Age-period-cohort analysis

Description

The package includes functions for age-period-cohort analysis. The statistical model is a generalized linear model (GLM) allowing for age, period and cohort factors, or a sub-set of the factors. The canonical parametrisation of Kuang, Nielsen and Nielsen (2008a) is used. The outline of an analysis is described below.

Details

The apc package uses the canonical parameters suggested by Kuang, Nielsen and Nielsen (2008a) and generalized by Nielsen (2014). These evolve around the second differences of age, period and cohort factors as well as an three parameters (level and two slopes) for a linear plane. The age, period and cohort factors themselves are not identifiable. They could be ad hoc identified by associating the levels and two slopes to the age, period and cohort factors in a particular way. This should be done with great care as such ad hoc identification easily masks which information is coming from the data and which information is coming from the choice of ad hoc identification scheme. An illustration is given below. A short description of the package can be found in Nielsen (2015).

A formal analysis of the identification of the age-period-cohort model can be found in Nielsen and Nielsen (2014). Forecasting is discussed in Kuang, Nielsen and Nielsen (2008b, 2011) and Martinez Miranda, Nielsen and Nielsen (2015). Methods for cross section data are introduced in Fannon, Monden and Nielsen (2019). Methods for panel data are introduced in Fannon (2020). For a recent overview see Fannon and Nielsen (2019). Methods for 2-sample analysis and mixed frequency age, period times scales are available for aggregate data, see Nielsen (2022a,b).

Inference. When analyzing aggregate data using an over-dispersed Poisson model or a log-normal model, inference is based on a Central Limit Theorem for infinitely divisible distributions developed in Harnau and Nielsen (2018) and Kuang and Nielsen (2020). This supports the situation where the data array has fixed dimensions but the information content in each cell is thought to be large.

The package covers age-period-cohort models for three types of data.

1. Tables of aggregate data.

2. Repeated cross sectional data.

3. Panel data.

Vignettes showing how to use the package and reproduce existing results are available on vignettes.

The apc package can be used as follows.

1. Aggregate data. For a vignette with an introduction to analysis of aggregate data, see see IntroductionAggregateData.pdf, IntroductionAggregateData.R. on vignettes.

   1. Organize the data in as an apc.data.list. Data are included in matrix format. Information needs to be given about the original data format. Optionally, information can be given about the labels for the time scales. Data in age-period format with mixed frequency is possible. That is age and period can be grouped differently. Choose options data.format and unit accordingly.

   2. Construct descriptive plots using apc.plot.data.all. This gives a series of descriptive plots. The plots can be called individually through

      1. Plot data sums using apc.plot.data.sums. Numerical values can be obtained through apc.data.sums.

      2. Sparsity plots of data using apc.plot.data.sparsity.

      3. Plot data using all combinations of two time scales using apc.plot.data.within.

   3. Get an deviance table for the age-period-cohort model through apc.fit.table. For two-sample models choose apc.fit.table.2s.

   4. Estimate a particular (sub-model of) age-period-cohort model through apc.fit.model. For two-sample models choose apc.fit.model.2s.

   5. Plot probability transforms of observed responses given fit using apc.plot.fit.pt.

   6. Plot estimated parameters through apc.plot.fit. For two-sample models choose apc.plot.fit.2s. Numerical values of certain transformations of the canonical parameter can be obtained through apc.identify. For mixed frequency data use apc.identify.mixed.

   7. Recursive analysis can be done by selecting a subset of the observations through apc.data.list.subset and then repeating analysis. This will reveal how sensitive the results are to particular age, period and cohort groups.

   8. Forecasting. Some functions have been been added for forecasting in from a Poisson response-only model with an age-cohort parametrization apc.forecast.ac and with an age-period parametrization apc.forecast.ap. See also the overview on apc.forecast

2. Repeated cross section and Panel Data. For a vignette with an introduction to analysis of repeated cross section data and panel data, see IntroductionIndividualData.pdf, IntroductionIndividualData.R on Vignettes. Further examples can be found in a second vignette, see IntroductionIndividualDataFurtherExample.pdf, IntroductionIndividualDataFurtherExample.R.

Data examples include

1. Aggregate data: 1-sample

   1. data.asbestos includes counts of deaths from mesothelioma in the UK. This dataset has no measure for exposure. It can be analysed using a Poisson model with an "APC" or an "AC" design. Source: Martinez Miranda, Nielsen and Nielsen (2015). Also used in Nielsen (2015).

   2. data.Italian.bladder.cancer includes counts of deaths from bladder cancer in the Italy. This dataset includes a measure for exposure. It can be analysed using a Poisson model with an "APC" or an "AC" design. Source: Clayton and Schifflers (1987a).

   3. data.Belgian.lung.cancer includes counts of deaths from lung cancer in the Belgium. This dataset includes a measure for exposure. It can be analysed using a Poisson model with an "APC", "AC", "AP" or "Ad" design. Source: Clayton and Schifflers (1987a).

   4. data.Japanese.breast.cancer includes counts of deaths from breast cancer in the Japan. This dataset includes a measure for exposure. It can be analysed using a Poisson model with an "APC" design. Source: Clayton and Schifflers (1987b).

2. Aggregate data: 2-sample & mixed frequency

   1. data.Swiss.suicides includes mixed-frequency counts of suicides for women and men in Switzerland. This is used as illustration in Nielsen (2022b), see vignettes. Source: Riebler, Held, Rue and Bopp (2012).

3. Repeated cross section data

   1. Wage data from the package ISLR

4. Panel data

   1. PSID7682 data from the package AER. These are panel data on earnings for 595 individuals for the years 1976-1982.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 29 Jan 2015 updated 7 July 2025.

References

Clayton, D. and Schifflers, E. (1987a) Models for temperoral variation in cancer rates. I: age-period and age-cohort models. Statistics in Medicine 6, 449-467.

Clayton, D. and Schifflers, E. (1987b) Models for temperoral variation in cancer rates. II: age-period-cohort models. Statistics in Medicine 6, 469-481.

Fannon, Z. (2020). D.Phil. thesis. University of Oxford.

Fannon, Z., Monden, C. and Nielsen, B. (2018) Age-period cohort modelling and covariates, with an application to obesity in England 2001-2014. Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2018/2018W05_obesity.pdf. Supplement Code for replication: https://www.nuffield.ox.ac.uk/economics/Papers/2018/2018W05_obesityReplication.zip.

Fannon, Z. and Nielsen, B. (2019) Age-period-cohort models. Oxford Research Encyclopedia of Economics and Finance. Oxford University Press. Download: doi:10.1093/acrefore/9780190625979.013.495. Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2018/2018W04_age_period_cohort_models.pdf.

Harnau, J. and Nielsen, B. (2018) Over-dispersed age-period-cohort models. Journal of the American Statistical Association 113, 1722-1732. Download: Article, doi:10.1080/01621459.2017.1366908. Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2017/HarnauNielsen2017apcDP.pdf.

Kuang, D. and Nielsen, B. (2020) Generalized Log-Normal Chain-Ladder. Scandinavian Actuarial Journal 2020, 553–576. Download: Open access: doi:10.1080/03461238.2019.1696885. Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2018/2018W02_KuangNielsen2018GLNCL.pdf.

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008a) Identification of the age-period-cohort model and the extended chain ladder model. Biometrika 95, 979-986. Download: doi:10.1093/biomet/asn026. Nuffield Discussion Paper: http://www.nuffield.ox.ac.uk/economics/papers/2007/w5/KuangNielsenNielsen07.pdf.

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008b) Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika 95, 987-991. Download: doi:10.1093/biomet/asn038. Nuffield Discussion Paper: http://www.nuffield.ox.ac.uk/economics/papers/2008/w9/KuangNielsenNielsen_Forecast.pdf.

Kuang, D., Nielsen, B. and Nielsen, J.P. (2011) Forecasting in an extended chain-ladder-type model. Journal of Risk and Insurance 78, 345-359. Download: doi:10.1111/j.1539-6975.2010.01395.x. Nuffield Discussion Paper: http://www.nuffield.ox.ac.uk/economics/papers/2010/w5/Forecast24jun10.pdf.

Martinez Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2015) Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality. Journal of the Royal Statistical Society A 178, 29-55. Download: doi:10.1111/rssa.12051. Nuffield Discussion paper: http://www.nuffield.ox.ac.uk/economics/papers/2013/Asbestos8mar13.pdf.

Nielsen, B. (2015) apc: An R package for age-period-cohort analysis. R Journal 7, 52-64. Download: Open access.

Nielsen, B. (2014) Deviance analysis of age-period-cohort models. Download: Nuffield Discussion paper: http://www.nuffield.ox.ac.uk/economics/papers/2014/apc_deviance.pdf.

Nielsen, B. (2022a) Age-period-cohort analysis of mixed frequency data. Download: Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2022/2022-W02apc_mixed.pdf.

Nielsen, B. (2022b) Two-sample age-period-cohort models with an application to Swiss suicide rates. Download: Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2022/2022-W03apc_2sample.pdf.

Nielsen, B. and Nielsen, J.P. (2014) Identification and forecasting in mortality models. The Scientific World Journal. vol. 2014, Article ID 347043, 24 pages. Download: doi:10.1155/2014/347043.

Riebler, A. and Held, L. and Rue, H. and Bopp, M. (2012) Gender-specific differences and the impact of family integration on time trends in age-stratified Swiss suicide rates. Journal of the Royal Statistical Society, Series A, 175, 479-490.

See Also

Vignettes are available on Vignettes.

Further information, including minor upgrades and a python version can be found on apc development web page.

Examples

#	see vignettes

Internal apc Functions

Description

Internal apc functions

Details

These are not to be called by the user.

function.coin is used the coin problem that is relevant for mixed arrays. Given coins of value G,H, it returns the Frobenius number, the Sylvester number and the number of Non-Representable amounts.

function.coin.array is used the coin problem that is relevant for mixed arrays. Given r.g G-coins and r.h H-coins, it returns an array of all possible amounts that can be paid.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 25 Jun 2025 (1 Feb 2016)

Arrange data as an apc.data.list

Description

This is step 1 of the apc analysis.

The apc package is aimed at range of data types. This analysis and labelling of parameters depends on the choice data type. In order to keep track of this choice the data first has to be arranged as an apc.data.list. The function purpose of this function is to aid the user in constructing a list with the right information.

Age period cohort analysis is used in two situations. A dose-response situation, where both doses (exposure, risk set, cases) and responses (counts of deaths, outcomes) are available. And a response situation where only a response is available. If the aim is to directly model mortality ratios (counts of death divided by exposure) this will be thought of a response

The apc.data.list gives sufficient information for the further analysis. It is sufficient to store this information. It has 2 obligatory arguments, which are a response matrix and a character indicating the data format. It also has some further optional arguments, which have certain default values. Some times it may be convenient to add further arguments to the apc.data.list. This will not affect the apc analysis.

Mixed frequency available for AP and PA formats. This is indicated through unit and by chosing data.format as APm or PAm.

apc.data.list generates default row and column names for the response and dose matrices when these are not provided by the user.

Usage

apc.data.list(response, data.format, dose=NULL,
					age1=NULL, per1=NULL, coh1=NULL, unit=NULL,
					per.zero=NULL, per.max=NULL,
					time.adjust=NULL, label=NULL,
					n.decimal=NULL, add.names.to.data.matrix=TRUE)

Arguments

Details

If the user does not set values for any of age1, per1, coh1, unit then the value is set to unit.

The user can set values of age1, per1, coh1 that are incongruent. The functions only use two these that are relevant for the chosen data.format. Example: the data.format may be "AC" and the user sets age1, per1, but age1, coh1 are relevant for this data format. The apc.data.list then sets coh1=unit, by default, while ignoring the value for per1. Other commands such as apc.data.list.subset or apc.fit.table, will internally, as default option, call the function apc.get.index. That function will, in this example, set per1 according to the values of age1 and coh1.

If the user does not set a value for time.adjust this is set equal to unit when the user does not specify at least two age1, per1, coh1. Otherwise it is set to 0. The former choice matches the values in the theory papers, where indices count 1,2,... to follow standard notation for row/column indices for matrices, so that age+coh=per+unit. The latter choice seeks to match a real time scale the user sets according to age+coh=per.

Default for unit was NULL in apc_2.0.1 and earlier and it is now 1.

Value

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 23 Jun 2025 (17 Nov 2016)

References

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008a) Identification of the age-period-cohort model and the extended chain ladder model. Biometrika 95, 979-986. Download: doi:10.1093/biomet/asn026; Earlier version Nuffield DP.

Nielsen, B. (2014) Deviance analysis of age-period-cohort models. Download: Nuffield DP.

Nielsen, B. (2015) apc: An R package for age-period-cohort analysis. R Journal 7, 52-64. Download: Open access.

See Also

The below example shows how the data.Japanese.breast.cancer data.list was generated. Other provided data sets include data.asbestos data.Belgian.lung.cancer data.Italian.bladder.cancer.

A subset of the data can be selected using apc.data.list.subset.

Examples

###############
#	Artificial data
#	(1) Generate a 5x7 matrix and make arbitrary decisions for rest

response <- matrix(data=seq(1:35),nrow=5,ncol=7)
data.list	<- apc.data.list(response=response,data.format="AP",
					age1=25,per1=1955,coh1=NULL,unit=5,
					per.zero=NULL,per.max=NULL)
data.list

#	(2) Mixed frequency data.
#	Age moves in steps of 2x3=6. Period moves in steps of 2x2=4.

data.list	<- apc.data.list(response=response,data.format="APm",
					age1=25,per1=1955,coh1=NULL,unit=c(2,3,2),
					per.zero=NULL,per.max=NULL)
data.list					

#	(3) Chain Ladder data

k			<- 5
v.response 	<- seq(1:(k*(k+1)/2))
data.list	<- apc.data.list(response=vector.2.triangle(v.response,k),
							data.format="CL",age1=2001)
data.list

###############
#	Japanese breast cancer
#	This is the code used to generate the data.Japanese.breast.cancer
v.rates		<- c( 0.44, 0.38, 0.46, 0.55, 0.68,
			 	  1.69, 1.69, 1.75, 2.31, 2.52,
				  4.01, 3.90, 4.11, 4.44, 4.80,
				  6.59, 6.57, 6.81, 7.79, 8.27,
				  8.51, 9.61, 9.96,11.68,12.51,
				 10.49,10.80,12.36,14.59,16.56,
				 11.36,11.51,12.98,14.97,17.79,
				 12.03,10.67,12.67,14.46,16.42,
				 12.55,12.03,12.10,13.81,16.46,
				 15.81,13.87,12.65,14.00,15.60,
				 17.97,15.62,15.83,15.71,16.52)
v.cases		<- c(   88,   78,  101,  127,  179,
				   299,  330,  363,  509,  588,
				   596,  680,  798,  923, 1056,
				   874,  962, 1171, 1497, 1716,
				  1022, 1247, 1429, 1987, 2398,
				  1035, 1258, 1560, 2079, 2794,
				   970, 1087, 1446, 1828, 2465,
				   820,  861, 1126, 1549, 1962,
				   678,  738,  878, 1140, 1683,
				   640,  628,  656,  900, 1162,
				   497,  463,  536,  644,  865)				 
#	see also example below for generating labels

rates	<- matrix(data=v.rates,nrow=11, ncol=5,byrow=TRUE)
cases	<- matrix(data=v.cases,nrow=11, ncol=5,byrow=TRUE)

# 	A data list is now constructed as follows
#	note that list entry rates is redundant,
#	but included since it represents original data

data.Japanese.breast.cancer	<- apc.data.list(response=cases,
			dose=cases/rates,data.format="AP",
			age1=25,per1=1955,coh1=NULL,unit=5,
			per.zero=NULL,per.max=NULL,time.adjust=0,
			label="Japanese breast cancer")

#	or when exploiting the default values

data.Japanese.breast.cancer	<- apc.data.list(response=cases,
			dose=cases/rates,data.format="AP",
			age1=25,per1=1955,unit=5,
			label="Japanese breast cancer")

###################################################
# 	Code for generating labels

row.names <- paste(as.character(seq(25,75,by=5)),"-",as.character(seq(29,79,by=5)),sep="")
col.names <- paste(as.character(seq(1955,1975,by=5)),"-",as.character(seq(1959,1979,by=5)),sep="")

Cut age, period and cohort groups from data set.

