Hierarchical Regression — Sequential blocks with ΔR² + F-change

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Enters predictors in researcher-specified ORDERED BLOCKS and tests the incremental variance (ΔR²) added by each block BEYOND the preceding blocks.

Reports per-block R², adjusted R², ΔR², F-change with df (df₁ = block size, df₂ = residual df), p-change, plus the final-block coefficient table with full diagnostics. Block order is THEORETICAL, not algorithmic — this is NOT stepwise regression. The standard tool in psychology / education research for testing whether a focal predictor adds explanatory power beyond established controls.

Worked example

Does personality add predictive power beyond study time?

Block 1 enters study time; Block 2 adds conscientiousness. Hierarchical regression tests whether the second block improves prediction of exam score.

Result

Study time alone gave R² = .18. Adding conscientiousness raised it to R² = .37 — a significant gain, ΔR² = .20, F-change(1, 87) = 27.1, p < .001.

How you'd report it (APA)

Adding conscientiousness significantly improved prediction of exam score beyond study time, ΔR² = .20, F(1, 87) = 27.1, p < .001.

Try it yourself: Load this ready-made sample and follow the run above.

When to use it

  • Demographic / clinical controls → focal predictor → moderator
    Exam score ~ Block 1: age + prior GPA (R² = .42); Block 2: hours studied (R² = .47, ΔR² = .05, F-change = 14.3, p < .001) — study time matters even after ability proxies.
  • Incremental validity of new psychometric measure
    Job performance ~ Block 1: cognitive ability + conscientiousness (R² = .31); Block 2: new emotional intelligence scale (R² = .35, ΔR² = .04, p = .002) — EI adds incremental validit

When NOT to — use instead

Hypotheses

H₀: ΔR²_block_b = 0 for each block b ≥ 2 (the focal hypothesis), AND β_j = 0 for every coefficient (per-coefficient tests).
Hₐ: at least one block contributes ΔR² > 0 beyond preceding blocks.

Parameter tested: ΔR² (incremental) per block; vector of population coefficients β

Assumptions (and what to do if they fail)

Residuals of the FINAL block are approximately normally distributed. Under CLT this matters less as n grows (n ≥ 100 usually robust).medium

Check: Shapiro-Wilk p-value on final-block residuals.

If violated: Non-normal residuals affect small-n CIs / p-values; CLT protects for n ≥ 100.

Constant residual variance in the final block. Heteroscedasticity inflates SEs.high

Check: Breusch-Pagan p-value.

If violated: Heteroscedasticity inflates SEs — F-change tests and per-coefficient inference become unreliable.

Residuals are independent (no autocorrelation).high

Check: Durbin-Watson statistic in [1.5, 2.5] is OK.

If violated: Autocorrelated residuals bias SEs and the F-change test.

No single case dominates the final-model fit (Cook's D > 4/n threshold).high

Check: Max Cook's D reported.

If violated: Influential cases can flip the sign of ΔR² conclusions block-by-block.

VIF for each predictor in the final model is moderate (< 5 preferred; < 10 tolerable).medium

Check: Max VIF across predictors.

If violated: High VIF inflates SEs — significant ΔR² with no significant β shows shared variance among predictors in the same block.

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