Summary Statistics

Study guides: Exam MAS-II, Exam DISC-DA, Exam PCPA

Summary statistics condense a data set, or a fitted model, into a few numbers a reader can take in at once. For data they measure location, spread, shape and association, usually tabulated by group; for a fitted model they measure fit against complexity — the log-likelihood, AIC, BIC and, in a Linear Mixed Model, the estimated variance components.

$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$

$s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}$

Describing data — the tabular half of presenting data, beside graphs:

Location: the mean $\bar{x}$ and the median. For right-skewed severity the mean sits well above the median, and the gap is itself a finding
Spread: the standard deviation $s$ (Sample Variance), the interquartile range $Q_3 - Q_1$ , the range, and the coefficient of variation $s/\bar{x}$ for comparing spread across variables on different scales
Shape and completeness: percentiles (the five-number summary behind a Box Plot), the share of zeros, the largest values, and the count of missing values in each column
Categorical variables and association: counts, proportions and — in insurance — exposure by level; the correlation matrix for numeric predictors and two-way tables for categorical ones
Insurance one-way tables show exposure, claim count, frequency, severity and pure premium by level, with pure premium $=$ frequency $\times$ severity. Each is a ratio of sums, not an average of ratios, and each level's figure is confounded with everything correlated with it — the reason final relativities come from a multivariate model
Use a table when the reader needs exact values, a graph when the message is a shape or a comparison

Summarizing a fitted mixed model:

$-2 \times$ log-likelihood, by REML or ML, and the likelihood ratio test between nested models. Use REML to compare random effects or covariance structures with identical fixed effects; use ML to compare fixed effects
AIC $= -2\ell + 2k$ and BIC $= -2\ell + k\ln n$ for choices that are not nested. BIC's penalty per parameter is heavier once $\ln n > 2$ , i.e. $n \geq 8$ , so it favours smaller models
Variance components and the implied intraclass correlation. A component estimated at zero means the grouping explains nothing. Testing whether a variance is zero puts the null on the boundary of the parameter space, so the naive $\chi^2$ p-value is too large — for a single variance component it should be halved
Fixed-effect tests — a $t$ or $F$ statistic per term — guide variable selection, alongside the diagnostic plots that check the assumptions behind them

A One-Way Table by Vehicle Use Personal auto experience by vehicle use:

Use Exposure Claims Losses

Pleasure 6,000 300 1,500,000

Commute 3,000 195 1,053,000

Business 1,000 90 540,000

Build the frequency, severity and pure premium table, with pure premium relativities to Pleasure, and the all-uses totals.

Answer

Use Frequency Severity Pure premium Relativity

Pleasure 0.0500 5,000 250.00 1.000

Commute 0.0650 5,400 351.00 1.404

Business 0.0900 6,000 540.00 2.160

All 0.0585 5,287 309.30

The Business relativity decomposes as $1.80$ in frequency times $1.20$ in severity, which is $2.16$ . The overall frequency is $585/10{,}000 = 0.0585$ , not the average of the three rows ( $0.0683$ ), because Pleasure carries $60\%$ of the exposure. Before quoting $2.16$ as a rating factor, check what else Business use travels with — annual mileage, vehicle type — since a one-way table credits all of it to vehicle use.

Use	Exposure	Claims	Losses
Pleasure	6,000	300	1,500,000
Commute	3,000	195	1,053,000
Business	1,000	90	540,000

Use	Frequency	Severity	Pure premium	Relativity
Pleasure	0.0500	5,000	250.00	1.000
Commute	0.0650	5,400	351.00	1.404
Business	0.0900	6,000	540.00	2.160
All	0.0585	5,287	309.30

Choosing a Random-Effects Structure from Summary Statistics Loss ratios for $500$ policy-years are fitted by REML with identical fixed effects. Model 1 has a random intercept per policyholder ( $2$ covariance parameters); Model 2 adds a random slope on year and its covariance with the intercept ( $4$ ). The $-2$ REML log-likelihoods are $2{,}431.6$ and $2{,}420.2$ . Compare them by likelihood ratio test, AIC and BIC.
Answer
 $\begin{align*} \text{LRT} &= 2{,}431.6 - 2{,}420.2 \\ &= 11.4 \\ \text{AIC}_1 &= 2{,}431.6 + 2(2) \\ &= 2{,}435.6 \\ \text{AIC}_2 &= 2{,}420.2 + 2(4) \\ &= 2{,}428.2 \\ \text{BIC}_1 &= 2{,}431.6 + 2\ln 500 \\ &= 2{,}444.0 \\ \text{BIC}_2 &= 2{,}420.2 + 4\ln 500 \\ &= 2{,}445.1 \end{align*}$ 
The LRT of $11.4$ on $2$ degrees of freedom exceeds $\chi^2_{0.05,\,2} = 5.99$ ; because the slope variance is tested on its boundary, the true p-value is smaller still than the naive one, so the conclusion only strengthens. AIC also prefers Model 2. BIC, charging $\ln 500 = 6.21$ per parameter rather than $2$ , narrowly prefers Model 1. When the criteria split, look at the size of the estimated slope variance and at the residual plots by policyholder before adding structure.

How it appears on the exam

DISC-DA: Section 8.1 covers traditional data analysis. Questions give a small table of policy or claim data and ask for a mean, median, standard deviation or a one-way frequency, severity or pure premium, or ask what a gap between mean and median says about skewness.

MAS-II: Questions compare fitted mixed models by log-likelihood, AIC and BIC, as in the second example.

Exam traps

Average of ratios: An overall frequency is total claims over total exposure, not the average of the levels' frequencies.

Mean of skewed losses: For right-skewed severity the mean sits well above the median; reporting only the mean overstates the typical claim.

One-way tables confound: A level's relativity in a one-way table includes the effect of everything correlated with it.