0410 - Update existing assumption with glmnet

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
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1	Female	58	58	validation	1.2049934	1	46.576992
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1	Male	145	145	training	0.0407330	0	1.000000
2	Female	1	1	testing	3.4391701	4	26.956758
2	Female	1	1	training	16.8199108	25	131.837114
2	Female	1	1	validation	4.8690097	4	38.164066
2	Female	2	2	testing	8.4215361	7	76.585180
2	Female	2	2	training	33.4366359	33	304.071696
2	Female	2	2	validation	12.5127439	14	113.790492
2	Female	3	3	testing	10.1954511	8	122.776112
2	Female	3	3	training	39.7703114	30	478.923802
2	Female	3	3	validation	13.6052486	9	163.837726
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2	Female	4	4	training	56.4465036	41	1058.041304
2	Female	4	4	validation	19.8016876	26	371.165654
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2	Female	5	5	training	43.5650085	44	1107.256538
2	Female	5	5	validation	15.3210388	9	389.402436
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2	Female	6	6	training	39.2160866	37	1143.759634
2	Female	6	6	validation	12.7613105	8	372.190932
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2	Female	7	7	training	33.3735919	31	1112.638504
2	Female	7	7	validation	11.4659107	14	382.260734
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2	Female	8	8	training	28.0607463	24	1056.344916
2	Female	8	8	validation	10.2420852	7	385.562612
2	Female	9	9	testing	6.9291008	8	282.566706
2	Female	9	9	training	26.4248450	24	1077.597462
2	Female	9	9	validation	8.5575731	7	348.975334
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2	Female	10	10	training	20.8227947	21	875.790492
2	Female	10	10	validation	6.7019021	9	281.876772
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2	Female	11	11	training	19.6170129	17	836.439384
2	Female	11	11	validation	6.3254713	6	269.708408
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2	Female	12	12	training	18.9162928	13	811.753544
2	Female	12	12	validation	6.3377458	6	271.971240
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2	Female	13	13	training	18.9048324	16	820.166266
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2	Female	14	14	validation	5.6368334	4	252.478430
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2	Female	15	15	training	15.9919037	12	753.979428
2	Female	15	15	validation	5.6878510	6	268.168364
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2	Female	16	16	training	15.3167149	8	760.172458
2	Female	16	16	validation	4.7968684	5	238.069802
2	Female	17	17	testing	4.5943883	1	236.689934
2	Female	17	17	training	14.5099008	16	747.509186
2	Female	17	17	validation	4.8741537	4	251.102656
2	Female	18	18	testing	4.4047840	4	231.562612
2	Female	18	18	training	14.1388528	10	743.289494
2	Female	18	18	validation	4.5248141	5	237.872678
2	Female	19	19	testing	4.7373500	3	250.018474
2	Female	19	19	training	14.2159404	15	750.260734
2	Female	19	19	validation	4.4948387	5	237.219692
2	Female	20	20	testing	3.9555386	5	207.215598
2	Female	20	20	training	14.0645156	18	736.786398
2	Female	20	20	validation	4.3314779	7	226.909626
2	Female	21	21	testing	4.3789007	4	226.299780
2	Female	21	21	training	13.7123155	10	708.646794
2	Female	21	21	validation	4.6472578	4	240.168364
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2	Female	22	22	training	12.0635549	10	613.172458
2	Female	22	22	validation	3.8416815	6	195.266926
2	Female	23	23	testing	7.1837645	7	359.026662
2	Female	23	23	training	11.2084907	9	560.172458
2	Female	23	23	validation	3.3596218	3	167.905532
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2	Female	24	24	training	11.6964440	9	576.975334
2	Female	24	24	validation	3.9455053	4	194.628320
2	Female	25	25	testing	3.0696014	2	150.788496
