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Probability, Statistics and Financial Data / Linear Regression and Analysis of Variance

One-Way Analysis of Variance

A regression predictor is not always a number. "Which sector does this stock belong to?", "which execution venue filled this order?", "which trading desk booked this P&L?" are all categorical predictors, and the question a desk asks of them is structurally the same question The ANOVA F-Test asked of a continuous one: does group membership explain a meaningful share of the variation in the outcome? This note builds the one-way analysis of variance, the classical tool for comparing I≥2I\ge2 group means at once, and shows it is not really a new technique — it is multiple regression with a design matrix built from group-indicator columns rather than continuous measurements, so every result already proved (the sum-of-squares decomposition, the FF-test, the exact null distribution) transfers with only the interpretation changing. The note closes with the practical next question a significant ANOVA always raises: which groups differ, answered by the Tukey and Bonferroni post-hoc procedures.
Learning objectives. By the end of this note you should be able to:
  • state the balanced one-way ANOVA model and its parameter constraints;
  • derive the between/within sum-of-squares decomposition and the FF-test for equality of means, and state their exact null distribution;
  • recognise one-way ANOVA as a special case of multiple regression with dummy-variable predictors;
  • apply Tukey's method and the Bonferroni correction to identify which specific pairs of means differ, and explain why a battery of unadjusted pairwise tt-tests inflates the overall Type I error rate.

In this lesson

  1. The model
  2. The sum-of-squares decomposition and the FF-test
  3. One-way ANOVA is multiple regression in disguise
  4. Post-hoc comparisons: which groups differ?

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

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