R: Life Modelling Recipes#

By Pat Reen - originally published on his GitHub site here, which has the original code.

Overview#

Background#

This section sets out a few practical recipes for modelling with (life) insurance data. Insurance events are typically of low probability - these recipes consider some of the limitations of “small data” model fitting (where the observations of interest are sparse) and other topics for insurance like comparisons to standard tables. Themes

  • Common data transforms, summary stats, and simple visualisations

  • Regression

    • Grouped vs ungrouped data

    • Choice of: response distribution, link (and offsets), explanatory variables

    • Modelling variance to industry / reference (A/E or A - E)

    • Model selection: stepwise regression, likelihood tests, model evaluation

    • Predictions, confidence intervals and visualisations

As with other recipes, the code can be found on GitHub.

Libraries#

A list of packages used in the recipe book.

library(rmdformats) # theme for the HTML doc
library(bookdown)   # bibliography formatting
library(kableExtra) # formatting tables
library(scales)     # data formatting  
library(dplyr)      # tidyverse: data manipulation
library(tidyr)      # tidyverse: tidy messy data
library(corrplot)   # correlation matrix visualisation, optional
library(ggplot2)    # tidyverse: graphs
library(pdp)        # tidyverse: arranging plots
library(GGally)     # tidyverse: extension of ggplot2
library(ggthemes)   # tidyverse: additional themes for ggplot, optional
library(plotly)     # graphs, including 3D 
library(caret)      # sampling techniques
library(broom)      # tidyverse: visualising statistical objects
library(pROC)       # visualising ROC curves
library(lmtest)     # tests for regression models

# packages below have some interaction with earlier packages, not always needed
library(arm)        # binned residual plot
library(msme)       # statistical tests, pearson dispersion
library(MASS)       # statistics

Further reading#

A few books and articles of interest:

  • R Markdown Cookbook has everything you need to know to set up an r markdown document like this one.

  • Generalised Linear Models for Insurance Data is a great book introducing GLMs in the context of insurance, considering problems specific to insurance data.

  • Tidyverse documentation full set of documentation for Tidyverse packages (packages for data science) e.g. dplyr for data manipulation; tidyr for tidying up messy data; ggplot for visualisation.

Data Simulation#

This section sets out a method for generating dummy data. The simulated data is intended to reflect typical data used in an analysis of disability income incidence experience and is used throughout this analysis. Replace this data with your actual data.

More detail on the techniques used can be found in the section on data manipulation.

Simulating policies#

We start by simulating a mix of 200k policies over 3 years. Some simplifying assumptions e.g. nil lapse / new business (no allowance for part years of exposure), no indexation. Mix of business assumptions for benefit period, waiting period and occupation taken taken from James Louw. 2012, with the remainder based on an anecdotal view of industry mix not intended to be reflective of any one business.

# set the seed value (for the random number generator) so that the simulated data frame can be replicated later
set.seed(10)
# create 200k policies
n <- 200000