Description

For a recursive analysis it is useful to be able to cut age, period and cohort groups from a data set. Function returns an apc.data.list with data.format "trapezoid".

When used with default values the function turns an apc.data.list into a new apc.data.list with data.format "trapezoid" without reducing dataset.

Usage

apc.data.list.subset(apc.data.list,
							age.cut.lower=0,age.cut.upper=0,
							per.cut.lower=0,per.cut.upper=0,
							coh.cut.lower=0,coh.cut.upper=0,
							apc.index=NULL,
							suppress.warning=FALSE)

Arguments

Value

Arguments: Notes

If apc.index is supplied then the input can be simplified. It suffices to write apc.data.list = list(response=response,data.format=data.format,dose=dose), where dose could be dose=NULL. Likewise apc.index does not need to be a full apc.index list. It suffices to construct a list with entries age.max, per.max, coh.max, age1, per1, coh1, unit, per.zero, index.trap, index.data.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 4 Dec 2013 recoded 26 Apr 2017

See Also

The below example uses artificial data. For an example using data.asbestos see apc.plot.fit.

Examples

###############
#	Artificial data
#	Generate a 5x7 matrix and make arbitrary decisions for rest

response <- matrix(data=seq(1:35),nrow=5,ncol=7)
data.list	<- list(response=response,dose=NULL,data.format="AP",
					age1=25,per1=1955,coh1=NULL,unit=5,
					per.zero=NULL,per.max=NULL,time.adjust=0)
data.list

apc.data.list.subset(data.list,1,1,0,0,0,0)

Computes age, period and cohort sums of a matrix

Description

Computes age, period and cohort sums of a matrix. This is the same as taking column, row and diagonal sums. The match between the age, period and cohort sums and column, row and diagonal sums depends on the data format

Usage

apc.data.sums(apc.data.list,data.type="r",
			average=FALSE,keep.incomplete=TRUE,apc.index=NULL)

Arguments

Value

Arguments: Notes

If apc.index is supplied then the input can be simplified. For instance if data.type="r" then, for the first argument, it suffices to write apc.data.list = list(response=response). Likewise apc.index does not need to be a full apc.index list. It suffices to construct a list with entries age.max, per.max, coh.max, index.trap, index.data, per.zero.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 15 Aug 2018 (15 Dec 2013)

See Also

The example below uses Japanese breast cancer data, see data.Japanese.breast.cancer

Examples

#####################
#  EXAMPLE with artificial data
#  generate a 3x4 matrix in "AP" data.format with the numbers 1..12

m.data  	<- matrix(data=seq(length.out=12),nrow=3,ncol=4)
m.data
data.list	<- apc.data.list(m.data,"AP")
apc.data.sums(data.list)

#	$sums.age
#	 [1] 22 26 30
#	$sums.per
#	[1]  6 15 24 33
#	$sums.coh
#	[1]  3  8 15 24 18 10

apc.data.sums(data.list,average=TRUE)
#	$sums.age
#	[1] 5.5 6.5 7.5
#	$sums.per
#	[1]  2  5  8 11
#	$sums.coh
#	[1]  3  4  5  8  9 10

apc.data.sums(data.list,keep.incomplete=FALSE)
#	$sums.age
#	 [1] 22 26 30
#	$sums.per
#	[1]  6 15 24 33
#	$sums.coh
#	[1]  NA NA 15 24 NA NA

#####################
#	EXAMPLE with Japanese breast cancer data

data.list	<- data.Japanese.breast.cancer()	#	function gives data list
apc.data.sums(data.list)

#	$sums.age
#	[1]  573 2089 4053 6220 8083 8726 7796 6318 5117 3986 3005
#	$sums.per
#	[1]  7519  8332 10064 13183 16868
#	$sums.coh
#	[1]  497 1103 1842 2858 4474 5550 6958 7471 7531 6931 5111 3080 1666  715  179

#	Compare with the response matrix

data.list$response

#	      1955-1959 1960-1964 1965-1969 1970-1974 1975-1979
#	25-29        88        78       101       127       179
#	30-34       299       330       363       509       588
#	35-39       596       680       798       923      1056
#	40-44       874       962      1171      1497      1716
#	45-49      1022      1247      1429      1987      2398
#	50-54      1035      1258      1560      2079      2794
#	55-59       970      1087      1446      1828      2465
#	60-64       820       861      1126      1549      1962
#	65-69       678       738       878      1140      1683
#	70-74       640       628       656       900      1162
#	75-79       497       463       536       644       865

Fits an age period cohort model

Description

apc.fit.model fits the age period cohort model as a Generalized Linear Model using glm.fit. The model is parametrised in terms of the canonical parameter introduced by Kuang, Nielsen and Nielsen (2008), see also the implementation in Martinez Miranda, Nielsen and Nielsen (2015). This parametrisation has a number of advantages: it is freely varying, it is the canonical parameter of a regular exponential family, and it is invariant to extentions of the data matrix. Other parametrizations can be computed using apc.identify.

apc.fit.model can be be used for all three age period cohort factors, or for submodels with fewer of these factors.

apc.fit.model can be used either for mortality rates through a dose-response model or for mortality counts through a pure response model without doses/exposures.

The GLM families include Poisson regressions (with log link) and Normal/Gaussian least squares regressions.

apc.fit.table produces a deviance table for 15 combinations of the three factors and linear trends: "APC", "AP", "AC", "PC", "Ad", "Pd", "Cd", "A", "P", "C", "t", "tA", "tP", "tC", "1".

The functions can now handle mixed frequency data. Such data are coded through apc.data.list. For details, see Nielsen (2022a).

Usage

	apc.fit.model(apc.data.list,model.family,model.design,apc.index=NULL,
			replicate.version.1.3.1=FALSE)
		apc.fit.table(apc.data.list,model.family,model.design.reference="APC",
			apc.index=NULL,digits=3)

Arguments

Value

apc.fit.table produces a deviance table. There are 15 rows corresponding to all possible design choices. The columns are as follows.

apc.fit.model returns a list. The entries are as follows.

Note

For gaussian families deviance is defined differently in apc and glm. Here it is -2 log likelihood. In glm it is RSS.

The values for apc.fit.model include the apc.data.list and the apc.index returned by apc.get.index.

For the poisson.response the inference is conditional on the level, see Martinez Miranda, Nielsen and Nielsen (2015). The coefficients.canonical computed by apc are therefore different from the default coefficients computed by glm.

For the od.poisson.response an asymptotic theory is used that mimics the conditioning for poisson.response. The asymptotic distribution are, however, asymptotically t or F distributed, see Harnau and Nielsen (2017).

For the log.normal.response standard normal theory applies for quantities on the log scale including estimators. An asymptotic theory for quantities on the original scale is provided in Kuang and Nielsen (2018).

For coefficients the 3rd and 4th columns have headings t value and Pr(>|t|) for od.poisson.response to indicate an asymptotic t theory and otherwise z value and Pr(>|z|) to indicate an asymptotic normal theory. The labels are inherited from glm.fit.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 28 Jun 2025 (27 Aug 2014)

References

Harnau, J. and Nielsen (2016) Over-dispersed age-period-cohort models. To appear in Journal of the American Statistical Association. Download: Nuffield DP

Kuang, D, Nielsen B (2018) Generalized log-normal chain-ladder. mimeo Nuffield Collge.

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008a) Identification of the age-period-cohort model and the extended chain ladder model. Biometrika 95, 979-986. Download: doi:10.1093/biomet/asn026; Earlier version Nuffield DP.

Martinez Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2015) Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality. Journal of the Royal Statistical Society A 178, 29-55. Download: doi:10.1111/rssa.12051, Nuffield DP.

Nielsen, B. (2022a) Age-period-cohort analysis of mixed frequency data. Download Nuffield DP.

See Also

The fit is done using glm.fit.

The examples below use Italian bladder cancer data, see data.Italian.bladder.cancer and Belgian lung cancer data, see data.Belgian.lung.cancer.

In example 3 the design matrix is called is called using apc.get.design.

Examples

#####################
#	EXAMPLE 1 with Italian bladder cancer data

data.list	<- data.Italian.bladder.cancer()	#	function gives data list
apc.fit.table(data.list,"poisson.dose.response")

#	       -2logL df.residual prob(>chi_sq) LR.vs.APC df.vs.APC prob(>chi_sq)       aic
#	APC    33.179          27         0.191        NA        NA            NA   487.624
#	AP    512.514          40         0.000   479.335        13         0.000   940.958
#	AC     39.390          30         0.117     6.211         3         0.102   487.835
#	PC   1146.649          36         0.000  1113.470         9         0.000  1583.094
#	Ad    518.543          43         0.000   485.364        16         0.000   940.988
#	Pd   4041.373          49         0.000  4008.194        22         0.000  4451.818
#	Cd   1155.629          39         0.000  1122.450        12         0.000  1586.074
#	A    2223.800          44         0.000  2190.621        17         0.000  2644.245
#	P   84323.944          50         0.000 84290.765        23         0.000 84732.389
#	C   23794.205          40         0.000 23761.026        13         0.000 24222.650
#	t    4052.906          52         0.000  4019.727        25         0.000  4457.351
#	tA   5825.158          53         0.000  5791.979        26         0.000  6227.602
#	tP  84325.758          53         0.000 84292.579        26         0.000 84728.203
#	tC  33446.796          53         0.000 33413.617        26         0.000 33849.241
#	1   87313.678          54         0.000 87280.499        27         0.000 87714.123
#
#	Table suggests that "APC" and "AC" fit equally well.  Try both

fit.apc	<- apc.fit.model(data.list,"poisson.dose.response","APC")
fit.ac	<- apc.fit.model(data.list,"poisson.dose.response","AC")

#	Compare the estimates: They are very similar

fit.apc$coefficients.canonical
fit.ac$coefficients.canonical

#####################
#	EXAMPLE 2 with Belgian lung cancer data
#	This example illustrates how to find the linear predictors 

data.list	<- data.Belgian.lung.cancer()

#	Get an APC fit

fit.apc	<- apc.fit.model(data.list,"poisson.dose.response","APC")

#	The linear predictor of the fit is a vector.
#	But, we would like it in the same format as the data.
#	Thus create matrix of same dimension as response data
#	This can be done in two ways

m.lp	<- data.list$response	#	using original information	
m.lp	<- fit.apc$response		# 	using information copied when fitting

#	the fit object index.data is used to fill linear predictor in
#	vector format into matrix format

m.lp[fit.apc$index.data]	<-fit.apc$linear.predictors
exp(m.lp)

#####################
#	EXAMPLE 3 with Belgian lung cancer data
#	This example illustrates how apc.fit.model works.

data.list	<- data.Belgian.lung.cancer()

#	Vectorise data
index		<- apc.get.index(data.list)
v.response	<- data.list$response[index$index.data]
v.dose		<- data.list$dose[index$index.data]

#	Get design
m.design	<- apc.get.design(index,"APC")$design

#	Fit using glm.fit from stats package
fit.apc.glm	<- glm.fit(m.design,v.response,family=poisson(link="log"),offset=log(v.dose))

#	Get canonical coefficients
v.cc		<- fit.apc.glm$coefficients

#	Find linear predictors and express in matrix form
m.fit		<- data.list$response			#	create matrix
m.fit[index$index.data]		<- m.design 
m.fit.offset		<- m.fit + log(data.list$dose)	#	add offset
exp(m.fit.offset)

#	Compare with linear.predictors from glm.fit
#	difference should be zero
sum(abs(m.fit.offset[index$index.data]-fit.apc.glm$linear.predictors))

#####################
#	EXAMPLE 4 with Taylor-Ashe loss data
#	This example illustrates the over-dispersed poisson response model.

data <- data.loss.TA()
fit.apc.od <- apc.fit.model(data,"od.poisson.response","APC")
fit.apc.od$coefficients.canonical[1:5,]
fit.apc.no.od <- apc.fit.model(data,"poisson.response","APC")
fit.apc.no.od$coefficients.canonical[1:5,]

Fits an age period cohort model for 2 samples

Description

Generalizes apc.fit.model to a 2 sample model. For an application, see the vignette ReproducingN2025.pdf, ReproducingN2025.R on vignette.

Usage

 apc.fit.model.2s(apc.data.list.1,apc.data.list.2,model.family,
            model.design.common="APC",model.design.difference="APC",
            gls.weight=c(1,1),apc.index=NULL,time.series=NULL)
apc.fit.table.2s(apc.data.list.1,apc.data.list.2,model.family,
            restrict="difference",model.design.reference.common="APC",
            model.design.reference.difference="APC",
            gls.weight=c(1,1),time.series=NULL,digits=3)

Arguments

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 7 Jul 2025

References

Nielsen, B. (2022) Two-sample age-period-cohort models with an application to Swiss suicide rates. Download: Nuffield Discussion Paper 2022-W03.

Forecasts from age-period-cohort models.

Description

In general forecasts from age-period-cohort models require extrapolation of the estimated parameters. This has to be done without introducing identifications problems, see Kuang, Nielsen and Nielsen (2008b,2011). There are many different possibilities for extrapolation for the different sub-models. The extrapolation results in point forecasts. Distribution forecasts should be build on top of these, see Martinez Miranda, Nielsen and Nielsen (2015) and Harnau and Nielsen (2016). At present three experimental functions apc.forecast.ac, apc.forecast.apc and apc.forecast.ap are available.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 10 Sep 2016 (1 Feb 2016)

References

Harnau, J. and Nielsen (2016) Over-dispersed age-period-cohort models. To appear in Journal of the American Statistical Association. Download: Nuffield DP

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008b) Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika 95, 987-991. Download: doi:10.1093/biomet/asn038; Earlier version Nuffield DP.

Kuang, D., Nielsen B. and Nielsen J.P. (2011) Forecasting in an extended chain-ladder-type model. Journal of Risk and Insurance 78, 345-359. Download: doi:10.1111/j.1539-6975.2010.01395.x; Earlier version: Nuffield DP.

