Introduction

The goal of this R script is to provide an example of how to update an existing assumption using a penalized regression. For illustration purposes we will start with an assumption that varies by duration and gender. We will then use the penalized regression to update the assumption to better fit the data.

To begin let’s first load the packages and data. We’ll load in the starting expected assumption (E) data that we initially explored in the “0250 - PoissonGLM Offset 2018.R” program and tag it onto our CTR data which was processed from the “0150 - Separate data into train val test.R” program. To perform this, we’ll have to filter the CTR dataset with the same filters from earlier. We’ll then follow up by joining the starting E to the CTR dataset by claim duration and gender.

# 4.1.0 - Load the Packages and Explore the Data 
library(dplyr)
library(glmnet)
library(stringr)
library(Matrix)
library(doParallel)
library(ggplot2)
library(plotly)
library(scales)
library(kableExtra)
library(knitr)

# Initialize file paths for working directories
data_stored = "C:\\ILTCI_Workshop\\Data"
data_output = "C:\\ILTCI_Workshop\\Output"


# Load in existing assumption (E) data
load(paste0(data_stored, "\\", "smoothed_assumptions_20180301.RData"))

# Load prior processed data from the "0150 - Separate data into train val test.R" program 
load(paste0(data_output, "\\", "ctr_data.RData"))

# Apply filters used earlier
ctr_data <- ctr_data %>%
  filter(GroupIndicator == "Ind", 
         ClaimType != "Unk",
         Cov_Type_Bucket == "Comprehensive"
         )

# Join our smoothed assumption to the data
ctr_data_bench <- ctr_data %>% 
  left_join(smoothed_assumptions,
            by = c("ClaimDuration", "Gender")
            ) %>% 
  # Calculate the expected claim terminations
  mutate(exp_terms = smoothed*Exposure)

#Remove object from environment and garbage collect to save memory
invisible(rm(ctr_data))

Note: The ‘invisible’ function surpess the output. Also notice how we can directly load a file by specifying the path it is located instead of changing our work directory.

Using the kable() function from the kableExtra package we can explore the aggregated dataset we created within this document via a scroll box. This gives the reader of such a document the ability to dig deeper into the data if they are interested without overwhelming them with rows of data if they are not interested.

#Take a peak at the first 6 row
head(ctr_data_bench) %>%
kable("html") %>%
  kable_styling() %>%
  scroll_box(width = "900px", height = "250px")
fake_claim_id GroupIndicator Gender IncurredAgeBucket Incurred_Year ClaimType Region StateAbbr Diagnosis_Category TQ_Status Cov_Type_Bucket Infl_Rider_Bucket EP_Bucket Max_Ben_Bucket ClaimDuration Exposure Terminations start_duration end_duration incurred_date study_start_date current_date Sample V1 smoothed exp_terms
1 Ind Female 70 to 74 1991 NH South VA Injury Q Comprehensive GPO 90/100 5 + 103 1 0 102 108 1991-07-01 2000-01-01 2000-02-01 training 353 0.025871 0.025871
1 Ind Female 70 to 74 1991 NH South VA Injury Q Comprehensive GPO 90/100 5 + 104 1 0 102 108 1991-07-01 2000-01-01 2000-03-01 training 354 0.025871 0.025871
1 Ind Female 70 to 74 1991 NH South VA Injury Q Comprehensive GPO 90/100 5 + 106 1 0 102 108 1991-07-01 2000-01-01 2000-05-01 training 356 0.025871 0.025871
1 Ind Female 70 to 74 1991 NH South VA Injury Q Comprehensive GPO 90/100 5 + 107 1 0 102 108 1991-07-01 2000-01-01 2000-06-01 training 357 0.025871 0.025871
4 Ind Male 60 to 64 1991 HHC Northeast CT Stroke Q Comprehensive None 90/100 3 to 4 104 1 0 101 138 1991-07-01 2000-01-01 2000-04-01 validation 104 0.040733 0.040733
4 Ind Male 60 to 64 1991 HHC Northeast CT Stroke Q Comprehensive None 90/100 3 to 4 109 1 0 101 138 1991-07-01 2000-01-01 2000-09-01 validation 109 0.040733 0.040733

Data Exploration:

Now that we have the expected terminations tagged on the data let’s explore the fit of the assumption. Currently the assumption varies just by ‘gender’ and ‘claim duration’ so let’s explore that fit initially.

# https://protect-us.mimecast.com/s/8cD4CVOZrvuk1mkQIG0QUt?domain=4.1.1.1 - View overall AtoE on training set

AtoE_train_all <- ctr_data_bench %>%
  filter(Sample == "training") %>%
  summarise(Termination = comma(sum(Terminations)),
            Rate = percent(sum(Terminations)/sum(Exposure)),
            AtoE = round(sum(Terminations)/sum(exp_terms), 2))

kable(AtoE_train_all,"html") %>%
  kable_styling() 
Termination Rate AtoE
12,449 2.63% 0.81
# https://protect-us.mimecast.com/s/4eBMCW6BvPfXvZXoSxU00P?domain=4.1.1.2 - View overall AtoE grouped by gender on training set
AtoE_train_gender <- ctr_data_bench %>%
  filter(Sample == "training") %>%
  group_by(Gender) %>%
  summarise(Termination = comma(sum(Terminations)),
            Rate = percent(sum(Terminations)/sum(Exposure)),
            AtoE = round(sum(Terminations)/sum(exp_terms), 2))

kable(AtoE_train_gender,"html") %>%
  kable_styling() 
Gender Termination Rate AtoE
Female 7,603 2.33% 0.87
Male 4,846 3.29% 0.74

Next, we’ll create a plot which will allow us to see how well the E fits the training data by duration. We have made this plot interactive by wrapping it in the ggplotly() function from the plotly package. With such a plot you are able to zoom in and out as well as hover over points on the graph to view the underlying data.

# https://protect-us.mimecast.com/s/nd-kCXDVwQtD3jDrFDVWac?domain=4.1.1.4 
# Determine how well the E fits training data dependent by only duration
duration <- ctr_data_bench %>%
  filter(Sample == "training") %>%
  group_by(ClaimDuration) %>%
  summarise(actual_hz_rate = sum(Terminations)/sum(Exposure),
            exp_hz_rate = sum(exp_terms)/sum(Exposure),
            terms = sum(Terminations),
            exposure = sum(Exposure)) %>% data.frame()

# Plot Hazard Rate vs Claim Duration
ggplotly(ggplot() + 
           geom_line(data = duration, aes(x = ClaimDuration, y = exp_hz_rate))+ 
           geom_line(data = duration, linetype = "dotted",
                     aes(x = ClaimDuration, y = actual_hz_rate))+
           ggtitle("Raw Hazard rates by duration"))

We’ll now look at a plot that utilizies both duration and gender.

# https://protect-us.mimecast.com/s/aSM-C1w7q6SBKxBWi1-hdY?domain=4.1.1.5 - Let's see how well the E fits the training data by duration and gender
gender_duration <- ctr_data_bench %>%
  filter(Sample == "training") %>%
  group_by(ClaimDuration,
           Gender) %>%
  summarise(actual_hz_rate = sum(Terminations)/sum(Exposure),
            exp_hz_rate = sum(exp_terms)/sum(Exposure),
            terms = sum(Terminations),
            exposure = sum(Exposure)) %>% data.frame()

# Plot Hazard Rate vs. Gender and Duration
ggplotly(ggplot() + 
           geom_line(data = gender_duration, aes(x = ClaimDuration, 
                                                 y = exp_hz_rate, col=c(Gender)))+ 
           geom_line(data = gender_duration, linetype = "dotted",
                     aes(x = ClaimDuration, y = actual_hz_rate, col=c(Gender)))+
           ggtitle("Raw Hazard rates by duration and gender"))

Pre-processing Data

Now that we have explored the data we see that the fit by duration and gender is off. Let’s update the fit based on the training data using a penalized glm. We will select the optimal penalty value using a 10-fold cross-validation (CV). To do this first we need create random fold IDs to assign observations randomly to the 10 different folds.

# https://protect-us.mimecast.com/s/raf_C2kJr8tZAWZyiX23I3?domain=4.1.2.1 Use a variable as a place holder for the number of folds
nfolds <- 10

# Grab list of distinct claim IDs since we will create the fold by claim ID 
# in order to avoid splitting one claim into two or more folds
distinct_claim_ids <- ctr_data_bench %>%  
  distinct(fake_claim_id)

# Use seed to make results reproducible and sample 10 folds
set.seed(28)
folds <- data.frame(distinct_claim_ids,
                    fold = sample(nfolds, 
                                  size = nrow(distinct_claim_ids), 
                                  replace = TRUE
                                  )
                    )

# Join folds assignments back to the data
ctr_data_bench <- ctr_data_bench %>%
  left_join(folds, by="fake_claim_id")

rm(distinct_claim_ids,
   folds)

head(ctr_data_bench) %>%
kable("html") %>%
  kable_styling() %>%
  scroll_box(width = "900px", height = "250px")
fake_claim_id GroupIndicator Gender IncurredAgeBucket Incurred_Year ClaimType Region StateAbbr Diagnosis_Category TQ_Status Cov_Type_Bucket Infl_Rider_Bucket EP_Bucket Max_Ben_Bucket ClaimDuration Exposure Terminations start_duration end_duration incurred_date study_start_date current_date Sample V1 smoothed exp_terms fold
1 Ind Female 70 to 74 1991 NH South VA Injury Q Comprehensive GPO 90/100 5 + 103 1 0 102 108 1991-07-01 2000-01-01 2000-02-01 training 353 0.025871 0.025871 1
1 Ind Female 70 to 74 1991 NH South VA Injury Q Comprehensive GPO 90/100 5 + 104 1 0 102 108 1991-07-01 2000-01-01 2000-03-01 training 354 0.025871 0.025871 1
1 Ind Female 70 to 74 1991 NH South VA Injury Q Comprehensive GPO 90/100 5 + 106 1 0 102 108 1991-07-01 2000-01-01 2000-05-01 training 356 0.025871 0.025871 1
1 Ind Female 70 to 74 1991 NH South VA Injury Q Comprehensive GPO 90/100 5 + 107 1 0 102 108 1991-07-01 2000-01-01 2000-06-01 training 357 0.025871 0.025871 1
4 Ind Male 60 to 64 1991 HHC Northeast CT Stroke Q Comprehensive None 90/100 3 to 4 104 1 0 101 138 1991-07-01 2000-01-01 2000-04-01 validation 104 0.040733 0.040733 1
4 Ind Male 60 to 64 1991 HHC Northeast CT Stroke Q Comprehensive None 90/100 3 to 4 109 1 0 101 138 1991-07-01 2000-01-01 2000-09-01 validation 109 0.040733 0.040733 1

A quick glimpse indicates that we have successfully added a random fold ID. We can verify that each fold has approximately the same amount of observations.

# Verify that each fold has about the same amount of data

round(table(ctr_data_bench$fold)/(nrow(ctr_data_bench)), digits = 2)
## 
##   1   2   3   4   5   6   7   8   9  10 
## 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

Now that we have the K-fold ID’s, we’re almost ready to build our model.

