Home / DOE / Additional Information / The S-Hat Model
The S-Hat Model¶
A quantitative output can get two regressions rather than one. The Y-Hat model predicts the mean of the response. The S-Hat model predicts its standard deviation.
What it needs¶
Replicated runs. A standard deviation cannot be measured from a single observation, so the S-Hat model is fitted only where the design repeats its design points.
The mechanism is the grouping described on Covariate Pattern: runs whose factor settings are identical form a group, a standard deviation is computed within each group, and those standard deviations become the response of a second regression against the same factors.
The standard deviation is the ordinary sample standard deviation, with an n - 1 divisor. A group with fewer than two responses produces no value and drops out.
The two models share one coding¶
The standard deviation model reuses the coding of the response model. It does not recompute a centre and scale from its own subset of rows, so the two models are on the same coded scale and their coefficients are comparable.
Full details: Design Coding.
Weighted S-Hat¶
Use weighted S-Hat regression on the Regression settings page of the Options dialog looks as though it decides whether the second regression is weighted. It does not: the second regression is always fitted unweighted. With the setting on, and only where the groups hold different numbers of runs, Quantum XL prepares a weight per group and the sheet's heading block says S-hat regression weighted by df, but the fit itself ignores the weights, so the coefficients are the same either way. See DOE Options, which records it as a product question.
Where S-Hat appears¶
| Where | What it is |
|---|---|
| The regression sheet's prediction block | the S-Hat row, beside Y-Hat |
| PI Lower and PI Upper | computed from it |
| Cp, Cpk and the DPM estimate | computed from Y-Hat and S-Hat together |
| Interaction, main effects and thumbnail charts | selectable as the value to plot, as S-Hat or S-bar |
| Optimize | usable as the sigma behind a Cpk or DPM goal |
The prediction interval on the regression sheet is three standard deviations either side of the predicted mean, using the predicted standard deviation at those same settings:
The sheet labels this a 99 percent interval. It is a three sigma interval on the fitted standard deviation, not an interval derived from the \(t\) distribution and the residual degrees of freedom.
When the response has no standard deviation model, or every term has been removed from it, that cell holds the standard error of the response regression instead of a prediction. It is then a single constant, so the interval stops varying with the set points. The software says so in the S-Hat cell's own comment.
The capability figures beside the prediction treat the predicted mean and predicted standard deviation as the mean and standard deviation of a normal distribution:
A third figure, Cp written with a subscript p, sits above Cpk on the sheet and needs both limits:
With only one specification limit set, Cp is not reported, and Cpk and DPM use that limit's term alone. The tail areas are multiplied by one million with no bound applied to the result.
The same S-Hat substitution applies here. With no standard deviation model, or none left after terms have been removed, all three figures are computed from the response regression's standard error, so they stop varying with the set points.
Full details: Prediction Equation.
S-Hat and S-bar are not the same number¶
Both describe the spread, and which one a chart shows depends on where the chart got its data.
| Source | Value | What it is |
|---|---|---|
| A regression sheet | S-Hat | the fitted standard deviation from the S-Hat model, available at any factor setting |
| A design sheet | S-bar | the average of the within-group standard deviations at each factor setting, available only where runs exist |
See Chart Source.
When there is no S-Hat model¶
With no replication there is nothing to fit, and the consequences show up in three places at once: no S-Hat model, a lack-of-fit block whose cells read NA on the regression sheet, and no S-bar available on the design-source charts. All three have the same cause.
Wherever an S-Hat prediction is needed and there is none, the response model's standard error is used instead. That covers the regression sheet as much as Optimize: the S-Hat row of the prediction block, the prediction interval, Cp, Cpk and DPM are all built from it, so they still report numbers. Because it is a single constant rather than a prediction, those figures stop varying with the set points. The same substitution happens when an S-Hat model exists but every term has been removed from it. The software says so in the S-Hat cell's own comment.