Martinez Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2015) Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality. Journal of the Royal Statistical Society A 178, 29-55. Download: doi:10.1111/rssa.12051, Nuffield DP.

Forecast for responses model with AC or CL structure.

Description

Computes forecasts for a model with AC or Chain Ladder structure. Forecasts of the linear predictor are given for all models. Distributions forecasts are provided for a Poisson response model (using Martinez Miranda, Nielsen and Nielsen, 2015), for an over-dispersed Poisson response model (using Harnau and Nielsen, 2017) and for a log normal response model (using Kuang and Nielsen, 2018) This is done for the triangle which shares age and cohort indices with the data.

Usage

apc.forecast.ac(apc.fit,sum.per.by.age=NULL,
			sum.per.by.coh=NULL, quantiles=NULL, suppress.warning=TRUE)

Arguments

Details

The default output only reports standard errors. By setting the argument quantiles to, for instance, quantiles=c(0.05,0.50,0.95) forecast quantiles are reported.

Poisson response forecast errors. The asymptotic theory for the Poisson forecast standard errors is presented in Martinez Miranda, Nielsen and Nielsen (2015). The sampling theory is based on multinomial model, conditional on the total number of outcomes. Asymptotically this gives a normal theory. There are two independent contributions to the forecast error: a process error and an estimation error. The empirical example of that paper uses the data data.asbestos. The results of that paper are reproduced in the vignette ReproducingMMNN2015.pdf, ReproducingMMNN2015.R on Vignettes.

Overdispersed Poisson response forecast errors. The asymptotic theory for the overdispersed Poission forecast standard errors is presented in Harnau and Nielsen (2018). The sampling theory is based on infinitely devisible distributions, with the compound Poisson distribution as a special case. This results in scale nuisance parameter, which is estimated by the deviance of the AC model divided by the degrees of freedom. Asymptotically this gives a t/F theory. There are three independent contributions to the forecast error: a process error, an estimation error and a sampling error for the overall mean.

Generalized log normal forecast errors. Uses the asymptotic theory present in Kuang and Nielsen (2018). The sampling theory is based on infinitely devisible distributions, using small sigma asymptotics. There are two independent contributions to the forecast error: a process error and an estimation error.

The examples below are based on the smaller data reserving sets data.loss.VNJ, data.loss.TA. See also data.loss.XL.

Value

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 18 November 2019 (2 Mar 2016)

References

Harnau, J. and Nielsen (2018) Over-dispersed age-period-cohort models. Journal of the American Statistical Association 113, 1722-1732. Download: Nuffield DP

Martinez Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2015) Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality. Journal of the Royal Statistical Society A 178, 29-55. Download: doi:10.1111/rssa.12051, Nuffield DP.

Martinez Miranda, M.D., Nielsen, B., Nielsen, J.P. and Verrall, R. (2011) Cash flow simulation for a model of outstanding liabilities based on claim amounts and claim numbers. ASTIN Bulletin 41, 107-129.

Kuang, D, Nielsen B (2018) Generalized log-normal chain-ladder. mimeo Nuffield Collge.

See Also

The example below uses Japanese breast cancer data, see data.Japanese.breast.cancer

Examples

#####################
#  EXAMPLE with reserving data: data.loss.VNJ()
#  Data used in Martinez Miranda, Nielsen, Nielsen and Verrall (2011)
#  Point forecasts are the Chain-Ladder forecasts	
#  *NOTE* Data are over-dispersed,
#		so distribution forecast are *NOT* reliable
#  The same could be done data.asbestos(),
#		which are not over-dispersed
#		see vignette.

data	<- data.loss.VNJ()
fit.ac	<- apc.fit.model(data,"poisson.response","AC")
forecast	<- apc.forecast.ac(fit.ac)

#	forecasts by "policy-year"
forecast$response.forecast.coh
#	          forecast         se    se.proc     se.est
#	coh_2     1684.763   57.69067   41.04586   40.53949
#	coh_3    29379.085  220.53214  171.40328  138.76362
#	coh_4    60637.929  313.33867  246.24770  193.76066
#	coh_5   101157.697  385.69930  318.05298  218.18857
#	coh_6   173801.522  501.42184  416.89510  278.60786
#	coh_7   249348.589  595.21937  499.34816  323.94060
#	coh_8   475991.739  864.06580  689.92155  520.20955
#	coh_9   763918.643 1182.70450  874.02440  796.78810
#	coh_10 1459859.526 2216.80272 1208.24647 1858.58945

#	forecasts of "cash-flow"
forecast$response.forecast.per
#	reproduces Table 6 of MMNNV (2011)
#	         forecast         se    se.proc   se.est
#	per_11 1353858.32 1456.92459 1163.55417 876.7958
#	per_12  754180.12 1017.37629  868.43544 529.9758
#	per_13  488612.42  816.62860  699.00817 422.2202
#	per_14  318043.00  664.36135  563.95302 351.1880
#	per_15  184610.86  508.97704  429.66366 272.8494
#	per_16  115022.56  414.64945  339.14976 238.5615
#	per_17   63145.15  320.93564  251.28700 199.6360
#	per_18   35812.79  255.08766  189.24267 171.0466
#	per_19    2494.27   78.10439   49.94266  60.0502

#	forecast  of "total reserve"
#	reproduces Table 6 of MMNNV (2011)
forecast$response.forecast.all
#	    forecast       se  se.proc   se.est
#	all  3315779 3182.737 1820.928 2610.371

#####################
#	Forecast of cashflows for 7th cohort (policy year)
#	Note a series of warnings are given because 
#		this is done by truncating the data
#		which generates the warnings associated
#		with apc.data.list.subset()
forecast<- apc.forecast.ac(fit.ac,sum.per.by.coh=7)
forecast$response.forecast.per.by.coh
#	         forecast        se   se.proc    se.est
#	per_11 102975.337 355.97444 320.89771 154.08590
#	per_12  58061.306 267.24671 240.95914 115.58329
#	per_13  40466.866 226.40049 201.16378 103.87646
#	per_14  21615.765 170.90637 147.02301  87.13910
#	per_15  24410.927 194.70158 156.23997 116.17994
#	per_16   1818.389  61.09857  42.64257  43.75668
#
#	This can also be intercept corrected
#		Such intercept corrections are useful when
#		analysing data.asbestos().
#		Unclear if they are useful for
#		reserving.
forecast$intercept.correction.per.by.coh
#   > [1] 1.241798
forecast$response.forecast.per.by.coh.ic
#	         forecast        se   se.proc    se.est
#	per_11 127874.573 355.97444 320.89771 154.08590
#	per_12  72100.417 267.24671 240.95914 115.58329
#	per_13  50251.675 226.40049 201.16378 103.87646
#	per_14  26842.415 170.90637 147.02301  87.13910
#	per_15  30313.441 194.70158 156.23997 116.17994
#	per_16   2258.071  61.09857  42.64257  43.75668

#####################
#	Forecast of cashflows cumulated for
#		6th and 7th cohort (policy year)
forecast<- apc.forecast.ac(fit.ac,sum.per.by.coh=c(6,7))
forecast$response.forecast.per.by.coh.ic
#	         forecast        se   se.proc    se.est
#	per_11 226219.380 460.52781 414.62816 200.42295
#	per_12 139628.153 366.48699 325.74697 167.93339
#	per_13  87022.435 295.86605 257.16360 146.29970
#	per_14  66584.160 277.64858 224.94656 162.75067
#	per_15  34962.678 206.77289 163.00324 127.22018
#	per_16   2392.759  61.09857  42.64257  43.75668

#####################
#  EXAMPLE with reserving data: data.loss.TA()
#  Data used in Harnau and Nielsen (2016)
data	<- data.loss.TA()
fit.ac	<- apc.fit.model(data,"od.poisson.response","AC")
forecast	<- apc.forecast.ac(fit.ac,quantiles=c(0.01,0.05,0.5,0.95,0.99))
forecast$response.forecast.all
#	    forecast      se se.proc  se.est  tau.est 
#	all 18680856 2675417 1007826 2474680 134561.2
#	...
#	 t-0.010  t-0.050  t-0.500  t-0.950  t-0.990 
#	12158931 14160544 18680856 23201167 25202781
#	...
#	 G-0.010  G-0.050  G-0.500  G-0.950  G-0.990 
#	12760202 14398564 18553290 23417098 25792423 
forecast$response.forecast.per

#####################
#  EXAMPLE with reserving data: data.loss.XL()
#  see helpfile for data.loss.XL 	

Forecast for Poisson response model with AP structure.

Description

Computes forecasts for a model with AP structure. The data can have any form allowed in, see apc.data.list. These are all special cases of generalised trapezoids. If the "lower triangle" with the largest (age,coh) values are not observed, they can be forecast using this function. The function extrapolates the AP model to the lower triangle where per.zero+per.max < per <= age.max+coh.max-1. The estimates of the age parameters can be used for the lower triangle. The estimates of the period parameters need to be extrapolated for the lower triangle. Thus, the function extrapolates per.forecast.J=age.max+coh.max-1-per.zero-per.max period values. The extrapolation method has to chosen so as not to introduce an identification problem, see Kuang, Nielsen and Nielsen (2008b,2011). Two such extrapolation methods are implemented in this function: "I0" and "I1". The default is to report the linear predictor.

If the model.family="binominal.dose.response", that is a logistic model, then forecasts of dose, response and survival probability are given for lower triangle.

Usage

apc.forecast.ap(apc.fit,extrapolation.type="I0",suppress.warning=TRUE)

Arguments

Details

When model.family=binomial.dose.response forecasts are made by the component method, see Cox (1976). It is intended to be used for a population analysis situation where the response equals cohort-decrease of dose. For cell in forecast array with index (age,cohort) then: Survival probability is survival=1/(1+exp(predictor_(a,c))). Dose is dose_(a,c)=max(0,dose_(a-1,c)-response_(a-1,c)). Response is response_(a,c)=dose_(a,c)*(1-survival_(a,c)).

Value

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 2 May 2016 (2 Mar 2016)

References

Cox, P.R. (1976) Demography. 5th Edition. Cambridge: Cambridge University Press. (page 324).

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008b) Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika 95, 987-991. Download: doi:10.1093/biomet/asn038; Earlier version Nuffield DP.

Kuang, D., Nielsen B. and Nielsen J.P. (2011) Forecasting in an extended chain-ladder-type model. Journal of Risk and Insurance 78, 345-359. Download: doi:10.1111/j.1539-6975.2010.01395.x; Earlier version: Nuffield DP.

Forecast models with APC structure.

Description

Computes forecasts for a model with APC structure. Forecasts of the linear predictor are given for all models. This is done for the triangle which shares age and cohort indices with the data.

Usage

apc.forecast.apc(apc.fit,extrapolation.type="I0",
suppress.warning=TRUE)

Arguments

Details

The example below is based on the smaller data reserving sets data.loss.TA.

Value

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 10 Sep 2016

References

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008b) Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika 95, 987-991. Download: doi:10.1093/biomet/asn038; Earlier version Nuffield DP.

See Also

The example below uses Taylor and Ashe reserving see data.loss.TA

Examples

#####################
#  EXAMPLE with reserving data: data.loss.TA()

data	<- data.loss.TA()
fit.apc	<- apc.fit.model(data,"poisson.response","APC")
forecast	<- apc.forecast.apc(fit.apc)

#	forecasts by "policy-year"
forecast$response.forecast.coh
#	         forecast
#	coh_2    91718.82
#	coh_3   464661.38
#	coh_4   704591.94
#	coh_5  1025337.23
#	coh_6  1503253.81
#	coh_7  2330768.44
#	coh_8  4115906.56
#	coh_9  4257958.30
#	coh_10 4567231.84
#	forecasts of "cash-flow"
forecast$response.forecast.per
#	        forecast
#	per_11 5274762.58
#	per_12 4213526.23
#	per_13 3188451.80
#	per_14 2210649.45
#	per_15 1644203.06
#	per_16 1236495.32
#	per_17  764552.75
#	per_18  444205.71
#	per_19   84581.44
#	forecast  of "total reserve"
forecast$response.forecast.all
#	    forecast
#	all 19061428

Create design matrices

Description

Functions to create the apc design matrix for the canonical parameters. Based on Nielsen (2014b), which generalises introduced by Kuang, Nielsen and Nielsen (2008). In normal use these function are needed for internal use by apc.fit.model.

The resulting function design matrix is collinear, so a sub-set of the columns have to be selected. The columns are: intercept, age/period/cohort slopes, age/period/cohort double differences. Thus, there are three slopes instead of two. Before use, one has to select which parameters are needed. This should include at either one/two of age/cohort slopes or period slope or no slope.

The functions can now handle mixed frequency data. Such data are coded through apc.data.list. For details, see Nielsen (2022a).

Usage

apc.get.design(apc.index,model.design)
apc.get.design.collinear(apc.index)

Arguments

Value

apc.get.design returns a list with

apc.get.design.collinear returns a collinear design matrix for the unrestricted "APC" model. It has an extra column. The columns 2-4 are linear trends in age, period and cohort directions. At most two of these should be used. They are selected by slopes.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 1 Mar 2015

References

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008a) Identification of the age-period-cohort model and the extended chain ladder model. Biometrika 95, 979-986. Download: doi:10.1093/biomet/asn026; Earlier version Nuffield DP.

Nielsen, B. (2014b) Deviance analysis of age-period-cohort models.

Nielsen, B. (2022a) Age-period-cohort analysis of mixed frequency data. Download Nuffield DP.

See Also

The vignette NewDesign.pdf, NewDesign.R on Vignettes.