Before we do so, we first have to do some more data processing to prepare our data for modeling. With this model we will only adjust by gender and granular monthly claim durations. Later we can try incorporating additional variables and claim duration bucketing. We are also going to create a new claim duration variable that limits our durations to 160 months. This was done since our test data contains later durations, which are not in our training data. If we didn’t do this then we wouldn’t have predictions for the later durations in the test data, which would cause the code to break when we score the test data at the end.

# 4.1.2 
# Create column where if Claim Duration is over 160, replace Duration with 160
ctr_data_bench <- ctr_data_bench %>%
  mutate(ClaimDuration_v2_ = ifelse(ClaimDuration > 160, 160, ClaimDuration))

# select the variables needed to train and test the model and then aggregate the data 
# to speed up training.
agg_data <- ctr_data_bench %>%
  group_by(fold,
           Gender,
           ClaimDuration_v2_,
           ClaimDuration,
           Sample
           ) %>%
  summarise(exp_terms = sum(exp_terms),
            Terminations = sum(Terminations),
            Exposure = sum(Exposure)
            ) %>% data.frame()

rm(ctr_data_bench)

kable(agg_data,"html") %>%
  kable_styling() %>%
  scroll_box(width = "900px", height = "250px")
fold Gender ClaimDuration_v2_ ClaimDuration Sample exp_terms Terminations Exposure
1 Female 1 1 testing 3.1253265 4 24.496802
1 Female 1 1 training 16.5031879 22 129.354590
1 Female 1 1 validation 6.1883021 7 48.504888
1 Female 2 2 testing 7.1220752 8 64.767924
1 Female 2 2 training 34.6015256 30 314.665166
1 Female 2 2 validation 13.0765596 10 118.917814
1 Female 3 3 testing 7.7869221 6 93.772018
1 Female 3 3 training 36.8045353 39 443.209202
1 Female 3 3 validation 14.0745040 18 169.488614
1 Female 4 4 testing 12.2810002 11 230.196818
1 Female 4 4 training 60.1266571 49 1127.022626
1 Female 4 4 validation 20.1958404 19 378.553710
1 Female 5 5 testing 9.6139943 7 244.351108
1 Female 5 5 training 48.3902244 46 1229.895144
1 Female 5 5 validation 15.5888591 14 396.209406
1 Female 6 6 testing 8.4512867 12 246.486618
1 Female 6 6 training 37.3381868 37 1088.989612
1 Female 6 6 validation 12.6152911 14 367.932194
1 Female 7 7 testing 7.2712924 9 242.416816
1 Female 7 7 training 34.4968349 28 1150.086178
1 Female 7 7 validation 11.0150003 12 367.227880
1 Female 8 8 testing 5.7104960 4 214.971240
1 Female 8 8 training 29.2533437 23 1101.240162
1 Female 8 8 validation 10.2151392 5 384.548232
1 Female 9 9 testing 5.8675043 8 239.275114
1 Female 9 9 training 26.0535916 21 1062.457858
1 Female 9 9 validation 9.4381490 12 384.884960
1 Female 10 10 testing 11.0880613 9 466.355202
1 Female 10 10 training 20.6711070 19 869.410624
1 Female 10 10 validation 6.9863357 4 293.839824
1 Female 11 11 testing 11.2393315 12 479.227880
1 Female 11 11 training 20.0699879 18 855.753544
1 Female 11 11 validation 6.5288430 3 278.379868
1 Female 12 12 testing 10.8406305 14 465.203214
1 Female 12 12 training 19.6452411 21 843.034850
1 Female 12 12 validation 6.3909071 7 274.252546
1 Female 13 13 testing 5.5078613 2 238.952766
1 Female 13 13 training 19.8660225 15 861.866486
1 Female 13 13 validation 6.7718248 5 293.788496
1 Female 14 14 testing 5.2283639 5 234.182744
1 Female 14 14 training 17.5270091 19 785.049230
1 Female 14 14 validation 6.1170030 2 273.985620
1 Female 15 15 testing 5.2618655 1 248.084182
1 Female 15 15 training 15.9332383 17 751.213500
1 Female 15 15 validation 5.8026548 7 273.581086
1 Female 16 16 testing 4.9660041 6 246.464050
1 Female 16 16 training 15.5215972 16 770.340822
1 Female 16 16 validation 5.1482969 3 255.511284
1 Female 17 17 testing 4.3555171 7 224.383962
1 Female 17 17 training 15.4129310 16 794.030756
1 Female 17 17 validation 4.8709650 4 250.938386
1 Female 18 18 testing 4.3649430 10 229.468144
1 Female 18 18 training 14.2545082 12 749.369582
1 Female 18 18 validation 4.5397350 3 238.657080
1 Female 19 19 testing 4.1915932 5 221.215598
1 Female 19 19 training 13.8300551 12 729.895246
1 Female 19 19 validation 4.9594734 5 261.741262
1 Female 20 20 testing 4.2782483 6 224.121130
1 Female 20 20 training 13.7466267 14 720.133412
1 Female 20 20 validation 4.6900533 4 245.694028
1 Female 21 21 testing 4.2190149 2 218.036948
1 Female 21 21 training 13.5460724 5 700.055422
1 Female 21 21 validation 4.8454862 3 250.412722
1 Female 22 22 testing 7.2082381 7 366.383962
1 Female 22 22 training 11.5060169 16 584.833632
1 Female 22 22 validation 4.0958679 7 208.186838
1 Female 23 23 testing 6.8869164 8 344.190932
1 Female 23 23 training 11.9922105 9 599.340822
1 Female 23 23 validation 3.8894287 7 194.383962
1 Female 24 24 testing 6.9458443 5 342.632414
1 Female 24 24 training 11.5768930 12 571.077990
1 Female 24 24 validation 3.7268009 3 183.839824
1 Female 25 25 testing 2.8847581 3 141.708408
1 Female 25 25 training 11.4267130 16 561.316156
1 Female 25 25 validation 3.9199971 5 192.562612
1 Female 26 26 testing 3.2045952 4 158.314160
1 Female 26 26 training 10.5910959 11 523.223786
1 Female 26 26 validation 3.3725165 4 166.609846
1 Female 27 27 testing 3.1887327 4 159.412722
1 Female 27 27 training 10.8302482 4 541.431196
1 Female 27 27 validation 3.6711457 5 183.529758
1 Female 28 28 testing 3.2202280 2 163.000000
1 Female 28 28 training 10.1981521 9 516.205312
1 Female 28 28 validation 3.4048876 4 172.347014
1 Female 29 29 testing 2.4567875 3 125.365488
1 Female 29 29 training 9.9397025 10 507.205312
1 Female 29 29 validation 2.9868321 3 152.412722
1 Female 30 30 testing 2.8487011 2 145.624226
1 Female 30 30 training 9.5118711 10 486.242260
1 Female 30 30 validation 3.2694248 3 167.131416
1 Female 31 31 testing 2.5168682 1 128.182744
1 Female 31 31 training 9.5707919 6 487.435290
1 Female 31 31 validation 2.9639576 3 150.952766
1 Female 32 32 testing 2.5893340 2 130.919912
1 Female 32 32 training 9.7543303 9 493.190932
1 Female 32 32 validation 2.8182228 2 142.492810
1 Female 33 33 testing 2.6362727 1 132.051328
1 Female 33 33 training 9.2405848 9 462.862392
1 Female 33 33 validation 2.7652393 2 138.511284
1 Female 34 34 testing 4.9426675 3 244.989714
1 Female 34 34 training 6.9561905 6 344.792590
1 Female 34 34 validation 2.6754036 1 132.609846
1 Female 35 35 testing 4.9247676 3 241.398342
1 Female 35 35 training 6.9504989 5 340.694028
1 Female 35 35 validation 2.6715674 3 130.952766
1 Female 36 36 testing 4.7194144 3 228.698122
1 Female 36 36 training 7.2436172 4 351.018474
1 Female 36 36 validation 2.4560652 2 119.018474
1 Female 37 37 testing 2.4321369 2 116.492810
1 Female 37 37 training 6.8379515 8 327.519472
1 Female 37 37 validation 2.6454181 3 126.708408
1 Female 38 38 testing 2.5952951 2 122.854204
1 Female 38 38 training 7.1469731 7 338.318254
1 Female 38 38 validation 2.5256737 0 119.558518
1 Female 39 39 testing 2.4373729 1 114.018474
1 Female 39 39 training 6.9999791 7 327.453764
1 Female 39 39 validation 2.6090912 5 122.051328
1 Female 40 40 testing 2.4155431 8 111.665268
1 Female 40 40 training 6.5381046 6 302.242260
1 Female 40 40 validation 2.3413194 2 108.234072
1 Female 41 41 testing 2.5864192 2 118.149890
1 Female 41 41 training 6.5387110 4 298.694028
1 Female 41 41 validation 2.5054177 4 114.449670
1 Female 42 42 testing 2.4386949 1 110.084182
1 Female 42 42 training 6.4412004 3 290.759736
1 Female 42 42 validation 2.1434277 2 96.755642
1 Female 43 43 testing 2.1405309 0 95.478430
1 Female 43 43 training 6.7291522 3 300.153984
1 Female 43 43 validation 2.2824105 4 101.806970
1 Female 44 44 testing 2.0668442 2 91.098562
1 Female 44 44 training 6.3907479 5 281.679648
1 Female 44 44 validation 2.2856178 0 100.741262
1 Female 45 45 testing 2.2121274 1 96.347014
1 Female 45 45 training 5.9596516 8 259.566706
1 Female 45 45 validation 1.9941725 2 86.854204
1 Female 46 46 testing 3.4283356 3 147.544138
1 Female 46 46 training 5.1439826 2 221.379868
1 Female 46 46 validation 1.6685068 2 71.806970
1 Female 47 47 testing 3.3419784 4 142.121130
1 Female 47 47 training 4.9336109 5 209.806970
1 Female 47 47 validation 1.4139902 0 60.131416
1 Female 48 48 testing 3.1669900 9 133.077990
1 Female 48 48 training 4.7523185 1 199.694028
1 Female 48 48 validation 1.7494215 3 73.511284
1 Female 49 49 testing 1.3041460 1 54.149890
1 Female 49 49 training 4.4659249 5 185.431196
1 Female 49 49 validation 1.5734712 3 65.332634
1 Female 50 50 testing 1.6817370 0 69.000000
1 Female 50 50 training 4.9721918 3 204.004094
1 Female 50 50 validation 1.6466038 0 67.558518
1 Female 51 51 testing 1.7509313 1 70.985620
1 Female 51 51 training 4.6801580 3 189.741262
1 Female 51 51 validation 1.5612514 1 63.295686
1 Female 52 52 testing 1.5648661 1 62.689934
1 Female 52 52 training 4.7547226 6 190.478430
1 Female 52 52 validation 1.6065891 1 64.361394
1 Female 53 53 testing 1.4225520 1 56.314160
1 Female 53 53 training 4.2776151 6 169.336728
1 Female 53 53 validation 1.7682700 0 70.000000
1 Female 54 54 testing 1.3079634 1 51.164270
1 Female 54 54 training 4.9410433 1 193.281306
1 Female 54 54 validation 1.7127880 0 67.000000
1 Female 55 55 testing 1.3452920 1 52.000000
1 Female 55 55 training 4.1043512 8 158.646794
1 Female 55 55 validation 1.7054140 1 65.919912
1 Female 56 56 testing 1.3389702 0 51.755642
1 Female 56 56 training 4.0691306 4 157.285400
1 Female 56 56 validation 1.3711630 0 53.000000
1 Female 57 57 testing 1.3962900 3 53.971240
1 Female 57 57 training 4.1971577 4 162.234072
1 Female 57 57 validation 1.6240292 2 62.774116
1 Female 58 58 testing 2.2942312 5 88.679648
1 Female 58 58 training 3.2959225 5 127.398342
1 Female 58 58 validation 1.2049934 1 46.576992
1 Female 59 59 testing 2.1159502 1 81.788496
1 Female 59 59 training 3.1673116 3 122.427102
1 Female 59 59 validation 1.1900660 0 46.000000
1 Female 60 60 testing 2.5898617 6 100.106750
1 Female 60 60 training 2.9862146 3 115.427102
1 Female 60 60 validation 1.1743946 2 45.394248
1 Female 61 61 testing 1.0089690 0 39.000000
1 Female 61 61 training 2.9582716 5 114.347014
1 Female 61 61 validation 0.9432555 1 36.459956
1 Female 62 62 testing 1.0348400 0 40.000000
1 Female 62 62 training 2.8471379 2 110.051328
1 Female 62 62 validation 1.0407898 1 40.229978
1 Female 63 63 testing 1.0802602 1 41.755642
1 Female 63 63 training 2.6270484 0 101.544138
1 Female 63 63 validation 1.1468237 0 44.328540
1 Female 64 64 testing 0.9407056 1 36.361394
1 Female 64 64 training 2.6634910 2 102.952766
1 Female 64 64 validation 1.0259683 0 39.657080
1 Female 65 65 testing 1.1269024 1 43.558518
1 Female 65 65 training 2.5642568 1 99.117036
1 Female 65 65 validation 0.8613927 1 33.295686
1 Female 66 66 testing 0.8804640 1 34.032854
1 Female 66 66 training 2.5189425 1 97.365488
1 Female 66 66 validation 0.8406215 1 32.492810
1 Female 67 67 testing 0.9526051 1 36.821350
1 Female 67 67 training 2.7908800 4 107.876772
1 Female 67 67 validation 0.7740580 2 29.919912
1 Female 68 68 testing 1.0543892 3 40.755642
1 Female 68 68 training 2.3233961 2 89.806970
1 Female 68 68 validation 0.9572270 0 37.000000
1 Female 69 69 testing 0.7863296 0 30.394248
1 Female 69 69 training 2.4425515 2 94.412722
1 Female 69 69 validation 0.7243880 0 28.000000
1 Female 70 70 testing 1.3135772 3 50.774116
1 Female 70 70 training 1.8882110 4 72.985620
1 Female 70 70 validation 0.6794457 0 26.262832
1 Female 71 71 testing 1.1617510 2 44.905532
1 Female 71 71 training 1.7537562 2 67.788496
1 Female 71 71 validation 0.8079508 1 31.229978
1 Female 72 72 testing 1.3351983 4 51.609846
1 Female 72 72 training 1.7978485 1 69.492810
1 Female 72 72 validation 0.5432910 0 21.000000
1 Female 73 73 testing 0.4398070 2 17.000000
1 Female 73 73 training 1.6557440 0 64.000000
1 Female 73 73 validation 0.6154322 0 23.788496
1 Female 74 74 testing 0.4656780 0 18.000000
1 Female 74 74 training 1.2761787 0 49.328540
1 Female 74 74 validation 0.5517907 1 21.328540
1 Female 75 75 testing 0.4101641 1 15.854204
1 Female 75 75 training 1.6596218 2 64.149890
1 Female 75 75 validation 0.5950330 0 23.000000
1 Female 76 76 testing 0.3851431 0 14.887058
1 Female 76 76 training 1.5357386 1 59.361394
1 Female 76 76 validation 0.4915490 0 19.000000
1 Female 77 77 testing 0.2069680 0 8.000000
1 Female 77 77 training 1.3398202 1 51.788496
1 Female 77 77 validation 0.4915490 0 19.000000
1 Female 78 78 testing 0.3104520 0 12.000000
1 Female 78 78 training 1.3296206 2 51.394248
1 Female 78 78 validation 0.5628402 1 21.755642
1 Female 79 79 testing 0.4398070 0 17.000000
1 Female 79 79 training 1.3978840 3 54.032854
1 Female 79 79 validation 0.3783434 1 14.624226
1 Female 80 80 testing 0.3363230 0 13.000000
1 Female 80 80 training 1.3452920 0 52.000000
1 Female 80 80 validation 0.5267696 1 20.361394
1 Female 81 81 testing 0.5174200 0 20.000000
1 Female 81 81 training 1.5374386 1 59.427102
1 Female 81 81 validation 0.6379033 1 24.657080
1 Female 82 82 testing 0.4860772 1 18.788496
1 Female 82 82 training 0.9750763 2 37.689934
1 Female 82 82 validation 0.3791933 0 14.657080
1 Female 83 83 testing 0.7761300 0 30.000000
1 Female 83 83 training 0.9572270 0 37.000000
1 Female 83 83 validation 0.2706095 1 10.459956
1 Female 84 84 testing 0.6734960 1 26.032854
1 Female 84 84 training 0.7999290 2 30.919912
1 Female 84 84 validation 0.3104520 0 12.000000
1 Female 85 85 testing 0.3363230 0 13.000000
1 Female 85 85 training 0.6510248 1 25.164270
1 Female 85 85 validation 0.3104520 0 12.000000
1 Female 86 86 testing 0.2845810 0 11.000000
1 Female 86 86 training 0.9148346 0 35.361394
1 Female 86 86 validation 0.2845810 0 11.000000
1 Female 87 87 testing 0.4215857 0 16.295686
1 Female 87 87 training 0.8028510 1 31.032854
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10 Male 116 116 validation 0.0407330 0 1.000000
10 Male 117 117 testing 0.0407330 0 1.000000
10 Male 117 117 training 0.0814660 0 2.000000
10 Male 117 117 validation 0.0407330 0 1.000000
10 Male 118 118 training 0.0608066 1 1.492810
10 Male 118 118 validation 0.0407330 0 1.000000
10 Male 119 119 testing 0.1221990 0 3.000000
10 Male 119 119 validation 0.0407330 0 1.000000
10 Male 120 120 testing 0.0814660 0 2.000000
10 Male 120 120 training 0.0407330 0 1.000000
10 Male 121 121 testing 0.0407330 0 1.000000
10 Male 121 121 training 0.0407330 0 1.000000
10 Male 122 122 testing 0.0407330 0 1.000000
10 Male 122 122 training 0.0814660 0 2.000000
10 Male 123 123 training 0.0814660 0 2.000000
10 Male 123 123 validation 0.0407330 0 1.000000
10 Male 124 124 training 0.0407330 0 1.000000
10 Male 124 124 validation 0.0407330 0 1.000000
10 Male 125 125 training 0.0621449 1 1.525664
10 Male 125 125 validation 0.0407330 0 1.000000
10 Male 126 126 training 0.0407330 0 1.000000
10 Male 128 128 testing 0.0407330 0 1.000000
10 Male 129 129 training 0.0407330 0 1.000000
10 Male 131 131 testing 0.0814660 0 2.000000
10 Male 131 131 training 0.0407330 0 1.000000
10 Male 133 133 training 0.0407330 0 1.000000
10 Male 134 134 testing 0.0407330 0 1.000000
10 Male 136 136 training 0.0407330 0 1.000000
10 Male 137 137 testing 0.0407330 0 1.000000
10 Male 139 139 testing 0.0407330 0 1.000000
10 Male 139 139 training 0.0407330 0 1.000000
10 Male 140 140 testing 0.0407330 0 1.000000
10 Male 140 140 training 0.0407330 0 1.000000
10 Male 141 141 training 0.0407330 0 1.000000
10 Male 142 142 training 0.0407330 0 1.000000
10 Male 145 145 training 0.0407330 0 1.000000
10 Male 146 146 training 0.0407330 0 1.000000
10 Male 147 147 training 0.0407330 0 1.000000
10 Male 150 150 training 0.0407330 0 1.000000
10 Male 152 152 training 0.0407330 0 1.000000
10 Male 156 156 testing 0.0407330 0 1.000000
10 Male 158 158 testing 0.0407330 0 1.000000
10 Male 159 159 testing 0.0407330 0 1.000000
10 Male 160 160 testing 0.0407330 0 1.000000
10 Male 160 161 testing 0.0407330 0 1.000000
10 Male 160 164 testing 0.0407330 0 1.000000
10 Male 160 165 testing 0.0407330 0 1.000000
10 Male 160 168 testing 0.0407330 0 1.000000