2	Female	25	25	training	11.3627167	15	558.172458
2	Female	25	25	validation	3.6620442	5	179.891152
2	Female	26	26	testing	3.1989839	5	158.036948
2	Female	26	26	training	11.0342166	9	545.114938
2	Female	26	26	validation	3.3442941	3	165.215598
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2	Female	27	27	training	11.1795816	10	558.895246
2	Female	27	27	validation	3.4254828	4	171.248452
2	Female	28	28	testing	2.7354149	2	138.459956
2	Female	28	28	training	10.3571333	8	524.252546
2	Female	28	28	validation	3.2918282	2	166.624226
2	Female	29	29	testing	2.6463191	4	135.036948
2	Female	29	29	training	10.3413803	8	527.702216
2	Female	29	29	validation	3.0179380	0	154.000000
2	Female	30	30	testing	2.5799746	4	131.887058
2	Female	30	30	training	10.3469716	8	528.932194
2	Female	30	30	validation	2.9616544	5	151.398342
2	Female	31	31	testing	2.3096321	2	117.628320
2	Female	31	31	training	10.1273052	4	515.778210
2	Female	31	31	validation	3.3862512	1	172.459956
2	Female	32	32	testing	2.2764194	1	115.098562
2	Female	32	32	training	9.5607334	7	483.402436
2	Female	32	32	validation	2.9907421	2	151.215598
2	Female	33	33	testing	2.4156440	1	121.000000
2	Female	33	33	training	9.9864677	10	500.223786
2	Female	33	33	validation	2.8591562	4	143.215598
2	Female	34	34	testing	5.2357228	7	259.515378
2	Female	34	34	training	7.1728129	4	355.529758
2	Female	34	34	validation	2.6142158	4	129.576992
2	Female	35	35	testing	4.7535165	4	233.004094
2	Female	35	35	training	7.4177946	7	363.599560
2	Female	35	35	validation	2.5213876	2	123.591372
2	Female	36	36	testing	4.8838245	7	236.665268
2	Female	36	36	training	7.2620915	8	351.913720
2	Female	36	36	validation	2.4458956	1	118.525664
2	Female	37	37	testing	2.7179553	1	130.182744
2	Female	37	37	training	7.4014866	5	354.511284
2	Female	37	37	validation	2.5577906	3	122.511284
2	Female	38	38	testing	2.5045487	1	118.558518
2	Female	38	38	training	7.4428961	11	352.326442
2	Female	38	38	validation	2.7719295	2	131.215598
2	Female	39	39	testing	2.4923738	3	116.591372
2	Female	39	39	training	7.5577130	4	353.544138
2	Female	39	39	validation	2.5134434	2	117.576992
2	Female	40	40	testing	2.1863420	2	101.069802
2	Female	40	40	training	6.8043959	8	314.552326
2	Female	40	40	validation	2.2725589	2	105.055422
2	Female	41	41	testing	2.2357588	1	102.131416
2	Female	41	41	training	7.1570978	5	326.942480
2	Female	41	41	validation	2.1485992	2	98.149890
2	Female	42	42	testing	2.5026519	5	112.971240
2	Female	42	42	training	6.8711598	2	310.168364
2	Female	42	42	validation	2.1692198	4	97.919912
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2	Female	43	43	validation	2.4003979	3	107.069802
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2	Female	44	44	training	6.7485381	5	297.449670
2	Female	44	44	validation	2.2245885	3	98.051328
2	Female	45	45	testing	1.9335901	2	84.215598
2	Female	45	45	training	6.4495910	2	280.905532
2	Female	45	45	validation	2.3080692	1	100.525664
2	Female	46	46	testing	4.2907870	5	184.661174
2	Female	46	46	training	5.2910326	2	227.708408
2	Female	46	46	validation	1.4173960	0	61.000000
2	Female	47	47	testing	3.9471879	6	167.858298
2	Female	47	47	training	5.0369416	4	214.201218
2	Female	47	47	validation	1.6888790	1	71.821350
2	Female	48	48	testing	3.4599450	7	145.388056
2	Female	48	48	training	4.8653958	4	204.445576
2	Female	48	48	validation	1.6584810	2	69.689934
2	Female	49	49	testing	1.6617960	0	69.000000
2	Female	49	49	training	5.0147631	4	208.219692
2	Female	49	49	validation	1.7321192	1	71.919912
2	Female	50	50	testing	1.8279750	0	75.000000
2	Female	50	50	training	4.8111898	3	197.398342
2	Female	50	50	validation	1.5994593	1	65.624226
2	Female	51	51	testing	1.4804157	2	60.018474
2	Female	51	51	training	4.7294900	3	191.741262
