Response Variable

Study guide: Exam MAS-I

The response variable — also the dependent variable, target or outcome — is the quantity $Y$ a model explains or predicts from its predictors. Its data type and the way its variance behaves determine the model: in a Generalized Linear Model, the response distribution and the link function.

$g\big(E[Y_i]\big) = \mathbf{x}_i^\top\boldsymbol\beta$

$\mathrm{Var}(Y_i) = \phi\,V(\mu_i)$

$\mu_i = E[Y_i]$ ; $V$ is the variance function fixed by the chosen distribution and $\phi$ is the dispersion parameter. Choosing a distribution is choosing how the variance grows with the mean
Continuous, roughly symmetric, constant variance — Normal with the identity link: ordinary Linear Regression
Claim counts — Poisson ( $V(\mu) = \mu$ ), or negative binomial if overdispersed, with a log link and log exposure as an offset: Poisson Regression
Claim severity — positive and right-skewed, with standard deviation roughly proportional to the mean: Gamma ( $V(\mu) = \mu^2$ ) with a log link, weighted by claim count. The inverse Gaussian ( $V(\mu) = \mu^3$ ) suits a heavier tail
Pure premium or loss cost — mostly zeros plus a skewed positive amount: the Tweedie with power $1 < p < 2$ and a log link
Binary response — claim or no claim, renew or lapse, fraud referral: binomial with the logit link, which is Logistic Regression. The fitted value is a probability; a linear model on a $0/1$ response can predict values outside $[0, 1]$
A GLM links the mean and leaves $Y$ itself untransformed. Least squares on $\ln Y$ is a different model: $E[\ln Y] \neq \ln E[Y]$ , so exponentiating its fitted values understates the mean

Matching Each Response to a Model Choose a distribution and link for each response: (a) number of claims per policy-year; (b) average severity per claim; (c) whether a policy renews; (d) loss cost per car-year, $92\%$ of which are zero.

Answer

(a) Poisson with a log link and $\ln(\text{exposure})$ as an offset; switch to negative binomial if the residual deviance is far above its degrees of freedom

(b) Gamma with a log link, weighted by the number of claims behind each average — an average of many claims is less variable than one claim

(c) Binomial with a logit link: logistic regression, whose fitted values are renewal probabilities

(d) Tweedie with $1 < p < 2$ and a log link. The point mass at zero rules out the Gamma, and the continuous positive part rules out the Poisson; the Tweedie is a compound Poisson–Gamma that has both

In every case the log (or logit) link keeps the fitted mean in the range the response can take, and makes the coefficients multiplicative.

A Binary Response: Probabilities and Odds A logistic model for whether a claim is referred to the special investigations unit has $\hat\eta = -1.5 + 0.8\,I_{\text{attorney}}$ . Find the referral probability with and without attorney involvement, the odds ratio, and the ratio of the probabilities.
Answer
 $\begin{align*} \hat\pi_0 &= \frac{1}{1 + e^{1.5}} \\ &= 0.182 \\ \hat\pi_1 &= \frac{1}{1 + e^{0.7}} \\ &= 0.332 \\ \text{Odds ratio} &= e^{0.8} \\ &= 2.23 \end{align*}$ 
The probability ratio is $0.332/0.182 = 1.82$ . Attorney involvement multiplies the odds of referral by $2.23$ but the probability by only $1.82$ . The two agree only when the probability is small, so an odds ratio read as a relative risk overstates the effect whenever the outcome is common.