Fit GLMNET:

Now we have our data ready let’s get ready for the regression function in the glmnet package. It’s a bit different than what we’ve seen so far, glmnet() uses x, y, and offset input data instead of formulas like in glm(). As a result, we will have to create a few different data sets.

# 4.1.3 
# x: Select our independent variables 
ind_vars <- agg_data %>% 
  filter(Sample == "training") %>%
  select(Gender,
         ClaimDuration_v2_
         ) 

# y: Grab our terminations
terminations <- agg_data %>% 
  filter(Sample == "training") %>%
  select(Terminations)

# offset: create expected termination offset
offset_cal <- agg_data %>% 
  filter(Sample == "training") %>%
  select(exp_terms) %>% 
  log() %>% # take the log since we are using a log-link function in the GLM
  as.matrix() 

Next we need to prep our independent variables more for modeling. For all categorical variables we will build indicators for every level. This is different than what is typically done with a non-penalized glm where one would code one of the levels as a reference level. This is known as one-hot encoding (building a full suite of dummy variables). Doing so then allows the penalization to determine which variables should be squeezed into the reference level. You can choose a reference level yourself if you want, but this is not necessary with a penalized regression. It can actually do the math when there is perfect multicollinearity, unlike an OLS regression. This method of one-hot encoding is also applicable to other machine learning methods such as tree based models.

We will also store the data in a sparse matrix via sparse.model.matrix() from the Matrix package, which is an effective way to store data. It works by only storing the non-zero elements of the data, saving a lot of space. Although not every package works with the sparse matrix format, glmnet is a package that does work with this type of data.