2	Female	51	51	validation	1.4460252	1	58.624226
2	Female	52	52	testing	1.6974160	0	68.000000
2	Female	52	52	training	5.0432463	2	202.036948
2	Female	52	52	validation	1.5366237	1	61.558518
2	Female	53	53	testing	1.4784168	1	58.525664
2	Female	53	53	training	4.8295707	6	191.186838
2	Female	53	53	validation	1.5902498	2	62.952766
2	Female	54	54	testing	1.6947829	1	66.295686
2	Female	54	54	training	4.6451937	3	181.708408
2	Female	54	54	validation	1.5109003	3	59.102656
2	Female	55	55	testing	1.7631058	2	68.149890
2	Female	55	55	training	5.3618306	2	207.252546
2	Female	55	55	validation	1.3915622	0	53.788496
2	Female	56	56	testing	1.6298730	0	63.000000
2	Female	56	56	training	4.7195716	3	182.427102
2	Female	56	56	validation	1.4746470	0	57.000000
2	Female	57	57	testing	1.4162112	3	54.741262
2	Female	57	57	training	4.7565980	5	183.858298
2	Female	57	57	validation	1.3130992	0	50.755642
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2	Female	59	59	validation	1.0089690	0	39.000000
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2	Female	62	62	training	3.3023502	7	127.646794
2	Female	62	62	validation	1.0730885	1	41.478430
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2	Female	66	66	validation	0.7439372	0	28.755642
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2	Female	68	68	validation	0.8194783	3	31.675554
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4	Female	1	1	validation	6.0793219	9	47.650684
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4	Female	2	2	training	34.8817369	40	317.213398
4	Female	2	2	validation	12.3244320	6	112.077990
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4	Female	3	3	training	39.9583935	25	481.188732
4	Female	3	3	validation	14.1863614	19	170.835628
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4	Female	4	4	training	58.4889177	63	1096.324606
4	Female	4	4	validation	18.5452830	16	347.615426
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4	Female	5	5	training	45.8483111	46	1165.289392
4	Female	5	5	validation	16.3856955	19	416.461952
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4	Female	6	6	training	37.6052310	35	1096.778108
4	Female	6	6	validation	13.4331802	12	391.786398
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4	Female	7	7	validation	11.4662192	7	382.271020
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4	Female	8	8	training	28.1572378	31	1059.977330
4	Female	8	8	validation	10.3043226	7	387.905532
4	Female	9	9	testing	6.6790469	6	272.369582
4	Female	9	9	training	26.9420231	16	1098.687836
4	Female	9	9	validation	9.2140276	10	375.745356
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4	Female	10	10	training	20.8847491	22	878.396244
4	Female	10	10	validation	7.2443079	4	304.689934
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4	Female	11	11	training	20.0587672	19	855.275114
4	Female	11	11	validation	6.7787835	6	289.036948
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4	Female	12	12	training	19.4213029	13	833.425004
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4	Female	13	13	validation	7.1936348	6	312.088276
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4	Female	14	14	validation	6.3816599	6	285.839824
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4	Female	15	15	training	17.3973383	15	820.242260
4	Female	15	15	validation	6.1889208	5	291.792590
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4	Female	16	16	training	15.3134875	13	760.012282
4	Female	16	16	validation	5.3715496	2	266.591372
4	Female	17	17	testing	4.4162215	3	227.511284
4	Female	17	17	training	15.2888917	13	787.640602
4	Female	17	17	validation	5.6286317	4	289.971240
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4	Female	18	18	training	15.1288952	12	795.336728
4	Female	18	18	validation	5.0680542	6	266.431196