Examples

#####################
#	EXAMPLE 1 with Belgian lung cancer data
#	This example illustrates how apc.fit.model works.

data.list	<- data.Belgian.lung.cancer()

#	Vectorise data
index		<- apc.get.index(data.list)
v.response	<- data.list$response[index$index.data]
v.dose		<- data.list$dose[index$index.data]

#	Get design
m.design.apc	<- apc.get.design(index,"APC")$design

#	Fit using glm.fit from stats package
fit.apc.glm	<- glm.fit(m.design.apc,v.response,family=poisson(link="log"),offset=log(v.dose))
fit.apc.glm$deviance

#	Compare with standard output from apc.fit.model
apc.fit.model(data.list,"poisson.dose.response","APC")$deviance

#####################
#	EXAMPLE 2 with Belgian lung cancer data
#	The age-drift model gives a good fit.
#	This fit can be refined to a cubic or quadratic age effect.
#	The latter is not precoded so one will have to work directly with the design matrix.
#	SEE ALSO VIGNETTE

data.list	<- data.Belgian.lung.cancer()

#	Vectorise data
index		<- apc.get.index(data.list)
v.response	<- data.list$response[index$index.data]
v.dose		<- data.list$dose[index$index.data]

#	Get design matrix for "Ad"
m.design.ad	<- apc.get.design(index,"Ad")$design

#	Modify design matrix for cubic or quadratic age effect
#	Note this implies a linear or constant double difference
#	Quadractic age effect: restrict double differences to be equal
p	<- ncol(m.design.ad)
m.rest.q	<- matrix(data=0,nrow=p,ncol=4)
m.rest.q[1,1]	<- 1
m.rest.q[2,2]	<- 1
m.rest.q[3,3]	<- 1
m.rest.q[4:p,4]	<- 1
m.design.adq	<- m.design.ad %*% m.rest.q
#	Cubic age effect: restrict double differences to be linear
m.rest.c	<- matrix(data=0,nrow=p,ncol=5)
m.rest.c[1,1]	<- 1
m.rest.c[2,2]	<- 1
m.rest.c[3,3]	<- 1
m.rest.c[4:p,4]	<- 1
m.rest.c[4:p,5]	<- seq(1,p-3)
m.design.adc	<- m.design.ad %*% m.rest.c

#	Poisson regression for dose-response and with log link
fit.ad	<- glm.fit(m.design.ad,v.response,family=poisson(link="log"),offset=log(v.dose))
fit.adc	<- glm.fit(m.design.adc,v.response,family=poisson(link="log"),offset=log(v.dose))
fit.adq	<- glm.fit(m.design.adq,v.response,family=poisson(link="log"),offset=log(v.dose))

#	Deviance tests
fit.adc$deviance - fit.ad$deviance 
fit.adq$deviance - fit.ad$deviance 
#	Degrees of freedom
ncol(m.design.ad) - ncol(m.design.adc)
ncol(m.design.ad) - ncol(m.design.adq)

Get indices for mapping data into trapezoid formation

Description

This function does the internal book keeping between the original data format and the trapezoid format. It creates index matrices to transform data between original format, trapezoid format and a vector, as well as values to keep track of the labels for the time scales.

The generalized trapezoids are introduced in Kuang, Nielsen and Nielsen (2008), see also Nielsen (2014).

Usage

apc.get.index(apc.data.list)

Arguments

Value

A list containing the following values.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 31 Mar 2015

References

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008a) Identification of the age-period-cohort model and the extended chain ladder model. Biometrika 95, 979-986. Download: doi:10.1093/biomet/asn026; Earlier version Nuffield DP.

Nielsen, B. (2014) Deviance analysis of age-period-cohort models. Nuffield DP.

Examples

################
#	Artificial data

###############
#	Artificial data
#	Generate a 3x5 matrix and make arbitrary decisions for rest

response <- matrix(data=seq(1:15),nrow=3,ncol=5)
data.list	<- list(response=response,dose=NULL,data.format="AP",
					age1=25,per1=1955,coh1=NULL,
					unit=5,per.zero=NULL,per.max=NULL,time.adjust=0)
apc.get.index(data.list)

Imposing hypotheses on age-period-cohort models.

Description

apc has a set of standard hypotheses that can be imposed on the age-period-cohort model. A deviance table can be found on apc.fit.table, while fits of restricted models can be found using apc.fit.model.

Other linear hypotheses can be imposed using a little bit of coding, see the vignette NewDesign.pdf, NewDesign.R on Vignettes.

For over-dispersed Poisson models for responses and no doses the theory is worked out in Harnau and Nielsen (2017).

In general forecasts from age-period-cohort models require extrapolation of the estimated parameters. This has to be done without introducing identifications problems, see Kuang, Nielsen and Nielsen (2008b,2011). There are many different possibilities for extrapolation for the different sub-models. The extrapolation results in point forecasts. Distribution forecasts should be build on top of these, see Martinez Miranda, Nielsen and Nielsen (2015) and Harnau and Nielsen (2016). At present three experimental functions apc.forecast.ac, apc.forecast.apc and apc.forecast.ap are available.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 10 Sep 2016 (1 Feb 2016)

References

Harnau, J. and Nielsen (2016) Over-dispersed age-period-cohort models. To appear in Journal of the American Statistical Association. Download: Nuffield DP

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008b) Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika 95, 987-991. Download: doi:10.1093/biomet/asn038; Earlier version Nuffield DP.

Kuang, D., Nielsen B. and Nielsen J.P. (2011) Forecasting in an extended chain-ladder-type model. Journal of Risk and Insurance 78, 345-359. Download: doi:10.1111/j.1539-6975.2010.01395.x; Earlier version: Nuffield DP.

Martinez Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2015) Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality. Journal of the Royal Statistical Society A 178, 29-55. Download: doi:10.1111/rssa.12051, Nuffield DP.

Identification of time effects

Description

Computes ad hoc identified time effects.

Usage

apc.identify(apc.fit.model)

Arguments

Details

Forms ad hoc identified time effects from the canonical parameter. These are used either indirectly by apc.plot.fit or they are computed directly with this command.

The ad hoc identifications are based on Nielsen (2014b). For details see also the vignette Identification.pdf, Identification.R on Vignettes or in the notes below.

For model designs of any type two ad hoc identified time effects.

(1) The type "sum.sum" (same as "ss.dd") gives double sums anchored in the middle of the first period diagonal.

(2) The type "detrend" gives double sums that start in zero and end in zero.

For model designs with only two time effects, that is "AC", "AP", "PC" there is a further ad hoc identification.

(3) The type "demean" gives single sums of single differences. Derived from "detrend" where the linear trends are attributed to the double sums of double differences. Level unchanged.

(4) The type "dif" gives the single differences derived from "demean". Could also have been chosen as canonical parametrisation for these models.

Value

Note

* indicates that values only implemented for designs "AC", "AP", "PC".

The differences are not identified for design "APC". An arbitrary level can be moved between differences for age, period and cohort.

The differences are not identified for designs "Ad", "Pd", "Cd". These models have two linear trends and one set of double differences. In the model "Ad", as an example, one linear trend will be associated with age, but it is arbitrary whether the second linear trend should be associated with period or cohort. The slope of the age trend will depend on that arbitrary choice. In turn the level of the age differences will be arbitrary.

(1) The type "sum.sum" (same as "ss.dd") gives double sums anchored to be zero in the three points where age=cohort=U, age=U+1,cohort=U age=U,cohort=U+1 with apc.fit.model$U and where U is the integer value of (per.zero+3)/2 This corresponds to the representation in Nielsen (2014b). The linear plane is parametrised in terms of a level, which is the value of the predictor at age=cohort=U; an age slope, which is the difference of the values of the predictor at age=U+1,cohort=U and age=cohort=U; an cohort slope, which is the difference of the values of the predictor at age=U,cohort=U+1 and age=cohort=U.

(2) The type "detrend" gives double sums that start in zero and end in zero. The linear plane is parametrised in terms of a level, which is the value of the predictor at age=cohort=1, which is usually outside the index set for the data; while age and cohort slopes are adjusted for the ad hoc identification of the time effects.

(3) Subsumes var.apc.identify from apc.indiv (25 Sep 2020)

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> & Zoe Fannon 25 Sep 2020 (12 Apr 2015)

References

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008a) Identification of the age-period-cohort model and the extended chain ladder model. Biometrika 95, 979-986. Download: doi:10.1093/biomet/asn026; Earlier version Nuffield DP.

Nielsen, B. (2014b) Deviance analysis of age-period-cohort models. Work in progress.

See Also

The vignette Identification.pdf.

Examples

########################
#	Belgian lung cancer
# 	first an example with APC design, note that demean and dif not defined.

data.list	<- data.Belgian.lung.cancer()

fit.apc	<- apc.fit.model(data.list,"poisson.dose.response","APC")
fit.apc$coefficients.canonical
id.apc	<- apc.identify(fit.apc)
id.apc$coefficients.ssdd
id.apc$coefficients.detrend
id.apc$coefficients.demean
id.apc$coefficients.dif

fit.ap	<- apc.fit.model(data.list,"poisson.dose.response","AP")
fit.ap$coefficients.canonical
id.ap	<- apc.identify(fit.ap)
id.ap$coefficients.ssdd
id.ap$coefficients.detrend
id.ap$coefficients.demean
id.ap$coefficients.dif

Identification of time effects for mixed frequency data

Description

Generalizes apc.identify to mixed frequency data.

Usage

apc.identify.mixed(apc.fit.model)

Arguments

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 3 Jul 2025

References

Nielsen, B. (2022) Age-period-cohort analysis of mixed frequency data . Download: Nuffield Discussion Paper 2022-W02.

Implements direct tests between APC models

Description

This function allows the user to directly compare any of the APC model, its submodels, or the TS model to any smaller model. For example, the function can be used to compare the TS to the Ad model or the Ad model to the A model. Comparisons are by likelihood ratio or Wald tests.

Usage

	apc.indiv.compare.direct(data, big.model, small.model, unit=1,
        			dep.var, covariates=NULL, model.family,
                    n.coh.excl.start=0, n.coh.excl.end=0,
                    n.age.excl.start=0, n.age.excl.end=0,
                    n.per.excl.start=0, n.per.excl.end=0,
                    NR.controls=NULL, test, dist,
                    wt.var=NULL, plmmodel="notplm", id.var=NULL)
        apc.indiv.waldtest.fullapc(data, dist="F", big.model="APC", 
                    small.model, dep.var, covariates=NULL,
                    model.family="gaussian", unit=1, 
                    n.coh.excl.start=0, n.coh.excl.end=0,
                    n.age.excl.start=0, n.age.excl.end=0,
                    n.per.excl.start=0, n.per.excl.end=0,
                    existing.big.model.fit=NULL, 
                    existing.small.model.fit=NULL,
                    existing.collinear=NULL,
					plmmodel = "notplm", id.var=NULL, wt.var=NULL)
        apc.indiv.waldtest.TS(data, dist="F", small.model="APC",
                    dep.var, covariates=NULL,
                    model.family="gaussian", unit=1, 
                    n.coh.excl.start=0, n.coh.excl.end = 0,
                    n.age.excl.start=0, n.age.excl.end = 0,
                    n.per.excl.start=0, n.per.excl.end = 0,
                    existing.small.model.fit=NULL, 
                    existing.big.model.fit=NULL, 
                    existing.collinear=NULL)
        apc.indiv.LRtest.fullapc(data,  big.model="APC", 
                    small.model,
                    dep.var, covariates=NULL, 
                    model.family="binomial", unit=1,
                    n.coh.excl.start=0, n.coh.excl.end=0,
                    n.age.excl.start=0, n.age.excl.end=0,
                    n.per.excl.start=0, n.per.excl.end=0,
                    existing.big.model.fit=NULL,
                    existing.small.model.fit=NULL,
                    existing.collinear=NULL)
       apc.indiv.LRtest.TS(data, small.model="APC", dep.var, covariates=NULL,
                    model.family="binomial", unit=1, 
                    n.coh.excl.start=0, n.coh.excl.end=0,
                    n.age.excl.start=0, n.age.excl.end=0,
                    n.per.excl.start=0, n.per.excl.end=0,
                    existing.small.model.fit=NULL, 
                    existing.big.model.fit=NULL, 
                    existing.collinear=NULL,
                    NR.controls=NULL)                                      

Arguments

Details

These functions are designed to facilitate direct comparison between sub-models. The functions are used to construct the rows of tables in apc.indiv.model.table but can also more helpfully be used to compare nested sub-models that gain similar levels of suport from such a table, e.g. PC to P.

Value

Author(s)

Zoe Fannon <zoe.fannon@economics.ox.ac.uk> 26 Jun 2020

References

Fannon, Z. (2018) apc.indiv: R tools to estimate age-period-cohort models with repeated cross section data. Mimeo. University of Oxford.

Fannon, Z., Monden, C. and Nielsen, B. (2018) Age-period-cohort modelling and covariates, with an application to obesity in England 2001-2014. Mimeo. University of Oxford.

See Also

For model estimation: apc.indiv.est.model. The data in these examples are the Wage data from the package ISLR and the PSID7682 data from the package AER.

For examples, see the vignette IntroductionIndividualData.pdf, IntroductionIndividualData.R on Vignettes. Further examples in the vignette IntroductionIndividualDataFurtherExamples.pdf, IntroductionIndividualDataFurtherExamples.R.

Examples

#### see vignettes

Estimate a single APC model

Description

The function apc.indiv.est.model is used to estimate any of: the APC model, any APC submodel, or the time-saturated model. To estimate the APC model or a submodel, it calls apc.indiv.design.collinear, apc.indiv.design.model, and apc.indiv.fit.model in that order. To estimate the time-saturated (TS) model it calls either apc.indiv.estimate.TS or apc.indiv.logit.TS, depending on the selected model.family. These functions can also be called directly by the user.

Usage

  apc.indiv.est.model(data, unit = 1,
                      n.coh.excl.start=0, n.coh.excl.end=0,
                      n.per.excl.start=0, n.per.excl.end=0,
                      n.age.excl.start=0, n.age.excl.end=0,  
                      model.design = "APC", dep.var = NULL, 
                      covariates = NULL, model.family = "gaussian",
                      NR.controls = NULL, 
                      existing.collinear = NULL,
                      existing.design = NULL,
					  plmmodel = "notplm", id.var = NULL,
                      wt.var = NULL)					  
  apc.indiv.design.collinear(data, unit = 1,
                             n.coh.excl.start = 0, n.coh.excl.end = 0,
                             n.per.excl.start = 0, n.per.excl.end = 0,
                             n.age.excl.start = 0, n.age.excl.end = 0)
  apc.indiv.design.model(apc.indiv.design.collinear,
                         model.design = "APC", dep.var = NULL, 
                         covariates = NULL, plmmodel = "notplm",
                         wt.var = NULL, id.var = NULL)
  apc.indiv.fit.model(apc.indiv.design.model, model.family = "gaussian", DV = NULL)
  apc.indiv.estimate.TS(data, dep.var, covariates = NULL)
  apc.indiv.logit.TS(data, dep.var, covariates = NULL, NR.controls = NULL)

Arguments

Details

The casual user should start with the general function apc.indiv.est.model for analysis. The underlying functions should be employed if the user needs to run many models using the same relatively large dataset, in which case time can be saved by running apc.indiv.design.collinear just once and using apc.indiv.design.model and apc.indiv.fit.model to estimate each of the models.

The time-saturated (TS) binomial model is estimated by a customized Newton-Rhapson iteration. Aspects of this iteration can be controlled by specifying the NR.controls option of apc.indiv.est.model or of apc.indiv.logit.TS. NR.controls is a named list of length 8. In order, the elements are: maxit.loop, maxit.linesearch, tolerance, init, inv.tol, d1.tol, custom.kappa, custom.zeta. maxit.loop sets the maximum number of Newton-Rhapson iterations, and has a default of 10. maxit.linesearch sets the maximum number of linesearch iterations within each Newton-Rhapson iteration, and has a default of 20. tolerance sets the condition for convergence, i.e. the tolerated difference between likelihoods from one Newton-Rhapson iteration to the next; the default is .002. init sets the starting values for the iteration. The default is "ols", meaning that estimates from the linear probability model are the starting values; one can also use "zero" to set the starting values to zero, or use "custom" and specify custom starting values using custom.kappa and custom.zeta. inv.tolsets the tolerance of small values when inverting a matrix (using solve), and the default is the machine precision. d1.tol sets the magnitude of norm of first derivative to be tolerated in Newton-Rhapson iteration, and has a default of .002. custom.kappa is used to specify custom starting values for the TS indicator parameters, while custom.zeta is used to specify custom starting values for parameters on any covariates.

