following components:
label
data label
age
vector of fitted ages
year
vector of fitted fitted years
<series>
matrix of observed (source) mortality rates used for fitting. It is named the same way as the chosen series
ax
parameter estimates of (mean) age-specific mortality rates across the entire fitting period
bx
parameter estimates of age-specific interaction effect between age and period
kt
parameter estimates of year-specific period trend of mortality rates
df
degree of freedom of the fitted GLM model
residuals
residuals of the fitted model in the form of a functional time series object
fitted
fitted values of the fitted model in the form of a functional time series object
varprop
percent of variance
y
source mortality data in the form of a functional time series object
mdev
mean deviance of total and base lack of fit (see also lca)
model
string expression of the fitted model
adjust
type of error structure (e.g. "poisson" or "gaussian")
call
copy of the R call to the model
conv.iter
number of iterations used to reach convergence
bx0
parameter estimates of age-specific interaction effect between age and cohort (only applies to the age-period-cohort model)
itx
parameter estimates of year-specific cohort trend of mortality rates (only applies to the age-period-cohort model)
Author(s)
Zoltan Butt, Steven Haberman and Han Lin Shang
References
Renshaw, A. E. and Haberman, S. (2003a), “Lee-Carter mortality forecasting: a parallel generalised linear modelling approach for England and Wales mortality projections", Journal of the Royal Statistical Society, Series C, 52(1), 119-137.
Renshaw, A. E. and Haberman, S. (2003b), “Lee-Carter mortality forecasting with age specific enhancement", Insurance: Mathematics and Economics, 33, 255-272.
Renshaw, A. E. and Haberman, S. (2006), “A cohort-based extension to the Lee-Carter model for mortality reduction factors", Insurance: Mathematics and Economics, 38, 556-570.
Renshaw, A. E. and Haberman, S. (2008), “On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling", Insurance: Mathematics and Economics, 42(2), 797-816.
Renshaw, A. E. and Haberman, S. (2009), “On age-period-cohort parametric mortality rate projections", Insurance: Mathematics and Economics, 45(2), 255-270.
See Also
Examples
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# standard LC model with Gaussian errors (corresponding to SVD graduation):
# correct 0 or missing mortality rates before graduation
mod6g <- lca.rh(dd.cmi.pens, mod='lc', error='gauss', max=110, interpolate=TRUE)
# AP LC model with Poisson errors
mod6p <- lca.rh(dd.cmi.pens, mod='lc', error='pois', interpolate=TRUE)
# Model Summary, Coefficients and Plotting:
mod6p; coef(mod6p); plot(mod6p)
# Comparison with standard fitting method
# Standard LC model (with Gaussian errors) - SVD fit (demography package)
modlc <- lca(dd.cmi.pens, interp=TRUE, adjust='none')
# Gaussian (SVD) - Gaussian (iterative)
round(modlc$ax-mod6g$ax, 4)
round(modlc$bx-mod6g$bx, 4)
round(modlc$kt-mod6g$kt, 4)
# -------------------------------------------------- #
# APC LC model fitted to restricted age range with 'deviance' residuals
# the remaining 0/NA values reestimated:
# WARNING: for proper fit recommend dec=6, but it can lead to slow convergence!
mod1 <- lca.rh(dd.cmi.pens, age=60:100, mod='m', interpolate=TRUE, res='dev', dec=1)
Example output
Loading required package: demography
Loading required package: forecast
This is demography 1.20
Loading required package: rainbow
Loading required package: MASS
Loading required package: pcaPP
Loading required package: date
Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl_init' failed, running with rgl.useNULL = TRUE
3: .onUnload failed in unloadNamespace() for 'rgl', details:
call: fun(...)
error: object 'rgl_quit' not found
Original sample: Mortality data for CMI
Series: male
Years: 1983 - 2003
Ages: 50 - 108
Applied sample: Mortality data for CMI (Corrected: interpolate)
Series: male
Years: 1983 - 2003
Ages: 50 - 108
Fitting model: [ LC = a(x)+b1(x)*k(t) ]
- with gaussian error structure -
Note: 109 cells have 0/NA deaths and 0 have 0/NA exposure
out of a total of 1239 data cells.
Starting values are:
age age.c bx1.c per per.c
1 50 -3.462 0.017 1983 0
2 51 -4.095 0.017 1984 0
3 52 -4.404 0.017 1985 0
4 53 -4.615 0.017 1986 0
5 54 -4.480 0.017 1987 0
6 55 -4.504 0.017 1988 0
7 56 -4.770 0.017 1989 0
8 57 -4.576 0.017 1990 0
9 58 -4.814 0.017 1991 0
10 59 -4.502 0.017 1992 0
11 60 -4.652 0.017 1993 0
12 61 -4.627 0.017 1994 0
13 62 -4.637 0.017 1995 0
14 63 -4.394 0.017 1996 0
15 64 -4.117 0.017 1997 0
16 65 -4.083 0.017 1998 0
17 66 -4.042 0.017 1999 0
18 67 -3.952 0.017 2000 0
19 68 -3.831 0.017 2001 0
20 69 -3.725 0.017 2002 0
21 70 -3.599 0.017 2003 0
22 71 -3.436 0.017
23 72 -3.346 0.017
24 73 -3.228 0.017
25 74 -3.115 0.017
26 75 -3.002 0.017
27 76 -2.902 0.017
28 77 -2.810 0.017
29 78 -2.707 0.017
30 79 -2.604 0.017
31 80 -2.519 0.017
32 81 -2.399 0.017
33 82 -2.311 0.017
34 83 -2.225 0.017
35 84 -2.141 0.017
36 85 -2.045 0.017
37 86 -1.979 0.017
38 87 -1.874 0.017
39 88 -1.807 0.017
40 89 -1.725 0.017
41 90 -1.664 0.017
42 91 -1.590 0.017
43 92 -1.552 0.017
44 93 -1.440 0.017
45 94 -1.359 0.017
46 95 -1.299 0.017
47 96 -1.284 0.017
48 97 -1.245 0.017
49 98 -1.276 0.017
50 99 -1.261 0.017
51 100 -1.315 0.017
52 101 -1.268 0.017
53 102 -1.476 0.017
54 103 -1.247 0.017
55 104 -0.889 0.017
56 105 -1.098 0.017
57 106 -0.795 0.017
58 107 0.105 0.017
59 108 -0.237 0.017
Iterative fit:
#iter Dev non-conv
1 251.9952 0
2 116.7591 0
3 113.7653 0
4 113.6234 0
5 113.6131 0
6 113.6124 0
7 113.6123 0
8 113.6123 0
Iterations finished in: 8 steps
Updated values are:
age age.c bx1.c per per.c
1 50 -3.46190 0.09761 1983 17.71325
2 51 -4.09482 0.03689 1984 17.8319
3 52 -4.40400 0.02929 1985 17.04362
4 53 -4.61490 0.02405 1986 13.65193
5 54 -4.48012 0.01451 1987 13.78856
6 55 -4.50413 0.03160 1988 11.94677
7 56 -4.77048 0.00670 1989 10.52122
8 57 -4.57601 0.02193 1990 9.06468
9 58 -4.81427 0.01332 1991 8.60141
10 59 -4.50233 -0.00138 1992 6.17926
11 60 -4.65201 0.01202 1993 -0.93023
12 61 -4.62690 0.00793 1994 1.10729
13 62 -4.63747 0.03909 1995 -1.74929
14 63 -4.39403 0.00933 1996 -4.68995
15 64 -4.11729 0.02059 1997 -9.25547
16 65 -4.08299 0.01571 1998 -7.27498
17 66 -4.04168 0.01426 1999 -11.05124
18 67 -3.95211 0.01691 2000 -15.99149
19 68 -3.83069 0.01697 2001 -19.78654
20 69 -3.72485 0.01442 2002 -27.60037
21 70 -3.59892 0.01720 2003 -29.12033
22 71 -3.43584 0.01535
23 72 -3.34567 0.01441
24 73 -3.22840 0.01314
25 74 -3.11466 0.01253
26 75 -3.00222 0.01140
27 76 -2.90200 0.01226
28 77 -2.80974 0.01309
29 78 -2.70686 0.01101
30 79 -2.60402 0.01118
31 80 -2.51932 0.01082
32 81 -2.39878 0.01084
33 82 -2.31114 0.00992
34 83 -2.22452 0.01023
35 84 -2.14145 0.01059
36 85 -2.04520 0.00914
37 86 -1.97936 0.00932
38 87 -1.87386 0.00920
39 88 -1.80746 0.00938
40 89 -1.72537 0.00886
41 90 -1.66449 0.00760
42 91 -1.58971 0.00711
43 92 -1.55192 0.00784
44 93 -1.43969 0.00711
45 94 -1.35939 0.01022
46 95 -1.29881 0.01039
47 96 -1.28441 0.00648
48 97 -1.24456 0.01226
49 98 -1.27575 0.00832
50 99 -1.26096 0.01686
51 100 -1.31451 0.01010
52 101 -1.26812 0.01127
53 102 -1.47610 0.00288
54 103 -1.24738 0.01420
55 104 -0.88950 0.04912
56 105 -1.09801 0.02799
57 106 -0.79478 0.06768
58 107 0.10536 0.00000
59 108 -0.23740 0.05096
total sums are:
b0 b1 itx kt
0 1 0 0
Warning message:
A total of 152 0/NA central mortality rates are re-estimated by the "interpolate" method.
