Description Usage Arguments Details Value References See Also Examples
Fit univariate penalized composite link model (PCLM) to ungroup binned count data, e.g. age-at-death distributions grouped in age classes.
1 2 3 4 5 6 7 8 9 10 |
x |
Vector containing the starting values of the input intervals/bins.
For example: if we have 3 bins |
y |
Vector with counts to be ungrouped. It must have the same dimension
as |
nlast |
Length of the last interval. In the example above |
offset |
Optional offset term to calculate smooth mortality rates. A vector of the same length as x and y. See \insertCiterizzi2015;textualungroup for further details. |
out.step |
Length of estimated intervals in output. Values between 0.1 and 1 are accepted. Default: 1. |
ci.level |
Level of significance for computing confidence intervals.
Default: |
verbose |
Logical value. Indicates whether a progress bar should be
shown or not.
Default: |
control |
List with additional parameters:
|
The PCLM method is based on the composite link model, which extends standard generalized linear models. It implements the idea that the observed counts, interpreted as realizations from Poisson distributions, are indirect observations of a finer (ungrouped) but latent sequence. This latent sequence represents the distribution of expected means on a fine resolution and has to be estimated from the aggregated data. Estimates are obtained by maximizing a penalized likelihood. This maximization is performed efficiently by a version of the iteratively reweighted least-squares algorithm. Optimal values of the smoothing parameter are chosen by minimizing Bayesian or Akaike's Information Criterion.
The output is a list with the following components:
input |
A list with arguments provided in input. Saved for convenience. |
fitted |
The fitted values of the PCLM model. |
ci |
Confidence intervals around fitted values. |
goodness.of.fit |
A list containing goodness of fit measures: standard errors, AIC and BIC. |
smoothPar |
Estimated smoothing parameters: |
bins.definition |
Additional values to identify the bins limits and location in input and output objects. |
deep |
A list of objects created in the fitting process. Useful in diagnosis of possible issues. |
call |
An unevaluated function call, that is, an unevaluated expression which consists of the named function applied to the given arguments. |
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 | # Data
x <- c(0, 1, seq(5, 85, by = 5))
y <- c(294, 66, 32, 44, 170, 284, 287, 293, 361, 600, 998,
1572, 2529, 4637, 6161, 7369, 10481, 15293, 39016)
offset <- c(114, 440, 509, 492, 628, 618, 576, 580, 634, 657,
631, 584, 573, 619, 530, 384, 303, 245, 249) * 1000
nlast <- 26 # the size of the last interval
# Example 1 ----------------------
M1 <- pclm(x, y, nlast)
ls(M1)
summary(M1)
fitted(M1)
plot(M1)
# Example 2 ----------------------
# ungroup even in smaller intervals
M2 <- pclm(x, y, nlast, out.step = 0.5)
head(fitted(M1))
plot(M1, type = "s")
# Note, in example 1 we are estimating intervals of length 1. In example 2
# we are estimating intervals of length 0.5 using the same aggregate data.
# Example 3 ----------------------
# Do not optimise smoothing parameters; choose your own. Faster.
M3 <- pclm(x, y, nlast, out.step = 0.5,
control = list(lambda = 100, kr = 10, deg = 10))
plot(M3)
summary(M2)
summary(M3) # not the smallest BIC here, but sometimes is not important.
# Example 4 -----------------------
# Grouped x & grouped offset (estimate death rates)
M4 <- pclm(x, y, nlast, offset)
plot(M4, type = "s")
# Example 5 -----------------------
# Grouped x & ungrouped offset (estimate death rates)
ungroupped_Ex <- pclm(x, y = offset, nlast, offset = NULL)$fitted # ungroupped offset data
M5 <- pclm(x, y, nlast, offset = ungroupped_Ex)
|
Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl.init' failed, running with 'rgl.useNULL = TRUE'.