4	Female	19	19	testing	4.1564597	2	219.361394
4	Female	19	19	training	13.5508153	14	715.158078
4	Female	19	19	validation	5.0271727	5	265.314160
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4	Female	20	20	training	14.3316053	14	750.778210
4	Female	20	20	validation	5.1369788	8	269.106750
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4	Female	21	21	training	14.1869663	13	733.176552
4	Female	21	21	validation	5.0622297	7	261.613940
4	Female	22	22	testing	7.8132440	5	397.135510
4	Female	22	22	training	11.4334211	13	581.143698
4	Female	22	22	validation	3.9871559	5	202.661174
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4	Female	23	23	training	12.5018446	7	624.811064
4	Female	23	23	validation	3.9617820	1	198.000000
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4	Female	24	24	training	11.8662377	10	585.351108
4	Female	24	24	validation	4.0913224	3	201.821350
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4	Female	25	25	training	11.9151143	13	585.307968
4	Female	25	25	validation	3.9199971	4	192.562612
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4	Female	26	26	validation	3.7598575	7	185.745356
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4	Female	27	27	training	10.9132984	20	545.583082
4	Female	27	27	validation	3.7531293	3	187.628320
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4	Female	28	28	training	10.6587467	10	539.519472
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4	Female	37	37	training	7.6914206	4	368.398342
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4	Female	39	39	validation	2.3652090	4	110.642700
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4	Female	41	41	training	6.9485715	5	317.416816
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4	Female	42	42	training	7.3077601	8	329.876772
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9	Male	145	145	testing	0.0407330	0	1.000000
9	Male	146	146	testing	0.0407330	0	1.000000
9	Male	147	147	testing	0.0407330	0	1.000000
9	Male	148	148	testing	0.0407330	0	1.000000
9	Male	152	152	testing	0.0407330	0	1.000000
9	Male	154	154	testing	0.0407330	0	1.000000
10	Female	1	1	testing	3.6356505	7	28.496802
10	Female	1	1	training	15.7612843	12	123.539432
10	Female	1	1	validation	5.7067966	6	44.730772
10	Female	2	2	testing	8.0844217	5	73.519472
10	Female	2	2	training	38.3547185	27	348.796582
10	Female	2	2	validation	12.4906173	14	113.589274
10	Female	3	3	testing	6.9416769	7	83.593368
10	Female	3	3	training	39.0732447	33	470.529554
10	Female	3	3	validation	13.4094961	13	161.480426
10	Female	4	4	testing	13.9279457	13	261.067398
10	Female	4	4	training	60.3889214	46	1131.938546
10	Female	4	4	validation	18.4545762	15	345.915206
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10	Female	5	5	training	47.7363879	33	1213.277110
10	Female	5	5	validation	15.2181108	15	386.786398
10	Female	6	6	testing	9.9547754	9	290.336728
10	Female	6	6	training	38.9277799	32	1135.351006
10	Female	6	6	validation	11.7747324	4	343.416816
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10	Female	7	7	training	35.1473615	37	1171.774014
10	Female	7	7	validation	9.9954788	17	333.238166
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10	Female	8	8	training	30.0855550	28	1132.568702
10	Female	8	8	validation	9.0221042	11	339.636508
10	Female	9	9	testing	6.3219425	4	257.806970
10	Female	9	9	training	27.3958050	27	1117.192928
10	Female	9	9	validation	8.3127058	5	338.989714
10	Female	10	10	testing	12.7627806	12	536.792590
10	Female	10	10	training	20.0869591	18	844.841820
10	Female	10	10	validation	6.6468807	4	279.562612
10	Female	11	11	testing	11.1984937	12	477.486618
10	Female	11	11	training	20.9196895	18	891.983522
10	Female	11	11	validation	6.1040402	3	260.266926