Original sample: Mortality data for CMI
Series: male
Years: 1983 - 2003
Ages: 50 - 108
Applied sample: Mortality data for CMI (Corrected: interpolate)
Series: male
Years: 1983 - 2003
Ages: 50 - 100
Fitting model: [ LC = a(x)+b1(x)*k(t) ]
- with poisson error structure and with deaths as weights -
Note: 45 cells have 0/NA deaths and 45 have 0/NA exposure
out of a total of 1071 data cells.
Starting values are:
age age.c bx1 bx1.c per per.c
1 50 -3.462 50 0.02 1983 0
2 51 -4.095 51 0.02 1984 0
3 52 -4.404 52 0.02 1985 0
4 53 -4.615 53 0.02 1986 0
5 54 -4.480 54 0.02 1987 0
6 55 -4.504 55 0.02 1988 0
7 56 -4.770 56 0.02 1989 0
8 57 -4.576 57 0.02 1990 0
9 58 -4.814 58 0.02 1991 0
10 59 -4.502 59 0.02 1992 0
11 60 -4.652 60 0.02 1993 0
12 61 -4.627 61 0.02 1994 0
13 62 -4.637 62 0.02 1995 0
14 63 -4.394 63 0.02 1996 0
15 64 -4.117 64 0.02 1997 0
16 65 -4.083 65 0.02 1998 0
17 66 -4.042 66 0.02 1999 0
18 67 -3.952 67 0.02 2000 0
19 68 -3.831 68 0.02 2001 0
20 69 -3.725 69 0.02 2002 0
21 70 -3.599 70 0.02 2003 0
22 71 -3.436 71 0.02
23 72 -3.346 72 0.02
24 73 -3.228 73 0.02
25 74 -3.115 74 0.02
26 75 -3.002 75 0.02
27 76 -2.902 76 0.02
28 77 -2.810 77 0.02
29 78 -2.707 78 0.02
30 79 -2.604 79 0.02
31 80 -2.519 80 0.02
32 81 -2.399 81 0.02
33 82 -2.311 82 0.02
34 83 -2.225 83 0.02
35 84 -2.141 84 0.02
36 85 -2.045 85 0.02
37 86 -1.979 86 0.02
38 87 -1.874 87 0.02
39 88 -1.807 88 0.02
40 89 -1.725 89 0.02
41 90 -1.664 90 0.02
42 91 -1.590 91 0.02
43 92 -1.552 92 0.02
44 93 -1.440 93 0.02
45 94 -1.359 94 0.02
46 95 -1.299 95 0.02
47 96 -1.284 96 0.02
48 97 -1.245 97 0.02
49 98 -1.276 98 0.02
50 99 -1.261 99 0.02
51 100+ -1.271 100 0.02
Iterative fit:
#iter Dev non-conv
1 11291.24 0
2 2097.723 0
3 1284.158 0
4 1256.141 0
5 1254.86 0
6 1254.775 0
7 1254.765 0
8 1254.763 0
9 1254.762 0
10 1254.762 0
11 1254.762 0
12 1254.762 0
13 1254.762 0
14 1254.762 0
Iterations finished in: 14 steps
Updated values are:
age age.c bx1 bx1.c per per.c
1 50 -3.66478 50 0.10966 1983 13.73515
2 51 -4.19899 51 0.04787 1984 11.98794
3 52 -4.63348 52 0.03683 1985 12.33066
4 53 -4.81217 53 0.01652 1986 9.7473
5 54 -4.66409 54 0.01276 1987 10.77193
6 55 -4.57660 55 0.01757 1988 8.89844
7 56 -4.91748 56 -0.00553 1989 6.71887
8 57 -4.49245 57 0.02375 1990 5.10403
9 58 -4.61074 58 0.02647 1991 5.85633
10 59 -4.46297 59 0.00683 1992 4.35955
11 60 -4.48528 60 0.01542 1993 3.39662
12 61 -4.55880 61 0.00805 1994 0.65272
13 62 -4.49516 62 0.02859 1995 -0.74961
14 63 -4.38858 63 0.01640 1996 -3.81511
15 64 -4.01845 64 0.02017 1997 -10.44243
16 65 -4.08439 65 0.02379 1998 -8.34926
17 66 -4.04477 66 0.02323 1999 -10.19638
18 67 -3.94810 67 0.02570 2000 -15.1888
19 68 -3.82265 68 0.02504 2001 -12.36537
20 69 -3.72129 69 0.02198 2002 -16.0094
21 70 -3.59880 70 0.02369 2003 -16.44316
22 71 -3.44047 71 0.02158
23 72 -3.34554 72 0.02069
24 73 -3.22789 73 0.01927
25 74 -3.11083 74 0.01824
26 75 -3.00329 75 0.01679
27 76 -2.89804 76 0.01696
28 77 -2.80524 77 0.01834
29 78 -2.70233 78 0.01647
30 79 -2.60582 79 0.01610
31 80 -2.52293 80 0.01604
32 81 -2.40074 81 0.01573
33 82 -2.31086 82 0.01487
34 83 -2.22338 83 0.01483
35 84 -2.13834 84 0.01442
36 85 -2.04378 85 0.01392
37 86 -1.98080 86 0.01402
38 87 -1.87325 87 0.01394
39 88 -1.80414 88 0.01371
40 89 -1.72358 89 0.01312
41 90 -1.66284 90 0.01118
42 91 -1.58319 91 0.01101
43 92 -1.54325 92 0.01284
44 93 -1.43955 93 0.00985
45 94 -1.35467 94 0.01552
46 95 -1.29103 95 0.01660
47 96 -1.26240 96 0.01172
48 97 -1.22455 97 0.01755
49 98 -1.25369 98 0.01164
50 99 -1.27202 99 0.02212
51 100+ -1.25330 100 0.02612
total sums are:
b0 b1 itx kt
0 1 0 0
Warning messages:
1: In lca.set(dat, year, age, series, max.age, interpolate) :
=> data above age 100 are grouped.