Value

.

Author(s)

Zoe Fannon <zoe.fannon@economics.ox.ac.uk> 26 Jun 2020

References

Fannon, Z. (2018) apc.indiv: R tools to estimate age-period-cohort models with repeated cross section data. Mimeo. University of Oxford.

Fannon, Z., Monden, C. and Nielsen, B. (2018) Age-period-cohort modelling and covariates, with an application to obesity in England 2001-2014. Mimeo. University of Oxford.

See Also

For model estimation: glm, svyglm, plm For model testing: apc.indiv.model.table, apc.indiv.compare.direct, waldtest, linearHypothesis. For plotting: apc.plot.fit. The data in these examples are the Wage data from the package ISLR and the PSID7682 data from the package AER.

For examples, see the vignette IntroductionIndividualData.pdf, IntroductionIndividualData.R on Vignettes. Further examples in the vignette IntroductionIndividualDataFurtherExamples.pdf, IntroductionIndividualDataFurtherExamples.R.

Examples

#### see vignettes

Generate table to select APC submodel

Description

These functions test, for a given choice of dependent variable and covariates, which of the TS, APC, and APC submodels provides the best fit to the data. Comparison is by Wald or likelihood ratio test and where appropriate by Akaike Information Criterion. A table is generated with these statistics for each model considered.

Usage

apc.indiv.model.table(data, dep.var, covariates = NULL,
			unit = 1, n.coh.excl.start = 0, n.coh.excl.end = 0,
			n.age.excl.start = 0, n.age.excl.end = 0,
			n.per.excl.start = 0, n.per.excl.end = 0,
			model.family, NR.controls = NULL,
			test, dist,
			TS=FALSE, wt.var=NULL, plmmodel="notplm",
			id.var=NULL)
	   apc.indiv.waldtable(data, dep.var, covariates = NULL, 
	   dist="F", unit = 1, model.family, 
	    	n.coh.excl.start = 0, n.coh.excl.end = 0,
			n.age.excl.start = 0, n.age.excl.end = 0,
			n.per.excl.start = 0, n.per.excl.end = 0,
			 wt.var=NULL, plmmodel="notplm",
			id.var=NULL)
			apc.indiv.waldtable.TS(data, dep.var, covariates=NULL, dist = "F",
                                unit=1, model.family = "gaussian",
                                n.coh.excl.start=0, n.coh.excl.end=0, 
                                n.age.excl.start=0, n.age.excl.end=0, 
                                n.per.excl.start=0, n.per.excl.end=0)
     apc.indiv.LRtable(data,  dep.var, covariates=NULL, 
                              model.family, unit=1, 
                              n.coh.excl.start=0, n.coh.excl.end=0, 
                              n.age.excl.start=0, n.age.excl.end=0,
                              n.per.excl.start=0, n.per.excl.end=0)
	   apc.indiv.LRtable.TS(data,  dep.var, covariates=NULL, 
                                 model.family, unit=1, 
                                 n.coh.excl.start=0, n.coh.excl.end=0, 
                                 n.age.excl.start=0, n.age.excl.end=0,
                                 n.per.excl.start=0, n.per.excl.end=0,
                                 NR.controls=NR.controls)

Arguments

Details

Each row of the table corresponds to a single sub-model of the APC model. The first three columns test the sub-model in question against the time-saturated model. The next three columns test the sub-model against the full APC model. The final two columns report the likelihood and AIC of the estimated sub-model. The model with the lowest AIC value which is also not rejected in tests against the APC and TS models should be selected.

Value

Author(s)

Zoe Fannon <zoe.fannon@economics.ox.ac.uk> 26 Jun 2020

References

Fannon, Z. (2018) apc.indiv: R tools to estimate age-period-cohort models with repeated cross section data. Mimeo. University of Oxford.

Fannon, Z., Monden, C. and Nielsen, B. (2018) Age-period-cohort modelling and covariates, with an application to obesity in England 2001-2014. Mimeo. University of Oxford.

See Also

For model estimation: apc.indiv.est.model For pairwise model comparison: apc.indiv.model.table, waldtest, linearHypothesis. The data in these examples are the Wage data from the package ISLR and the PSID7682 data from the package AER.

For examples, see the vignette IntroductionIndividualData.pdf, IntroductionIndividualData.R on Vignettes. Further examples in the vignette IntroductionIndividualDataFurtherExamples.pdf, IntroductionIndividualDataFurtherExamples.R.

Examples

#### see vignettes

Make all descriptive plots.

Description

Plots data sums using apc.plot.data.sums. Sparsity plots of data using apc.plot.data.sparsity. Plots data using all combinations of two time scales using apc.plot.data.within. Level plots of data using apc.plot.data.level. The latter plot is done for responses and if applicable also for doses and mortality rates.

Usage

apc.plot.data.all(apc.data.list,log ="",rotate=FALSE)

Arguments

Warning

A warning is produced if dimension is not divisible by thin, so that one group is smaller than other groups.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 25 Apr 2015

See Also

The example below uses Italian bladder cancer data, see data.Italian.bladder.cancer

Examples

#####################
#  EXAMPLE with artificial data
#  generate a 3x4 matrix in "AP" data.format with the numbers 1..12

m.data  	<- matrix(data=seq(length.out=12),nrow=3,ncol=4)
m.data
data.list	<- apc.data.list(m.data,"AP")
apc.plot.data.all(data.list,log="")				

#####################
#	EXAMPLE with Italian bladder cancer data
#
#	get data list, then make all descriptive plots.
# 	Note that warnings are given in relation to the data chosen thinning
#	This can be avoided by working with the individual plots, and in particular
#	with apc.plot.data.within where the thinning happens.
#
#	data.list	<- data.Italian.bladder.cancer()	
#	apc.plot.data.all(data.list)

Level plot of data matrix.

Description

This plot shows level plot of data matrix based on levelplot in the package lattice.

Usage

apc.plot.data.level(apc.data.list,data.type="r",
 						rotate=FALSE,apc.index=NULL,
						main=NULL,lab=NULL,
						contour=FALSE,colorkey=TRUE)

Arguments

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 26 Apr 2015

See Also

data.Japanese.breast.cancer for information on the data used in the example.

Examples

#####################
#	EXAMPLE with Japanese breast cancer data
#	Clayton and Shifflers (1987b) use APC design
#	Make a data list
#	Then plot data.
#	Note: No plot appears to have approximately parallel lines.

data.list	<- data.Japanese.breast.cancer()
apc.plot.data.level(data.list,"r")
dev.new()
apc.plot.data.level(data.list,"d",contour=TRUE)
												
#	It also works with a single argument, but then a default log scale is used.
# 	Note that warnings are given in relation to the data chosen thinning

apc.plot.data.within(data.list)

#####################
#	EXAMPLE with Italian bladder cancer data
#	Clayton and Shifflers (1987a) use AC design
#	Note: plot of within cohort against age appears to have approximately parallel lines.
#		  This is Figure 2 in Clayton and Shifflers (1987a)
#	Note: plot of within age against cohort appears to have approximately parallel lines.
#		  Indicates that interpretation should be done carefully.

data.list	<- data.Italian.bladder.cancer()	
apc.plot.data.within(data.list,"m",1,log="y")

#####################
#	EXAMPLE with asbestos data
#	Miranda Martinex, Nielsen and Nielsen (2014).
#	This is Figure 1d 

data.list	<- data.asbestos()	
apc.plot.data.within(data.list,type="l",lty=1)

This plot shows heat map of the sparsity of a data matrix.

Description

The plot shows where the data matrix is sparse.

Usage

apc.plot.data.sparsity(apc.data.list,
						data.type="a",swap.axes=FALSE,
						apc.index=NULL,
						sparsity.limits=c(1,2),
						cex=NULL,pch=15,
						main.outer=NULL)

Arguments

Details

The default values is used to highlight where a matrix of counts has values of zero and one. Estimation can be very noise in those areas.

Note

Note that the axes for plots grow from bottom left while axes for matrices grow from top left. The exception is when data.format="CL", in which case both grow from top left.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 25 Apr 2015 updated 27 Apr 2015

See Also

The example below uses asbestos data, see data.asbestos

Examples

#####################
#  EXAMPLE with artificial data
#  generate a 3x4 matrix in "AP" data.format with the numbers 1..12

m.data  	<- matrix(data=seq(length.out=12),nrow=3,ncol=4)
m.data
data.list	<- apc.data.list(m.data,"AP")
apc.plot.data.sparsity(data.list)

#####################
#	EXAMPLE with Japanese breast cancer data
#	get data list, then make sparsity plots.

data.list	<- data.asbestos()					
apc.plot.data.sparsity(data.list)

This plot shows sums of data matrix by age, period or cohort.

Description

Produces plots showing age, period and cohort sums. As a default this is done both for responses and dose, giving a total of six plots.

Usage

apc.plot.data.sums(apc.data.list,data.type="a",
            average=FALSE,keep.incomplete=TRUE,apc.index=NULL,
            type="o",log="",main.outer=NULL,main.sub=NULL,
            las=1,scale.response=1,scale.dose=1,scale.rate=1)

Arguments

Details

The data sums are computed using apc.data.sums. Then plotted as requested.

Note

Use apc.data.sums if numerical values needed.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 15 Aug 2018 (15 Dec 2013)

References

Martinez Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2015) Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality. Journal of the Royal Statistical Society A 178, 29-55. Download: doi:10.1111/rssa.12051, Nuffield DP.

See Also

The example below uses Japanese breast cancer data, see data.Japanese.breast.cancer

Examples

#####################
#  	EXAMPLE with artificial data
#  	Generate a 3x4 matrix in "AP" data.format with the numbers 1..12
#	Then make a data list
#	Then plot data sums.
#	Note only 3 plots are made as there are no doses

m.data  	<- matrix(data=seq(length.out=12),nrow=3,ncol=4)
m.data
data.list	<- apc.data.list(m.data,"AP")
apc.plot.data.sums(data.list)					 
apc.plot.data.sums(data.list,average=TRUE)
apc.plot.data.sums(data.list,keep.incomplete=FALSE)					 

#####################
#	EXAMPLE with Japanese breast cancer data
#	Make a data list
#	Then plot data sums for both responses and doses.

data.list	<- data.Japanese.breast.cancer()	
apc.plot.data.sums(data.list)					

# 	Or plot data sums for responses only

apc.plot.data.sums(data.list,data.type="r")		

#####################
#	EXAMPLE with asbestos data
#	Miranda Martinex, Nielsen and Nielsen (2013).
#	This is Figure 1,a-c 

data.list	<- data.asbestos()	
apc.plot.data.sums(data.list,type="l")

This plot shows time series of matrix within age, period or cohort.

Description

apc.plot.data.within produces plot showing time series of matrix within age, period or cohort against one of the other two indices. apc.plot.data.within.all.six produces all six plots in one panel plot.

These plots are sometimes used to gauge how many of the age, period, cohort factors are needed: If lines are parallel when dropping one index the corresponding factor may not be needed. In practice these plots should possibly be used with care, see Italian bladder cancer example below.

Usage

apc.plot.data.within(apc.data.list,
					data.type="r",plot.type="awc",
					average=FALSE,
					thin=NULL,apc.index=NULL,
					ylab=NULL,type="o",log="",legend=TRUE,
					lty=1:5,col=1:6,bty="n",main=NULL,
					x="topleft",return=FALSE)
apc.plot.data.within.all.six(apc.data.list,
					data.type="r",
					average=FALSE,
					thin=NULL,apc.index=NULL,
					ylab=NULL,type="o",log="",legend=TRUE,
					lty=1:5,col=1:6,bty="n",main.outer=NULL,
					x="topleft")

Arguments

Warning

A warning is produced if dimension is not divisible by thin, so that one group is smaller than other groups.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 17 Nov 2016 (25 Apr 2015)

References

Clayton, D. and Schifflers, E. (1987a) Models for temperoral variation in cancer rates. I: age-period and age-cohort models. Statistics in Medicine 6, 449-467.

Clayton, D. and Schifflers, E. (1987b) Models for temperoral variation in cancer rates. II: age-period-cohort models. Statistics in Medicine 6, 469-481.

Martinez Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2015) Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality. Journal of the Royal Statistical Society A 178, 29-55. Download: doi:10.1111/rssa.12051, Nuffield DP.

See Also

data.Japanese.breast.cancer, data.Italian.bladder.cancer and data.asbestos for information on the data used in the example.

Examples

#####################
#  	EXAMPLE with artificial data
#  	Generate a 3x4 matrix in "AP" data.format with the numbers 1..12
#	Then make a data list
#	Then plot data.
#  	Note: this deterministic matrix has neither age, period, or cohort factors,
#		 only linear trends.  Thus all 6 plots have parallel lines.	

m.data  	<- matrix(data=seq(length.out=12),nrow=3,ncol=4)
m.data
data.list	<- apc.data.list(m.data,"AP")
apc.plot.data.within(data.list,log="")

#	It also works with a single argument, but then a default log scale is used.

apc.plot.data.within(data.list)			

#####################
#	EXAMPLE with Japanese breast cancer data
#	Clayton and Shifflers (1987b) use APC design
#	Make a data list
#	Then plot data.
#	Note: No plot appears to have approximately parallel lines.

data.list	<- data.Japanese.breast.cancer()	
apc.plot.data.within(data.list,"m",1,log="y")
												
#	It also works with a single argument, but then a default log scale is used.
# 	Note that warnings are given in relation to the data chosen thinning

apc.plot.data.within(data.list)

#####################
#	EXAMPLE with Italian bladder cancer data
#	Clayton and Shifflers (1987a) use AC design
#	Note: plot of within cohort against age appears to have approximately parallel lines.
#		  This is Figure 2 in Clayton and Shifflers (1987a)
#	Note: plot of within age against cohort appears to have approximately parallel lines.
#		  Indicates that interpretation should be done carefully.

data.list	<- data.Italian.bladder.cancer()	
apc.plot.data.within(data.list,"m",1,log="y")

#####################
#	EXAMPLE with asbestos data
#	Miranda Martinex, Nielsen and Nielsen (2014).
#	This is Figure 1d 

data.list	<- data.asbestos()	
apc.plot.data.within(data.list,type="l",lty=1)

Plots of apc estimates

Description

Functions to plot the apc estimates found by apc.fit.model. The function apc.plot.fit detects the type of model.design and model.family from the fit values and makes appropriate plots.

Depending on the model.design the plot has up to 9 sub plots. The type of these can be chosen using type

Model designs of any type. If type is "detrend" or "sum.sum" the canonical age period cohort parametrisation is used. This involves double differences of the time effects. The first row of plots are double differences of the time effects. The next two rows of plots illustrate the representation theorem depending on the choice of type. In both cases the sum of the plots add up to the predictor.

