Regression & Correlation

Block-Significance F-Test (Type-III SS for a Factor Block)

Wald-form F-test for a group of related model-matrix columns — typically the n − 1 indicators that encode one categorical factor.

Statusvalidated

Version1.0.0

Minitab equivalentStat > DOE > Analyze (Adj SS column for the Categorical factor block)

Description

When a categorical factor with k levels expands via effect coding to k − 1 indicator columns, you want one ANOVA row per factor — not n − 1 separate rows per indicator. The block test bundles those columns into a single test with df = m (block size). Form: SS_block = β_block' [V_block]^(−1) β_block, with V_block = the corresponding sub-matrix of (X'X)^(−1). Reduces to the classical t² · MSE form for m = 1. Produces the same F as R's drop1(model, test='F') row for a multi-level factor.

Formulas

Wald quadratic

Q = β_{block}^{⊤} V_{block}^{- 1} β_{block}, V_{block} = (X^{'} X)_{block, block}^{- 1}

Quadratic form over β of the block, weighted by the (X'X)⁻¹ sub-matrix

Sum of squares

S S_{block} = Q, M S_{block} = S S_{block} / m, F = M S_{block} / MSE

Block SS, mean square over block df m, F against model MSE

Single-column collapse

m = 1 ⟹ S S_{block} = t_{j}^{2} \cdot MSE

For m = 1 the formula collapses to the usual t²·MSE

Assumptions

Block sub-matrix is non-singular (else block carries no information)
Standard OLS assumptions (homoscedasticity, independence)

Limitations

Not directly applicable to mixed-effects models (random effects)

References

Searle, S.R. (1987). Linear Models for Unbalanced Data, Wiley
Montgomery, D.C. (2017). Design and Analysis of Experiments, 9th Ed., Wiley — Chapter 3 (ANOVA — Type III SS)