2: A total of 62 0/NA central mortality rates are re-estimated by the "interpolate" method.
3: In lca.set(dat, year, age, series, max.age, interpolate) :
There are 45 cells with 0/NA exposures, which are ignored in the current analysis.
Try reducing the maximum age or choosing a different age range.
Alternatively, fit LC model with error= "gaussian" .
------------------------------------------------------------
Iterative Lee-Carter Family Regression:
Fitted Model: LC = a(x)+b1(x)*k(t)
------------------------------------------------------------
Call:
lca.rh(dat = dd.cmi.pens, error = "pois", model = "lc", interpolate = TRUE)
Error Structure: poisson
Data Source: CMI [male] over
calendar years: (1983 - 2003) and ages: (50 - 100)
Deviance convergence in: 14 iterations
dev dev.c df df.c
1 Mean deviance base 1.386 df base 905
2 Mean deviance total 1.733 df tot 969
$ax
50 51 52 53 54 55 56 57
-3.664778 -4.198990 -4.633480 -4.812166 -4.664094 -4.576603 -4.917484 -4.492450
58 59 60 61 62 63 64 65
-4.610738 -4.462966 -4.485284 -4.558802 -4.495163 -4.388582 -4.018450 -4.084386
66 67 68 69 70 71 72 73
-4.044770 -3.948101 -3.822646 -3.721286 -3.598799 -3.440473 -3.345543 -3.227889
74 75 76 77 78 79 80 81
-3.110830 -3.003294 -2.898043 -2.805242 -2.702330 -2.605824 -2.522928 -2.400737
82 83 84 85 86 87 88 89
-2.310857 -2.223376 -2.138335 -2.043778 -1.980805 -1.873255 -1.804143 -1.723578
90 91 92 93 94 95 96 97
-1.662836 -1.583192 -1.543246 -1.439546 -1.354666 -1.291026 -1.262399 -1.224546
98 99 100+
-1.253693 -1.272024 -1.253302
$bx1
50 51 52 53 54 55
0.109658844 0.047873209 0.036834894 0.016524002 0.012761338 0.017569759
56 57 58 59 60 61
-0.005533893 0.023754004 0.026469942 0.006825326 0.015416679 0.008054662
62 63 64 65 66 67
0.028587738 0.016402451 0.020166842 0.023793875 0.023225184 0.025701837
68 69 70 71 72 73
0.025043304 0.021976668 0.023693098 0.021583006 0.020694512 0.019268260
74 75 76 77 78 79
0.018242279 0.016793247 0.016959929 0.018339228 0.016473895 0.016101602
80 81 82 83 84 85
0.016035836 0.015730764 0.014872955 0.014828706 0.014422307 0.013920957
86 87 88 89 90 91
0.014015661 0.013941010 0.013707746 0.013120237 0.011179747 0.011013119
92 93 94 95 96 97
0.012843697 0.009849051 0.015523186 0.016599591 0.011720309 0.017548979
98 99 100
0.011636377 0.022116443 0.026117600
$kt
Time Series:
Start = 1983
End = 2003
Frequency = 1
1983 1984 1985 1986 1987 1988
13.7351468 11.9879431 12.3306557 9.7472977 10.7719269 8.8984391
1989 1990 1991 1992 1993 1994
6.7188685 5.1040278 5.8563274 4.3595468 3.3966200 0.6527240
1995 1996 1997 1998 1999 2000
-0.7496111 -3.8151063 -10.4424258 -8.3492618 -10.1963816 -15.1888032
2001 2002 2003
-12.3653698 -16.0094000 -16.4431642
attr(,"class")
[1] "coef"
Warning message:
In lca(dd.cmi.pens, interp = TRUE, adjust = "none") :
Replacing zero values with estimates
50 51 52 53 54 55 56 57 58 59
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
60 61 62 63 64 65 66 67 68 69
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
70 71 72 73 74 75 76 77 78 79
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
80 81 82 83 84 85 86 87 88 89
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
90 91 92 93 94 95 96 97 98 99
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
100 101 102 103 104 105 106 107 108
0.0431 -2.1938 -2.6187 -3.1566 -3.7254 -3.3821 -3.7093 -4.8758 -4.3386
Warning message:
In modlc$ax - mod6g$ax :
longer object length is not a multiple of shorter object length
50 51 52 53 54 55 56 57 58 59
0.0211 0.0141 0.0131 0.0121 0.0080 0.0128 0.0092 0.0092 -0.0016 0.0011
60 61 62 63 64 65 66 67 68 69
0.0050 0.0007 0.0049 0.0048 0.0059 0.0042 0.0044 0.0044 0.0043 0.0037
70 71 72 73 74 75 76 77 78 79
0.0033 0.0030 0.0031 0.0030 0.0029 0.0026 0.0023 0.0028 0.0028 0.0023
80 81 82 83 84 85 86 87 88 89
0.0028 0.0025 0.0026 0.0021 0.0019 0.0021 0.0022 0.0023 0.0017 0.0020
90 91 92 93 94 95 96 97 98 99
0.0017 0.0012 0.0009 0.0019 0.0024 0.0023 0.0007 0.0045 0.0030 0.0039
100 101 102 103 104 105 106 107 108
0.0100 0.1075 0.0481 0.0282 -0.0130 -0.0055 -0.0233 0.0159 -0.0198
Warning message:
In modlc$bx - mod6g$bx :
longer object length is not a multiple of shorter object length
Time Series:
Start = 1983
End = 2003
Frequency = 1
[1] -1.1770 -1.2322 -1.2443 -1.4224 -1.0046 -1.6144 -1.5137 -1.5358 -0.8667
[10] -3.1944 -3.9221 -3.5117 -1.9017 -2.5550 -1.9025 -0.6584 0.1459 0.6083
[19] 4.6862 11.9366 11.8800
Original sample: Mortality data for CMI
Series: male
Years: 1983 - 2003
Ages: 50 - 108
Applied sample: Mortality data for CMI (Corrected: interpolate)
Series: male
Years: 1983 - 2003
Ages: 60 - 100
Fitting model: [ M = a(x)+b0(x)*i(t-x)+b1(x)*k(t) ]
- with poisson error structure and with deaths as weights -
Note: 0 cells have 0/NA deaths and 0 have 0/NA exposure
out of a total of 861 data cells.
Automated start: initial values by glm fitting of factors i(t-x)+k(t).