"detrend"

The last row of plots are double sums of double differences detrend so that that each series starts in zero and ends in zero. The corresponding level and (up to) two linear trends are shown in the middle row of plots. The linear trends are identified to be 0 for age, period or cohort equal to its smallest value. See note 2 below.

"sum.sum"

The last row of plots are double sums of double differences anchored as in the derivation of Nielsen (2014b). The corresponding level and (up to) two linear trends are shown in the middle row of plots. The linear trends are identified to be 0 for the anchoring point U of age, period or cohort as described in Nielsen (2014b). See note 1 below.

Model designs with 2 factors. If type is "dif" the canonical two factor parametrisation is used. This involves single differences. It is only implemented for model.design of "AC", "AP", "PC". It does not apply for model.design of "APC" because single differences are not identified. It does not apply for the drift models where model.design is "Ad", "Pd", "Cd", "t" because it is not clear which time scale the second linear trend should be attributed to. It is not implemented for model.design of "tA, "tP", "tC", "1". The first row of plots are single differences of the time effects. The next two rows of plots illustrate the representation theorem. In the second row the level is given and in the third row plots of single sums of single differences are given, normalised to start in zero.

Appearance may vary. Note, the plots "detrend" and "dif" can give very different appearance of the time effects. The "dif" plots are dominated by linear trends. They can therefore be more difficult to interpret than the "detrend" plots, where linear trends are set aside.

Standard deviations. All plots include plots of 1 and 2 standard deviations. The only exception is the intercept in the case model.family is "poisson.response" as this uses a multinomial sampling scheme, where the intercept is set to increase in the asymptotic experiment. The default is to plot standard deviations around zero, so that they represent a test for zero values of the parameters. Using the argument sdv.at.zero the standard deviations can be centered around the estimates. This can give a very complicated appearance.

Values of coefficients. These can be found using apc.identify.

Usage

apc.plot.fit(apc.fit.model,scale=FALSE,
					sdv.at.zero=TRUE,type="detrend",
					include.linear.plane=TRUE,
					include.double.differences=TRUE,
					sub.plot=NULL,main.outer=NULL,main.sub=NULL,
					cex=NULL,cex.axis=NULL,cex.lab=NULL,cex.main=NULL,
					cex.main.outer=1.2,
					line.main=0.5,line.main.outer=NULL,
					las=NULL,mar=NULL,oma=NULL,mgp=c(2,1,0),
					vec.xlab=NULL)

Arguments

Note

(1) The type "sum.sum" (same as "ss.dd") gives double sums anchored to be zero in the three points where age=cohort=U, age=U+1,cohort=U age=U,cohort=U+1 with apc.fit.model$U and where U is the integer value of (per.zero+3)/2 This corresponds to the representation in Nielsen (2014b). The linear plane is parametrised in terms of a level, which is the value of the predictor at age=cohort=U; an age slope, which is the difference of the values of the predictor at age=U+1,cohort=U and age=cohort=U; an cohort slope, which is the difference of the values of the predictor at age=U,cohort=U+1 and age=cohort=U.

(2) The type "detrend" gives double sums that start in zero and end in zero. The linear plane is parametrised in terms of a level, which is the value of the predictor at age=cohort=1, which is usually outside the index set for the data; while age and cohort slopes are adjusted for the ad hoc identification of the time effects.

(3) The default of the titles main.sub are generated internally depending on model specification. In the case of model.design="APC" and a dose-response model family the default value is c(expression(paste("(a) ",Delta^2,alpha)), expression(paste("(b) ",Delta^2,beta)), expression(paste("(c) ",Delta^2,gamma)), "(d) first linear trend", "(e) level", "(f) second linear trend", expression(paste("(g) detrended ",Sigma^2,Delta^2,alpha)), expression(paste("(h) detrended ",Sigma^2,Delta^2,beta)), expression(paste("(i) detrended ",Sigma^2,Delta^2,gamma)))

(4) Default values of parameters changed (28 Sep 2020). The old appearance can be reproduced by setting cex.lab=1.5. For example:

data.list <- data.Italian.bladder.cancer()

fit.apc <- apc.fit.model(data.list,"poisson.dose.response","APC")

apc.plot.fit(fit.apc,cex.lab=1.5)

The code subsumes var.apc.plot.fit by Zoe Fannon.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> & Zoe Fannon 28 September 2020 (12 Apr 2015).

References

Kuang, D., Nielsen, B. and Nielsen, J.P. (2008a) Identification of the age-period-cohort model and the extended chain ladder model. Biometrika 95, 979-986. Download: doi:10.1093/biomet/asn026; Earlier version Nuffield DP.

Nielsen, B. (2014b) Deviance analysis of age-period-cohort models. Work in progress.

See Also

data.asbestos and data.Italian.bladder.cancer for information on the data used in the example.

Values of coefficients can be found using apc.identify.

Further information on the identification in the vignette Identification.pdf, Identification.R on Vignettes.

Examples

#####################
#	Example with Italian bladder cancer data
#	Note that the model.design "AC" cannot be rejected against "APC"
#		so there is little difference between the two plots of those fits.

data.list	<- data.Italian.bladder.cancer()
apc.fit.table(data.list,"poisson.dose.response")
fit.apc		<- apc.fit.model(data.list,"poisson.dose.response","APC")
apc.plot.fit(fit.apc)
#	now try an AC model
#	can use dev.new() to see both
fit.ac		<- apc.fit.model(data.list,"poisson.dose.response","AC")
apc.plot.fit(fit.ac)

#	to check the numerical values for the last two rows of plots use
apc.identify(fit.ac)$coefficients.detrend

#	to get only a sub plot and playing with titles
#	main.outer not used with individual plot
apc.plot.fit(fit.ac,sub.plot="a",main.outer="My outer title",main.sub="My sub title")
#	to play with
#		titles (main.outer/main.sub),
#		label orientation (las),
#		axis titles (vec.xlab)
apc.plot.fit(fit.ac,main.outer="My outer title",
				main.sub=c("1","2","3","4","5","6","7","8","9"),
				las=1,
				vec.xlab=c("a","b","c","d","e","f","g","h","i"))

Plots of apc estimates for 2 sample model

Description

Generalizes apc.plot.fit to a 2 sample model. For an application, see the vignette ReproducingN2025.pdf, ReproducingN2025.R on vignette.

Usage

 apc.plot.fit.2s(apc.fit.model.1,apc.fit.model.2,
            type="macro",which.plot=0,
            col=NULL,label=NULL,lty=NULL)

Arguments

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 3 Jul 2025

References

Nielsen, B. (2022) Two-sample age-period-cohort models with an application to Swiss suicide rates. Download: Nuffield Discussion Paper 2022-W03.

Make all fit plots.

Description

Plots estimates using apc.plot.fit. Probability transform plot of residuals using apc.plot.fit.pt. Level plot of residuals using apc.plot.fit.residuals. Level plot of fitted values using apc.plot.fit.fitted.values. Level plot of linear predictors using apc.plot.fit.linear.predictors. Level plots of responses and rates (if dose is availble) using apc.plot.data.level.

Usage

apc.plot.fit.all(apc.fit.model,log ="",rotate=FALSE)

Arguments

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 2t Apr 2015

See Also

The example below uses Italian bladder cancer data, see data.Italian.bladder.cancer

Examples

#####################
#	EXAMPLE with Italian bladder cancer data

#	get data list, then make all descriptive plots.
# 	Note that warnings are given in relation to the data chosen thinning
#	This can be avoided by working with the individual plots, and in particular
#	with apc.plot.data.within where the thinning happens.

data.list	<- data.Italian.bladder.cancer()
fit			<- apc.fit.model(data.list,"poisson.dose.response","APC")
apc.plot.fit.all(fit)

Plot probability transform of responses given fitted values

Description

Constructs probability transforms of responses given fitted values from apc.fit.model. The plot is given in the original coordinate system. Colours and symbols are used to indicate whether responses are central to the fitted distribution or in the tails of the fitted distribution.

Usage

apc.plot.fit.pt(apc.fit.model,
					   do.plot=TRUE,do.value=FALSE,
					   pch=c(21,24,25),
					   col=c("black","green","blue","red"),
					   bg=NULL,cex=NULL,main=NULL)

Arguments

Value

Vector of probability transforms. Only produced if do.value is set to TRUE. See example below.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 2 Dec 2013

See Also

data.Italian.bladder.cancer for information on the data used in the example.

Examples

#####################
#	Example with Italian bladder cancer data
#	HOW TO USE VALUE

data.list	<- data.Italian.bladder.cancer()
fit			<- apc.fit.model(data.list,"poisson.dose.response","APC")
v.pt		<- apc.plot.fit.pt(fit,do.value=TRUE)
m.pt		<- matrix(data=NA,nrow=fit$data.xmax,ncol=fit$data.ymax)
m.pt[fit$index.data]	<- v.pt
m.pt

#	            [,1]      [,2]       [,3]       [,4]      [,5]
#	 [1,] 0.63782311 0.5651585 0.33982477 0.91299734 0.5759652
#	 [2,] 0.82676269 0.8992667 0.26378120 0.28795884 0.3708787
#	 [3,] 0.54139571 0.2445995 0.51923747 0.63451773 0.7955547
#	 [4,] 0.87364488 0.8228499 0.07219437 0.38789788 0.5938305
#	 [5,] 0.86797473 0.3934085 0.34525271 0.38955656 0.5097203
#	 [6,] 0.65027598 0.8377994 0.29018594 0.03694977 0.7990229
#	 [7,] 0.43769468 0.1099946 0.50261364 0.56777485 0.8916552
#	 [8,] 0.67518708 0.5519831 0.67817803 0.19793887 0.5354669
#	 [9,] 0.02717016 0.2066092 0.77035122 0.89047749 0.5017919
#	[10,] 0.71037782 0.9464356 0.36897847 0.41790169 0.2080577
#	[11,] 0.50922468 0.3085978 0.55261186 0.77592343 0.3597815

Level plots of residuals / fitted values / linear predictors

Description

Level plots of residuals / fitted values / linear predictors. Returns residuals / fitted values / linear predictors as matrices when requested. The plots use apc.plot.data.level. They plot are given in the original coordinate system.

Usage

apc.plot.fit.residuals(apc.fit.model,
					rotate=FALSE,main=NULL,lab=NULL,
					contour=FALSE,colorkey=TRUE,return=FALSE)
	   apc.plot.fit.fitted.values(apc.fit.model,
					rotate=FALSE,main=NULL,lab=NULL,
					contour=FALSE,colorkey=TRUE,return=FALSE)
	   apc.plot.fit.linear.predictors(apc.fit.model,
					rotate=FALSE,main=NULL,lab=NULL,
					contour=FALSE,colorkey=TRUE,return=FALSE)

Arguments

Value

Matrix of the original format with residuals / fitted values /linear predictors as entries. Only produced if return is set to TRUE.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 26 Apr 2015

See Also

data.Italian.bladder.cancer for information on the data used in the example.

Examples

#####################
#	Example with Italian bladder cancer data

data.list	<- data.Italian.bladder.cancer()
fit			<- apc.fit.model(data.list,"poisson.dose.response","APC")
apc.plot.fit.fitted.values(fit,return=TRUE)

#       1955-1959   1960-1964   1965-1969   1970-1974   1975-1979
# 25-29   3.04200    3.368944    2.261518    2.327538   12.000000
# 30-34  13.11980   12.835733   13.955859   10.416142    9.672462
# 35-39  24.15536   33.591644   33.388355   37.542301   26.322340
# 40-44  69.89262   68.842728   96.652963   98.478793  113.132896
# 45-49 217.97285  189.375728  189.115063  272.281239  285.255119
# 50-54 450.44864  529.823519  462.504305  469.869189  701.354350
# 55-59 724.88451  904.298410 1069.452434  969.346982  966.017661
# 60-64 877.17820 1226.088350 1532.521380 1877.331703 1807.880364
# 65-69 950.36106 1296.011123 1798.196048 2336.012274 3028.419493
# 70-74 903.94495 1187.708772 1598.021907 2302.605072 3222.719298
# 75-79 831.00000  953.055049 1280.930166 1755.788768 2678.226017

Add connected line and standard deviation polygons to a plot

Description

Draws a line for point forecasts and adds shaded region for forecast distribution around it. This is added to a plot in the same way as lines and polygon add lines and polygons to a plot.

Usage

apc.polygon(m.forecast,x.origin=1,
	plot.se=TRUE,plot.se.proc=FALSE,plot.se.est=FALSE,
	unit=1,
	col.line=1,lty.line=1,lwd.line=1,
	q.se=c(2,2,2),
	angle.se=c(45,45,45),
	border.se=c(NA,NA,NA),
	col.se=gray(c(0.50,0.80,0.90)),
	density.se=c(NULL,NULL,NULL),
	lty.se=c(1,1,1))

Arguments

Details

The empirical example of Martinez Miranda, Nielsen and Nielsen (2015) uses the data data.asbestos. The results of that paper are reproduced in the vignette ReproducingMMNN2015.pdf, ReproducingMMNN2015.R on Vignettes. The function is used there.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 6 Jan 2016

References

Martinez Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2015) Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality. Journal of the Royal Statistical Society A 178, 29-55. Download: doi:10.1111/rssa.12051, Nuffield DP.

Test normality of residuals

Description

Function that tests normality of vector of residuals.

Usage

apc.test.normal.residuals(list_or_vector,remove.zeros=FALSE,tolerance=1e-10)

Arguments

Belgian lung cancer data

Description

Function that organises Belgian lung cancer data in apc.data.list format.

The data set is taken from table VIII of Clayton and Schifflers (1987a), which contains age-specific incidence rates (per 100,000 person-years observation) of lung cancer in Belgian females during the period 1955-1978. Numerators are also available. The original source was the WHO mortality database.

The data set is in "AP"-format. The original data set is unbalanced since the first four period groups cover 5 years, while the last covers 4 years. The primary data set has 4 period groups, which is obtained when using the function without arguments or with argument unbalanced=FALSE. The secondary data set has 5 uneven sized period groups, wwhich is obtained when using the function with argument unbalanced=TRUE. The apc.package is at present not aimed at such unbalanced data.

Usage

data.Belgian.lung.cancer(unbalanced = FALSE)

Arguments

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 8 Sep 2015 (24 Oct 2013)

Source

Table VIII of Clayton and Schifflers (1987a).

References

Clayton, D. and Schifflers, E. (1987a) Models for temperoral variation in cancer rates. I: age-period and age-cohort models. Statistics in Medicine 6, 449-467.

See Also

General description of apc.data.list format.

Examples

#########################
##	It is convient to construct a data variable

data	<- data.Belgian.lung.cancer()

##	To see the content of the data

data

Italian bladder cancer data

Description

Function that organises Italian bladder data in apc.data.list format.

The data set is taken from table IV of Clayton and Schifflers (1987a), which contains age-specific incidence rates (per 100,000 person-years observation) of bladder cancer in Italian males during the period 1955-1979. Numerators are also available. The original source was the WHO mortality database.

The data set is in "AP"-format.

Usage

data.Italian.bladder.cancer()

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 8 Sep 2015 (24 Oct 2013)

Source

Table IV of Clayton and Schifflers (1987a).