# data frame columns
# policy_year skewed to early years, but tail is fat
df <- data.frame(id = c(1:n), cal_year = 2018,policy_year = round(rweibull(n, scale=5, shape=2),0)) 
df <- df %>% mutate(sex = replicate(n,sample(c("m","f","u"), size=1, replace=TRUE, prob=c(.75,.20,.05))),
smoker = replicate(n,sample(c("n","s","u"), size=1, replace=TRUE, prob=c(.85,.1,.05))),
# mix of business for benefit_period, waiting_period, occupation taken from industry presentation
benefit_period = replicate(n,sample(c("a65","2yr","5yr"), size=1, replace=TRUE, prob=c(.76,.12,.12))),
waiting_period = replicate(n,sample(c("14d","30d","90d","720d"), size=1, replace=TRUE, prob=c(.04,.7,.15,.11))),
occupation = replicate(n,sample(c("prof","sed","techn","blue","white"), size=1, replace=TRUE, prob=c(.4,.2,.2,.1,.1))),
# age and policy year correlated; age normally distributed around 40 + policy_year (where policy_year is distributed around 5 years), floored at 25, capped at 60
age = round(pmax(pmin(rnorm(n,mean = 40+policy_year, sd = 5),60),25),0),
# sum_assured, age and occupation are correlated; sum assured normally distributed around some mean (dependent on age rounded to 10 and on occupation), floored at 500
sum_assured = 
  round(
    pmax(
      rnorm(n,mean = (round(age,-1)*100+ 1000) * 
      case_when(occupation %in% c("white","prof") ~ 1.5, occupation %in% c("sed") ~ 1.3 , TRUE ~ 1), 
      sd = 2000),500),
      0)
  )
# generate 3 years of exposure for the 200k policies => assume no lapses or new business
df2 <- df %>% mutate(cal_year=cal_year+1,policy_year=policy_year+1,age=age+1)
df3 <- df2 %>% mutate(cal_year=cal_year+1,policy_year=policy_year+1,age=age+1)
df <- rbind(df,df2,df3)

Expected claim rate#

Set p values from which to simulate claims. The crude p values below were derived from the Society of Actuaries Analysis of USA Individual Disability Claim Incidence Experience from 2006 to 2014 (SOA, 2019), with some allowance for Australian industry differentials (Ian Welch. 2020).

# by cause, age and sex, based upon polynomials fitted to crude actual rates
# sickness
f_sick_age_m <- function(age) {-0.0000003*age^3 + 0.000047*age^2 - 0.00203*age + 0.02715}
f_sick_age_f <- function(age) {-0.0000002*age^3 + 0.000026*age^2 - 0.00107*age + 0.01550} 	  	 	  
f_sick_age_u <- function(age) {f_sick_age_f(age)*1.2}
f_sick_age   <- function(age,sex) {case_when(sex == "m" ~ f_sick_age_m(age), sex == "f" ~ f_sick_age_f(age), sex == "u" ~ f_sick_age_u(age))}

# accident
f_acc_age_m <- function(age) {-0.00000002*age^3 + 0.000004*age^2 - 0.00020*age + 0.00340}
f_acc_age_f <- function(age) {-0.00000004*age^3 + 0.000007*age^2 - 0.00027*age + 0.00374} 	  	 	  
f_acc_age_u <- function(age) {f_sick_age_f(age)*1.2}
f_acc_age   <- function(age,sex) {case_when(sex == "m" ~ f_acc_age_m(age), sex == "f" ~ f_acc_age_f(age), sex == "u" ~ f_acc_age_u(age))}

# smoker, wp and occ based upon ratio of crude actual rates by category
# occupation adjustment informed by FSC commentary on DI incidence experience
f_smoker   <- function(smoker) {case_when(smoker == "n" ~ 1, smoker == "s" ~ 1.45, smoker == "u" ~ 0.9)}
f_wp   <- function(waiting_period) {case_when(waiting_period == "14d" ~ 1.4, waiting_period == "30d" ~ 1, waiting_period == "90d" ~ 0.3, waiting_period == "720d" ~ 0.2)}
f_occ_sick   <- function(occupation) {case_when(occupation == "prof" ~ 1, occupation == "sed" ~ 1, occupation == "techn" ~ 1, occupation == "blue" ~ 1, occupation == "white" ~ 1)}
f_occ_acc   <- function(occupation) {case_when(occupation == "prof" ~ 1, occupation == "sed" ~ 1, occupation == "techn" ~ 4.5, occupation == "blue" ~ 4.5, occupation == "white" ~ 1)}