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10	Female	12	12	training	20.9040336	25	897.053324
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10	Female	13	13	training	20.0803824	17	871.166266
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10	Female	14	14	training	18.1587386	16	813.344916
10	Female	14	14	validation	5.7579989	4	257.905532
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10	Female	15	15	training	16.5582153	5	780.679648
10	Female	15	15	validation	5.1082127	6	240.839824
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10	Female	16	16	training	15.9892858	10	793.552326
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10	Female	17	17	training	15.2433338	17	785.293588
10	Female	17	17	validation	5.0399244	4	259.642700
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10	Female	18	18	training	14.3471184	15	754.238166
10	Female	18	18	validation	4.5996522	5	241.806970
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10	Female	19	19	validation	4.1770803	8	220.449670
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10	Female	20	20	training	14.6197035	20	765.870580
10	Female	20	20	validation	4.2287816	3	221.529758
10	Female	21	21	testing	4.4179582	5	228.318254
10	Female	21	21	training	14.9786874	10	774.092370
10	Female	21	21	validation	4.3021365	4	222.332634
10	Female	22	22	testing	7.8510969	9	399.059516
10	Female	22	22	training	12.8005421	11	650.632414
10	Female	22	22	validation	3.1920760	3	162.248452
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10	Female	23	23	training	11.3505258	8	567.271020
10	Female	23	23	validation	3.5590544	4	177.872678
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10	Female	24	24	training	11.5618662	12	570.336728
10	Female	24	24	validation	3.6214448	1	178.642700
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10	Female	25	25	training	11.6016503	8	569.909626
10	Female	25	25	validation	3.3662619	1	165.361394
10	Female	26	26	testing	3.4359026	3	169.741262
10	Female	26	26	training	10.7282175	10	529.997902
10	Female	26	26	validation	3.3718515	4	166.576992
10	Female	27	27	testing	3.0011891	3	150.036948
10	Female	27	27	training	11.5649779	11	578.162172
10	Female	27	27	validation	3.3113794	2	165.544138
10	Female	28	28	testing	2.8303005	0	143.262832
10	Female	28	28	training	10.6079165	11	536.946574
10	Female	28	28	validation	3.3413603	3	169.131416
10	Female	29	29	testing	2.8438586	2	145.117036
10	Female	29	29	training	10.2231545	12	521.669362
10	Female	29	29	validation	3.1994617	1	163.262832
10	Female	30	30	testing	2.4914436	3	127.361394
10	Female	30	30	training	11.2848634	9	576.876772
10	Female	30	30	validation	2.9924234	4	152.971240
10	Female	31	31	testing	2.6633444	2	135.642700
10	Female	31	31	training	10.3294609	9	526.073896
10	Female	31	31	validation	3.1088613	4	158.332634
10	Female	32	32	testing	2.4217286	2	122.445576
10	Female	32	32	training	10.1138676	9	511.369582
10	Female	32	32	validation	3.0092205	3	152.149890
10	Female	33	33	testing	2.6814479	1	134.314160
10	Female	33	33	training	9.6349864	12	482.618034
10	Female	33	33	validation	2.8981412	4	145.168364
10	Female	34	34	testing	4.8592336	3	240.854204
10	Female	34	34	training	7.1661846	5	355.201218
10	Female	34	34	validation	2.3099750	4	114.496904
10	Female	35	35	testing	5.2048932	9	255.129318
10	Female	35	35	training	7.5668438	5	370.905532
10	Female	35	35	validation	2.2893104	2	112.215598
10	Female	36	36	testing	4.9900131	3	241.811064
10	Female	36	36	training	7.4496805	5	361.004094
10	Female	36	36	validation	2.3664447	4	114.675554
10	Female	37	37	testing	2.4171320	4	115.774116
10	Female	37	37	training	7.4238219	7	355.581086
10	Female	37	37	validation	2.2381470	5	107.201218
10	Female	38	38	testing	2.6410153	1	125.018474
10	Female	38	38	training	6.9057486	3	326.899340