References

Clayton, D. and Schifflers, E. (1987a) Models for temperoral variation in cancer rates. I: age-period and age-cohort models. Statistics in Medicine 6, 449-467.

See Also

General description of apc.data.list format.

Examples

#########################
##	It is convient to construct a data variable

data	<- data.Italian.bladder.cancer()

##	To see the content of the data

data

Japanese breast cancer data

Description

Function that organises Japanese breast data in apc.data.list format.

The data set is taken from table I of Clayton and Schifflers (1987b), which contains age-specific mortality rates (per 100,000 person-years observation) of breast cancer in Japan, during the period 1955-1979. Reported in 5 year age groups and 5 year period groups. Numbers of cases on which rates are based are also available. The original source was WHO mortality data base.

The data set is in "AP"-format.

Usage

data.Japanese.breast.cancer()

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 8 Sep 2015 (24 Oct 2013)

Source

Table I of Clayton and Schifflers (1987b)

References

Clayton, D. and Schifflers, E. (1987b) Models for temperoral variation in cancer rates. II: age-period-cohort models. Statistics in Medicine 6, 469-481.

See Also

General description of apc.data.list format.

Examples

#########################
##	It is convient to construct a data variable

data	<- data.Japanese.breast.cancer()

##	To see the content of the data

data

2-sample mortality data.

Description

Function that organises mortality data from Riebler and Held (2010) in apc.data.list format.

The data set is taken from the supplementary data of Riebler and Held (2010). Mortality data for women in Denmark and Norway

The original source was Jacobsen et al. (2004).

The data set is in "AP"-format.

Usage

data.RH.mortality.dk()
data.RH.mortality.no()

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 17 Sep 2016

Source

Riebler and Held (2010), supplementary material.

References

Jacobsen, R, von Euler, M, Osler, M, Lynge, E and Keiding, N (2004) Women's death in Scandinavia - what makes Denmark different? European Journal of Epidemiology 19, 117-121.

Riebler, A and Held, L. (2010) The analysis of heterogeneous time trends in multivariate age-period-cohort models. Biostatistics 11, 57–59. Download: doi:10.1093/biostatistics/kxp037, see supplementary material.

See Also

General description of apc.data.list format.

Examples

#########################
##	It is convient to construct a data variable

data	<- data.US.prostate.cancer()

##	To see the content of the data

data

Swiss suicide data

Description

Function that organises Swiss suicide data. Data are used in Nielsen (2022) to illustrate the 2-sample age-period-cohort analysis. The analysis is presented in a vignette ReproducingN2025.pdf, ReproducingN2025.R on vignette. Source: Riebler, Held, Rue and Bopp (2012).

Usage

data.Swiss.suicides()

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 7 Jul 2025

References

Nielsen, B. (2022) Two-sample age-period-cohort models with an application to Swiss suicide rates. Download: Nuffield Discussion Paper 2022-W03.

Riebler, A. and Held, L. and Rue, H. and Bopp, M. (2012) Gender-specific differences and the impact of family integration on time trends in age-stratified Swiss suicide rates. Journal of the Royal Statistical Society, Series A, 175, 479-490.

US prostate cancer data

Description

Function that organises US prostate data in apc.data.list format.

The data set is taken from table 2 of Holford (1983), which contains age-specific counts of deaths and midperiod population measured in 1000s, during the period 1935-1969. Reported in 5 year age groups and 5 year period groups.

The original source was Cancer deaths: National Center for Health Statistics, 1937-1973 Population 1935-60: Grove and Hetzel, 1968 Population 1960-69: Bureau of the Census, 1974

The data set is in "AP"-format.

Usage

data.US.prostate.cancer()

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 8 Sep 2015 (28 Apr 2015)

Source

Table 2 of Holford (1983)

References

Holford, T.R. (1983) The estimation of age, period and cohort effects for vital rates. Biometrics 39, 311-324.

See Also

General description of apc.data.list format.

Examples

#########################
##	It is convient to construct a data variable

data	<- data.US.prostate.cancer()

##	To see the content of the data

data

UK aids data

Description

Function that organises UK aids data in apc.data.list format.

The data set is taken from table 1 of De Angelis and Gilks (1994). The data are also analysed by Davison and Hinkley (1998, Example 7.4). The data are reporting delays for AIDS counting the number of cases by the date of diagnosis and length of reporting delay, measured by quarter.

The data set is in "trapezoid"-format. The original data set is unbalanced in various ways: first column covers a reporting delay of less than one month (or should it be less than one quarter?); last column covers a reporting delay of at least 14 quarters; last diagonal include incomplete counts. The default data set excludes the incomplete counts in the last diagonal, but includes the unbalanced first and last columns.

Usage

data.aids(all.age.groups = FALSE)

Arguments

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 7 Feb 2016

Source

Table 1 of De Angelis and Gilks (1994). Also analysed by Davison and Hinkley (1998, Example 7.4).

References

De Angelis, D. and Gilks, W.R. (1994) Estimating acquired immune deficiency syndrome incidence accounting for reporting delay. Journal of the Royal Statistical Sociey A 157, 31-40.

Davison, A.C. and Hinkley, D.V. (1998) Bootstrap methods and their application. Cambridge: Cambridge University Press.

See Also

General description of apc.data.list format.

Examples

#########################
##	It is convient to construct a data variable
data	<- data.Belgian.lung.cancer()
##	To see the content of the data
data

#########################
#	Forecast AIDS incidences by diagonsis year (cohort).
#	uses as poisson response model with an AC structure
#	although there is evidence of overdispersion and the
#	period effect appears significant.
#	The omission of the period effect follows
#	Davison and Hinkley and a parsimoneous model may be
#	advantageous when forecasting.
#
apc.fit.table(data.aids(),"poisson.response")
fit <- apc.fit.model(data.aids(),"poisson.response","AC")
forecast <- apc.forecast.ac(fit)
data.sums.coh <- apc.data.sums(data.aids())$sums.coh
forecast.total <- forecast$response.forecast.coh
forecast.total[,1]	<- forecast.total[,1]+data.sums.coh[25:38]
x	<- seq(1983.5,1992.75,by=1/4)
y	<- data.sums.coh
xlab<- "diagnosis year (cohort)"
ylab<- "diagnoses"
main<- "Davison and Hinkley, Fig 7.6, parametric version"
plot(x,y,xlim=c(1988,1993),ylim=c(200,600),xlab=xlab,ylab=ylab,main=main)
apc.polygon(forecast.total,x.origin=1989.25,unit=1/4)

Asbestos data

Description

Function that organises asbestos data in apc.data.list format.

Counts of mesothelioma deaths in the UK by age and period. Mesothelioma is most often caused by exposure to asbestos.

The data set is in "PA"-format.

data.asbestos is for men 1967-2012 data.asbestos.2013 is the same as data.asbestos.2013.men and is for men 1968-2013. data.asbestos.2013.women and is for women 1968-2013.

The primary data set includes ages 25-89, which is obtained when using the function without arguments or with argument all.age.groups=FALSE. The secondary data includes younger and older age groups, which is obtained when using the function with argument all.age.groups=TRUE. The apc.package is at present not aimed at such unbalanced data.

Usage

data.asbestos(all.age.groups = FALSE)
data.asbestos.2013(all.age.groups = FALSE)
data.asbestos.2013.women(all.age.groups = FALSE)
data.asbestos.2013.men(all.age.groups = FALSE)

Arguments

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 30 April 2016

Source

Data were prepared for the Asbestos Working Party by the UK Health and Safety Executive. An APC analysis of these data can be found in Martinez Miranda, Nielsen and Nielsen (2015). The results of that paper are reproduced in the vignette ReproducingMMNN2015.pdf, ReproducingMMNN2015.R on Vignettes. These data are also used in Nielsen (2015).

The updated data set data.asbestos.2013 is for 1968-2013 and has the same structure. This is analysed in Martinez-Miranda, Nielsen and Nielsen (2016).

References

Martinez Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2015) Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality. Journal of the Royal Statistical Society A 178, 29-55. Download: Nuffield DP.

Martinez-Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2016) A simple benchmark for mesothelioma projection for Great Britain. To appear in Occupational and Environmental Medicine. Download: Nuffield DP.

Nielsen, B. (2015) apc: An R package for age-period-cohort analysis. R Journal 7, 52-64. Download: Open access.

See Also

General description of apc.data.list format.

Examples

#########################
#	apc data list

data.list	<- data.asbestos()
objects(data.list)

#####################
#	Figure 1,a-c from
#	Miranda Martinex, Nielsen and Nielsen (2015).

data.list	<- data.asbestos()	
apc.plot.data.sums(data.list,type="l")

#####################
#	Figure 1,d from
#	Miranda Martinex, Nielsen and Nielsen (2015).
data.list	<- data.asbestos()	
apc.plot.data.within(data.list,type="l",lty=1)

Motor data

Description

Function that organises loss data in apc.data.list format.

The data set is taken from table 3.5 of Barnett & Zehnwirth (2000). Source of data unclear. It includes a run-off triangle: "response" (X) is paid amounts (units not reported) along with measures of exposure.

Data also analysed in e.g. Kuang, Nielsen, Nielsen (2011).

The data set is in "CL"-format.

At present apc.package does not have functions for either forecasting or for exploiting the counts. For this one can with advantage use the DCL.package.

Usage

data.loss.BZ

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 8 Sep 2015 (18 Mar 2015)

Source

Tables 1,2 of Verrall, Nielsen and Jessen (2010).

References

Barnett G, Zehnwirth B (2000) Best estimates for reserves. Proc. Casualty Actuar. Soc. 87, 245–321.

Kuang D, Nielsen B, Nielsen JP (2011) Forecasting in an extended chain-ladder-type model Journal of Risk and Insurance 78, 345-359

See Also

General description of apc.data.list format.

Examples

#########################
##	It is convient to construct a data variable

data	<- data.loss.BZ()

##	To see the content of the data

data

#########################
#	Fit geometric chain-ladder model

apc.fit.table(data,"log.normal.response")

Motor data

Description

Function that organises loss data in apc.data.list format.

The data set is taken from Table 1 of Verrall (1991), who attributes the data to Taylor and Ashe (1983). It includes a run-off triangle: "response" (X) is paid amounts (units not reported).

Data also analysed in various papers, e.g. England and Verrall (1999).

The data set is in "CL"-format.

At present apc.package does not have functions for either forecasting or for exploiting the counts. For this one can with advantage use the DCL.package.

Usage

data.loss.TA

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 8 Sep 2015 (18 Mar 2015)

Source

Tables 1 of Verrall (1991).

References

England, P., Verrall, R.J. (1999) Analytic and bootstrap estimates of prediction errors in claims reserving Insurance: Mathematics and Economics 25, 281-293

Taylor, G.C., Ashe, F.R. (1983) Second moments of estimates of outstanding claims Journal of Econometrics 23, 37-61

Verrall, R.J. (1991) On the estimation of reserves from loglinear models Insurance: Mathematics and Economics 10, 75-80

See Also

General description of apc.data.list format.

Examples

#########################
##	It is convient to construct a data variable

data	<- data.loss.TA()

##	To see the content of the data

data

#########################
#	Fit chain-ladder model

apc.fit.table(data,"poisson.response")

#	The overdispersed poisson model is experimental at the moment,
#	so not documented
apc.fit.table(data,"od.poisson.response")

Motor data

Description

Function that organises motor data in apc.data.list format.

The data set is taken from tables 1,2 of Verrall, Nielsen and Jessen (2010). Data from Codan, Danish subsiduary of Royal & Sun Alliance. It is a portfolio of third party liability from motor policies. The time units are in years. There are two run-off triangles: "response" (X) is paid amounts (units not reported) "counts" (N) is number of reported claims.

Data also analysed in e.g. Martinez Miranda, Nielsen, Nielsen and Verrall (2011) and Kuang, Nielsen, Nielsen (2015).

The data set is in "CL"-format.

At present apc.package does not have functions for either forecasting or for exploiting the counts. For this one can with advantage use the DCL.package.

Usage

data.loss.VNJ

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 18 Mar 2015 updated 4 Jan 2016

Source

Tables 1,2 of Verrall, Nielsen and Jessen (2010).

References

Verrall R, Nielsen JP, Jessen AH (2010) Prediction of RBNS and IBNR claims using claim amounts and claim counts ASTIN Bulletin 40, 871-887

Martinez Miranda, M.D., Nielsen, B., Nielsen, J.P. and Verrall, R. (2011) Cash flow simulation for a model of outstanding liabilities based on claim amounts and claim numbers. ASTIN Bulletin 41, 107-129

Kuang D, Nielsen B, Nielsen JP (2015) The geometric chain-ladder Scandinavian Acturial Journal 2015, 278-300.

See Also

General description of apc.data.list format.

Examples

#########################
##	It is convient to construct a data variable

data	<- data.loss.VNJ()

##	To see the content of the data

data

#########################
#	Fit chain-ladder model

fit.ac	<- apc.fit.model(data,"poisson.response","AC")
fit.ac$coefficients.canonical
id.ac	<- apc.identify(fit.ac)
id.ac$coefficients.dif

#########################
#	Compare output with	table 7.2 in
#	Kuang D, Nielsen B, Nielsen JP (2015)
#	               Estimate   Std. Error    z value      Pr(>|z|)
#	level       13.07063963 0.0000000000        Inf  0.000000e+00
#	D_age_2     -0.06543495 0.0006018694 -108.71950  0.000000e+00
#	D_age_3     -0.80332424 0.0008757527 -917.29576  0.000000e+00
#	D_age_4     -0.41906516 0.0012294722 -340.84965  0.000000e+00
#	D_age_5     -0.29097802 0.0015627740 -186.19329  0.000000e+00
#	D_age_6     -0.57299006 0.0021628918 -264.91850  0.000000e+00
#	D_age_7     -0.36101594 0.0030016569 -120.27222  0.000000e+00
#	D_age_8     -0.62706059 0.0046139466 -135.90547  0.000000e+00
#	D_age_9      0.12160793 0.0061126021   19.89463  4.529830e-88
#	D_age_10    -2.59708012 0.0245028290 -105.99103  0.000000e+00
#	D_cohort_2  -0.02591843 0.0009037977  -28.67724 7.334840e-181
#	D_cohort_3   0.18973130 0.0011301184  167.88621  0.000000e+00
#	D_cohort_4   0.12354693 0.0010508785  117.56539  0.000000e+00
#	D_cohort_5  -0.10114701 0.0010566534  -95.72392  0.000000e+00
#	D_cohort_6   0.03594882 0.0010913718   32.93912 6.056847e-238
#	D_cohort_7  -0.17175409 0.0011676536 -147.09336  0.000000e+00
#	D_cohort_8   0.20671145 0.0012098255  170.86055  0.000000e+00
#	D_cohort_9   0.04056617 0.0012325163   32.91329 1.418555e-237
#	D_cohort_10  0.06876759 0.0015336998   44.83771  0.000000e+00

#########################
#	Get deviance table.
#	APC strongly rejected => overdispersion?
#   AC (Chain-ladder) rejected against APC (inference invalid anyway)
#   	=> one should be careful with distribution forecasts

apc.fit.table(data,"poisson.response")

