Regression & Correlation

Lack-of-Fit Test (F-Test from Replicate Observations)

Compares model mis-specification against pure noise from replicated design points.

Statusvalidated

Version1.0.0

Minitab equivalentStat > DOE > Factorial > Analyze Factorial Design (Pure Error + Lack of Fit row)

Description

When a design contains replicates (multiple observations at the same design point), the residual sum of squares SSE can be decomposed into pure-error (SS_PE — within-replicate-group scatter, model-independent) and lack-of-fit (SS_LoF — anything the model fails to explain beyond pure noise). The F-test compares the mean lack-of-fit against pure error. A significant result means the model is mis-specified; a term type is missing (often a quadratic or interaction term). Required diagnostic in Six-Sigma RSM practice: without replicates, LoF cannot be assessed.

Formulas

Decomposition

SSE = S S_{PE} + S S_{L o F}

Residual decomposition into pure-error and lack-of-fit components

Pure error

S S_{PE} = g \sum i \in g \sum (y_{i} - \overset{y}{ˉ}_{g})^{2}, d f_{PE} = g \sum (n_{g} - 1)

Sum of within-group deviations from the group mean

Lack of fit

S S_{L o F} = SSE - S S_{PE}, d f_{L o F} = d f_{E} - d f_{PE}

Whatever pure error does not explain is model mis-specification

F-Test

F = \frac{M S _{L o F}}{M S _{PE}} = \frac{S S _{L o F} / d f _{L o F}}{S S _{PE} / d f _{PE}} \sim F_{d f_{L o F}, d f_{PE}}

Significant F → the model is mis-specified

Assumptions

At least one replicate group of size ≥ 2 (else df_PE = 0)
Number of unique design points exceeds the number of model parameters (else df_LoF ≤ 0)
Residuals are independent and approximately normal (standard ANOVA assumptions)

Limitations

Cannot run without replicates — the implementation returns null
When replicates agree exactly (MS_PE = 0): convention F = ∞, p = 0 if MS_LoF > 0; F = 0, p = 1 otherwise

References

Montgomery, D.C. (2017). Design and Analysis of Experiments, 9th Ed., Wiley — Chapter 11 (Response Surface Methods)
Myers, R.H., Montgomery, D.C., Anderson-Cook, C.M. (2016). Response Surface Methodology, 4th Ed., Wiley — Chapter 2.4 (Lack of Fit Testing)