Starting values are:
coh coh.c age age.c bx1.c bx0.c per per.c
1 1883 0.000 60 -3.962 1 1 1983 0
2 1884 0.000 61 -4.15 1 1 1984 -0.035
3 1885 0.000 62 -4.307 1 1 1985 -0.03
4 1886 0.304 63 -4.394 1 1 1986 -0.08
5 1887 0.180 64 -4.117 1 1 1987 -0.067
6 1888 0.315 65 -4.083 1 1 1988 -0.105
7 1889 0.294 66 -4.042 1 1 1989 -0.142
8 1890 0.126 67 -3.952 1 1 1990 -0.173
9 1891 0.230 68 -3.831 1 1 1991 -0.161
10 1892 0.157 69 -3.725 1 1 1992 -0.186
11 1893 0.132 70 -3.599 1 1 1993 -0.198
12 1894 0.170 71 -3.436 1 1 1994 -0.249
13 1895 0.168 72 -3.346 1 1 1995 -0.267
14 1896 0.236 73 -3.228 1 1 1996 -0.316
15 1897 0.218 74 -3.115 1 1 1997 -0.425
16 1898 0.188 75 -3.002 1 1 1998 -0.391
17 1899 0.216 76 -2.902 1 1 1999 -0.417
18 1900 0.204 77 -2.81 1 1 2000 -0.494
19 1901 0.233 78 -2.707 1 1 2001 -0.443
20 1902 0.242 79 -2.604 1 1 2002 -0.503
21 1903 0.245 80 -2.519 1 1 2003 -0.496
22 1904 0.241 81 -2.399 1 1
23 1905 0.237 82 -2.311 1 1
24 1906 0.250 83 -2.225 1 1
25 1907 0.235 84 -2.141 1 1
26 1908 0.255 85 -2.045 1 1
27 1909 0.261 86 -1.979 1 1
28 1910 0.278 87 -1.874 1 1
29 1911 0.241 88 -1.807 1 1
30 1912 0.256 89 -1.725 1 1
31 1913 0.264 90 -1.664 1 1
32 1914 0.282 91 -1.59 1 1
33 1915 0.294 92 -1.552 1 1
34 1916 0.302 93 -1.44 1 1
35 1917 0.298 94 -1.359 1 1
36 1918 0.301 95 -1.299 1 1
37 1919 0.205 96 -1.284 1 1
38 1920 0.280 97 -1.245 1 1
39 1921 0.277 98 -1.217 1 1
40 1922 0.254 99 -1.171 1 1
41 1923 0.260 100 -1.113 1 1
42 1924 0.259
43 1925 0.280
44 1926 0.217
45 1927 0.231
46 1928 0.146
47 1929 0.158
48 1930 0.196
49 1931 0.138
50 1932 0.162
51 1933 0.056
52 1934 0.099
53 1935 0.148
54 1936 0.307
55 1937 0.137
56 1938 0.046
57 1939 -0.069
58 1940 0.141
59 1941 0.000
60 1942 0.000
61 1943 0.000
Iterative fit:
#iter Dev non-conv
1 1227.449 0
2 986.0854 0
3 964.5798 0
4 960.5438 0
5 958.1041 0
6 956.3613 0
7 955.0191 0
8 953.9481 0
9 953.0771 0
10 952.3605 0
11 951.7657 0
12 951.2685 0
13 950.85 0
14 950.4958 0
15 950.1942 0
16 949.9359 0
17 949.7138 0
18 949.5217 0
19 949.3549 0
20 949.2094 0
21 949.0821 0
22 948.9703 0
23 948.8717 0
24 948.7845 0
25 948.7072 0
26 948.6385 0
27 948.5771 0
28 948.5223 0
Iterations finished in: 28 steps
Updated values are:
coh coh.c age age.c bx1.c bx0.c per per.c
1 1883 0.00000 60 -3.96154 0.037 -0.00448 1983 0
2 1884 0.00000 61 -4.14969 0.02236 -0.01502 1984 -1.49152
3 1885 0.00000 62 -4.30729 0.03346 0.02249 1985 -1.19678
4 1886 6.92759 63 -4.39403 0.02561 0.03314 1986 -3.3779
5 1887 6.02592 64 -4.11729 0.02747 0.04587 1987 -2.76767
6 1888 10.13724 65 -4.08299 0.0254 0.02975 1988 -4.39501
7 1889 9.33845 66 -4.04168 0.02472 0.02919 1989 -5.97238
8 1890 2.83107 67 -3.95211 0.02645 0.0308 1990 -7.33723
9 1891 7.30243 68 -3.83069 0.02534 0.02963 1991 -6.85736
10 1892 5.68111 69 -3.72485 0.0223 0.0252 1992 -7.95591
11 1893 5.72384 70 -3.59892 0.0248 0.02714 1993 -8.51816
12 1894 6.14052 71 -3.43584 0.02372 0.02472 1994 -10.71434
13 1895 6.26127 72 -3.34567 0.02371 0.02482 1995 -11.52257
14 1896 9.16262 73 -3.2284 0.02269 0.02316 1996 -13.64346
15 1897 8.62478 74 -3.11466 0.02197 0.02237 1997 -18.49419
16 1898 7.83571 75 -3.00222 0.02112 0.02071 1998 -17.10461
17 1899 8.72667 76 -2.902 0.02268 0.02242 1999 -18.17228
18 1900 8.56060 77 -2.80974 0.02444 0.02406 2000 -21.675
19 1901 9.90668 78 -2.70686 0.02272 0.02224 2001 -19.4563
20 1902 10.44614 79 -2.60402 0.02239 0.02136 2002 -22.16247
21 1903 10.50051 80 -2.51932 0.02258 0.02151 2003 -22.11091
22 1904 10.28958 81 -2.39878 0.0224 0.0216
23 1905 10.24765 82 -2.31114 0.02146 0.02076
24 1906 10.66324 83 -2.22452 0.02233 0.02198
25 1907 10.10317 84 -2.14145 0.02206 0.02227
26 1908 11.00683 85 -2.0452 0.02228 0.02231
27 1909 11.22488 86 -1.97936 0.02243 0.02256
28 1910 11.90284 87 -1.87386 0.02262 0.02332
29 1911 10.28136 88 -1.80746 0.02263 0.0243
30 1912 10.93245 89 -1.72537 0.02205 0.02411
31 1913 11.26405 90 -1.66449 0.01948 0.02205
32 1914 12.01282 91 -1.58971 0.01962 0.02319
33 1915 12.39963 92 -1.55192 0.02372 0.02904
34 1916 12.63777 93 -1.43969 0.01905 0.02304
35 1917 12.28156 94 -1.35939 0.03034 0.0384
36 1918 12.35943 95 -1.29881 0.03039 0.03871
37 1919 8.42015 96 -1.28441 0.01912 0.02546
38 1920 11.49265 97 -1.24456 0.03365 0.04627
39 1921 11.38516 98 -1.21707 0.0204 0.02246
40 1922 10.61342 99 -1.17088 0.03658 0.03639
41 1923 10.77340 100 -1.11272 0.02447 0.01069
42 1924 10.74548
43 1925 11.64160
44 1926 8.92009
45 1927 9.28655
46 1928 6.19111
47 1929 6.70444
48 1930 8.13581
49 1931 6.68522
50 1932 7.61467
51 1933 3.65491
52 1934 5.81890
53 1935 8.15327
54 1936 12.51210
55 1937 7.81903
56 1938 6.93578
57 1939 -1.01942
58 1940 23.09438
59 1941 0.00000
60 1942 0.00000
61 1943 0.00000
total sums are:
b0 b1 itx kt
1.0000 1.0000 505.3191 -224.9261
Warning messages:
1: A total of 1 0/NA central mortality rates are re-estimated by the "interpolate" method.
2: In lca.rh(dd.cmi.pens, age = 60:100, mod = "m", interpolate = TRUE, :
The cohorts outside [1886, 1940] were zero weighted (clipped).
3: In dpois(y, mu, log = TRUE) : non-integer x = 3.269435
ilc documentation built on May 2, 2019, 5:07 a.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
A purpose-built regression routine to fit any of the six variants of the class of Lee-Carter model structures using an iterative Newton-Raphson fitting method.
Usage
1 2 3 4 5 |
Arguments
dat |
source data object of |
year |
vector of years to be included in the regression (all available years by default) |
age |
vector of ages to be included in the regression (all available ages by default) |
series |
numerical index corresponding to the target series to be used from the source data |
max.age |
highest age to be used in the regression |
dec.conv |
number of decimal places used to achieve convergence. The lower the value the faster the convergence of the fitting algorithm. |
clip |
number of marginal birth cohorts to exclude from the regression (i.e., give 0 weights). It is only applicable to the first 5 models (see below) |
error |
type of error structure of the model choice (Poisson distribution of the errors by default) |
model |
a character (see usage) or a numeric value (1-6) to specify the model choice |
restype |
types of residuals, which also controls the type of the fitted value.
