In a generalized additive model, how does a shape function keep a non-linear effect readable?

## The form A linear model constrains every feature's contribution to a straight line: `score = b0 + b1*x1 + ... + bp*xp`. A generalized additive model (GAM) relaxes the *shape* while keeping the *additivity*: ``` g(E[y]) = b0 + f1(x1) + f2(x2) + ... + fp(xp) ``` Each `fj` is an arbitrary smooth one-dimensional function of a single feature, estimated from data. `g` is a link function chosen for the outcome type — the identity link for a continuous target, the log-odds link for a binary target, so that the additive score is on the log-odds scale and the predicted probability is its logistic transform. The `fj` are the **shape functions**. Fitting them is done with smoothers — splines with a roughness penalty, or many small one-feature learners summed — with a smoothing parameter controlling how wiggly each curve may get. That parameter is a bias-variance dial: too smooth and the curve degenerates towards a straight line and misses real structure; too wiggly and the curve chases noise, which shows up as implausible zig-zags a domain expert will reject on sight. ## Why additivity is what buys readability The curve is readable *because* the terms are summed. Under additivity, the effect of changing a feature from value `a` to value `b` is `fj(b) - fj(a)`, and that difference does not depend on any other feature's value. So a single two-dimensional plot — feature on the horizontal axis, contribution to the score on the vertical — is a complete and exact statement of what the model does with that feature. No caveats, no "holding everything else at its mean", no averaging over a distribution. Because the intercept and the curves are only identified up to a constant, shape functions are conventionally **centred** to average zero over the training data. Then `b0` is the baseline score for an average case and each curve reads as "how much this feature's value pushes the score above or below that baseline". ## What it buys over a single coefficient Consider a hospital model for 30-day readmission with age as an input. A linear term reports one number — say, risk rises steadily with age. The shape function can instead reveal risk climbing through middle age, a distinct bump in the early sixties, and a flattening or even a dip afterwards. That non-monotone structure is exactly what a single slope averages away. Two consequences follow: 1. **Better fit without losing the audit.** You get the non-linearity that a black box would have found, in a form a clinician can look at and either endorse or challenge. 2. **Detection of data problems.** Sharp steps or spikes in a shape function often mark an artefact rather than biology — an imputed placeholder value, a coding change, an age recorded as 0 for missing. A flexible curve makes those visible as an isolated spike; a linear coefficient absorbs them silently. ## The limits, and how to state them - **No interactions by default.** A pure GAM assumes the features act additively. If risk depends on age *only for patients with a particular comorbidity*, no set of independent curves can express that. The standard remedy is to add named pairwise terms, `f(xj, xk)`, which is honest but costs readability: a surface has to be read as a heat map rather than a curve, and adding many of them turns the model back into something nobody reads. - **Correlated features still split credit.** If two features carry the same information, the fit can distribute the effect between their two curves in ways that are jointly correct but individually misleading. Check pairwise correlation before reading any single curve as "the effect". - **Shape is not cause.** A curve tells you how the model's score moves with the feature in the data it was fitted on. It is not evidence that intervening on the feature moves the outcome. - **Uncertainty matters at the edges.** Regions with few training rows have poorly determined curves; a dramatic swing in the tail of an age curve at ages 95+ may rest on a handful of rows. Plot the curve with an uncertainty band and refuse to read the sparse ends. ## Interview framing Write the additive form, name the link, say the words "one function per feature, summed", and then make the readability argument explicitly: additivity is what lets a single curve be a complete statement of the feature's effect. Finish with the interaction limitation, unprompted — candidates who state the cost of their preferred model are the ones interviewers believe.