# anecdotal allowance for higher rates at larger policy size and for older policies
f_sa_sick <- function(sum_assured) {case_when(sum_assured<=6000 ~ 1, sum_assured>6000 & sum_assured<=10000 ~ 1.1, sum_assured>10000 ~ 1.3)}
f_sa_acc <- function(sum_assured) {case_when(sum_assured<=6000 ~ 1, sum_assured>6000 & sum_assured<=10000 ~ 1, sum_assured>10000 ~ 1)}
f_pol_yr_sick <- function(policy_year) {case_when(policy_year<=5 ~ 1, policy_year>5 & policy_year<=10 ~ 1.1, policy_year>10 ~ 1.3)}
f_pol_yr_acc <- function(policy_year) {case_when(policy_year<=5 ~ 1, policy_year>5 & policy_year<=10 ~ 1, policy_year>10 ~ 1)}

Expected claims#

Add the crude p values to the data and simulate 1 draw from a binomial with prob = p for each record. Gives us a vector of claim/no-claim for each policy. Some simplifying assumptions like independence of sample across years for each policy and independence of accident and sickness incidences.

# add crude expected
df$inc_sick_expected=f_sick_age(df$age,df$sex)*f_smoker(df$smoker)*f_wp(df$waiting_period)*f_occ_sick(df$occupation)*f_sa_sick(df$sum_assured)*f_pol_yr_sick(df$policy_year)
df$inc_acc_expected=f_acc_age(df$age,df$sex)*f_smoker(df$smoker)*f_wp(df$waiting_period)*f_occ_acc(df$occupation)*f_sa_acc(df$sum_assured)*f_pol_yr_acc(df$policy_year)
# add prediction
df$inc_count_sick = sapply(df$inc_sick_expected,function(z){rbinom(1,1,z)})
df$inc_count_acc = sapply(df$inc_acc_expected,function(z){rbinom(1,1,z)})*(1-df$inc_count_sick)
df$inc_count_tot = df$inc_count_sick + df$inc_count_acc
# add amounts prediction
df$inc_amount_sick = df$inc_count_sick * df$sum_assured
df$inc_amount_acc =  df$inc_count_acc * df$sum_assured
df$inc_amount_tot =  df$inc_count_tot * df$sum_assured

Grouped data#

The data generated above are records for each individual policy, however data like this is often grouped as it is easier to store and computation is easier (Piet de Jong, Gillian Z. Heller, 2008, p49, 105). Later we will consider the differences between a model on ungrouped vs grouped data.

# group data (see section on data manipulation below)
df_grp <- df %>% group_by(cal_year, policy_year, sex, smoker, benefit_period, waiting_period, occupation, age) %>% 
summarise(sum_assured=sum(sum_assured),inc_count_sick_exp=sum(inc_sick_expected),inc_count_acc_exp=sum(inc_acc_expected),        inc_count_sick=sum(inc_count_sick),inc_count_acc=sum(inc_count_acc),inc_count_tot=sum(inc_count_tot),inc_amount_sick=sum(inc_amount_sick),inc_amount_acc=sum(inc_amount_acc),inc_amount_tot=sum(inc_amount_tot), exposure=n(),.groups = 'drop') 

Check that the exposure for the grouped data is the same as the total on ungrouped:

# check count - same as total row count of the main df
sum(df_grp$exposure)
600000

And that the number of rows of data are significantly lower:

# number of rows of the grouped data is significantly lower
nrow(df_grp)
109590

Data Exploration#

The sections below rely heavily upon the dplyr package.

Data structure#

Looking at the metadata for the data frame and a sample of the contents.