10	Female	38	38	validation	2.2524368	0	106.624226
10	Female	39	39	testing	2.8015718	2	131.055422
10	Female	39	39	training	7.1794230	6	335.848012
10	Female	39	39	validation	2.1377000	0	100.000000
10	Female	40	40	testing	2.3554448	0	108.887058
10	Female	40	40	training	7.0244919	5	324.726882
10	Female	40	40	validation	2.2232097	2	102.774116
10	Female	41	41	testing	2.2376017	0	102.215598
10	Female	41	41	training	6.4677783	3	295.453764
10	Female	41	41	validation	2.1691414	3	99.088276
10	Female	42	42	testing	2.1078555	4	95.149890
10	Female	42	42	training	6.6268396	9	299.139604
10	Female	42	42	validation	1.9574700	0	88.361394
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10	Female	43	43	training	6.8008285	5	303.351108
10	Female	43	43	validation	1.9159268	1	85.459956
10	Female	44	44	testing	2.5671447	2	113.149890
10	Female	44	44	training	6.6294612	4	292.201218
10	Female	44	44	validation	2.0363760	1	89.755642
10	Female	45	45	testing	2.0619680	1	89.806970
10	Female	45	45	training	6.3434188	4	276.281306
10	Female	45	45	validation	1.8003561	2	78.412722
10	Female	46	46	testing	3.6962410	7	159.073896
10	Female	46	46	training	5.0058544	6	215.435290
10	Female	46	46	validation	1.4924478	1	64.229978
10	Female	47	47	testing	3.5821019	2	152.332634
10	Female	47	47	training	5.1156467	7	217.548232
10	Female	47	47	validation	1.8774335	4	79.839824
10	Female	48	48	testing	3.2893522	5	138.219692
10	Female	48	48	training	4.6906490	4	197.102656
10	Female	48	48	validation	1.5468700	0	65.000000
10	Female	49	49	testing	1.3451926	2	55.854204
10	Female	49	49	training	4.7202163	6	195.989714
10	Female	49	49	validation	1.6858800	1	70.000000
10	Female	50	50	testing	1.3162418	6	54.004094
10	Female	50	50	training	4.3117185	4	176.905532
10	Female	50	50	validation	1.5723335	0	64.511284
10	Female	51	51	testing	1.4444044	1	58.558518
10	Female	51	51	training	4.8929841	6	198.369582
10	Female	51	51	validation	1.3721281	1	55.628320
10	Female	52	52	testing	1.4228340	0	57.000000
10	Female	52	52	training	4.8567741	7	194.566706
10	Female	52	52	validation	1.5367259	4	61.562612
10	Female	53	53	testing	1.1202499	2	44.347014
10	Female	53	53	training	4.6079278	4	182.412722
10	Female	53	53	validation	1.6233434	1	64.262832
10	Female	54	54	testing	1.1587788	1	45.328540
10	Female	54	54	training	4.4063513	4	172.365488
10	Female	54	54	validation	1.3293280	0	52.000000
10	Female	55	55	testing	0.9830980	0	38.000000
10	Female	55	55	training	4.1522154	3	160.496904
10	Female	55	55	validation	1.5583157	2	60.234072
10	Female	56	56	testing	1.4229050	0	55.000000
10	Female	56	56	training	4.1835582	6	161.708408
10	Female	56	56	validation	1.4572757	0	56.328540
10	Female	57	57	testing	1.1311522	2	43.722788
10	Female	57	57	training	4.0741245	0	157.478430
10	Female	57	57	validation	1.3452920	0	52.000000
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10	Female	58	58	training	2.8764088	2	111.182744
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10	Female	59	59	training	3.3171717	5	128.219692
10	Female	59	59	validation	1.1175528	2	43.197124
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10	Female	60	60	training	2.6804903	2	103.609846
10	Female	60	60	validation	1.2218868	1	47.229978
10	Female	61	61	testing	1.1273804	2	43.576992
10	Female	61	61	training	3.1792111	2	122.887058
10	Female	61	61	validation	1.0865820	1	42.000000
10	Female	62	62	testing	0.8775420	3	33.919912
10	Female	62	62	training	2.6155209	1	101.098562
10	Female	62	62	validation	0.8749921	2	33.821350
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10	Female	63	63	training	2.5579351	4	98.872678
10	Female	63	63	validation	1.0632609	0	41.098562
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10	Female	64	64	training	2.5795562	3	99.708408
