Session Settings
# Graphs----
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legend_size = 11Reserving exercice on reporting delays in AIDS Data
Triangle
Reserving
Reporting Delays
England
AIDS
Introduction
In Brief
The reported number of Acquired Immune Deficiency Syndrome (AIDS) cases in England and Wales has recently been significantly underestimated due to substantial reporting delays. To address this issue, we will utilize the Chainladder package, specifically employing the MackChainLadder function.
This analysis is crucial for accurately understanding the progression and impact of AIDS, which is essential for effective resource allocation and public health planning. By applying this method to AIDS reports in England and Wales from July 1983 to December 1992, we aim to adjust for reporting delays, thereby providing a more accurate picture of the epidemic’s evolution during this period.
Required Packages
Show the code
required_libraries <- c(
"tidyverse",
"tidyr",
"ChainLadder",
"boot",
"dplyr"
)
invisible(lapply(required_libraries, library, character.only = TRUE))Data
The AIDS dataset consists of 570 rows and 6 columns, capturing critical information about AIDS cases in England and Wales. Although all AIDS cases are required to be reported to the Communicable Disease Surveillance Centre, there is often a significant delay between diagnosis and reporting. To accurately estimate the prevalence of AIDS, it is essential to account for cases that have been diagnosed but not yet reported.
This dataset, obtained from Angelis and Gilks (1994) and available in the boot package, records reported AIDS cases from July 1983 through December 1992. The data is organized by both the date of diagnosis and the delay in reporting, allowing for a more comprehensive analysis of the reporting delays and their impact on understanding the true prevalence of AIDS.
Dictionaries
The list of the 6 attributes from the aids dataset is reported in Table 1.
aids
| Column Name | Description | Notes |
|---|---|---|
| year | The year in which the diagnosis was made. | |
| quarter | The quarter of the year in which the diagnosis was made. | Values range from 1 to 4, representing Q1 to Q4. |
| delay | The time delay (in months) between diagnosis and reporting. | 0 indicates reporting within one month. Longer delays are grouped in 3-month intervals, with the value representing the midpoint of the interval (e.g., 2 means the report was delayed between 1 and 3 months). |
| dud | An indicator of censoring, where categories have incomplete information. | The recorded number is a lower bound only. |
| time | The number of quarters from July 1983 until the end of the quarter in which the cases were diagnosed. | |
| y | The number of AIDS cases reported. |
Importation and triangle transformation
Code for importing the dataset and transforming it into a triangle
data("aids")
aids_agg <- aids |>
group_by(year, quarter, delay) |>
summarise(cases = sum(y), .groups = 'drop') |>
mutate(year_quarter = paste(year, quarter, sep = "-"))
# Create a data frame in long format
aids_long <- aids_agg |>
select(year_quarter, delay, cases)
# Reshape the data into a wide format (triangle format)
triangle <- aids_long |>
pivot_wider(names_from = delay,
values_from = cases,
values_fill = list(cases = 0)) |>
arrange(year_quarter)
# Convert the wide format data frame to a matrix
triangle_matrix <- as.matrix(triangle |>
select(-year_quarter)) # Exclude the year_quarter column for matrix conversion
rownames(triangle_matrix) <- triangle$year_quarter
# Triangular shape
n_rows <- nrow(triangle_matrix)
n_cols <- ncol(triangle_matrix)
for (i in seq_len(n_cols-1)) {
triangle_matrix[(n_rows - i + 1):n_rows, i+1] <- NA
}
full_rows <- which(rowSums(is.na(triangle_matrix)) == 0)
full_triangle <- triangle_matrix[full_rows, ]
close_data <- colSums(full_triangle, na.rm = TRUE)
triangle_rows <- which(rowSums(is.na(triangle_matrix)) != 0)
triangle_matrix_final <- rbind(close_data, triangle_matrix[triangle_rows, ])
# Convert matrix to a 'triangle' object for chainladder
triangle_cl <- as.triangle(triangle_matrix_final)
print(triangle_cl)In this context, a triangle is a table used to display data over time, organized across two dimensions:
- Lines as Origin Year: Represents the year when the diagnosis occurred.
- Columns as Development Year: Indicates the number of months (+/- 1) that have passed since the origin year.
This table structure is instrumental in tracking and analyzing the progression of data from the time of origin over subsequent months. It allows for a clear visualization of how cases develop and accumulate over time, providing valuable insights for further analysis.