# MASS package interferes with select()
select <- dplyr::select
# glimpse() or str() returns detail on the structure of the data frame. Our data consists of 600k rows and 15 columns. The columns are policy ID, several explanatory variables like sex and smoker, expected counts of claim (inc_sick_expected and inc_acc_expected) and actual counts of claim (inc_count_sick/acc/tot).
glimpse(df)
Rows: 600,000
Columns: 18
$ id                <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1~
$ cal_year          <dbl> 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018~
$ policy_year       <dbl> 4, 5, 5, 3, 8, 6, 6, 6, 3, 5, 3, 4, 7, 4, 5, 5, 9, 6~
$ sex               <chr> "m", "f", "m", "m", "f", "f", "m", "m", "m", "m", "m~
$ smoker            <chr> "n", "n", "n", "n", "n", "n", "n", "n", "s", "n", "n~
$ benefit_period    <chr> "a65", "5yr", "a65", "a65", "a65", "2yr", "a65", "a6~
$ waiting_period    <chr> "90d", "30d", "30d", "30d", "30d", "14d", "720d", "3~
$ occupation        <chr> "techn", "blue", "blue", "prof", "sed", "prof", "tec~
$ age               <dbl> 41, 46, 41, 27, 53, 49, 54, 42, 43, 52, 36, 47, 47, ~
$ sum_assured       <dbl> 7119, 5582, 6113, 5147, 8864, 11209, 6378, 6780, 597~
$ inc_sick_expected <dbl> 0.000742731, 0.001828800, 0.002475770, 0.000698100, ~
$ inc_acc_expected  <dbl> 0.0007365330, 0.0100735200, 0.0024551100, 0.00052234~
$ inc_count_sick    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0~
$ inc_count_acc     <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0~
$ inc_count_tot     <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0~
$ inc_amount_sick   <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0~
$ inc_amount_acc    <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0~
$ inc_amount_tot    <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0~
# head() returns the first 6 rows of the data frame. Similar to head(), sample_n() returns rows from our data frame, however these are chosen randomly. e.g. sample_n(df,5,replace=FALSE)
head(df)
idcal_yearpolicy_yearsexsmokerbenefit_periodwaiting_periodoccupationagesum_assuredinc_sick_expectedinc_acc_expectedinc_count_sickinc_count_accinc_count_totinc_amount_sickinc_amount_accinc_amount_tot
1 2018 4 m n a65 90d techn 41 7119 0.0007427310.0007365330 0 0 0 0 0
2 2018 5 f n 5yr 30d blue 46 5582 0.0018288000.0100735200 0 0 0 0 0
3 2018 5 m n a65 30d blue 41 6113 0.0024757700.0024551100 0 0 0 0 0
4 2018 3 m n a65 30d prof 27 5147 0.0006981000.0005223400 0 0 0 0 0
5 2018 8 f n a65 30d sed 53 8864 0.0024788060.0031379200 0 0 0 0 0
6 2018 6 f n 2yr 14d prof 49 11209 0.0039363320.0036554560 0 0 0 0 0
# class() returns the class of a column.
class(df$benefit_period)
'character'

Factors#

From the above you’ll note that the categorical columns are stored as characters. Factorising these makes them easier to work with in our models e.g. for BP factorise a65|2yr|5yr as 1|2|3. Factors are stored as integers and have labels that tell us what they are, they can be ordered and are useful for statistical analysis.

table() returns a table of counts at each combination of column values. prop.table() converts these to a proportion. For example, applying this to the column “sex” shows us that ~75% of our data is “m” and that the other data are either “f” or “u” (unknown).

table(df$sex)
prop.table(table(df$sex))
     f      m      u 
120951 449487  29562 
       f        m        u 
0.201585 0.749145 0.049270 

We can then convert the columns to factors based upon the values of the column and ordering by frequency. Base level should be chosen such that it has sufficient observations for an intercept to be computed meaningfully.

df$sex <- factor(df$sex, levels = c("m","f","u"))
df$smoker <- factor(df$smoker, levels = c("n","s","u"))
df$benefit_period <- factor(df$benefit_period, levels = c("a65","2yr","5yr"))
df$waiting_period <- factor(df$waiting_period, labels = c("30d","14d","720d","90d"))
df$occupation <- factor(df$occupation, labels = c("prof", "sed","techn","white","blue"))