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10	Female	65	65	testing	1.0009473	2	38.689934
10	Female	65	65	training	2.6995616	2	104.347014
10	Female	65	65	validation	0.9898977	0	38.262832
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10	Female	66	66	training	2.5455576	2	98.394248
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10	Female	67	67	training	2.1528707	2	83.215598
10	Female	67	67	validation	1.0730885	2	41.478430
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10	Female	68	68	training	2.2101905	8	85.431196
10	Female	68	68	validation	0.9025631	1	34.887058
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10	Female	70	70	testing	1.3093273	2	50.609846
10	Female	70	70	training	1.7333570	0	67.000000
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10	Female	71	71	training	1.5137454	4	58.511284
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10	Male	8	8	training	27.1400009	31	587.433192
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10	Male	9	9	training	25.1691405	14	570.340822
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10	Male	67	67	validation	0.5702620	0	14.000000
10	Male	68	68	testing	0.4922250	2	12.084182
10	Male	68	68	training	1.1405240	0	28.000000
10	Male	68	68	validation	0.4560925	1	11.197124
10	Male	69	69	testing	0.3665970	0	9.000000
10	Male	69	69	training	1.2627230	0	31.000000
10	Male	69	69	validation	0.3665970	0	9.000000
10	Male	70	70	testing	0.9775920	0	24.000000
10	Male	70	70	training	0.8842486	2	21.708408
10	Male	70	70	validation	0.3258640	0	8.000000
10	Male	71	71	testing	0.7331940	0	18.000000
10	Male	71	71	training	0.5702620	0	14.000000
10	Male	71	71	validation	0.2851310	0	7.000000
10	Male	72	72	testing	0.7159636	3	17.576992
10	Male	72	72	training	0.6109950	1	15.000000
10	Male	72	72	validation	0.2851310	0	7.000000
10	Male	73	73	testing	0.3665970	0	9.000000
10	Male	73	73	training	0.6517280	1	16.000000
10	Male	73	73	validation	0.2443980	0	6.000000
10	Male	74	74	testing	0.3665970	0	9.000000
10	Male	74	74	training	0.6524805	1	16.018474
10	Male	74	74	validation	0.2671481	0	6.558518
10	Male	75	75	testing	0.4480630	0	11.000000
10	Male	75	75	training	0.5643233	1	13.854204
10	Male	75	75	validation	0.1656085	1	4.065708
10	Male	76	76	testing	0.2036650	0	5.000000
10	Male	76	76	training	0.5702620	1	14.000000
10	Male	76	76	validation	0.1629320	0	4.000000
10	Male	77	77	testing	0.2851310	0	7.000000
10	Male	77	77	training	0.2851310	0	7.000000
10	Male	77	77	validation	0.2443980	0	6.000000
10	Male	78	78	testing	0.0407330	0	1.000000
10	Male	78	78	training	0.3665970	0	9.000000
10	Male	78	78	validation	0.2130327	1	5.229978
10	Male	79	79	testing	0.1629320	0	4.000000
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10	Male	79	79	validation	0.1288902	1	3.164270
10	Male	80	80	testing	0.2036650	0	5.000000
10	Male	80	80	training	0.4073300	0	10.000000
10	Male	80	80	validation	0.0814660	0	2.000000
10	Male	81	81	testing	0.1221990	0	3.000000
10	Male	81	81	training	0.6784928	1	16.657080
10	Male	81	81	validation	0.1569933	1	3.854204
10	Male	82	82	testing	0.3258640	0	8.000000
10	Male	82	82	training	0.4073300	0	10.000000
10	Male	82	82	validation	0.1629320	0	4.000000
10	Male	83	83	testing	0.2851310	0	7.000000
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10	Male	83	83	validation	0.1629320	0	4.000000
10	Male	84	84	testing	0.3252783	2	7.985620
10	Male	84	84	training	0.3258640	0	8.000000
10	Male	84	84	validation	0.1221990	0	3.000000
10	Male	85	85	testing	0.0814660	0	2.000000
10	Male	85	85	training	0.3472759	0	8.525664
10	Male	85	85	validation	0.1275520	1	3.131416
10	Male	86	86	testing	0.0814660	0	2.000000
10	Male	86	86	training	0.3665970	0	9.000000
10	Male	86	86	validation	0.0814660	0	2.000000
10	Male	87	87	testing	0.1221990	0	3.000000
