5.2 Binomial & Logistic GLMs for Binary Outcomes

When the Target Is an Event, Not an Amount

Most of the GLM work in a rating context targets a number: a claim count, a severity, a pure premium. But a large share of P&C analytics targets an event that either happened or did not. The PCPA Project statement lists binomial alongside Gamma, Poisson, log-normal and Tweedie as families a candidate may need, so a binary target is squarely in scope.

Typical P&C binary targets:

• Will the policyholder renew?
• Will a newly opened claim exceed a threshold such as $100,000?
• Will a subrogation opportunity be realised?
• Should this claim be referred for special investigation?
• Will a quote convert to a bound policy?

The target $y_i$ takes the value 1 if the event occurred and 0 if it did not. The model's prediction, $\mu_i$, is the probability that the event occurs.

Why the Log Link Fails Here

In a GLM, the linear predictor $\eta_i = \mathbf{x}_i^T\boldsymbol{\beta}$ is unbounded — as coefficients and covariates move, it can take any value in $(-\infty, +\infty)$.

A probability cannot. It must satisfy $0 \le \mu_i \le 1$.

A log link gives $\mu_i = \exp(\eta_i)$, which is bounded below by 0 but unbounded above. A large enough linear predictor produces a fitted "probability" above 1, which is meaningless. What is needed is a link that stretches the interval $[0,1]$ across the whole real line.

The Logit Link

The standard choice, and the canonical link for the binomial family, is the logit:

$g(\mu) = \ln\left(\frac{\mu}{1-\mu}\right)$

As $\mu \to 0$ the logit goes to $-\infty$; as $\mu \to 1$ it goes to $+\infty$. The full model specification is:

$y_i \sim \text{Binomial}(\mu_i)$

$\ln\left(\frac{\mu_i}{1-\mu_i}\right) = \beta_0 + \beta_1 x_{i1} + \dots + \beta_p x_{ip}$

Inverting the link gives the logistic function, which converts the linear predictor back to a probability:

$\mu_i = \frac{1}{1 + e^{-\eta_i}}$

Three anchor points are worth committing to memory:

The probit and complementary log-log links are alternatives, but the logit is the default in actuarial practice and the one PCPA items are built around.

Reading the Coefficients: Odds Ratios

The quantity $\mu/(1-\mu)$ is the odds — the probability of occurrence divided by the probability of non-occurrence. Unlike a probability, odds are unbounded above, which is what makes the log of the odds a sensible linear scale.

Because the link is a logarithm, exponentiating the model equation turns it into a multiplicative model on the odds scale:

$\frac{\mu_i}{1-\mu_i} = e^{\beta_0} \cdot e^{\beta_1 x_{i1}} \cdots e^{\beta_p x_{ip}}$

So $e^{\beta_j}$ is an odds ratio:

• For a continuous predictor, a one-unit increase multiplies the odds by $e^{\beta_j}$. A coefficient of $0.24$ means the odds rise by $e^{0.24} - 1 \approx 27%$.
• For a categorical level, $e^{\beta_j}$ is the odds for that level relative to the base level. A coefficient of $0.24$ means that level's odds are about $27%$ higher than the base level's.

[!WARNING] An odds ratio is not a probability ratio, and it is not a relative risk. If the base probability is 0.02, the odds are 0.0204; multiplying by 1.271 gives 0.0259, a probability of 0.0253 — about a 26% rise, close to the odds ratio because the event is rare. But if the base probability is 0.60, the odds of 1.50 become 1.907, and the probability moves only from 0.600 to 0.656 — a rise of about 9%. Stating "27% more likely" for a common event is a real interpretation error.

A Worked Retention Example

A renewal model with a logit link produces:

For a five-year-tenure policyholder with no prior claim receiving a rate increase above 10%:

$\eta = 1.20 - 0.55 + 0.48 = 1.13$

$\mu = \frac{1}{1 + e^{-1.13}} = \frac{1}{1 + 0.3230} = 0.756$

Roughly a 76% chance of renewing. Note how the model arithmetic is additive on the logit scale and multiplicative on the odds scale — the same structural property that makes log-link severity and frequency models produce multiplicative rating factors.

Fitting and Response Formats

Binomial GLMs accept the target in more than one form, and getting this wrong silently changes the weights:

In Python, sm.GLM(y, X, family=sm.families.Binomial()) with freq_weights plays the same role; in SAS, PROC GENMOD or PROC LOGISTIC with DIST=BINOMIAL LINK=LOGIT and an EVENTS/TRIALS response.

[!WARNING] Check which outcome the software treats as the "event." PROC LOGISTIC by default models the probability of the lower ordered value, which frequently means it models the probability of non-renewal unless you specify DESCENDING or an explicit EVENT= option. Every sign in the coefficient table flips, and the model looks nonsensical until you notice.

Validating a Logistic Model

The usual GLM diagnostics apply — deviance, AIC and BIC, residual review — but two validation tools are specific to binary targets and appear directly in the assigned Monograph chapter:

• A quantile plot built by bucketing holdout records into quantiles of predicted probability and plotting the actual proportion of events against the average predicted probability. The same three criteria apply: accuracy, monotonicity, and vertical distance between the first and last quantiles.
• The confusion matrix and ROC curve, which convert probabilities into binary decisions at a chosen discrimination threshold. These are covered in detail in the validation chapter.