# 4.1.4 
# First we need to format all the categorical variables as factors to help the 
# sparse.model.matrix() function determine which variables to one-hot encode.
# Do this by listing out the variables we want to be factors
factor_vars <- c('Gender',
                 'ClaimDuration_v2_')

# Now update the variables in our list to be factors
ind_vars[, factor_vars] <- lapply(ind_vars[, factor_vars], as.factor)

We are now ready to create our model. We will store our glm formula in the glm_formula variable.

glm_formula <- formula(~ -1 # drop intercept since glmnet has a statment for this 
                       + Gender
                       + ClaimDuration_v2_:Gender
                       )

Here in one step we are one-hot encoding our factor variables we have stored in the factor_vars list, building out our model design matrix we have specified in the glm_formula from above and finally taking that design matrix and storing it as a sparse formatted matrix that only points to the rows and columns of the non-zero elements in our design matrix.

x_train_sp <- sparse.model.matrix(glm_formula,
                                  data = ind_vars,
                                  # This contrast call allows us to create indicators 
                                  # for every level of variables that have a factor format
                                  contrasts.arg = lapply(ind_vars[, sapply(ind_vars, is.factor)], 
                                                         contrasts, 
                                                         contrasts=FALSE)
                                  ) 

head(x_train_sp)
## 6 x 322 sparse Matrix of class "dgCMatrix"
##    [[ suppressing 322 column names 'GenderFemale', 'GenderMale', 'GenderFemale:ClaimDuration_v2_1' ... ]]
##                                                                          
## 1 1 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 2 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 3 1 . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 4 1 . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 5 1 . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . .
## 6 1 . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . .
##                                                                          
## 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
##                                                                          
## 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
##                                                                          
## 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
##                                                                          
## 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
##                                                                          
## 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
##                                                                          
## 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
##                                                                          
## 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
##                                                                      
## 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
# store y as sparse matrix
term_sp <- terminations %>% 
  as.matrix() %>% 
  Matrix(., sparse=TRUE)


# Grab fold IDs and store them as a vector since glmnet needs them in this format for 
# the k-fold CV
fold_ID <- agg_data %>%
  filter(Sample == "training") %>%
  select(., fold) %>% 
  t() %>% as.vector()

Hyperparameter Tuning

We will now explore fine tuning the hyperparameters. We want to check how our coefficients change as we vary the lambda penalty.

#  4.1.5 
# First we need to set alpha, the value that controls the blend between the ridge 
# and lasso penalties. Let's use an alpha = 0 since we are building a simple model 
# that adjusts by gender and duration and we believe that all coefficients in our 
# model will be non-zero.
my_alpha = 0

glmnet_model1 <- glmnet(x = x_train_sp,
                        y = term_sp,
                        family = "poisson",
                        offset = offset_cal,
                        alpha = my_alpha,
                        standardize = FALSE, # set to false since binary variables shouldn't be standardized, 
                                             # only continuous ones should be
                        intercept = FALSE # set intercept to false since the intercept term is not penalized in the model and 
                                          # we only want to adjust the assumption if there is credible enough data to do so
                        )

# plot coefficient path
plot.glmnet(glmnet_model1, xvar = "lambda")
title(main = "Coefficients Across the Penalty Path\n\n")

Unforunately, most of the time the default lambda sequence doesn’t show the full lambda path. Let’s grab the lambda sequence that was used above and store to use with the k-fold CV. We’ll stretch the lambda sequence a bit since usually the auto generated sequence doesn’t cover the full range of penalties (no penalty to full penalty).

#4.1.6

#Stretch out the sequence
my_lambda <- exp(
  seq(from=log(glmnet_model1$lambda[1]),
      to=log(glmnet_model1$lambda[1]*1e-8),
      length.out=100
  )
)


# Now let's view how our coefficients change with the full lambda sequence
glmnet_model2 <- glmnet(x = x_train_sp,
                        y = term_sp,
                        family = "poisson",
                        offset = offset_cal,
                        alpha = my_alpha,
                        lambda = my_lambda,
                        standardize = FALSE,
                        intercept = FALSE 
)

# plot coefficient path across the lambda penalties
plot.glmnet(glmnet_model2, xvar = "lambda", ylab = "Coefficients (Log Scale)")
title(main = "Coefficients Across the Penalty Path\n\n")

Note that this function plots the coefficients in log scale since they were fit using a log-link function. Below we have updated this graph by taking the exponential of the coefficient to put them on a multiplicative factor scale, which can be thought of AtoE adjustments being made to the starting E.

# Convert lambda coefficients back to original scale
factor_coef_data <- glmnet_model2
factor_coef_data$beta <- exp(factor_coef_data$beta)
plot.glmnet(factor_coef_data, xvar = "lambda", ylab = "Coefficients (Factor Scale)")
title(main = "Coefficients Across the Penalty Path\n\n")

# Save the coefficient path in a CSV for easier viewing
all_coeff <- coef(glmnet_model2, s=my_lambda) %>% 
  exp() %>% as.matrix() %>% data.frame() 

coeff_seq <- data.frame(all_coeff)

# Save as csv file
write.csv(coeff_seq, file = paste0(data_output, "\\", "0410_Full_coefficient_path.csv"))

Now let’s do a 10-fold cross-validation to select the optimal lambda. We’ll utilize parallel computing by creating new instances of R to process the cross-validation via the registerDoParallel() function in the doParallel Package.

Note, do not use more processes than there are folds. Also do not use more processes than number of cores you have on your computer; otherwise, this causes a traffic jam and slows things down.

# https://protect-us.mimecast.com/s/P9P5C31Jv6s9Vo9NFE9nvW?domain=4.1.7.1
registerDoParallel(6)

glmnet_model2_cv <- cv.glmnet(x = x_train_sp,
                             y = term_sp,
                             family = "poisson",
                             offset = offset_cal,
                             alpha = my_alpha,
                             foldid = fold_ID,
                             lambda = my_lambda,
                             parallel = TRUE,
                             standardize = FALSE,
                             intercept = FALSE
                             )

# plot cross-validation results
plot(glmnet_model2_cv)
title(main = "K-fold Cross-validation Performance Across the Penalty Path\n\n")

Next we’ll plot the coefficient path across the lambda penalties and add a red line that shows the optimal penalty chosen by the 10-fold CV.

# https://protect-us.mimecast.com/s/C33tC4xKw6t9GE9kF3Wtuw?domain=4.1.7.2
plot.glmnet(factor_coef_data, xvar = "lambda", ylab = "Coefficients (Factor Scale)")
abline(v = log(glmnet_model2_cv$lambda.min), col = "red")
title(main = "Coefficients Across the Penalty Path\n\n")

Let’s zoom in closer to the optimal penalty to get a better view of the coefficients. It appears all but two coefficients produce factors in the range of about 0.95 to 1.05. The two other coefficients have large downward adjustments less than 0.9.

plot.glmnet(factor_coef_data, xlim = c(-1.8, log(max(glmnet_model2_cv$lambda))),
            ylim = c(0.75, 1.1),
            xvar = "lambda", ylab = "Coefficients (Factor Scale)")
abline(v = log(glmnet_model2_cv$lambda.min), col = "red")
title(main = "Coefficients Across the Penalty Path\n\n")

Now let’s extract our coefficients and predictions that correspond to the lambda the produces the min cross-validation error.

# 4.1.8
Coeff_lambda_min <- coef(glmnet_model2_cv, s="lambda.min") %>% 
  as.matrix() %>% data.frame()

# This one corresponds to the lambda that is one standard error from the min cross-validation error
Coeff_lambda_1SE <- coef(glmnet_model2_cv, s="lambda.1se") %>% 
  as.matrix() %>% data.frame()

compare <- data.frame(Coeff_lambda_min[,-1],
                      lambda_min = exp(Coeff_lambda_min$X1),
                      lambda_1se = exp(Coeff_lambda_1SE$X1))

We can save the coefficients as a csv file so we can view it outside of R. Looking at the file we can see that the main adjustments were just based on gender with both getting a downward adjustment and male had the largest adjustment.

When models become more complex with additional variables and interactions, reviewing the relationships of the coefficients becomes challenging. One approach for reviewing the final model for reasonableness is to develop pseudo observations of every combination in the model and export predictions for each combination. Then from these predicted values you can create base tables of the assumptions to review the relationships across different cells. Another approach is to review the predictions on the historical experience for reasonableness, which we will do later in this program.