10	Male	87	87	training	0.2331063	1	5.722788
10	Male	87	87	validation	0.1221990	0	3.000000
10	Male	88	88	testing	0.1629320	0	4.000000
10	Male	88	88	training	0.2851310	0	7.000000
10	Male	88	88	validation	0.0814660	0	2.000000
10	Male	89	89	testing	0.1629320	0	4.000000
10	Male	89	89	training	0.3258640	0	8.000000
10	Male	89	89	validation	0.2036650	0	5.000000
10	Male	90	90	testing	0.0814660	0	2.000000
10	Male	90	90	training	0.2851310	0	7.000000
10	Male	90	90	validation	0.0814660	0	2.000000
10	Male	91	91	testing	0.0814660	0	2.000000
10	Male	91	91	training	0.1629320	0	4.000000
10	Male	91	91	validation	0.1221990	0	3.000000
10	Male	92	92	testing	0.1221990	0	3.000000
10	Male	92	92	training	0.4073300	0	10.000000
10	Male	92	92	validation	0.1221990	0	3.000000
10	Male	93	93	testing	0.1476256	1	3.624226
10	Male	93	93	training	0.1629320	0	4.000000
10	Male	93	93	validation	0.0814660	0	2.000000
10	Male	94	94	training	0.2443980	0	6.000000
10	Male	94	94	validation	0.0407330	0	1.000000
10	Male	95	95	testing	0.2036650	0	5.000000
10	Male	95	95	training	0.2851310	0	7.000000
10	Male	95	95	validation	0.1221990	0	3.000000
10	Male	96	96	testing	0.0814660	0	2.000000
10	Male	96	96	training	0.1221990	0	3.000000
10	Male	96	96	validation	0.0814660	0	2.000000
10	Male	97	97	training	0.1221990	0	3.000000
10	Male	97	97	validation	0.0814660	0	2.000000
10	Male	98	98	training	0.2443980	0	6.000000
10	Male	98	98	validation	0.0814660	0	2.000000
10	Male	99	99	testing	0.0814660	0	2.000000
10	Male	99	99	training	0.0814660	0	2.000000
10	Male	99	99	validation	0.0407330	0	1.000000
10	Male	100	100	training	0.2036650	0	5.000000
10	Male	101	101	testing	0.0407330	0	1.000000
10	Male	101	101	training	0.2591187	1	6.361394
10	Male	101	101	validation	0.0814660	0	2.000000
10	Male	102	102	testing	0.0814660	0	2.000000
10	Male	102	102	training	0.1221990	0	3.000000
10	Male	103	103	training	0.1629320	0	4.000000
10	Male	103	103	validation	0.0407330	0	1.000000
10	Male	104	104	testing	0.0814660	0	2.000000
10	Male	104	104	training	0.1221990	0	3.000000
10	Male	104	104	validation	0.1221990	0	3.000000
10	Male	105	105	testing	0.0814660	0	2.000000
10	Male	105	105	training	0.2036650	0	5.000000
10	Male	105	105	validation	0.1221990	0	3.000000
10	Male	106	106	testing	0.1629320	0	4.000000
10	Male	106	106	training	0.1629320	0	4.000000
10	Male	106	106	validation	0.0407330	0	1.000000
10	Male	107	107	testing	0.1221990	0	3.000000
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10	Male	108	108	testing	0.1221990	0	3.000000
10	Male	108	108	training	0.1221990	0	3.000000
10	Male	108	108	validation	0.0407330	0	1.000000
10	Male	109	109	testing	0.0814660	0	2.000000
10	Male	109	109	training	0.0407330	0	1.000000
10	Male	110	110	testing	0.1221990	0	3.000000
10	Male	110	110	training	0.0814660	0	2.000000
10	Male	111	111	testing	0.1629320	0	4.000000
10	Male	111	111	training	0.0814660	0	2.000000
10	Male	111	111	validation	0.0407330	0	1.000000
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10	Male	112	112	training	0.0814660	0	2.000000
10	Male	113	113	testing	0.0407330	0	1.000000
10	Male	113	113	training	0.0407330	0	1.000000
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10	Male	114	114	training	0.0407330	0	1.000000
10	Male	115	115	testing	0.0814660	0	2.000000
10	Male	115	115	training	0.1629320	0	4.000000
10	Male	115	115	validation	0.0407330	0	1.000000
10	Male	116	116	training	0.1629320	1	4.000000
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
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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
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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
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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"))