Overview
Purpose
The prediction and monitoring of the acquired immune deficiency syndrome (AIDS) epidemic rely heavily on the availability of reliable and complete data on AIDS diagnoses. However, a significant challenge in this context is the considerable delay that often occurs between the diagnosis of an AIDS case and its subsequent reporting to the epidemic monitoring center. In England and Wales, this reporting process is managed by the Public Health Laboratory Service AIDS Centre at the Communicable Disease Surveillance Centre. To effectively utilize AIDS reports, it is crucial to account for cases that have been diagnosed but have not yet been reported.
To address this issue, we will employ the Mack chain ladder method, a statistical tool originally developed for insurance claims forecasting. The method is particularly well-suited for analyzing right-truncated data, making it an ideal choice for correcting delays in reporting. The Mack chain ladder method enables robust predictions of future case reports by estimating the development pattern of reported cases over time. This approach not only adjusts for the reporting delay but also provides a measure of the uncertainty associated with these estimates.
By applying the Mack chain ladder method, we aim to achieve more accurate and timely insights into the progression of the AIDS epidemic. Enhanced prediction and monitoring capabilities are essential for effective public health planning, resource allocation, and the implementation of timely interventions to control the spread of the disease. While various statistical techniques have been developed to address this inferential challenge, the Mack chain ladder method is distinguished by its reliability and practicality in handling right-truncated data.
Show the code
triangle_cum <- incr2cum(triangle_cl)
print(triangle_cum) dev
origin 0 2 5 8 11 14 17 20 23 26 29 32 35
close_data 355 1455 1784 2000 2134 2245 2328 2397 2451 2490 2526 2546 2553
1989-3 34 138 167 198 216 224 230 237 240 248 248 250 251
1989-4 38 139 173 191 200 215 221 222 224 226 228 231 233
1990-1 31 155 202 226 237 252 260 266 271 274 277 281 NA
1990-2 32 164 200 210 219 226 232 236 240 245 245 NA NA
1990-3 49 156 207 224 239 247 256 258 259 260 NA NA NA
1990-4 44 197 238 254 265 271 276 283 285 NA NA NA NA
1991-1 41 178 207 240 247 258 264 268 NA NA NA NA NA
1991-2 56 180 219 233 245 252 262 NA NA NA NA NA NA
1991-3 53 228 263 280 293 304 NA NA NA NA NA NA NA
1991-4 63 198 222 245 257 NA NA NA NA NA NA NA NA
1992-1 71 232 280 305 NA NA NA NA NA NA NA NA NA
1992-2 95 273 312 NA NA NA NA NA NA NA NA NA NA
1992-3 76 257 NA NA NA NA NA NA NA NA NA NA NA
1992-4 67 NA NA NA NA NA NA NA NA NA NA NA NA
dev
origin 38 41
close_data 2573 2626
1989-3 253 NA
1989-4 NA NA
1990-1 NA NA
1990-2 NA NA
1990-3 NA NA
1990-4 NA NA
1991-1 NA NA
1991-2 NA NA
1991-3 NA NA
1991-4 NA NA
1992-1 NA NA
1992-2 NA NA
1992-3 NA NA
1992-4 NA NACode to create the following graph
ChainLadder::plot(as.triangle(triangle_cum[-1, -n_cols]))
Code to create the following graph
plot(as.triangle(triangle_cum[-1, -n_cols]), lattice=TRUE)
Mack chain-ladder
In 1993, Thomas Mack introduced a groundbreaking method in his paper (Mack 1993), which estimates the standard errors of the chain-ladder forecast without assuming a specific distribution, under three key conditions.
The Mack Chain-Ladder Model
Following the notation established by Mack in 1999 (Mack 1999), let \(C_{ik}\) denote the cumulative loss amounts of origin period (e.g., accident year) \(i=1,\ldots,m\), with losses known for development period (e.g., development year) \(k \le n+1-i.\)
To forecast the amounts \(C_{ik}\) for \(k > n+1-i\), the Mack chain-ladder model makes the following assumptions:
\[ \begin{aligned} \text{CL1:} & \quad E[ F_{ik}| C_{i1},C_{i2},\ldots,C_{ik} ] = f_k \quad \text{where} \quad F_{ik} = \frac{C_{i,k+1}}{C_{ik}} \\ \text{CL2:} & \quad Var\left( \frac{C_{i,k+1}}{C_{ik}} \Bigg| C_{i1},C_{i2}, \ldots,C_{ik} \right) = \frac{\sigma_k^2}{w_{ik} C_{ik}^\alpha} \\ \text{CL3:} & \quad \{C_{i1},\ldots,C_{in}\} \text{ and } \{C_{j1},\ldots,C_{jn}\} \text{ are independent for origin periods } i \neq j \end{aligned} \]
where \(w_{ik} \in [0,1]\) and \(\alpha \in \{0,1,2\}\) are parameters that adjust the variance structure. If these assumptions hold, the Mack chain-ladder model provides an unbiased estimator for Incurred But Not Reported (IBNR) claims.