write.csv(compare, file = paste0(data_output, "\\", "0410_Coeff_lambda_min_and_1se.csv"))
kable(compare,"html") %>%
  kable_styling() %>%
  scroll_box(width = "900px", height = "250px")
lambda_min lambda_1se
(Intercept) 1.0000000 1.0000000
GenderFemale 0.8810632 0.8981919
GenderMale 0.7611667 0.8043140
GenderFemale:ClaimDuration_v2_1 1.0006540 0.9987875
GenderMale:ClaimDuration_v2_1 0.9970163 0.9958662
GenderFemale:ClaimDuration_v2_2 0.9906548 0.9936536
GenderMale:ClaimDuration_v2_2 0.9908191 0.9928462
GenderFemale:ClaimDuration_v2_3 1.0187608 1.0041644
GenderMale:ClaimDuration_v2_3 1.0144947 0.9998356
GenderFemale:ClaimDuration_v2_4 0.9855326 0.9897228
GenderMale:ClaimDuration_v2_4 0.9963781 0.9901861
GenderFemale:ClaimDuration_v2_5 0.9956959 0.9949156
GenderMale:ClaimDuration_v2_5 0.9752755 0.9836276
GenderFemale:ClaimDuration_v2_6 1.0114052 1.0013934
GenderMale:ClaimDuration_v2_6 0.9973466 0.9926815
GenderFemale:ClaimDuration_v2_7 0.9920954 0.9944787
GenderMale:ClaimDuration_v2_7 1.0114120 0.9981660
GenderFemale:ClaimDuration_v2_8 1.0091299 1.0009487
GenderMale:ClaimDuration_v2_8 0.9778136 0.9872198
GenderFemale:ClaimDuration_v2_9 0.9766828 0.9896735
GenderMale:ClaimDuration_v2_9 0.9773153 0.9875406
GenderFemale:ClaimDuration_v2_10 1.0211722 1.0053623
GenderMale:ClaimDuration_v2_10 1.0251068 1.0040588
GenderFemale:ClaimDuration_v2_11 0.9895114 0.9948824
GenderMale:ClaimDuration_v2_11 0.9590809 0.9831702
GenderFemale:ClaimDuration_v2_12 0.9890215 0.9947269
GenderMale:ClaimDuration_v2_12 1.0347996 1.0073735
GenderFemale:ClaimDuration_v2_13 1.0083624 1.0011845
GenderMale:ClaimDuration_v2_13 1.0130108 1.0006858
GenderFemale:ClaimDuration_v2_14 1.0229575 1.0059751
GenderMale:ClaimDuration_v2_14 1.0150625 1.0015174
GenderFemale:ClaimDuration_v2_15 0.9917533 0.9959625
GenderMale:ClaimDuration_v2_15 0.9879101 0.9931457
GenderFemale:ClaimDuration_v2_16 0.9867007 0.9944374
GenderMale:ClaimDuration_v2_16 0.9777264 0.9902134
GenderFemale:ClaimDuration_v2_17 1.0091153 1.0016564
GenderMale:ClaimDuration_v2_17 0.9816234 0.9914739
GenderFemale:ClaimDuration_v2_18 0.9831526 0.9934484
GenderMale:ClaimDuration_v2_18 1.0088698 0.9999795
GenderFemale:ClaimDuration_v2_19 0.9976626 0.9981027
GenderMale:ClaimDuration_v2_19 0.9954840 0.9960295
GenderFemale:ClaimDuration_v2_20 1.0116093 1.0024764
GenderMale:ClaimDuration_v2_20 0.9912190 0.9947693
GenderFemale:ClaimDuration_v2_21 0.9882286 0.9951211
GenderMale:ClaimDuration_v2_21 0.9898276 0.9944401
GenderFemale:ClaimDuration_v2_22 1.0061811 1.0009291
GenderMale:ClaimDuration_v2_22 0.9796588 0.9919128
GenderFemale:ClaimDuration_v2_23 0.9724276 0.9904413
GenderMale:ClaimDuration_v2_23 1.0320646 1.0074708
GenderFemale:ClaimDuration_v2_24 1.0232501 1.0061355
GenderMale:ClaimDuration_v2_24 0.9864588 0.9940830
GenderFemale:ClaimDuration_v2_25 1.0200292 1.0051581
GenderMale:ClaimDuration_v2_25 1.0015898 0.9986198
GenderFemale:ClaimDuration_v2_26 1.0081372 1.0015551
GenderMale:ClaimDuration_v2_26 0.9987965 0.9978348
GenderFemale:ClaimDuration_v2_27 1.0088295 1.0017748
GenderMale:ClaimDuration_v2_27 1.0027380 0.9990283
GenderFemale:ClaimDuration_v2_28 0.9799981 0.9929979
GenderMale:ClaimDuration_v2_28 0.9937295 0.9964309
GenderFemale:ClaimDuration_v2_29 1.0055070 1.0008063
GenderMale:ClaimDuration_v2_29 0.9889871 0.9951033
GenderFemale:ClaimDuration_v2_30 1.0058233 1.0009106
GenderMale:ClaimDuration_v2_30 1.0107730 1.0014417
GenderFemale:ClaimDuration_v2_31 0.9876805 0.9954359
GenderMale:ClaimDuration_v2_31 0.9841875 0.9937612
GenderFemale:ClaimDuration_v2_32 0.9777275 0.9923938
GenderMale:ClaimDuration_v2_32 0.9886064 0.9950797
GenderFemale:ClaimDuration_v2_33 1.0056070 1.0008806
GenderMale:ClaimDuration_v2_33 1.0062210 1.0002519
GenderFemale:ClaimDuration_v2_34 0.9936569 0.9974803
GenderMale:ClaimDuration_v2_34 1.0247141 1.0057806
GenderFemale:ClaimDuration_v2_35 0.9985985 0.9989434
GenderMale:ClaimDuration_v2_35 0.9896322 0.9958126
GenderFemale:ClaimDuration_v2_36 0.9894509 0.9962376
GenderMale:ClaimDuration_v2_36 0.9886509 0.9955436
GenderFemale:ClaimDuration_v2_37 1.0173625 1.0044727
GenderMale:ClaimDuration_v2_37 0.9995243 0.9987327
GenderFemale:ClaimDuration_v2_38 0.9916992 0.9969096
GenderMale:ClaimDuration_v2_38 1.0037766 0.9999302
GenderFemale:ClaimDuration_v2_39 1.0228907 1.0060817
GenderMale:ClaimDuration_v2_39 0.9971171 0.9980922
GenderFemale:ClaimDuration_v2_40 0.9844606 0.9947976
GenderMale:ClaimDuration_v2_40 1.0037622 0.9999795
GenderFemale:ClaimDuration_v2_41 0.9982545 0.9989000
GenderMale:ClaimDuration_v2_41 0.9869953 0.9952328
GenderFemale:ClaimDuration_v2_42 0.9918690 0.9970480
GenderMale:ClaimDuration_v2_42 1.0063339 1.0007998
GenderFemale:ClaimDuration_v2_43 1.0125331 1.0030840
GenderMale:ClaimDuration_v2_43 1.0065085 1.0008777
GenderFemale:ClaimDuration_v2_44 1.0013071 0.9998213
GenderMale:ClaimDuration_v2_44 1.0064453 1.0008848
GenderFemale:ClaimDuration_v2_45 0.9922533 0.9971885
GenderMale:ClaimDuration_v2_45 0.9816965 0.9939201
GenderFemale:ClaimDuration_v2_46 0.9962139 0.9984729
GenderMale:ClaimDuration_v2_46 1.0031246 1.0001425
GenderFemale:ClaimDuration_v2_47 1.0141734 1.0036232
GenderMale:ClaimDuration_v2_47 1.0104936 1.0022527
GenderFemale:ClaimDuration_v2_48 0.9816165 0.9942664
GenderMale:ClaimDuration_v2_48 1.0017084 0.9997796
GenderFemale:ClaimDuration_v2_49 1.0145965 1.0037538
GenderMale:ClaimDuration_v2_49 1.0089943 1.0018654
GenderFemale:ClaimDuration_v2_50 0.9948165 0.9981064
GenderMale:ClaimDuration_v2_50 1.0060949 1.0010698
GenderFemale:ClaimDuration_v2_51 1.0111297 1.0027769
GenderMale:ClaimDuration_v2_51 0.9900753 0.9966252
GenderFemale:ClaimDuration_v2_52 1.0045759 1.0009019
GenderMale:ClaimDuration_v2_52 0.9894660 0.9964430
GenderFemale:ClaimDuration_v2_53 0.9978172 0.9989868
GenderMale:ClaimDuration_v2_53 0.9990447 0.9991358
GenderFemale:ClaimDuration_v2_54 0.9910841 0.9970554
GenderMale:ClaimDuration_v2_54 0.9927849 0.9974015
GenderFemale:ClaimDuration_v2_55 0.9998340 0.9995755
GenderMale:ClaimDuration_v2_55 1.0013392 0.9998117
GenderFemale:ClaimDuration_v2_56 0.9912019 0.9971161
GenderMale:ClaimDuration_v2_56 0.9874214 0.9959305
GenderFemale:ClaimDuration_v2_57 1.0046762 1.0009709
GenderMale:ClaimDuration_v2_57 0.9943383 0.9978942
GenderFemale:ClaimDuration_v2_58 0.9874826 0.9961756
GenderMale:ClaimDuration_v2_58 0.9966299 0.9986321
GenderFemale:ClaimDuration_v2_59 1.0016978 1.0001995
GenderMale:ClaimDuration_v2_59 0.9886770 0.9964393
GenderFemale:ClaimDuration_v2_60 0.9875636 0.9962137
GenderMale:ClaimDuration_v2_60 0.9861251 0.9957473
GenderFemale:ClaimDuration_v2_61 1.0028952 1.0005514
GenderMale:ClaimDuration_v2_61 0.9970406 0.9988131
GenderFemale:ClaimDuration_v2_62 1.0006032 0.9999139
GenderMale:ClaimDuration_v2_62 0.9956873 0.9984465
GenderFemale:ClaimDuration_v2_63 1.0179991 1.0047784
GenderMale:ClaimDuration_v2_63 0.9989383 0.9993684
GenderFemale:ClaimDuration_v2_64 0.9988029 0.9994178
GenderMale:ClaimDuration_v2_64 1.0072233 1.0016613
GenderFemale:ClaimDuration_v2_65 1.0037649 1.0008166
GenderMale:ClaimDuration_v2_65 0.9953841 0.9983883
GenderFemale:ClaimDuration_v2_66 0.9869215 0.9960998
GenderMale:ClaimDuration_v2_66 1.0021215 1.0002948
GenderFemale:ClaimDuration_v2_67 1.0025016 1.0004834
GenderMale:ClaimDuration_v2_67 0.9846730 0.9954538
GenderFemale:ClaimDuration_v2_68 1.0031306 1.0006533
GenderMale:ClaimDuration_v2_68 0.9853158 0.9956450
GenderFemale:ClaimDuration_v2_69 0.9856154 0.9957501
GenderMale:ClaimDuration_v2_69 1.0036847 1.0007198
GenderFemale:ClaimDuration_v2_70 0.9939178 0.9981432
GenderMale:ClaimDuration_v2_70 0.9961896 0.9987163
GenderFemale:ClaimDuration_v2_71 0.9961203 0.9987596
GenderMale:ClaimDuration_v2_71 0.9939675 0.9981293
GenderFemale:ClaimDuration_v2_72 1.0084797 1.0021920
GenderMale:ClaimDuration_v2_72 0.9956936 0.9986023
GenderFemale:ClaimDuration_v2_73 0.9960042 0.9987379
GenderMale:ClaimDuration_v2_73 1.0068814 1.0016951
GenderFemale:ClaimDuration_v2_74 0.9895484 0.9969446
GenderMale:ClaimDuration_v2_74 1.0032559 1.0007073
GenderFemale:ClaimDuration_v2_75 0.9996456 0.9997595
GenderMale:ClaimDuration_v2_75 1.0001375 0.9998637
GenderFemale:ClaimDuration_v2_76 0.9862025 0.9960268
GenderMale:ClaimDuration_v2_76 1.0001471 0.9998665
GenderFemale:ClaimDuration_v2_77 1.0072477 1.0018689
GenderMale:ClaimDuration_v2_77 1.0036503 1.0008487
GenderFemale:ClaimDuration_v2_78 0.9987075 0.9995151
GenderMale:ClaimDuration_v2_78 0.9934418 0.9980390
GenderFemale:ClaimDuration_v2_79 0.9997155 0.9997994
GenderMale:ClaimDuration_v2_79 0.9932397 0.9979800