[1] "bin.definition" "call" "ci" "deep"
[5] "fitted" "goodness.of.fit" "input" "smoothPar"
Penalized Composite Link Model (PCLM)
Call:
pclm(x = x, y = y, nlast = nlast)
PCLM Type : Univariate
Number of input groups : 19
Number of fitted values : 111
Length of estimate bins : 1
Smoothing parameter lambda : 0
B-splines intervals/knot (kr): 2
B-splines degree (deg) : 3
AIC : 39.97
BIC : 59.81
[0,1) [1,2) [2,3) [3,4) [4,5) [5,6)
292.254945 47.567040 12.031104 5.101512 3.653694 3.801854
[6,7) [7,8) [8,9) [9,10) [10,11) [11,12)
4.815457 6.352663 7.702012 8.349355 8.188017 7.844892
[12,13) [13,14) [14,15) [15,16) [16,17) [17,18)
7.985346 9.025702 11.568904 16.221280 23.582754 33.474380
[18,19) [19,20) [20,21) [21,22) [22,23) [23,24)
43.922241 52.410476 56.932803 57.933128 57.243741 56.233473
[24,25) [25,26) [26,27) [27,28) [28,29) [29,30)
55.838792 56.123535 56.825906 57.606178 58.115022 58.277037
[30,31) [31,32) [32,33) [33,34) [34,35) [35,36)
58.166052 58.010385 58.102700 58.696747 60.036110 62.332101
[36,37) [37,38) [38,39) [39,40) [40,41) [41,42)
65.807514 70.655621 77.070029 85.156572 94.920752 106.255362
[42,43) [43,44) [44,45) [45,46) [46,47) [47,48)
118.899308 132.628986 147.270502 162.850021 179.626932 197.884008
[48,49) [49,50) [50,51) [51,52) [52,53) [53,54)
217.908603 239.747591 263.142589 287.749013 313.174561 339.648841
[54,55) [55,56) [56,57) [57,58) [58,59) [59,60)
368.253616 401.033766 441.270637 492.262928 557.117381 637.375129
[60,61) [61,62) [62,63) [63,64) [64,65) [65,66)
731.255969 833.938220 936.476422 1029.409950 1105.860969 1163.069720
[66,67) [67,68) [68,69) [69,70) [70,71) [71,72)
1204.488941 1235.701765 1263.485202 1294.268889 1333.474545 1385.809083
[72,73) [73,74) [74,75) [75,76) [76,77) [77,78)
1454.810931 1543.184019 1651.740928 1780.116411 1925.536316 2084.766185
[78,79) [79,80) [80,81) [81,82) [82,83) [83,84)
2255.232116 2435.332561 2626.617487 2830.882343 3049.000300 3278.429175
[84,85) [85,86) [86,87) [87,88) [88,89) [89,90)
3508.095452 3720.392827 3888.451678 3983.178892 3979.668950 3861.687654
[90,91) [91,92) [92,93) [93,94) [94,95) [95,96)
3628.237713 3292.607167 2882.269884 2431.775913 1977.636196 1550.974452
[96,97) [97,98) [98,99) [99,100) [100,101) [101,102)
1174.299714 859.519261 609.244616 418.991695 280.160080 182.530429
[102,103) [103,104) [104,105) [105,106) [106,107) [107,108)
116.138627 72.331820 44.198811 26.560956 15.734729 9.210650
[108,109) [109,110) [110,111)
5.340380 3.074238 1.761168
[0,1) [1,2) [2,3) [3,4) [4,5) [5,6)
292.254945 47.567040 12.031104 5.101512 3.653694 3.801854
Penalized Composite Link Model (PCLM)
Call:
pclm(x = x, y = y, nlast = nlast, out.step = 0.5)
PCLM Type : Univariate
Number of input groups : 19
Number of fitted values : 222
Length of estimate bins : 0.5
Smoothing parameter lambda : 8
B-splines intervals/knot (kr): 2
B-splines degree (deg) : 3
AIC : 39.96
BIC : 59.8
Penalized Composite Link Model (PCLM)
Call:
pclm(x = x, y = y, nlast = nlast, out.step = 0.5, control = list(lambda = 100,
kr = 10, deg = 10))
PCLM Type : Univariate
Number of input groups : 19
Number of fitted values : 222
Length of estimate bins : 0.5
Smoothing parameter lambda : 100
B-splines intervals/knot (kr): 10
B-splines degree (deg) : 10
AIC : 255.87
BIC : 267.49
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