Intuition Behind the Method
The Chain-Ladder model is a powerful tool used to project future claims based on historical data. The core idea is that claims development patterns are relatively stable and predictable. The model assumes that the ratios of cumulative losses between successive development years are consistent, allowing these ratios to be used in estimating future losses.
Assumptions Explained:
CL1: Expected Future Ratio
This assumption posits that the expected future ratio \(F_{ik}\) of cumulative losses between successive development years is constant, given the known losses up to the current development year. Essentially, this means that the ratio of cumulative losses from one development year to the next is assumed to follow a consistent pattern, captured by a factor \(f_k\).CL2: Variance of Future Ratio
Here, the model specifies that the variance of the future ratio of cumulative losses is proportional to \(\frac{\sigma_k^2}{w_{ik} C_{ik}^\alpha}\). The term \(\sigma_k^2\) represents the variability, \(w_{ik}\) is a weight, and \(\alpha\) is a parameter that adjusts for different variance levels. This assumption is crucial for quantifying the uncertainty around the forecasts.CL3: Independence of Origin Periods
This assumption ensures that cumulative loss amounts from different origin periods (e.g., different accident years) are independent. This independence simplifies the estimation process and increases the robustness of the model.
Practical Application
The Mack Chain-Ladder model can be viewed as a weighted linear regression through the origin for each development period:
\[ \text{lm}(y \sim x + 0, \text{weights} = w/x^{2-\alpha}), \]
where \(y\) is the vector of claims at development period \(k+1\) and \(x\) is the vector of claims at development period \(k\).
The Mack method is implemented in the ChainLadder package via the function MackChainLadder. This implementation enables actuaries to perform robust reserving exercises, forecast future claims developments, and maintain the financial stability of insurance companies by ensuring they can meet their future claim obligations.
For a comprehensive understanding of this methodology, including its practical implications and applications, see Gesmann (2014).
As an example we apply the MackChainLadder function to our triangle Damage:
mack <- MackChainLadder(triangle_cum, est.sigma="Mack")
mack # same as summary(mack) MackChainLadder(Triangle = triangle_cum, est.sigma = "Mack")
Latest Dev.To.Date Ultimate IBNR Mack.S.E CV(IBNR)
close_data 2,626 1.000 2,626 0.00 0.00000 NaN
1989-3 253 0.980 258 5.21 0.00112 0.000215
1989-4 233 0.972 240 6.67 0.03291 0.004938
1990-1 281 0.969 290 8.99 1.10632 0.123017
1990-2 245 0.960 255 10.08 1.57793 0.156584
1990-3 260 0.949 274 13.88 3.06281 0.220653
1990-4 285 0.935 305 19.94 4.42964 0.222143
1991-1 268 0.918 292 23.97 5.23819 0.218506
1991-2 262 0.896 292 30.45 6.32616 0.207725
1991-3 304 0.867 351 46.59 7.51700 0.161331
1991-4 257 0.829 310 53.12 8.71038 0.163967
1992-1 305 0.783 390 84.52 11.62803 0.137573
1992-2 312 0.708 441 128.97 18.94355 0.146883
1992-3 257 0.584 440 183.05 26.87186 0.146798
1992-4 67 0.153 437 369.56 80.76600 0.218545
Totals
Latest: 6,215.00
Dev: 0.86
Ultimate: 7,200.02
IBNR: 985.02
Mack.S.E 92.13
CV(IBNR): 0.09# Displaying the Mack model's parameters
mack$f [1] 3.805395 1.211481 1.106679 1.058359 1.046333 1.033174 1.024588 1.018210
[9] 1.015739 1.011771 1.008844 1.003304 1.007846 1.020599 1.000000# Viewing the full triangle data from the Mack model
mack$FullTriangle dev
origin 0 2 5 8 11 14 17
close_data 355 1455.0000 1784.0000 2000.0000 2134.0000 2245.0000 2328.0000
1989-3 34 138.0000 167.0000 198.0000 216.0000 224.0000 230.0000
1989-4 38 139.0000 173.0000 191.0000 200.0000 215.0000 221.0000
1990-1 31 155.0000 202.0000 226.0000 237.0000 252.0000 260.0000
1990-2 32 164.0000 200.0000 210.0000 219.0000 226.0000 232.0000
