Forward Selection and Model Interpretation in Linear Regression#

Stage 5 · 📈 Regression · Lesson 35 of 56 · intermediate

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Important

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Building the model#

Put forward selection to work on a real regression and it produces a compact, fitted model — a handful of predictors, each earning its place. Starting from the intercept alone, the procedure admits features one at a time until none of the leftovers improve the fit, leaving a parsimonious equation that is far easier to reason about than one stuffed with every available column.

Reading the order of entry#

The order in which features enter is itself informative. The first variable admitted is the single strongest predictor of the outcome; the second adds the most on top of the first, and so on. This sequence gives a rough importance ranking — though a subtle one, because each entry is judged given those already in, so a feature’s rank reflects its added value, not its value in isolation. On the taxi data, distance might enter first, with duration adding power beyond it.

Comparing nested models#

Each step yields a slightly larger model nested inside the next, which invites comparison. Because adding any feature can only increase ordinary \(R^2\), that raw measure always favours the bigger model and cannot judge whether an addition is worthwhile. Penalised criteria fix this: adjusted \(R^2\), AIC and BIC all reward fit but charge for each extra parameter, so they rise only when a feature earns its complexity. These are the yardsticks forward selection actually optimises.

Interpret with care#

Interpret the result cautiously. Because selection is greedy and driven by the data, the chosen model can be unstable — a slightly different sample might select different features — and repeatedly testing many features inflates apparent significance, so p-values from the final model read optimistically. The coefficients are still interpreted the usual way (effect per unit, holding others fixed), but the honest test of the model is its performance on the held-out data, not the selection statistics.

Hint