Thus, in the cases of |
scale |
logical, if TRUE, re-scale the interaction parameters so that the k_t has drift parameter equal to 1 (see also |
interpolate |
logical, if TRUE, replace before regression all zero or missing values in the mortality rates of |
verbose |
logical, if TRUE, the program prints out the updated deviance values along with the starting and final parameter estimates |
spar |
numerical smoothing spline parameter in the interval (0,1] (with a recommended value of 0.6). If it is not NULL, the interaction effects (i.e. β_x^{(0,1)}) are smoothed out after the initial regression. Consequently, the period and/or cohort effects are adjusted (smoothed out) accordingly. |
Details
Implements the modelling approach proposed in Renshaw and Haberman (2006), which extends the basic Lee-Carter model within the GLM framework. The function makes use of tailored iterative Newton-Raphson fitting algorithms to estimate the graduation parameters of the six variants within this class of extended Lee-Carter models.
Value
A Lee-Carter type fitted object with the following components:
label |
data label |
age |
vector of fitted ages |
year |
vector of fitted fitted years |
<series> |
matrix of observed (source) mortality rates used for fitting. It is named the same way as the chosen series |
ax |
parameter estimates of (mean) age-specific mortality rates across the entire fitting period |
bx |
parameter estimates of age-specific interaction effect between age and period |
kt |
parameter estimates of year-specific period trend of mortality rates |
df |
degree of freedom of the fitted GLM model |
residuals |
residuals of the fitted model in the form of a functional time series object |
fitted |
fitted values of the fitted model in the form of a functional time series object |
varprop |
percent of variance |
y |
source mortality data in the form of a functional time series object |
mdev |
mean deviance of total and base lack of fit (see also |
model |
string expression of the fitted model |
adjust |
type of error structure (e.g. "poisson" or "gaussian") |
call |
copy of the R call to the model |
conv.iter |
number of iterations used to reach convergence |
bx0 |
parameter estimates of age-specific interaction effect between age and cohort (only applies to the age-period-cohort model) |
itx |
parameter estimates of year-specific cohort trend of mortality rates (only applies to the age-period-cohort model) |
Author(s)
Zoltan Butt, Steven Haberman and Han Lin Shang
References
Renshaw, A. E. and Haberman, S. (2003a), “Lee-Carter mortality forecasting: a parallel generalised linear modelling approach for England and Wales mortality projections", Journal of the Royal Statistical Society, Series C, 52(1), 119-137.
Renshaw, A. E. and Haberman, S. (2003b), “Lee-Carter mortality forecasting with age specific enhancement", Insurance: Mathematics and Economics, 33, 255-272.
Renshaw, A. E. and Haberman, S. (2006), “A cohort-based extension to the Lee-Carter model for mortality reduction factors", Insurance: Mathematics and Economics, 38, 556-570.
Renshaw, A. E. and Haberman, S. (2008), “On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling", Insurance: Mathematics and Economics, 42(2), 797-816.
Renshaw, A. E. and Haberman, S. (2009), “On age-period-cohort parametric mortality rate projections", Insurance: Mathematics and Economics, 45(2), 255-270.
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | # standard LC model with Gaussian errors (corresponding to SVD graduation):
# correct 0 or missing mortality rates before graduation
mod6g <- lca.rh(dd.cmi.pens, mod='lc', error='gauss', max=110, interpolate=TRUE)
# AP LC model with Poisson errors
mod6p <- lca.rh(dd.cmi.pens, mod='lc', error='pois', interpolate=TRUE)
# Model Summary, Coefficients and Plotting:
mod6p; coef(mod6p); plot(mod6p)
# Comparison with standard fitting method
# Standard LC model (with Gaussian errors) - SVD fit (demography package)
modlc <- lca(dd.cmi.pens, interp=TRUE, adjust='none')
# Gaussian (SVD) - Gaussian (iterative)
round(modlc$ax-mod6g$ax, 4)
round(modlc$bx-mod6g$bx, 4)
round(modlc$kt-mod6g$kt, 4)
# -------------------------------------------------- #
# APC LC model fitted to restricted age range with 'deviance' residuals
# the remaining 0/NA values reestimated:
# WARNING: for proper fit recommend dec=6, but it can lead to slow convergence!
mod1 <- lca.rh(dd.cmi.pens, age=60:100, mod='m', interpolate=TRUE, res='dev', dec=1)
|
Example output
Loading required package: demography
Loading required package: forecast
This is demography 1.20
Loading required package: rainbow
Loading required package: MASS
Loading required package: pcaPP
Loading required package: date
Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl_init' failed, running with rgl.useNULL = TRUE
3: .onUnload failed in unloadNamespace() for 'rgl', details:
call: fun(...)
error: object 'rgl_quit' not found
Original sample: Mortality data for CMI
Series: male
Years: 1983 - 2003
Ages: 50 - 108
Applied sample: Mortality data for CMI (Corrected: interpolate)
Series: male
Years: 1983 - 2003
Ages: 50 - 108
Fitting model: [ LC = a(x)+b1(x)*k(t) ]
- with gaussian error structure -
Note: 109 cells have 0/NA deaths and 0 have 0/NA exposure
out of a total of 1239 data cells.
Starting values are:
age age.c bx1.c per per.c
1 50 -3.462 0.017 1983 0
2 51 -4.095 0.017 1984 0
3 52 -4.404 0.017 1985 0
4 53 -4.615 0.017 1986 0
5 54 -4.480 0.017 1987 0
6 55 -4.504 0.017 1988 0
7 56 -4.770 0.017 1989 0
8 57 -4.576 0.017 1990 0
9 58 -4.814 0.017 1991 0
10 59 -4.502 0.017 1992 0
11 60 -4.652 0.017 1993 0
12 61 -4.627 0.017 1994 0
13 62 -4.637 0.017 1995 0
14 63 -4.394 0.017 1996 0
15 64 -4.117 0.017 1997 0
16 65 -4.083 0.017 1998 0
17 66 -4.042 0.017 1999 0
18 67 -3.952 0.017 2000 0
19 68 -3.831 0.017 2001 0
20 69 -3.725 0.017 2002 0
21 70 -3.599 0.017 2003 0
22 71 -3.436 0.017
23 72 -3.346 0.017
24 73 -3.228 0.017
25 74 -3.115 0.017
26 75 -3.002 0.017
27 76 -2.902 0.017
28 77 -2.810 0.017
29 78 -2.707 0.017
30 79 -2.604 0.017
31 80 -2.519 0.017
32 81 -2.399 0.017
33 82 -2.311 0.017
34 83 -2.225 0.017
35 84 -2.141 0.017
36 85 -2.045 0.017