GenderFemale:ClaimDuration_v2_80 0.9869265 0.9962499
GenderMale:ClaimDuration_v2_80 0.9924097 0.9977695
GenderFemale:ClaimDuration_v2_81 1.0033399 1.0007995
GenderMale:ClaimDuration_v2_81 0.9993781 0.9996733
GenderFemale:ClaimDuration_v2_82 1.0059864 1.0015549
GenderMale:ClaimDuration_v2_82 0.9961805 0.9988366
GenderFemale:ClaimDuration_v2_83 0.9995988 0.9998071
GenderMale:ClaimDuration_v2_83 0.9961337 0.9988230
GenderFemale:ClaimDuration_v2_84 0.9989395 0.9996325
GenderMale:ClaimDuration_v2_84 0.9952809 0.9986063
GenderFemale:ClaimDuration_v2_85 0.9923524 0.9978110
GenderMale:ClaimDuration_v2_85 0.9999422 0.9998951
GenderFemale:ClaimDuration_v2_86 0.9929214 0.9979721
GenderMale:ClaimDuration_v2_86 0.9994578 0.9997553
GenderFemale:ClaimDuration_v2_87 1.0010241 1.0002061
GenderMale:ClaimDuration_v2_87 0.9977250 0.9992844
GenderFemale:ClaimDuration_v2_88 0.9998988 0.9999012
GenderMale:ClaimDuration_v2_88 0.9950017 0.9985564
GenderFemale:ClaimDuration_v2_89 0.9992325 0.9997147
GenderMale:ClaimDuration_v2_89 1.0005666 1.0000751
GenderFemale:ClaimDuration_v2_90 0.9980731 0.9994001
GenderMale:ClaimDuration_v2_90 0.9960608 0.9988327
GenderFemale:ClaimDuration_v2_91 0.9924656 0.9978560
GenderMale:ClaimDuration_v2_91 0.9948850 0.9985225
GenderFemale:ClaimDuration_v2_92 0.9942808 0.9983684
GenderMale:ClaimDuration_v2_92 0.9983388 0.9994619
GenderFemale:ClaimDuration_v2_93 0.9985814 0.9995426
GenderMale:ClaimDuration_v2_93 0.9981963 0.9994207
GenderFemale:ClaimDuration_v2_94 1.0007509 1.0001590
GenderMale:ClaimDuration_v2_94 1.0002728 1.0000197
GenderFemale:ClaimDuration_v2_95 1.0073965 1.0019753
GenderMale:ClaimDuration_v2_95 0.9960462 0.9988594
GenderFemale:ClaimDuration_v2_96 1.0032194 1.0008360
GenderMale:ClaimDuration_v2_96 0.9993283 0.9997772
GenderFemale:ClaimDuration_v2_97 1.0025053 1.0006377
GenderMale:ClaimDuration_v2_97 0.9966399 0.9990313
GenderFemale:ClaimDuration_v2_98 0.9943467 0.9984114
GenderMale:ClaimDuration_v2_98 0.9966399 0.9990313
GenderFemale:ClaimDuration_v2_99 0.9998556 0.9999195
GenderMale:ClaimDuration_v2_99 0.9985340 0.9995482
GenderFemale:ClaimDuration_v2_100 1.0048086 1.0012680
GenderMale:ClaimDuration_v2_100 0.9966399 0.9990313
GenderFemale:ClaimDuration_v2_101 0.9962188 0.9989247
GenderMale:ClaimDuration_v2_101 1.0002389 1.0000100
GenderFemale:ClaimDuration_v2_102 0.9974377 0.9992548
GenderMale:ClaimDuration_v2_102 0.9987754 0.9996179
GenderFemale:ClaimDuration_v2_103 1.0003831 1.0000664
GenderMale:ClaimDuration_v2_103 0.9972857 0.9992180
GenderFemale:ClaimDuration_v2_104 0.9984461 0.9995366
GenderMale:ClaimDuration_v2_104 0.9999228 0.9999484
GenderFemale:ClaimDuration_v2_105 0.9973939 0.9992536
GenderMale:ClaimDuration_v2_105 0.9970786 0.9991581
GenderFemale:ClaimDuration_v2_106 0.9996325 0.9998777
GenderMale:ClaimDuration_v2_106 0.9978700 0.9993867
GenderFemale:ClaimDuration_v2_107 1.0013861 1.0003549
GenderMale:ClaimDuration_v2_107 0.9994618 0.9998157
GenderFemale:ClaimDuration_v2_108 1.0073194 1.0019686
GenderMale:ClaimDuration_v2_108 0.9999839 0.9999659
GenderFemale:ClaimDuration_v2_109 0.9960433 0.9988987
GenderMale:ClaimDuration_v2_109 1.0024139 1.0006346
GenderFemale:ClaimDuration_v2_110 0.9964904 0.9990236
GenderMale:ClaimDuration_v2_110 0.9983452 0.9995238
GenderFemale:ClaimDuration_v2_111 0.9982769 0.9995108
GenderMale:ClaimDuration_v2_111 0.9980506 0.9994388
GenderFemale:ClaimDuration_v2_112 0.9997768 0.9999178
GenderMale:ClaimDuration_v2_112 0.9983452 0.9995238
GenderFemale:ClaimDuration_v2_113 0.9997760 0.9999176
GenderMale:ClaimDuration_v2_113 1.0003825 1.0000805
GenderFemale:ClaimDuration_v2_114 1.0001353 1.0000174
GenderMale:ClaimDuration_v2_114 0.9985811 0.9995918
GenderFemale:ClaimDuration_v2_115 1.0003773 1.0000846
GenderMale:ClaimDuration_v2_115 1.0017391 1.0004411
GenderFemale:ClaimDuration_v2_116 0.9984954 0.9995717
GenderMale:ClaimDuration_v2_116 1.0000150 0.9999749
GenderFemale:ClaimDuration_v2_117 0.9980834 0.9994569
GenderMale:ClaimDuration_v2_117 0.9986401 0.9996088
GenderFemale:ClaimDuration_v2_118 0.9975236 0.9993118
GenderMale:ClaimDuration_v2_118 1.0005777 1.0001366
GenderFemale:ClaimDuration_v2_119 1.0018691 1.0004982
GenderMale:ClaimDuration_v2_119 0.9989943 0.9997108
GenderFemale:ClaimDuration_v2_120 0.9997443 0.9999189
GenderMale:ClaimDuration_v2_120 1.0009235 1.0002360
GenderFemale:ClaimDuration_v2_121 0.9977859 0.9993849
GenderMale:ClaimDuration_v2_121 0.9990534 0.9997278
GenderFemale:ClaimDuration_v2_122 1.0015881 1.0004204
GenderMale:ClaimDuration_v2_122 1.0006864 1.0001679
GenderFemale:ClaimDuration_v2_123 0.9977518 0.9993754
GenderMale:ClaimDuration_v2_123 0.9990534 0.9997278
GenderFemale:ClaimDuration_v2_124 0.9979591 0.9994331
GenderMale:ClaimDuration_v2_124 0.9994672 0.9998469
GenderFemale:ClaimDuration_v2_125 0.9981386 0.9994831
GenderMale:ClaimDuration_v2_125 1.0030626 1.0008204
GenderFemale:ClaimDuration_v2_126 0.9983056 0.9995295
GenderMale:ClaimDuration_v2_126 0.9994081 0.9998298
GenderFemale:ClaimDuration_v2_127 1.0002076 1.0000475
GenderMale:ClaimDuration_v2_127 0.9995855 0.9998809
GenderFemale:ClaimDuration_v2_128 0.9984491 0.9995694
GenderMale:ClaimDuration_v2_128 0.9995855 0.9998809
GenderFemale:ClaimDuration_v2_129 1.0000179 0.9999949
GenderMale:ClaimDuration_v2_129 1.0013049 1.0003454
GenderFemale:ClaimDuration_v2_130 1.0026347 1.0007099
GenderMale:ClaimDuration_v2_130 0.9996447 0.9998979
GenderFemale:ClaimDuration_v2_131 0.9989563 0.9997104
GenderMale:ClaimDuration_v2_131 0.9997039 0.9999149
GenderFemale:ClaimDuration_v2_132 1.0007971 1.0002109
GenderMale:ClaimDuration_v2_132 0.9994672 0.9998469
GenderFemale:ClaimDuration_v2_133 1.0027429 1.0007398
GenderMale:ClaimDuration_v2_133 1.0032872 1.0008846
GenderFemale:ClaimDuration_v2_134 0.9989840 0.9997181
GenderMale:ClaimDuration_v2_134 0.9997631 0.9999319
GenderFemale:ClaimDuration_v2_135 0.9989563 0.9997104
GenderMale:ClaimDuration_v2_135 1.0016613 1.0004476
GenderFemale:ClaimDuration_v2_136 0.9988260 0.9996742
GenderMale:ClaimDuration_v2_136 0.9995264 0.9998639
GenderFemale:ClaimDuration_v2_137 0.9991300 0.9997587
GenderMale:ClaimDuration_v2_137 0.9997039 0.9999149
GenderFemale:ClaimDuration_v2_138 0.9989128 0.9996983
GenderMale:ClaimDuration_v2_138 0.9998223 0.9999489
GenderFemale:ClaimDuration_v2_139 0.9994342 0.9998431
GenderMale:ClaimDuration_v2_139 0.9998223 0.9999489
GenderFemale:ClaimDuration_v2_140 0.9991734 0.9997707
GenderMale:ClaimDuration_v2_140 0.9997631 0.9999319
GenderFemale:ClaimDuration_v2_141 0.9991734 0.9997707
GenderMale:ClaimDuration_v2_141 0.9997631 0.9999319
GenderFemale:ClaimDuration_v2_142 0.9994777 0.9998552
GenderMale:ClaimDuration_v2_142 0.9998223 0.9999489
GenderFemale:ClaimDuration_v2_143 0.9993038 0.9998069
GenderMale:ClaimDuration_v2_143 0.9997039 0.9999149
GenderFemale:ClaimDuration_v2_144 0.9993038 0.9998069
GenderMale:ClaimDuration_v2_144 1.0016574 1.0004465
GenderFemale:ClaimDuration_v2_145 0.9993473 0.9998190
GenderMale:ClaimDuration_v2_145 0.9997039 0.9999149
GenderFemale:ClaimDuration_v2_146 0.9993473 0.9998190
GenderMale:ClaimDuration_v2_146 0.9998223 0.9999489
GenderFemale:ClaimDuration_v2_147 0.9994777 0.9998552
GenderMale:ClaimDuration_v2_147 0.9998223 0.9999489
GenderFemale:ClaimDuration_v2_148 0.9993038 0.9998069
GenderMale:ClaimDuration_v2_148 0.9998815 0.9999660
GenderFemale:ClaimDuration_v2_149 1.0013161 1.0003546
GenderMale:ClaimDuration_v2_149 0.9998815 0.9999660
GenderFemale:ClaimDuration_v2_150 1.0015824 1.0004283
GenderMale:ClaimDuration_v2_150 0.9998815 0.9999660
GenderFemale:ClaimDuration_v2_151 0.9996082 0.9998914
GenderMale:ClaimDuration_v2_151 0.9999408 0.9999830
GenderFemale:ClaimDuration_v2_152 0.9995647 0.9998793
GenderMale:ClaimDuration_v2_152 0.9998815 0.9999660
GenderFemale:ClaimDuration_v2_153 0.9996517 0.9999034
GenderMale:ClaimDuration_v2_153 0.9999408 0.9999830
GenderFemale:ClaimDuration_v2_154 0.9997388 0.9999276
GenderMale:ClaimDuration_v2_154 1.0000000 1.0000000
GenderFemale:ClaimDuration_v2_155 0.9998258 0.9999517
GenderMale:ClaimDuration_v2_155 1.0000000 1.0000000
GenderFemale:ClaimDuration_v2_156 0.9998258 0.9999517
GenderMale:ClaimDuration_v2_156 1.0000000 1.0000000
GenderFemale:ClaimDuration_v2_157 0.9996952 0.9999155
GenderMale:ClaimDuration_v2_157 0.9999408 0.9999830
GenderFemale:ClaimDuration_v2_158 1.0017908 1.0004859
GenderMale:ClaimDuration_v2_158 0.9999408 0.9999830
GenderFemale:ClaimDuration_v2_159 0.9997823 0.9999396
GenderMale:ClaimDuration_v2_159 0.9999408 0.9999830
GenderFemale:ClaimDuration_v2_160 1.0003288 1.0000712
GenderMale:ClaimDuration_v2_160 1.0000000 1.0000000