# do the same for the grouped data
df_grp$sex <- factor(df_grp$sex, levels = c("m","f","u"))
df_grp$smoker <- factor(df_grp$smoker, levels = c("n","s","u"))
df_grp$benefit_period <- factor(df_grp$benefit_period, levels = c("a65","2yr","5yr"))
df_grp$waiting_period <- factor(df_grp$waiting_period, labels = c("30d","14d","720d","90d"))
df_grp$occupation <- factor(df_grp$occupation, labels = c("prof", "sed","techn","white","blue"))

If the column is already a factor, you can extract the levels to show what order they will be used in our models

table(levels(df$sex))
f m u 
1 1 1 

Selection methods#

table() is a method of summarizing data, returning a count at each combination of values in a column. sample() and sample_n() are other examples of selection methods. This section (not exhaustive) looks at a few more selection methods in dplyr.

not_run_sample_only <- function(){
# data subsets:  e.g. select from df where age <25 or >60 
    subset(df, age <25 | age > 60) 
#	dropping columns:
		# exclude columns
		mycols <- names(df) %in% c("cal_year", "smoker")
		new_df <- df[!mycols]
		# exclude 3rd and 5th column
		new_df <- df[c(-3,-5)]
		# delete columns v1 and v2 from new_df
		new_df$v3 <- new_df$v5 <- NULL
#	keeping columns: 
		# select variables v1, v2, v3
		mycols <- names(df) %in% c("cal_year", "smoker")
		new_df <- df[!mycols]
		# select 1st and 5th to 7th variables
    new_df <- df[c(1,5:7)]
}

Manipulation methods#

We might want to modify our data frame to prepare it for fitting our models. The section below looks at a few simple data manipulations. Here we also introduce the infix operator (%>%); this operator passes the argument to the left of it over to the code on the right, so df %>% “operation” passes the data frame “df” over to the operation on the right.

not_run_sample_only <- function(){
# create a copy of the dataframe to work from
  new_df <- df
# simple manipulations
	# select as in the selection methods section, but using infix
	new_df %>% select(id, age) # or a range using select(1:5) or select(contains("sick")) or select(starts_with("inc")); others e.g. ends_with(), last_col(), select(-age)
	# replace values in a column
	replace(new_df$sex,new_df$sex=="u","m") # no infix in base r
	# Rename, id to pol_id
	new_df %>% rename(pol_id = id)  #or (reversing the renaming)
	new_df %>% select(pol_id = id)  
	# alter data
	new_df <- new_df %>% mutate(inc_tot_expected = inc_acc_expected + inc_sick_expected) # need to assign the output back to the data frame
	# transmute - select and mutate simultaneously 
	new_df2 <- new_df %>% transmute(id, age, birth_year = cal_year - age)
	# sort
	new_df %>% arrange(desc(age))
	# filter
	new_df %>% filter(benefit_period == "a65", age <65) # or
	new_df %>% filter(benefit_period %in% c("a65","5yr"))
# aggregations
	# group by, also ungroup()
	new_df %>% group_by(sex) %>% # can add a mutate to group by which will aggregate only to the level specified in the group_by e.g. 
	mutate(sa_by_sex = sum(sum_assured)) # adds a new column with the total sum assured by sex.
	# after doing this, ungroup() in order to apply future operations to all records individually
	# count, sorting by most frequent and weighting by another column
	new_df %>% count(sex, wt= sum_assured, sort=TRUE)  # counts the number of entries for each value of sex, weighted by sum assured
	# summarize takes many observations and turns them into one observation. mean(), median(), min(), max(), and n() for the size of the group
	new_df %>% summarize(total = sum(sum_assured), min_age = min(age), max_age = max(age), max(inc_tot_expected)) 
	new_df %>% group_by(sex) %>% summarise(n = n())
	table(new_df$sex) # returns count by sex; no infix in base r
# outliers
	new_df %>% top_n(10, inc_tot_expected) # also operates on grouped table - returns top n per group
# window functions
	# lag - offset vector by 1 e.g. v <- c(1,3,6,14); so - lag(v) = NA 1 3 6
	new_df %>% arrange(id,age) %>% mutate(ifelse(id==lag(id),age - lag(age),1))
}

Missing data#

By default, the regression model will exclude any observation with missing values on its predictors. Missing values can be treated as a separate category for categorical data. For missing numeric data, imputation is a potential solution. In the example below we replace missing age with an average and add an indicator to the data to flag records that have been imputed.