37 86 -1.979 0.017
38 87 -1.874 0.017
39 88 -1.807 0.017
40 89 -1.725 0.017
41 90 -1.664 0.017
42 91 -1.590 0.017
43 92 -1.552 0.017
44 93 -1.440 0.017
45 94 -1.359 0.017
46 95 -1.299 0.017
47 96 -1.284 0.017
48 97 -1.245 0.017
49 98 -1.276 0.017
50 99 -1.261 0.017
51 100 -1.315 0.017
52 101 -1.268 0.017
53 102 -1.476 0.017
54 103 -1.247 0.017
55 104 -0.889 0.017
56 105 -1.098 0.017
57 106 -0.795 0.017
58 107 0.105 0.017
59 108 -0.237 0.017
Iterative fit:
#iter Dev non-conv
1 251.9952 0
2 116.7591 0
3 113.7653 0
4 113.6234 0
5 113.6131 0
6 113.6124 0
7 113.6123 0
8 113.6123 0
Iterations finished in: 8 steps
Updated values are:
age age.c bx1.c per per.c
1 50 -3.46190 0.09761 1983 17.71325
2 51 -4.09482 0.03689 1984 17.8319
3 52 -4.40400 0.02929 1985 17.04362
4 53 -4.61490 0.02405 1986 13.65193
5 54 -4.48012 0.01451 1987 13.78856
6 55 -4.50413 0.03160 1988 11.94677
7 56 -4.77048 0.00670 1989 10.52122
8 57 -4.57601 0.02193 1990 9.06468
9 58 -4.81427 0.01332 1991 8.60141
10 59 -4.50233 -0.00138 1992 6.17926
11 60 -4.65201 0.01202 1993 -0.93023
12 61 -4.62690 0.00793 1994 1.10729
13 62 -4.63747 0.03909 1995 -1.74929
14 63 -4.39403 0.00933 1996 -4.68995
15 64 -4.11729 0.02059 1997 -9.25547
16 65 -4.08299 0.01571 1998 -7.27498
17 66 -4.04168 0.01426 1999 -11.05124
18 67 -3.95211 0.01691 2000 -15.99149
19 68 -3.83069 0.01697 2001 -19.78654
20 69 -3.72485 0.01442 2002 -27.60037
21 70 -3.59892 0.01720 2003 -29.12033
22 71 -3.43584 0.01535
23 72 -3.34567 0.01441
24 73 -3.22840 0.01314
25 74 -3.11466 0.01253
26 75 -3.00222 0.01140
27 76 -2.90200 0.01226
28 77 -2.80974 0.01309
29 78 -2.70686 0.01101
30 79 -2.60402 0.01118
31 80 -2.51932 0.01082
32 81 -2.39878 0.01084
33 82 -2.31114 0.00992
34 83 -2.22452 0.01023
35 84 -2.14145 0.01059
36 85 -2.04520 0.00914
37 86 -1.97936 0.00932
38 87 -1.87386 0.00920
39 88 -1.80746 0.00938
40 89 -1.72537 0.00886
41 90 -1.66449 0.00760
42 91 -1.58971 0.00711
43 92 -1.55192 0.00784
44 93 -1.43969 0.00711
45 94 -1.35939 0.01022
46 95 -1.29881 0.01039
47 96 -1.28441 0.00648
48 97 -1.24456 0.01226
49 98 -1.27575 0.00832
50 99 -1.26096 0.01686
51 100 -1.31451 0.01010
52 101 -1.26812 0.01127
53 102 -1.47610 0.00288
54 103 -1.24738 0.01420
55 104 -0.88950 0.04912
56 105 -1.09801 0.02799
57 106 -0.79478 0.06768
58 107 0.10536 0.00000
59 108 -0.23740 0.05096
total sums are:
b0 b1 itx kt
0 1 0 0
Warning message:
A total of 152 0/NA central mortality rates are re-estimated by the "interpolate" method.
Original sample: Mortality data for CMI
Series: male
Years: 1983 - 2003
Ages: 50 - 108
Applied sample: Mortality data for CMI (Corrected: interpolate)
Series: male
Years: 1983 - 2003
Ages: 50 - 100
Fitting model: [ LC = a(x)+b1(x)*k(t) ]
- with poisson error structure and with deaths as weights -
Note: 45 cells have 0/NA deaths and 45 have 0/NA exposure
out of a total of 1071 data cells.
Starting values are:
age age.c bx1 bx1.c per per.c
1 50 -3.462 50 0.02 1983 0
2 51 -4.095 51 0.02 1984 0
3 52 -4.404 52 0.02 1985 0
4 53 -4.615 53 0.02 1986 0
5 54 -4.480 54 0.02 1987 0
6 55 -4.504 55 0.02 1988 0
7 56 -4.770 56 0.02 1989 0
8 57 -4.576 57 0.02 1990 0
9 58 -4.814 58 0.02 1991 0
10 59 -4.502 59 0.02 1992 0
11 60 -4.652 60 0.02 1993 0
12 61 -4.627 61 0.02 1994 0
13 62 -4.637 62 0.02 1995 0
14 63 -4.394 63 0.02 1996 0
15 64 -4.117 64 0.02 1997 0
16 65 -4.083 65 0.02 1998 0
17 66 -4.042 66 0.02 1999 0
18 67 -3.952 67 0.02 2000 0
19 68 -3.831 68 0.02 2001 0
20 69 -3.725 69 0.02 2002 0
21 70 -3.599 70 0.02 2003 0
22 71 -3.436 71 0.02
23 72 -3.346 72 0.02
24 73 -3.228 73 0.02
25 74 -3.115 74 0.02
26 75 -3.002 75 0.02
27 76 -2.902 76 0.02
28 77 -2.810 77 0.02
29 78 -2.707 78 0.02
30 79 -2.604 79 0.02
31 80 -2.519 80 0.02
32 81 -2.399 81 0.02
33 82 -2.311 82 0.02
34 83 -2.225 83 0.02
35 84 -2.141 84 0.02
36 85 -2.045 85 0.02
37 86 -1.979 86 0.02
38 87 -1.874 87 0.02
39 88 -1.807 88 0.02
40 89 -1.725 89 0.02
41 90 -1.664 90 0.02
42 91 -1.590 91 0.02
43 92 -1.552 92 0.02
44 93 -1.440 93 0.02
45 94 -1.359 94 0.02
46 95 -1.299 95 0.02
47 96 -1.284 96 0.02
48 97 -1.245 97 0.02
49 98 -1.276 98 0.02
50 99 -1.261 99 0.02
51 100+ -1.271 100 0.02
Iterative fit:
#iter Dev non-conv
1 11291.24 0
2 2097.723 0
3 1284.158 0
4 1256.141 0
5 1254.86 0
6 1254.775 0
7 1254.765 0
8 1254.763 0
9 1254.762 0
10 1254.762 0
11 1254.762 0
12 1254.762 0
13 1254.762 0
14 1254.762 0
Iterations finished in: 14 steps
Updated values are:
age age.c bx1 bx1.c per per.c
1 50 -3.66478 50 0.10966 1983 13.73515
2 51 -4.19899 51 0.04787 1984 11.98794
3 52 -4.63348 52 0.03683 1985 12.33066
4 53 -4.81217 53 0.01652 1986 9.7473
5 54 -4.66409 54 0.01276 1987 10.77193
6 55 -4.57660 55 0.01757 1988 8.89844
7 56 -4.91748 56 -0.00553 1989 6.71887
8 57 -4.49245 57 0.02375 1990 5.10403
9 58 -4.61074 58 0.02647 1991 5.85633
10 59 -4.46297 59 0.00683 1992 4.35955
11 60 -4.48528 60 0.01542 1993 3.39662
12 61 -4.55880 61 0.00805 1994 0.65272
13 62 -4.49516 62 0.02859 1995 -0.74961
14 63 -4.38858 63 0.01640 1996 -3.81511
15 64 -4.01845 64 0.02017 1997 -10.44243
16 65 -4.08439 65 0.02379 1998 -8.34926
17 66 -4.04477 66 0.02323 1999 -10.19638
18 67 -3.94810 67 0.02570 2000 -15.1888
19 68 -3.82265 68 0.02504 2001 -12.36537
20 69 -3.72129 69 0.02198 2002 -16.0094
21 70 -3.59880 70 0.02369 2003 -16.44316
22 71 -3.44047 71 0.02158
23 72 -3.34554 72 0.02069
24 73 -3.22789 73 0.01927
25 74 -3.11083 74 0.01824
26 75 -3.00329 75 0.01679
27 76 -2.89804 76 0.01696
28 77 -2.80524 77 0.01834
29 78 -2.70233 78 0.01647
30 79 -2.60582 79 0.01610
31 80 -2.52293 80 0.01604
32 81 -2.40074 81 0.01573
33 82 -2.31086 82 0.01487
34 83 -2.22338 83 0.01483
35 84 -2.13834 84 0.01442
36 85 -2.04378 85 0.01392
37 86 -1.98080 86 0.01402
38 87 -1.87325 87 0.01394
39 88 -1.80414 88 0.01371
40 89 -1.72358 89 0.01312
41 90 -1.66284 90 0.01118
42 91 -1.58319 91 0.01101
43 92 -1.54325 92 0.01284
44 93 -1.43955 93 0.00985
45 94 -1.35467 94 0.01552
46 95 -1.29103 95 0.01660
47 96 -1.26240 96 0.01172
48 97 -1.22455 97 0.01755
49 98 -1.25369 98 0.01164
50 99 -1.27202 99 0.02212
51 100+ -1.25330 100 0.02612
total sums are:
b0 b1 itx kt
0 1 0 0
Warning messages:
1: In lca.set(dat, year, age, series, max.age, interpolate) :
=> data above age 100 are grouped.