We can see that the female coefficient makes a large downward adjustment of 0.88. For illustration purposes we intentionally made the female E about 15% higher than the actual termination rates in the training data.

compare[2,]$lambda_min
## [1] 0.8810632
1/compare[2,]$lambda_min-1
## [1] 0.1349923

We can see that the male coefficient makes a large downward adjustment of 0.76. For illustration purposes we intentionally made the male E about 30% higher than the actual.

compare[3,]$lambda_min
## [1] 0.7611667
1/compare[3,]$lambda_min-1
## [1] 0.3137726

The remaining durational adjustments are pretty small with adjustments in the range of 0.95 to 1.05, which shows that the penalization with the k-fold CV helped balance the bias-variance trade-off to produce factor adjustments that are not overfit to the data (e.g., produce credibility adjusted coefficients).

Now let’s calculate the fit on our various data sets across the full lambda path. Typically we would not do such a test on all data sets since testing the various penalties on the data sets would bias our model. And most of the time you may not have enough data to afford to have data held out for a validation/test set. We are providing this illustration to help show how the k-fold cross-validation is a good proxy of how your model will perform on an out-of-sample data set that was not used to train the model. It also illustrated how the choice of the selecting the lambda.min and lambda.1se can involve some judgment, but usually both get you in the same ballpark.

devi = function(type) {
  # Select our independent variables
  agg_vars <- agg_data %>% 
    filter(Sample == type) 
  
  # Now update the variables in our list to be factors
  agg_vars[, factor_vars] <- lapply(agg_vars[, factor_vars], as.factor)
  
  x_vars <- sparse.model.matrix(glm_formula,
                                 data = agg_vars,
                                 contrasts.arg = lapply(agg_vars[, sapply(agg_vars, 
                                                                          is.factor)], 
                                                        contrasts, contrasts=FALSE)
                                ) 
  
  # Grab our terminations
  y <- agg_data %>% 
    filter(Sample == type) %>%
    select(Terminations)
  
  # create exposure offset
  offset <- agg_data %>% 
    filter(Sample == type) %>%
    select(exp_terms) %>% 
    log() %>% # take the log since we are using a log-link function in the GLM
    as.matrix()
  
  # predict at every lambda level
  eta <- predict(glmnet_model2, newx=x_vars, newoffset=offset, type='link')
  
  # Poisson deviance formula taken from glmnet code
  # https://protect-us.mimecast.com/s/lLOCC5y0x7fpO3pNF9BV_g?domain=github.com  
  deveta = y$Terminations * eta - exp(eta)
  devy = y$Terminations * log(y$Terminations) - y$Terminations
  devy[y$Terminations == 0] = 0
  
  apply(2*(devy - deveta), 2, mean, na.rm = TRUE)
}

agg_data %>% distinct(Sample)
##       Sample
## 1    testing
## 2   training
## 3 validation
deviance_cal = devi("training")
deviance_val = devi("validation")
deviance_test = devi("testing")

min_lambdas <- data.frame(train.min = which.min(deviance_cal),
                          cv.min = which.min(glmnet_model2_cv$cvm),
                          val.min = which.min(deviance_val),
                          test.min = which.min(deviance_test))

row.names(min_lambdas) <- "Lowest Error Lambda"

Let’s plot the training, validation, testing and 10-fold CV error. We will drop the values on the y-axis since we only care about the location of the lowest error on these curves when trying to select the lambda values. We’ll include lines that indicate the best lambda for each set.

# https://protect-us.mimecast.com/s/-gngC68VyWS0x70wivgzPf?domain=4.1.9.2 
plot(glmnet_model2_cv, 
     ylab='Error', 
     ylim = c(min(deviance_cal), max(glmnet_model2_cv$cvm)), 
     yaxt='n')
title(main = "Performance Across the Penalty Path for Various Data Sets\n\n")

#Add red line
abline(v = log(glmnet_model2$lambda[min_lambdas$cv.min]), col = "red")

# https://protect-us.mimecast.com/s/MthPC73Jz9hV92V7FPWOdQ?domain=4.1.9.3 - Add on the training error curve
points(log(glmnet_model2$lambda), deviance_cal, col="blue", pch="*")  
legend("topleft", c("10 fold CV", "Train"), pch="*", 
       col=c("red", "blue"), lty=1:3, cex=0.8) 

#add line
abline(v = log(glmnet_model2$lambda[min_lambdas$train.min]), col = "blue")


#add line
legend("topleft", c("10 fold CV", "Train", "Validation"), pch="*", 
       col=c("red", "blue", "green"), lty=1:3, cex=0.8) 
abline(v = log(glmnet_model2$lambda[min_lambdas$val.min]), col = "green")


#add line
legend("topleft", c("10 fold CV", "Train", "Validation", "Test"), pch="*", 
       col=c("red", "blue", "green", "orange"), lty=1:3, cex=0.8)
abline(v = log(glmnet_model2$lambda[min_lambdas$test.min]), col = "orange")

min_lambdas
##                     train.min cv.min val.min test.min
## Lowest Error Lambda       100     41      45       42

As you can see with the blue vertical line, the best fit based on the training error suggests using no penalty, which will always be the case when looking at the training error. Looking at the lambda index chosen for each we can see that the 10-fold CV did a good job at picking a lambda value that worked well on the validation and test sets. If we based our decision on selecting the penalty that produced the lowest training error then we would have no penalty at all and would have overfit the data. This highlights the ability of the penalized GLMs ability to automate the bias-variance trade-off.

Further Testing Fit

After reviewing our results now let’s test the fit further. First we’ll re-score all the data.

# https://protect-us.mimecast.com/s/mOirC82YA3CPxWP4IyToqa?domain=4.1.10.1 
# create exposure offset
offset <- agg_data %>%
  mutate(LnExp_Term = log(exp_terms)) %>%
  select(LnExp_Term) %>% 
  as.matrix() 

# Convert our variables in our factor_vars list to factors
agg_data[, factor_vars] <- lapply(agg_data[, factor_vars], as.factor)



# create our sparse matrix of indicator variables
x_all <- sparse.model.matrix(glm_formula,
                             data = agg_data,
                             contrasts.arg = lapply(agg_data[, sapply(agg_data, is.factor)], 
                                                    contrasts,
                                                    contrasts=FALSE)
                             ) 


# Make some predictions at lambda.1se, lambda.min and no penalty
lambda_1se <- predict(glmnet_model2_cv,
                      newx = x_all,
                      newoffset = offset,
                      type = 'response',
                      s = "lambda.1se")

lambda_min <- predict(glmnet_model2_cv,
                      newx = x_all,
                      newoffset = offset,
                      type = 'response',
                      s = "lambda.min")

no_penalty <- predict(glmnet_model2_cv,
                      newx = x_all,
                      newoffset = offset,
                      type = 'response',
                      s = 0)


predictions <- data.frame(agg_data,
                          lambda_1se = lambda_1se[,1],
                          lambda_min = lambda_min[,1],
                          no_penalty = no_penalty[,1])
rm(lambda_min,
   no_penalty,
   lambda_1se)

head(predictions) %>%
kable("html") %>%
  kable_styling() %>%
  scroll_box(width = "900px", height = "250px")
fold Gender ClaimDuration_v2_ ClaimDuration Sample exp_terms Terminations Exposure lambda_1se lambda_min no_penalty
1 Female 1 1 testing 3.125326 4 24.49680 2.803739 2.755411 2.761546
1 Female 1 1 training 16.503188 22 129.35459 14.805057 14.549862 14.582254
1 Female 1 1 validation 6.188302 7 48.50489 5.551543 5.455851 5.467998
1 Female 2 2 testing 7.122075 8 64.76792 6.356393 6.216357 6.122601
1 Female 2 2 training 34.601526 30 314.66517 30.881572 30.201233 29.745733
1 Female 2 2 validation 13.076560 10 118.91781 11.670720 11.413607 11.241465
# https://protect-us.mimecast.com/s/l51wC9rXB9SNPAN7tx8qz4?domain=4.1.10.2
AtoE_sample <- predictions %>%
  group_by(Sample) %>%
  summarise(Termination = comma(sum(Terminations)),
            Rate = percent(sum(Terminations)/sum(Exposure)),
            AtoE_bench = round(sum(Terminations)/sum(exp_terms), 2),
            AtoE_lambda_1se = round(sum(Terminations)/sum(lambda_1se), 2),
            AtoE_lambda_min = round(sum(Terminations)/sum(lambda_min), 2),
            AtoE_no_penalty = round(sum(Terminations)/sum(no_penalty), 2))

kable(AtoE_sample, "html") %>%
  kable_styling()
Sample Termination Rate AtoE_bench AtoE_lambda_1se AtoE_lambda_min AtoE_no_penalty
testing 4,344 2.5% 0.82 0.96 0.99 1.01
training 12,449 2.63% 0.81 0.95 0.98 1.00
validation 4,158 2.64% 0.82 0.96 0.99 1.01

The overall AtoEs on our three training sets indicate the no penalty and lambda.min are all close to 1.0. We will always see the no penalty having and AtoE of 1.0 on the training slice of data since it is fit 100% to that data. It is typical that the AtoE for lambda.min and lambda.1se on the training set differ from 1.0, which shows the penalization in action as it controls for overfitting. Although looking at the AtoE in overall aggregate doesn’t tell the full story since in aggregate the AtoE deviations cancel out across various cells.

AtoE_sample_gender <- predictions %>%
  group_by(Sample,
           Gender) %>%
  summarise(Termination = comma(sum(Terminations)),
            Rate = percent(sum(Terminations)/sum(Exposure)),
            AtoE_bench = round(sum(Terminations)/sum(exp_terms), 2),
            AtoE_lambda_1se = round(sum(Terminations)/sum(lambda_1se), 2),
            AtoE_lambda_min = round(sum(Terminations)/sum(lambda_min), 2),
            AtoE_no_penalty = round(sum(Terminations)/sum(no_penalty), 2))

kable(AtoE_sample_gender,"html") %>%
  kable_styling() %>%
  scroll_box(width = "900px", height = "250px")
Sample Gender Termination Rate AtoE_bench AtoE_lambda_1se AtoE_lambda_min AtoE_no_penalty
testing Female 2,648 2.19% 0.88 0.98 1.00 1.01
testing Male 1,696 3.2% 0.75 0.94 0.99 1.02
training Female 7,603 2.33% 0.87 0.97 0.99 1.00
training Male 4,846 3.29% 0.74 0.92 0.97 1.00
validation Female 2,582 2.38% 0.89 0.99 1.01 1.02
validation Male 1,576 3.23% 0.73 0.91 0.96 0.99

Looking at a more granular cut by gender we can see in general the AtoEs for lambda.min on the validation and test sets are closer to 1.0 than the no penalty prediction. However, we can see that the lambda.min AtoE for Males on the validation set is worse than the no penalty (0.96 vs 0.99), but keep in mind again the within cell AtoE deviations canceling out are at play here. One metric that overcomes deviation canceling out is mean squared error (MSE), which we will explore next to help us get a better idea of the model’s performance.

# 4.1.11
MSE_sample <- predictions %>%
  group_by(Sample) %>%
  summarise(Termination = comma(sum(Terminations)),
            Rate = percent(sum(Terminations)/sum(Exposure)),
            mse_bench = round(sum((Terminations-exp_terms)^2)/sum(Exposure), 4),
            mse_lambda_1se = round(sum((Terminations-lambda_1se)^2)/sum(Exposure), 4),
            mse_lambda_min = round(sum((Terminations-lambda_min)^2)/sum(Exposure), 4),
            mse_no_penalty = round(sum((Terminations-no_penalty)^2)/sum(Exposure), 4))

kable(MSE_sample,"html") %>%
  kable_styling() %>%
  scroll_box(width = "900px", height = "250px")
Sample Termination Rate mse_bench mse_lambda_1se mse_lambda_min mse_no_penalty
testing 4,344 2.5% 0.0297 0.0245 0.0243 0.0254
training 12,449 2.63% 0.0478 0.0263 0.0251 0.0237
validation 4,158 2.64% 0.0340 0.0272 0.0270 0.0274

By taking a look at the mean squared error (MSE), we can see the no penalty performs better on the training data because MSE is lower, but lambda.min performs better on both the validation and testing. This is what we expected and again confirms the penalization is working correctly. We can also look at MSE by gender.