# find the average age among non-missing values
summary(df$age)
# impute missing age values with the mean age
df$imputed_age <- ifelse(is.na(df$age)==TRUE,round(mean(df$age, na.rm=TRUE),2),df$age)
# create missing value indicator for age
df$missing_age <- ifelse(is.na(df$age)==TRUE,1,0)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  25.00   42.00   45.00   45.44   49.00   62.00 

Review exposure data#

The tables and graphs that follow look at:

  • the mix of business over rating factors using some of the selection methods described: These are all consistent with the simulation specification.

  • the correlation of ordered numerical rating factors: age and sum assured as well as age and policy year are positively correlated.

Data might need to be transformed in order to make the data more suitable to the assumptions within the model. Not considered here.

Look at distribution by single rating factors.

Benefit period mix#

df %>% count(benefit_period, wt = NULL, sort = TRUE) %>% 
   mutate(freq = percent(round(n / sum(n),2))) %>% 
   format(n, big.mark=",") 
benefit_periodnfreq
a65 456,55276%
2yr 72,159 12%
5yr 71,289 12%

Waiting period mix#

df %>% count(waiting_period, wt = NULL, sort = TRUE) %>% 
   mutate(freq = percent(round(n / sum(n),2))) %>% 
   format(n, big.mark=",")
waiting_periodnfreq
14d 420,00970.0%
90d 90,717 15.0%
720d 65,814 11.0%
30d 23,460 4.0%

Occupation mix#

df %>% count(occupation, wt = NULL, sort = TRUE) %>% 
   mutate(freq = percent(round(n / sum(n),2))) %>% 
   format(n, big.mark=",")
occupationnfreq
sed 240,34840%
techn 120,71720%
white 119,84720%
blue 59,613 10%
prof 59,475 10%

Consider a histogram to show the distribution of numeric data.

hist(df$age, main = "Histogram of age", xlab = "Age", ylab = "Frequency")
hist(df$sum_assured, main = "Histogram of sum assured", xlab = "Sum assured", ylab = "Frequency")
hist(df$policy_year, main = "Histogram of policy year", xlab = "Policy year", ylab = "Frequency")
../_images/c2d061545905f2b15e82df82cc734a1764eef773efec4d1d8b98193eda69598b.png ../_images/b1fa44c577d0847ed6f9a29bcb8a63b3675ae79e7d4b124d1825d47bd8897eaf.png ../_images/fe322a7c77c825ce8f5e64a39a5e6b496540cd93c742b28d398b17c3ef1e152c.png

Consider the correlation of ordered numeric explanatory variables.

# correlation of ordered numeric explanatory variables
#	pairs() gives correlation matrix and plots; test on a random sample from our data
df_sample <- sample_n(df,10000,replace=FALSE) 
df_sample %>% select(age,policy_year,sum_assured) %>% pairs
# or cor() to return just the correlation matrix
cor <-df_sample %>% select(age,policy_year,sum_assured) %>% cor
cor
# corrplot() is an alternative to visualise a correlation matrix
corrplot(cor, 
  addCoef.col = "black", # add coefficient of correlation
  method="color", 
  sig.level = 0.01, insig = "blank", 
  tl.col="black", # tl stands for text label
  tl.srt=45
  ) 
agepolicy_yearsum_assured
age1.00000000.43334340.2937906
policy_year0.43334341.00000000.1249400
sum_assured0.29379060.12494001.0000000
../_images/70ab12d4674c9f2f664e6941a21219814a7a2e2713d34c5205e7d9d58e3fabfd.png ../_images/58f54acbde5d9a9a91f07e2f867a116c06a1b4f43280ed12540261a0aead45e5.png

ggpairs() similarly shows correlations for ordered numeric data as well as other summary stats:

#ggpairs() similarly shows correlations for ordered numeric data as well as other summary stats
df_sample %>% select(age,policy_year,sum_assured,sex, smoker) %>% 
ggpairs(columns = 1:3, aes(color = sex, alpha = 0.5),
        upper = list(continuous = wrap("cor", size = 2.5)),
        lower = list(continuous = "smooth"))

df_sample %>% select(age,policy_year,sum_assured,sex, smoker) %>% 
ggpairs(columns = c("sum_assured", "smoker"), aes(color = sex, alpha = 0.5))
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
../_images/6c7b837c2867de70c5dcb6716e6e686e7543c0125f5aef7a14752778811a4bbe.png ../_images/db7970ac5f732a64a746ad32a115fb40847029cdb115428f6d9cc94dd462ae61.png

Review summary statistics for subsets of data.

head(aggregate(df$sum_assured~df$age,data=df,mean))
df$agedf$sum_assured
25 3584.319
26 4431.558
27 4735.551
28 5066.777
29 5068.259
30 5204.272

Data format#

There are two main formats for structured data - long and wide. For regression, the structure of data informs the model structure. For counts data:

  • the long format corresponds to Bernoulli (claim or no claim for each observation) and allows for predictor variables by observation;

  • the wide format correspond to Binomial (count of claims per exposure). Wide format structures can include matrix of successes and failures or a proportion of successes and corresponding weights / number of observations/exposure for each line.

There are several tidyverse functions that can help with restructuring data, for example, convert data into wide format e.g.separate into a separate column for each value of sex:

Wide data format example#

head(spread(df_sample, sex, inc_count_tot, fill = 0))
idcal_yearpolicy_yearsmokerbenefit_periodwaiting_periodoccupationagesum_assuredinc_sick_expected...inc_count_sickinc_count_accinc_amount_sickinc_amount_accinc_amount_totimputed_agemissing_agemfu
186530 2019 6 n a65 14d sed 45 7850 0.004401375... 0 0 0 0 0 45 0 0 0 0
89986 2020 4 n a65 14d techn 48 8359 0.005302440... 0 0 0 0 0 48 0 0 0 0
12688 2019 6 n 5yr 14d white 48 9391 0.005832684... 0 0 0 0 0 48 0 0 0 0
16974 2019 6 s a65 14d white 49 2687 0.008345518... 0 0 0 0 0 49 0 0 0 0
70839 2020 7 n a65 14d prof 41 3597 0.002475770... 0 0 0 0 0 41 0 0 0 0
120872 2018 4 n a65 14d prof 46 5177 0.004021200... 0 0 0 0 0 46 0 0 0 0

gather() converts data into long format; also pivot_longer() and pivot_wider().

Visualisation methods#

Introduction to ggplot#

This section sets out some simple visualisation methods using ggplot(). ggplot() Initializes a ggplot object. It can be used to declare the input data frame for a graphic and to specify the set of plot aesthetics intended to be common throughout all subsequent layers unless specifically overridden. The form of ggplot is:

ggplot(data = df, mapping = aes(x,y, other aesthetics), ...)

Examples below use ggplot to explore the exposure data.

# data argument passes the data frame to be visualised
# mapping argument defines a list of aesthetics for the visualisation - all subsequent layers use those unless overridden
# typically, the dependent variable is mapped onto the the y-axis and the independent variable is mapped onto the x-axis.
ggplot(data=df_sample, mapping=aes(x=age, y=sum_assured)) + # the '+' adds the layer below
# add subsequent visualisation layers, e.g. geom_point() for scatterplot
geom_point() +
# add a layer to change axis labels
# could add a layer to specify axis limits with ylim() and xlim() 
labs(x="Age", y="Sum assured", title = "Sum Assured by age")
../_images/c6d26874c7846967cc219e240cf10d7aa03cbef7a5119740e69ef59cf82f8eea.png

Layers#

The aesthetics input has a number of different options, for example x and y (axes), colour, size, fill, labels, alpha (transparency), shape, line type / width. You can change the aesthetics of each layer or default to the base layer. You can change the general look and feel of charts with a themes layer e.g. colour palette (see more in the next section).