1990-3 49 156.0000 207.0000 224.0000 239.0000 247.0000 256.0000
1990-4 44 197.0000 238.0000 254.0000 265.0000 271.0000 276.0000
1991-1 41 178.0000 207.0000 240.0000 247.0000 258.0000 264.0000
1991-2 56 180.0000 219.0000 233.0000 245.0000 252.0000 262.0000
1991-3 53 228.0000 263.0000 280.0000 293.0000 304.0000 314.0850
1991-4 63 198.0000 222.0000 245.0000 257.0000 268.9076 277.8284
1992-1 71 232.0000 280.0000 305.0000 322.7993 337.7556 348.9604
1992-2 95 273.0000 312.0000 345.2840 365.4343 382.3659 395.0506
1992-3 76 257.0000 311.3507 344.5654 364.6737 381.5701 394.2284
1992-4 67 254.9615 308.8810 341.8323 361.7811 378.5435 391.1014
dev
origin 20 23 26 29 32 35
close_data 2397.0000 2451.0000 2490.0000 2526.0000 2546.0000 2553.0000
1989-3 237.0000 240.0000 248.0000 248.0000 250.0000 251.0000
1989-4 222.0000 224.0000 226.0000 228.0000 231.0000 233.0000
1990-1 266.0000 271.0000 274.0000 277.0000 281.0000 281.9283
1990-2 236.0000 240.0000 245.0000 245.0000 247.1668 247.9834
1990-3 258.0000 259.0000 260.0000 263.0606 265.3871 266.2639
1990-4 283.0000 285.0000 289.4858 292.8934 295.4838 296.4600
1991-1 268.0000 272.8802 277.1752 280.4380 282.9182 283.8529
1991-2 268.4421 273.3304 277.6325 280.9006 283.3849 284.3211
1991-3 321.8077 327.6678 332.8251 336.7429 339.7212 340.8435
1991-4 284.6597 289.8432 294.4052 297.8708 300.5052 301.4980
1992-1 357.5407 364.0514 369.7814 374.1343 377.4432 378.6901
1992-2 404.7642 412.1348 418.6216 423.5494 427.2954 428.7070
1992-3 403.9218 411.2771 417.7504 422.6679 426.4061 427.8147
1992-4 400.7178 408.0148 414.4368 419.3153 423.0238 424.4213
dev
origin 38 41
close_data 2573.0000 2626.0000
1989-3 253.0000 258.2114
1989-4 234.8281 239.6652
1990-1 284.1403 289.9932
1990-2 249.9290 255.0772
1990-3 268.3530 273.8806
1990-4 298.7860 304.9405
1991-1 286.0800 291.9728
1991-2 286.5519 292.4544
1991-3 343.5177 350.5936
1991-4 303.8635 310.1226
1992-1 381.6613 389.5229
1992-2 432.0706 440.9706
1992-3 431.1713 440.0528
1992-4 427.7513 436.5623The Mack model factors start at 3.8 for the initial periods and gradually decrease to around 1 in the later periods. These factors represent the development factors for each period, indicating the expected growth in cumulative reports from one period to the next. This trend reflects the typical pattern where most reports occur early in the development process, with the rate of increase diminishing over time.
The triangular data illustrates how reports develop over time for each origin period. For instance, in the 4th quarter of 1992, the number of reported cases of acquired immune deficiency syndrome (AIDS) begins at 67 in the 1st period and grows to 437 cases by the 41st period.
Overall, this output offers a comprehensive view of how reports develop over time and highlights the associated development factors used in the chain ladder method.
References
Angelis, D. De, and W. R. Gilks. 1994. “Estimating Acquired Immune Deficiency Syndrome Accounting for Reporting Delay.” Journal of the Royal Statistical Society, A 157: 31–40.
Gesmann, Markus. 2014. “Claims Reserving and IBNR.” In Computational Actuarial Science with R, 545–84. Chapman; Hall/CRC.
Mack, T. 1993. “Distribution-Free Calculation of the Standard Error of Chain-Ladder Reserve Estimates.” ASTIN Bulletin 23 (2): 213–25.
———. 1999. “The Standard Error of Chain-Ladder Reserve Estimates: Recursive Calculation and Inclusion of a Tail-Factor.” ASTIN Bulletin 29: 361–66.
See also
For more similar triangles datasets, see nortritpl8800 (import with data("ausprivauto0405")): Australian liabilty insurance triangles dataset, sgautoprop9701: Singapore general liability triangles dataset (import with data("norauto")), swtri1auto: Switzerland general liability triangles dataset (import with data("beMTPL16")), or usautotri9504 (import with data("pg17trainpol")): US Automobile triangles dataset.