Replicates at identical predictor settings provide a pure-error reference for lack-of-fit analysis. Unique settings alone cannot supply that same comparison.
The statistical question
A regression lack-of-fit test compares model-related discrepancy with variability measured among replicates at the same predictor settings. Such replication permits residual variation to be partitioned into pure error and lack of fit. The test assesses aspects of the functional model, not every aspect of model quality. Its degrees of freedom depend on total observations, unique setting combinations and fitted parameters.
An illustrative review example
Suppose a fictional study measures three distinct setting combinations four times each, giving 12 observations. With three unique combinations and a two-parameter model, pure-error degrees of freedom are 12 minus three, or nine; lack-of-fit degrees of freedom are three minus two, or one. Replacing the 12 observations with 12 unique settings eliminates those within-setting replicates. The arithmetic does not establish a significant lack of fit or a preferred model.
Prepare the evidence for discussion
Record exact predictor combinations and replicate identities before fitting the model. Ask whether nominally identical runs actually changed product, lot or measurement conditions enough to affect their interpretation. Retain residual plots as well as the test output and its assumptions. If replication is absent, describe that limitation instead of manufacturing a pure-error estimate from arbitrary neighboring settings. Plan any additional confirmation with the trial owner.
Replicates provide a model-independent reference
This blank worksheet is for your own project. It contains no H M machine trial result.
| Question to resolve | Evidence to retain |
|---|---|
| Setting identity | Exact predictor combinations |
| Replicates | Repeated observations at each setting |
| Model size | Number of fitted parameters |
| Partition | Pure-error and lack-of-fit degrees of freedom |
Customer Questions
What supplies the pure-error reference?
Variation among replicated observations under the same predictor-setting combination supplies that reference.
Do unique settings provide identical replication evidence?
No. Different predictor settings do not create within-setting replicate variation.
Does a lack-of-fit test assess every model issue?
It addresses functional adequacy under its assumptions; other diagnostic questions remain.
Primary References
These references support the technical principles discussed in this guide. The worked examples and review questions are educational.
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