2: A total of 62 0/NA central mortality rates are re-estimated by the "interpolate" method.
3: In lca.set(dat, year, age, series, max.age, interpolate) :
There are 45 cells with 0/NA exposures, which are ignored in the current analysis.
Try reducing the maximum age or choosing a different age range.
Alternatively, fit LC model with error= "gaussian" .
------------------------------------------------------------
Iterative Lee-Carter Family Regression:
Fitted Model: LC = a(x)+b1(x)*k(t)
------------------------------------------------------------
Call:
lca.rh(dat = dd.cmi.pens, error = "pois", model = "lc", interpolate = TRUE)
Error Structure: poisson
Data Source: CMI [male] over
calendar years: (1983 - 2003) and ages: (50 - 100)
Deviance convergence in: 14 iterations
dev dev.c df df.c
1 Mean deviance base 1.386 df base 905
2 Mean deviance total 1.733 df tot 969
$ax
50 51 52 53 54 55 56 57
-3.664778 -4.198990 -4.633480 -4.812166 -4.664094 -4.576603 -4.917484 -4.492450
58 59 60 61 62 63 64 65
-4.610738 -4.462966 -4.485284 -4.558802 -4.495163 -4.388582 -4.018450 -4.084386
66 67 68 69 70 71 72 73
-4.044770 -3.948101 -3.822646 -3.721286 -3.598799 -3.440473 -3.345543 -3.227889
74 75 76 77 78 79 80 81
-3.110830 -3.003294 -2.898043 -2.805242 -2.702330 -2.605824 -2.522928 -2.400737
82 83 84 85 86 87 88 89
-2.310857 -2.223376 -2.138335 -2.043778 -1.980805 -1.873255 -1.804143 -1.723578
90 91 92 93 94 95 96 97
-1.662836 -1.583192 -1.543246 -1.439546 -1.354666 -1.291026 -1.262399 -1.224546
98 99 100+
-1.253693 -1.272024 -1.253302
$bx1
50 51 52 53 54 55
0.109658844 0.047873209 0.036834894 0.016524002 0.012761338 0.017569759
56 57 58 59 60 61
-0.005533893 0.023754004 0.026469942 0.006825326 0.015416679 0.008054662
62 63 64 65 66 67
0.028587738 0.016402451 0.020166842 0.023793875 0.023225184 0.025701837
68 69 70 71 72 73
0.025043304 0.021976668 0.023693098 0.021583006 0.020694512 0.019268260
74 75 76 77 78 79
0.018242279 0.016793247 0.016959929 0.018339228 0.016473895 0.016101602
80 81 82 83 84 85
0.016035836 0.015730764 0.014872955 0.014828706 0.014422307 0.013920957
86 87 88 89 90 91
0.014015661 0.013941010 0.013707746 0.013120237 0.011179747 0.011013119
92 93 94 95 96 97
0.012843697 0.009849051 0.015523186 0.016599591 0.011720309 0.017548979
98 99 100
0.011636377 0.022116443 0.026117600
$kt
Time Series:
Start = 1983
End = 2003
Frequency = 1
1983 1984 1985 1986 1987 1988
13.7351468 11.9879431 12.3306557 9.7472977 10.7719269 8.8984391
1989 1990 1991 1992 1993 1994
6.7188685 5.1040278 5.8563274 4.3595468 3.3966200 0.6527240
1995 1996 1997 1998 1999 2000
-0.7496111 -3.8151063 -10.4424258 -8.3492618 -10.1963816 -15.1888032
2001 2002 2003
-12.3653698 -16.0094000 -16.4431642
attr(,"class")
[1] "coef"
Warning message:
In lca(dd.cmi.pens, interp = TRUE, adjust = "none") :
Replacing zero values with estimates
50 51 52 53 54 55 56 57 58 59
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
60 61 62 63 64 65 66 67 68 69
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
70 71 72 73 74 75 76 77 78 79
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
80 81 82 83 84 85 86 87 88 89
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
90 91 92 93 94 95 96 97 98 99
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
100 101 102 103 104 105 106 107 108
0.0431 -2.1938 -2.6187 -3.1566 -3.7254 -3.3821 -3.7093 -4.8758 -4.3386
Warning message:
In modlc$ax - mod6g$ax :
longer object length is not a multiple of shorter object length
50 51 52 53 54 55 56 57 58 59
0.0211 0.0141 0.0131 0.0121 0.0080 0.0128 0.0092 0.0092 -0.0016 0.0011
60 61 62 63 64 65 66 67 68 69
0.0050 0.0007 0.0049 0.0048 0.0059 0.0042 0.0044 0.0044 0.0043 0.0037
70 71 72 73 74 75 76 77 78 79
0.0033 0.0030 0.0031 0.0030 0.0029 0.0026 0.0023 0.0028 0.0028 0.0023
80 81 82 83 84 85 86 87 88 89
0.0028 0.0025 0.0026 0.0021 0.0019 0.0021 0.0022 0.0023 0.0017 0.0020
90 91 92 93 94 95 96 97 98 99
0.0017 0.0012 0.0009 0.0019 0.0024 0.0023 0.0007 0.0045 0.0030 0.0039
100 101 102 103 104 105 106 107 108
0.0100 0.1075 0.0481 0.0282 -0.0130 -0.0055 -0.0233 0.0159 -0.0198
Warning message:
In modlc$bx - mod6g$bx :
longer object length is not a multiple of shorter object length
Time Series:
Start = 1983
End = 2003
Frequency = 1
[1] -1.1770 -1.2322 -1.2443 -1.4224 -1.0046 -1.6144 -1.5137 -1.5358 -0.8667
[10] -3.1944 -3.9221 -3.5117 -1.9017 -2.5550 -1.9025 -0.6584 0.1459 0.6083
[19] 4.6862 11.9366 11.8800
Original sample: Mortality data for CMI
Series: male
Years: 1983 - 2003
Ages: 50 - 108
Applied sample: Mortality data for CMI (Corrected: interpolate)
Series: male
Years: 1983 - 2003
Ages: 60 - 100
Fitting model: [ M = a(x)+b0(x)*i(t-x)+b1(x)*k(t) ]
- with poisson error structure and with deaths as weights -
Note: 0 cells have 0/NA deaths and 0 have 0/NA exposure
out of a total of 861 data cells.
Automated start: initial values by glm fitting of factors i(t-x)+k(t).