You can add more layers to the base plot, for example

  • Geometries (geom_), for example

    • point - scatterplot,

    • line,

    • histogram,

    • bar/ column,

    • boxplot,

    • density,

    • jitter - adds random noise to separate points,

    • count - counts the number of observations at each location, then maps the count to point area,

    • abline - adds a reference line - vertical, horizontal or diagonal,

    • curve - adds a curved line to the chart between specified points,

    • text - add a text layer to label data points.

  • Statistics (stat_)

    • smooth (curve fitted to the data),

    • bin (e.g. for histogram).

A note on overlapping points: these can be adjusted for by adding noise and transparency to your points:

  • within an existing geom e.g. geom_point(position=”*”) with options including: identity (default = position is as per data), dodge (dodge overlapping objects side-to-side), jitter (random noise), jitterdodge, and nudge (nudge points a fixed distance) e.g. geom_bar(position = “dodge”) or geom_bar(position=position_dodge(width=0.2)).

  • or use geom_* with arguments e.g. geom_jitter(alpha = 0.2, shape=1). Shape choice might help, shape = 1 gives hollow circles.

Or alternatively count overlapping points with geom_count().

A full list of layers is available here.

ggplot(data=df_sample, aes(x=age, y=sum_assured)) +
geom_point() + 
# separate overlapping points
geom_jitter(alpha = 0.2, width = 0.2) +
# add a smoothing line
geom_smooth(method = "glm", se=FALSE) 
`geom_smooth()` using formula 'y ~ x'
../_images/f749c6f3ce498f0b70d64eba01cd37db1cd819f9867d793e99364df1339c0a83.png

Themes#

You can add a themes layer to your graph (ggplot_themes), for example

  • theme_gray() |Gray background and white grid lines.

  • theme_bw() |White background and gray grid lines.

  • theme_linedraw() |A theme with only black lines of various widths on white backgrounds.

  • theme_light() |A theme similar to theme_linedraw() but with light grey lines and axes, to direct more attention towards the data.

  • theme_dark() |Similar to theme_light() but with a dark background.

  • Others |e.g. theme_minimal() and theme_classic()

Other packages like ggthemes carry many more options. Example of added themes layer below. See also these examples these examples from ggthemes.

# add an occupation group to the data
df_sample <- df_sample %>% mutate(occ_group = factor(case_when(occupation %in% c("white","prof","sed") ~ "WC", TRUE ~ "BC")))

# vary colour by occupation
ggplot(data=df_sample, aes(x=age, y=sum_assured, color=occ_group)) +
# jitter and fit a smoothed line as below
geom_jitter(alpha = 0.2, width = 0.2) +
geom_smooth(method = "glm", se=FALSE) +
# add labels
labs(x="Age", y="Sum assured", title = "Sum Assured by age") +
# adding theme and colour palette layers
theme_pander() + scale_color_gdocs()
`geom_smooth()` using formula 'y ~ x'
../_images/bbf7f2bba4aadbbdbc8953d7545b0ea6bdc7f8dae4831e311685df8ec2a46b16.png

3D visualisations with plotly#

ggplot does not cater to 3D visualisations, but this can be done through plotly simply.

plot_base <- plot_ly(data=df_sample, z= ~sum_assured, x= ~age, y=~policy_year, opacity=0.6) %>%
add_markers(color = ~occ_group,colors= c("blue", "red"), marker=list(size=2)) 
# show graph
# plot_base

# show graph in Jupyter notebooks
embed_notebook(plot_base)