MSE_sample_gender <- predictions %>%
  group_by(Sample,
           Gender) %>%
  summarise(Termination = comma(sum(Terminations)),
            Rate = percent(sum(Terminations)/sum(Exposure)),
            mse_bench = round(sum((Terminations-exp_terms)^2)/sum(Exposure), 4),
            mse_lambda_1se = round(sum((Terminations-lambda_1se)^2)/sum(Exposure), 4),
            mse_lambda_min = round(sum((Terminations-lambda_min)^2)/sum(Exposure), 4),
            mse_no_penalty = round(sum((Terminations-no_penalty)^2)/sum(Exposure), 4))

kable(MSE_sample_gender,"html") %>%
  kable_styling() %>%
  scroll_box(width = "900px", height = "250px")
Sample Gender Termination Rate mse_bench mse_lambda_1se mse_lambda_min mse_no_penalty
testing Female 2,648 2.19% 0.0225 0.0204 0.0203 0.0211
testing Male 1,696 3.2% 0.0462 0.0338 0.0335 0.0354
training Female 7,603 2.33% 0.0307 0.0218 0.0213 0.0203
training Male 4,846 3.29% 0.0859 0.0364 0.0336 0.0311
validation Female 2,582 2.38% 0.0270 0.0249 0.0249 0.0253
validation Male 1,576 3.23% 0.0496 0.0324 0.0315 0.0322

By viewing more granular by Gender we can see for that every cell in the validation and testing set the MSE is lower for the lambda.min prediction than the no penalty prediction. This again confirms the penalization is working correctly and shows that AtoEs sometimes do not show the full picture as deviations within cells can cancel out. In general you will witness a lower MSE for the lambda.min prediction than the no penalty prediction when viewing results on the hold-out data sets.

# https://protect-us.mimecast.com/s/k1obC0R3p8fglQgPsNuQXH?domain=4.1.12.1
agg_results <- predictions %>%
  group_by(ClaimDuration, Sample) %>%
  summarise(actual_hz_rate = sum(Terminations)/sum(Exposure),
            bench = sum(exp_terms)/sum(Exposure),
            no_penalty = sum(no_penalty)/sum(Exposure),
            lambda_min = sum(lambda_min)/sum(Exposure),
            lambda_1se = sum(lambda_1se)/sum(Exposure)
            )


#plot

ggplotly(ggplot() + 
           geom_line(data = agg_results[agg_results$Sample=="training",], 
                     linetype = "dotted", 
                     aes(x = ClaimDuration, y = actual_hz_rate, col="actual")
                     )+ 
           geom_line(data = agg_results[agg_results$Sample=="training",], 
                     linetype = "dashed", 
                     aes(x = ClaimDuration, y = bench, col="bench")
                     )+
           geom_line(data = agg_results[agg_results$Sample=="training",], 
                     aes(x = ClaimDuration, y = no_penalty, col="no_penalty")
                     )+
           geom_line(data = agg_results[agg_results$Sample=="training",], 
                     aes(x = ClaimDuration, y = lambda_min, col="lambda_min")
                     )+
           geom_line(data = agg_results[agg_results$Sample=="training",], 
                     aes(x = ClaimDuration, y = lambda_1se, col="lambda_1se")
                     )+
           ggtitle("Actual vs Predicted Hazard Rates on Training Set")
         )

Here we can see that no penalty is quite volatile while the lambda_min is much smoother.

# https://protect-us.mimecast.com/s/lJutCgJXKpSwnKwQhAoteS?domain=4.1.12.3 - View results on validation set
ggplotly(ggplot() + 
           geom_line(data = agg_results[agg_results$Sample=="validation",], 
                     linetype = "dotted", 
                     aes(x = ClaimDuration, y = actual_hz_rate, col="actual")
                     )+ 
           geom_line(data = agg_results[agg_results$Sample=="validation",], 
                     linetype = "dashed", 
                     aes(x = ClaimDuration, y = bench, col="bench")
                     )+
           geom_line(data = agg_results[agg_results$Sample=="validation",], 
                     aes(x = ClaimDuration, y = no_penalty, col="no_penalty")
                     )+
           geom_line(data = agg_results[agg_results$Sample=="validation",], 
                     aes(x = ClaimDuration, y = lambda_min, col="lambda_min")
                     )+
           geom_line(data = agg_results[agg_results$Sample=="validation",], 
                     aes(x = ClaimDuration, y = lambda_1se, col="lambda_1se")
                     )+
           ggtitle("Actual vs Predicted Hazard Rates on Validation Set")
         )
# https://protect-us.mimecast.com/s/1OjoCjRJMYfGkDGNIEReCP?domain=4.1.12.4 - View results on testing set
ggplotly(ggplot() + 
           geom_line(data = agg_results[agg_results$Sample=="testing",], 
                     linetype = "dotted", 
                     aes(x = ClaimDuration, y = actual_hz_rate, col="actual")
                     )+ 
           geom_line(data = agg_results[agg_results$Sample=="testing",], 
                     linetype = "dashed", 
                     aes(x = ClaimDuration, y = bench, col="bench")
                     )+
           geom_line(data = agg_results[agg_results$Sample=="testing",], 
                     aes(x = ClaimDuration, y = no_penalty, col="no_penalty")
                     )+
           geom_line(data = agg_results[agg_results$Sample=="testing",], 
                     aes(x = ClaimDuration, y = lambda_min, col="lambda_min")
                     )+
           geom_line(data = agg_results[agg_results$Sample=="testing",], 
                     aes(x = ClaimDuration, y = lambda_1se, col="lambda_1se")
                     )+
           ggtitle("Actual vs Predicted Hazard Rates on Testing Holdout")
         )
# https://protect-us.mimecast.com/s/xs-oCkRMK2fXZAXGSlP_sQ?domain=4.1.12.5 - View results on testing set for Males
agg_results_m <- predictions %>%
  filter(Gender == "Male") %>%
  group_by(ClaimDuration, Sample) %>%
  summarise(actual_hz_rate = sum(Terminations)/sum(Exposure),
            bench = sum(exp_terms)/sum(Exposure),
            no_penalty = sum(no_penalty)/sum(Exposure),
            lambda_min = sum(lambda_min)/sum(Exposure),
            lambda_1se = sum(lambda_1se)/sum(Exposure))

ggplotly(ggplot() + 
           geom_line(data = agg_results_m[agg_results_m$Sample=="testing",], 
                     linetype = "dotted", 
                     aes(x = ClaimDuration, y = actual_hz_rate, col="actual")
                     )+ 
           geom_line(data = agg_results_m[agg_results_m$Sample=="testing",], 
                     linetype = "dashed", 
                     aes(x = ClaimDuration, y = bench, col="bench")
                     )+
           geom_line(data = agg_results_m[agg_results_m$Sample=="testing",], 
                     aes(x = ClaimDuration, y = no_penalty, col="no_penalty")
                     )+
           geom_line(data = agg_results_m[agg_results_m$Sample=="testing",], 
                     aes(x = ClaimDuration, y = lambda_min, col="lambda_min")
                     )+
           geom_line(data = agg_results_m[agg_results_m$Sample=="testing",], 
                     aes(x = ClaimDuration, y = lambda_1se, col="lambda_1se")
                     )+
           ggtitle("Actual vs Predicted Hazard Rates for Males on Testing Holdout")
         )
# https://protect-us.mimecast.com/s/yigsClY6LRtPDmP4IQ_ZG5?domain=4.1.12.6 - View results on testing set for Females
agg_results_f <- predictions %>%
  filter(Gender == "Female") %>%
  group_by(ClaimDuration, Sample) %>%
  summarise(actual_hz_rate = sum(Terminations)/sum(Exposure),
            bench = sum(exp_terms)/sum(Exposure),
            no_penalty = sum(no_penalty)/sum(Exposure),
            lambda_min = sum(lambda_min)/sum(Exposure),
            lambda_1se = sum(lambda_1se)/sum(Exposure))

ggplotly(ggplot() + 
           geom_line(data = agg_results_f[agg_results_f$Sample=="testing",], 
                     linetype = "dotted", 
                     aes(x = ClaimDuration, y = actual_hz_rate, col="actual")
                     )+ 
           geom_line(data = agg_results_f[agg_results_f$Sample=="testing",], 
                     linetype = "dashed", 
                     aes(x = ClaimDuration, y = bench, col="bench")
                     )+
           geom_line(data = agg_results_f[agg_results_f$Sample=="testing",], 
                     aes(x = ClaimDuration, y = no_penalty, col="no_penalty")
                     )+
           geom_line(data = agg_results_f[agg_results_f$Sample=="testing",], 
                     aes(x = ClaimDuration, y = lambda_min, col="lambda_min")
                     )+
           geom_line(data = agg_results_f[agg_results_f$Sample=="testing",], 
                     aes(x = ClaimDuration, y = lambda_1se, col="lambda_1se")
                     )+
           ggtitle("Actual vs Predicted Hazard Rates for Females on Testing Holdout")
         )

So far we have only adjusted the existing assumption in the “smoothed_assumptions_20180301.RData” file to better fit the experience by gender and duration. In the next program we will explore additional variables to vary the assumption by. Therefore, let’s tag that updated lambda.min assumption onto the expected assumption file so we can join it onto our full data set.

# 4.1.13
# create exposure offset
offset <- smoothed_assumptions %>%
  mutate(LnExp_Term = log(smoothed)) %>%
  select(LnExp_Term) %>% 
  as.matrix() 

smoothed_assumptions <- smoothed_assumptions %>%
  mutate(ClaimDuration_v2_ = ifelse(ClaimDuration > 160, 160, ClaimDuration))
  
# Convert our variables in our factor_vars list to factors
smoothed_assumptions[, factor_vars] <- lapply(smoothed_assumptions[, factor_vars], as.factor)


# create our sparse matrix of indicator variables
x_all <- sparse.model.matrix(glm_formula,
                             data = smoothed_assumptions,
                             contrasts.arg = lapply(smoothed_assumptions[, sapply(smoothed_assumptions, is.factor)], 
                                                    contrasts,
                                                    contrasts=FALSE)
                             ) 


# adjust the smoothed starting assumption
lambda_min <- predict(glmnet_model2_cv,
                      newx = x_all,
                      newoffset = offset,
                      type = 'response',
                      s = "lambda.min")

# Attach to the data
smoothed_assump_adj <- data.frame(smoothed_assumptions,
                                  smooth_adj = lambda_min[,1]) %>%
  select(-ClaimDuration_v2_)

# Save as an Rdata file
save(smoothed_assump_adj, file = paste0(data_output, "\\", "smoothed_assump_adj.RData"))