#########################
#	        -2logL df.residual prob(>chi_sq)  LR.vs.APC df.vs.APC prob(>chi_sq)        aic
#	APC   176030.0          28             0         NA        NA            NA   176841.7
#	AP    305784.6          36             0   129754.6         8             0   306580.3
#	AC    374155.2          36             0   198125.2         8             0   374950.9
#	PC    553555.1          36             0   377525.0         8             0   554350.7
#	Ad    486013.4          44             0   309983.4        16             0   486793.0
#	Pd    710009.6          44             0   533979.6        16             0   710789.3
#	Cd    780859.4          44             0   604829.4        16             0   781639.1
#	A     575389.6          45             0   399359.6        17             0   576167.3
#	P    9483688.1          45             0  9307658.0        17             0  9484465.7
#	C    7969034.0          45             0  7793004.0        17             0  7969811.7
#	t     898208.1          52             0   722178.1        24             0   898971.7
#	tA    987389.4          53             0   811359.4        25             0   988151.1
#	tP   9690623.4          53             0  9514593.4        25             0  9691385.1
#	tC   8079187.6          53             0  7903157.6        25             0  8079949.3
#	1   10815443.5          54             0 10639413.5        26             0 10816203.2


#########################
#	Fit geometric chain-ladder model

fit.ac	<- apc.fit.model(data,"log.normal.response","AC")
fit.ac$coefficients.canonical
id.ac	<- apc.identify(fit.ac)
id.ac$coefficients.dif

#########################
#	Compare output with	table 7.2 in
#	Kuang D, Nielsen B, Nielsen JP (2015)
#	                 Estimate Std. Error     t value     Pr(>|t|)
#	level       13.0846325168  0.1322711 98.92285585 0.000000e+00
#	D_age_2     -0.0721758004  0.1291053 -0.55904595 5.761304e-01
#	D_age_3     -0.8180698189  0.1350216 -6.05880856 1.371335e-09
#	D_age_4     -0.3945325384  0.1433094 -2.75301253 5.904964e-03
#	D_age_5     -0.3354312554  0.1538274 -2.18056918 2.921530e-02
#	D_age_6     -0.6322104515  0.1673396 -3.77800844 1.580875e-04
#	D_age_7     -0.3020293471  0.1854134 -1.62895114 1.033234e-01
#	D_age_8     -0.5225495852  0.2112982 -2.47304367 1.339678e-02
#	D_age_9      0.0078494549  0.2531172  0.03101115 9.752607e-01
#	D_age_10    -2.5601846890  0.3415805 -7.49511273 6.624141e-14
#	D_cohort_2  -0.1025686798  0.1291053 -0.79445748 4.269292e-01
#	D_cohort_3   0.0820931043  0.1350216  0.60799994 5.431875e-01
#	D_cohort_4   0.3800465893  0.1433094  2.65193088 8.003292e-03
#	D_cohort_5  -0.0920821506  0.1538274 -0.59860701 5.494350e-01
#	D_cohort_6  -0.0530061052  0.1673396 -0.31675768 7.514275e-01
#	D_cohort_7  -0.2053813051  0.1854134 -1.10769405 2.679940e-01
#	D_cohort_8   0.2705853742  0.2112982  1.28058555 2.003393e-01
#	D_cohort_9  -0.0009224552  0.2531172 -0.00364438 9.970922e-01
#	D_cohort_10  0.0736954734  0.3415805  0.21574845 8.291838e-01

#########################
#	Get deviance table.
#	AC marginally rejected against APC

apc.fit.table(data,"log.normal.response")

#########################
#	     -2logL df.residual LR.vs.APC df.vs.APC prob(>chi_sq)     aic
#	APC -28.528          28        NA        NA            NA  27.472
#	AP   -3.998          36    24.530         8         0.002  36.002
#	AC   -9.686          36    18.842         8         0.016  30.314
#	PC   31.722          36    60.250         8         0.000  71.722
#	Ad    6.251          44    34.779        16         0.004  30.251
#	Pd   41.338          44    69.866        16         0.000  65.338
#	Cd   38.919          44    67.447        16         0.000  62.919
#	A    12.765          45    41.292        17         0.001  34.765
#	P   171.283          45   199.811        17         0.000 193.283
#	C   162.451          45   190.979        17         0.000 184.451
#	t    46.300          52    74.827        24         0.000  54.300
#	tA   49.541          53    78.069        25         0.000  55.541
#	tP  171.770          53   200.298        25         0.000 177.770
#	tC  163.280          53   191.808        25         0.000 169.280
#	1   182.166          54   210.694        26         0.000 186.166

US Casualty data, XL Group

Description

Function that organises US Casualty data from XL Group in apc.data.list format.

The data set is taken from table 1.1 Kuang and Nielsen (2020). Data are for US Casualty data from the XL Group. They are gross paid and reported loss and allocated loss adjustment expense in 1000 USD.

The data set is in "CL"-format.

Usage

data.loss.XL

Value

The value is a list in apc.data.list format.

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 26 August 2020 (10 Mar 2018)

Source

Table 1.1 of Kuang and Nielsen (2020) and in turn from XL Group Ltd.

References

Kuang, D. and Nielsen B. (2020) Generalized log-normal chain-ladder. Scandinavian Actuarial Journal 2020, 553-576. Download: Open access. Earlier version: Nuffield DP.

See Also

General description of apc.data.list format.

For explanation for Chain Ladder forecast, see apc.forecast.ac.

The analysis in Kuang and Nielsen (2020) is reproduced in the vignette ReproducingKN2020.pdf, ReproducingKN2020.R on Vignettes.

Examples

#########################
##	It is convenient to construct a data variable for paid data

data	<- data.loss.XL()
##	To see the content of the data
data

#########################
#	Get deviance table.
#	reproduce Table 4.1 in Kuang and Nielsen (2018).

apc.fit.table(data,"log.normal.response")
apc.fit.table(data,"log.normal.response",model.design.reference="AC")

#########################
#	> apc.fit.table(data,"log.normal.response")
#	     -2logL df.residual LR vs.APC df vs.APC prob(>chi_sq) F vs.APC prob(>F)     aic
#	APC 170.003         153       NaN       NaN           NaN      NaN      NaN 286.003
#	AP  243.531         171    73.527        18         0.000    3.564    0.000 323.531
#	AC  179.873         171     9.869        18         0.936    0.409    0.984 259.873
#	PC  633.432         171   463.428        18         0.000   68.736    0.000 713.432
#	Ad  258.570         189    88.567        36         0.000    2.230    0.000 302.570
#	Pd  643.892         189   473.888        36         0.000   36.340    0.000 687.892
#	Cd  649.142         189   479.139        36         0.000   37.368    0.000 693.142
#	A   357.359         190   187.355        37         0.000    5.956    0.000 399.359
#	P   644.176         190   474.172        37         0.000   35.412    0.000 686.176
#	C   672.392         190   502.388        37         0.000   41.099    0.000 714.392
#	t   664.488         207   494.484        54         0.000   27.015    0.000 672.488
#	tA  681.993         208   511.989        55         0.000   29.072    0.000 687.993
#	tP  664.746         208   494.742        55         0.000   26.560    0.000 670.746
#	tC  686.181         208   516.178        55         0.000   29.713    0.000 692.181
#	1   690.399         209   520.396        56         0.000   29.830    0.000 694.399
#
#	> apc.fit.table(data,"log.normal.response",model.design.reference="AC")
#	    -2logL df.residual LR vs.AC df vs.AC prob(>chi_sq) F vs.AC prob(>F)     aic
#	AC 179.873         171      NaN      NaN           NaN     NaN      NaN 259.873
#	Ad 258.570         189   78.698       18             0   4.319        0 302.570
#	Cd 649.142         189  469.269       18             0  79.257        0 693.142
#	A  357.359         190  177.486       19             0  11.955        0 399.359
#	C  672.392         190  492.519       19             0  84.930        0 714.392
#	t  664.488         207  484.615       36             0  42.993        0 672.488
#	tA 681.993         208  502.120       37             0  45.869        0 687.993
#	tC 686.181         208  506.308       37             0  46.886        0 692.181
#	1  690.399         209  510.526       38             0  46.670        0 694.399
 	
 
#########################
#	Fit log normal chain-ladder model
#	reproduce Table 4.2 in Kuang and Nielsen (2018).

fit.ac	<- apc.fit.model(data,"log.normal.response","AC")
id.ac	<- apc.identify(fit.ac)
id.ac$coefficients.dif
fit.ac$s2		
fit.ac$RSS

#########################
#	> id.ac$coefficients.dif
#	                  Estimate Std. Error     t value     Pr(>|t|)
#	level          7.660055032  0.1377951 55.59016605 0.000000e+00
#	D_age_1998     2.272100342  0.1335080 17.01846386 5.992216e-65
#	D_age_1999     0.932530550  0.1362610  6.84370899 7.716860e-12
#	D_age_2000     0.235606356  0.1398301  1.68494782 9.199864e-02
#	D_age_2001     0.088886609  0.1438733  0.61781154 5.366996e-01
#	D_age_2002    -0.176044303  0.1483681 -1.18653717 2.354102e-01
#	D_age_2003    -0.144445459  0.1533567 -0.94189218 3.462478e-01
#	D_age_2004    -0.427608601  0.1589136 -2.69082462 7.127565e-03
#	D_age_2005    -0.300527594  0.1651428 -1.81980421 6.878883e-02
#	D_age_2006    -0.399729999  0.1721838 -2.32153023 2.025824e-02
#	D_age_2007    -0.189656058  0.1802245 -1.05233225 2.926471e-01
#	D_age_2008    -0.242063670  0.1895226 -1.27722853 2.015216e-01
#	D_age_2009    -0.260459607  0.2004421 -1.29942545 1.937980e-01
#	D_age_2010    -0.555317528  0.2135164 -2.60081872 9.300158e-03
#	D_age_2011    -0.303234088  0.2295651 -1.32090683 1.865324e-01
#	D_age_2012     0.405830766  0.2499291  1.62378389 1.044219e-01
#	D_age_2013    -0.895278068  0.2769988 -3.23206421 1.228994e-03
#	D_age_2014     0.116668873  0.3156054  0.36966685 7.116307e-01
#	D_age_2015    -0.383048241  0.3777268 -1.01408813 3.105407e-01
#	D_age_2016    -0.273419402  0.5083832 -0.53782152 5.907003e-01
#	D_cohort_1998  0.288755900  0.1335080  2.16283663 3.055375e-02
#	D_cohort_1999  0.163424236  0.1362610  1.19934721 2.303930e-01
#	D_cohort_2000 -0.264981486  0.1398301 -1.89502518 5.808907e-02
#	D_cohort_2001  0.149829430  0.1438733  1.04139815 2.976908e-01
#	D_cohort_2002 -0.374386828  0.1483681 -2.52336417 1.162380e-02
#	D_cohort_2003 -0.198735893  0.1533567 -1.29590632 1.950078e-01
#	D_cohort_2004 -0.008807130  0.1589136 -0.05542087 9.558032e-01
#	D_cohort_2005 -0.005337953  0.1651428 -0.03232325 9.742143e-01
#	D_cohort_2006 -0.132272851  0.1721838 -0.76820710 4.423642e-01
#	D_cohort_2007 -0.021862643  0.1802245 -0.12130783 9.034472e-01
#	D_cohort_2008 -0.472602270  0.1895226 -2.49364600 1.264386e-02
#	D_cohort_2009 -0.437572798  0.2004421 -2.18303804 2.903301e-02
#	D_cohort_2010  0.295511564  0.2135164  1.38402260 1.663515e-01
#	D_cohort_2011  0.310545832  0.2295651  1.35275725 1.761332e-01
#	D_cohort_2012 -0.268692406  0.2499291 -1.07507473 2.823413e-01
#	D_cohort_2013  0.142131410  0.2769988  0.51311192 6.078730e-01
#	D_cohort_2014  0.201777590  0.3156054  0.63933494 5.226051e-01
#	D_cohort_2015 -0.092672697  0.3777268 -0.24534320 8.061907e-01
#	D_cohort_2016  0.872997251  0.5083832  1.71720334 8.594203e-02	
#	> fit.ac$s2	
#	[1] 0.1693316
#	> fit.ac$RSS
#	[1] 28.9557
#	> fit.ac$RSS

forecast <- apc.forecast.ac(fit.ac,quantiles=c(0.995))
forecast$response.forecast.coh

#########################
#	> forecast$response.forecast.coh
#			 forecast         se    se.proc      se.est     t-0.995
#	coh_2    1871.073   1026.463   707.4405    743.7428    4544.891
#	coh_3    5099.330   1874.681  1375.8435   1273.3744    9982.659
#	coh_4    7171.317   2123.128  1622.5220   1369.3412   12701.822
#	coh_5   11699.350   2984.949  2274.8292   1932.6338   19474.801
#	coh_6   13717.388   3345.138  2654.4080   2035.6984   22431.090
#	coh_7   14343.522   3188.410  2471.3130   2014.5886   22648.964
#	coh_8   18377.001   3834.057  2910.9751   2495.2390   28364.281
#	coh_9   25488.052   5241.618  3976.5389   3414.9225   39141.867
#	coh_10  30524.942   6213.652  4662.3320   4107.5694   46710.794
#	coh_11  40078.245   8115.990  5976.5789   5490.8835   61219.471
#	coh_12  32680.319   6603.511  4727.4210   4610.6241   49881.712
#	coh_13  28509.077   5895.265  4143.1332   4193.8760   43865.568
#	coh_14  51760.526  11013.030  7540.3989   8026.7807   80448.208
#	coh_15  98747.731  22063.641 14798.3216  16365.0210  156220.991
#	coh_16 100330.677  23254.845 14704.7084  18015.5316  160906.889
#	coh_17 149813.314  36629.836 21310.2885  29792.8931  245229.846
#	coh_18 221549.649  58610.037 29815.3239  50459.7158  374222.093
#	coh_19 229480.904  69931.745 29102.9866  63588.2473  411645.102
#	coh_20 575343.178 235016.967 70362.1087 224236.8135 1187535.497

Triangular matrices used in reserving

Description

Triangular matrices are used for reserving in general insurance. A matrix is triangular if it is square and it has NAs in lower triangle where row+col>dim. The apc package uses incremental triangles.

The function is.triangle tests if an object is a triangular matrix.

The function triangle.cumulative forms the cumulative version of an incremental matrix by taking partial sums in each row.

The function triangle.incremental forms the incremental version of an cumulative matrix by taking differences in each row.

The function vector.2.triangle turns a k*(k+1)/2 vector into a triangular matrix of dimension k.

Usage

is.triangle(m)
triangle.cumulative(m)
triangle.incremental(m)
vector.2.triangle(v,k)

Arguments

Author(s)

Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 21 Nov 2019 (7 Feb 2015)

Examples

#########################

m <- vector.2.triangle(1:10,4)
m
is.triangle(m)
triangle.cumulative(m)
triangle.incremental(m)