Starting values are:
coh coh.c age age.c bx1.c bx0.c per per.c
1 1883 0.000 60 -3.962 1 1 1983 0
2 1884 0.000 61 -4.15 1 1 1984 -0.035
3 1885 0.000 62 -4.307 1 1 1985 -0.03
4 1886 0.304 63 -4.394 1 1 1986 -0.08
5 1887 0.180 64 -4.117 1 1 1987 -0.067
6 1888 0.315 65 -4.083 1 1 1988 -0.105
7 1889 0.294 66 -4.042 1 1 1989 -0.142
8 1890 0.126 67 -3.952 1 1 1990 -0.173
9 1891 0.230 68 -3.831 1 1 1991 -0.161
10 1892 0.157 69 -3.725 1 1 1992 -0.186
11 1893 0.132 70 -3.599 1 1 1993 -0.198
12 1894 0.170 71 -3.436 1 1 1994 -0.249
13 1895 0.168 72 -3.346 1 1 1995 -0.267
14 1896 0.236 73 -3.228 1 1 1996 -0.316
15 1897 0.218 74 -3.115 1 1 1997 -0.425
16 1898 0.188 75 -3.002 1 1 1998 -0.391
17 1899 0.216 76 -2.902 1 1 1999 -0.417
18 1900 0.204 77 -2.81 1 1 2000 -0.494
19 1901 0.233 78 -2.707 1 1 2001 -0.443
20 1902 0.242 79 -2.604 1 1 2002 -0.503
21 1903 0.245 80 -2.519 1 1 2003 -0.496
22 1904 0.241 81 -2.399 1 1
23 1905 0.237 82 -2.311 1 1
24 1906 0.250 83 -2.225 1 1
25 1907 0.235 84 -2.141 1 1
26 1908 0.255 85 -2.045 1 1
27 1909 0.261 86 -1.979 1 1
28 1910 0.278 87 -1.874 1 1
29 1911 0.241 88 -1.807 1 1
30 1912 0.256 89 -1.725 1 1
31 1913 0.264 90 -1.664 1 1
32 1914 0.282 91 -1.59 1 1
33 1915 0.294 92 -1.552 1 1
34 1916 0.302 93 -1.44 1 1
35 1917 0.298 94 -1.359 1 1
36 1918 0.301 95 -1.299 1 1
37 1919 0.205 96 -1.284 1 1
38 1920 0.280 97 -1.245 1 1
39 1921 0.277 98 -1.217 1 1
40 1922 0.254 99 -1.171 1 1
41 1923 0.260 100 -1.113 1 1
42 1924 0.259
43 1925 0.280
44 1926 0.217
45 1927 0.231
46 1928 0.146
47 1929 0.158
48 1930 0.196
49 1931 0.138
50 1932 0.162
51 1933 0.056
52 1934 0.099
53 1935 0.148
54 1936 0.307
55 1937 0.137
56 1938 0.046
57 1939 -0.069
58 1940 0.141
59 1941 0.000
60 1942 0.000
61 1943 0.000
Iterative fit:
#iter Dev non-conv
1 1227.449 0
2 986.0854 0
3 964.5798 0
4 960.5438 0
5 958.1041 0
6 956.3613 0
7 955.0191 0
8 953.9481 0
9 953.0771 0
10 952.3605 0
11 951.7657 0
12 951.2685 0
13 950.85 0
14 950.4958 0
15 950.1942 0
16 949.9359 0
17 949.7138 0
18 949.5217 0
19 949.3549 0
20 949.2094 0
21 949.0821 0
22 948.9703 0
23 948.8717 0
24 948.7845 0
25 948.7072 0
26 948.6385 0
27 948.5771 0
28 948.5223 0
Iterations finished in: 28 steps
Updated values are:
coh coh.c age age.c bx1.c bx0.c per per.c
1 1883 0.00000 60 -3.96154 0.037 -0.00448 1983 0
2 1884 0.00000 61 -4.14969 0.02236 -0.01502 1984 -1.49152
3 1885 0.00000 62 -4.30729 0.03346 0.02249 1985 -1.19678
4 1886 6.92759 63 -4.39403 0.02561 0.03314 1986 -3.3779
5 1887 6.02592 64 -4.11729 0.02747 0.04587 1987 -2.76767
6 1888 10.13724 65 -4.08299 0.0254 0.02975 1988 -4.39501
7 1889 9.33845 66 -4.04168 0.02472 0.02919 1989 -5.97238
8 1890 2.83107 67 -3.95211 0.02645 0.0308 1990 -7.33723
9 1891 7.30243 68 -3.83069 0.02534 0.02963 1991 -6.85736
10 1892 5.68111 69 -3.72485 0.0223 0.0252 1992 -7.95591
11 1893 5.72384 70 -3.59892 0.0248 0.02714 1993 -8.51816
12 1894 6.14052 71 -3.43584 0.02372 0.02472 1994 -10.71434
13 1895 6.26127 72 -3.34567 0.02371 0.02482 1995 -11.52257
14 1896 9.16262 73 -3.2284 0.02269 0.02316 1996 -13.64346
15 1897 8.62478 74 -3.11466 0.02197 0.02237 1997 -18.49419
16 1898 7.83571 75 -3.00222 0.02112 0.02071 1998 -17.10461
17 1899 8.72667 76 -2.902 0.02268 0.02242 1999 -18.17228
18 1900 8.56060 77 -2.80974 0.02444 0.02406 2000 -21.675
19 1901 9.90668 78 -2.70686 0.02272 0.02224 2001 -19.4563
20 1902 10.44614 79 -2.60402 0.02239 0.02136 2002 -22.16247
21 1903 10.50051 80 -2.51932 0.02258 0.02151 2003 -22.11091
22 1904 10.28958 81 -2.39878 0.0224 0.0216
23 1905 10.24765 82 -2.31114 0.02146 0.02076
24 1906 10.66324 83 -2.22452 0.02233 0.02198
25 1907 10.10317 84 -2.14145 0.02206 0.02227
26 1908 11.00683 85 -2.0452 0.02228 0.02231
27 1909 11.22488 86 -1.97936 0.02243 0.02256
28 1910 11.90284 87 -1.87386 0.02262 0.02332
29 1911 10.28136 88 -1.80746 0.02263 0.0243
30 1912 10.93245 89 -1.72537 0.02205 0.02411
31 1913 11.26405 90 -1.66449 0.01948 0.02205
32 1914 12.01282 91 -1.58971 0.01962 0.02319
33 1915 12.39963 92 -1.55192 0.02372 0.02904
34 1916 12.63777 93 -1.43969 0.01905 0.02304
35 1917 12.28156 94 -1.35939 0.03034 0.0384
36 1918 12.35943 95 -1.29881 0.03039 0.03871
37 1919 8.42015 96 -1.28441 0.01912 0.02546
38 1920 11.49265 97 -1.24456 0.03365 0.04627
39 1921 11.38516 98 -1.21707 0.0204 0.02246
40 1922 10.61342 99 -1.17088 0.03658 0.03639
41 1923 10.77340 100 -1.11272 0.02447 0.01069
42 1924 10.74548
43 1925 11.64160
44 1926 8.92009
45 1927 9.28655
46 1928 6.19111
47 1929 6.70444
48 1930 8.13581
49 1931 6.68522
50 1932 7.61467
51 1933 3.65491
52 1934 5.81890
53 1935 8.15327
54 1936 12.51210
55 1937 7.81903
56 1938 6.93578
57 1939 -1.01942
58 1940 23.09438
59 1941 0.00000
60 1942 0.00000
61 1943 0.00000
total sums are:
b0 b1 itx kt
1.0000 1.0000 505.3191 -224.9261
Warning messages:
1: A total of 1 0/NA central mortality rates are re-estimated by the "interpolate" method.
2: In lca.rh(dd.cmi.pens, age = 60:100, mod = "m", interpolate = TRUE, :
The cohorts outside [1886, 1940] were zero weighted (clipped).
3: In dpois(y, mu, log = TRUE) : non-integer x = 3.269435
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