§10.1

Logistic Regression for Churn Scoring

Logistic regression appeared in Part III as a tool for isolating effects: a coefficient on a treatment, holding controls constant, that we asked whether to interpret causally. In Part IV the same machinery does a different job. We no longer ask what the coefficient on a feature means in a causal sense. We ask how well the model, taken as a whole, ranks future customers by their likelihood of doing something we care about.

That shift in stance is the entire content of this article. The model is identical. The questions it has to answer are different. Once a manager internalizes the move from "what is the effect of X on Y?" to "given X, what does this unit look like?", logistic regression becomes a remarkably useful first model — interpretable enough to inspect, calibrated enough to threshold, fast enough to retrain weekly.

The Executive Question

Of all our active customers, which ones are most likely to churn in the next 60 days — and how confident is the model in that order?

The deliverable is not "which customers will churn." Nobody knows that. The deliverable is a sortable score — a probability between 0 and 1 attached to every customer — that a retention team can use to prioritize action.

From Linear Combination to Probability

Logistic regression starts with the linear combination Part III used:

Linear predictor

\eta_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \cdots + \beta_p x_{ip}

For a customer with features $x_i$ , the predictor $\eta_i$ can be any real number. To turn it into a probability, logistic regression passes it through the logistic function:

Logistic transform

\Pr(Y_i = 1 \mid x_i) = \frac{1}{1 + e^{-\eta_i}}

The shape is an S-curve flattened to the interval (0, 1). Two consequences matter for the manager:

The output is a probability, with all the usual interpretive meaning. A score of 0.82 means the model believes 82 of every 100 customers with these features churn in the relevant window.
The score is monotone in $\eta$ . Whatever changes raise $\eta$ raise the probability, in the same order. That is what makes the score a useful ranking even when its calibration is imperfect.

Reading Coefficients in Log-Odds

Logistic regression's coefficients have a fixed interpretation in log-odds. A coefficient of $\beta_1 = 0.4$ on tickets_last_30d means a one-ticket increase multiplies the odds of churn by $e^{0.4} \approx 1.49$ , holding the other features constant.

A few rules that hold whenever coefficients are interpreted this way:

The transformation is multiplicative on odds, not on probabilities. A 50% boost in odds is roughly a 50% boost in probability when probabilities are small, but tapers off as they approach 1.
"Holding other features constant" still has to be true. If a coefficient is reported on a model with collinear features (e.g., trailing-30-day spend and trailing-90-day spend), the coefficient's marginal interpretation breaks down.
The coefficients are predictive, not causal. A high coefficient on support_tickets_last_30d does not imply that reducing ticket counts will reduce churn. It implies the model uses tickets to rank customers, and that ranking is useful for targeting offers.

This last point is the bridge back to Part III. Causal effects and predictive associations can have the same sign and entirely different magnitudes. The model is a useful sorter, not a verdict on causal mechanism.

The Score Distribution and the Threshold

The most useful single chart for a logistic model is the distribution of predicted probabilities, split by the realized outcome on the held-out set.

From probability score to action — pick a threshold, sort, intervene

stayed (label = 0) churned (label = 1) action zone

Figure 1. Predicted-probability distributions on the held-out set for customers who churned (red) and stayed (blue). The shaded zone marks customers above a 0.5 decision threshold — the population a default retention offer would target.

Three things to notice in this kind of plot:

Separation. The two distributions should be visibly offset. If they fully overlap, the model has not learned to distinguish the groups; threshold choice will not save it.
Overlap. A well-fit model still has overlap in the middle. Every customer in the overlap zone is genuinely ambiguous; the score is doing its job by reflecting that ambiguity.
Skew. Many real churn models have skewed score distributions — a small high-risk tail and a long low-risk body. This is fine, but it means the choice of threshold is not symmetric around 0.5. The mass of customers, and therefore the cost of a one-size offer, lives in the body.

The 0.5 threshold has no special status. It is a default the textbook used to avoid choosing. The choice we do care about — the threshold that maximizes expected profit — is the subject of §10.2.

When Is Logistic Regression the Right First Model?

The honest answer is: almost always, when the model has to be defended to a non-technical audience. Three features make it durable:

Calibration is straightforward. Out of the box the model produces probabilities, not arbitrary scores. They may need recalibration, but the units are interpretable.
Coefficients are inspectable. A manager can read the sign and rough size of each effect. Engineering errors and leakage often show up here first.
Speed. A logistic model with a thoughtful feature set retrains in seconds. The cost of trying ten variants is low.

Where it falls short: when interactions and non-linearities dominate, a linear-on-log-odds model leaves predictive power on the table. That is where trees and ensembles come in, in §10.4. The right move is usually to start with logistic and switch only when the marginal accuracy gain pays for the loss of interpretability.