Interval Estimates

(1-α)*100% Confidence Interval

\(\hat{y}\pm t_{(1-\frac{\alpha}{2}, n-p)}\cdot SE\)

where the standard error of the design at \(x_0\) is,

\(SE = s \cdot \sqrt{x_0 (X^T X)^{-1}x_0^T}\)

(1-α)*100% Prediction Interval

\(\hat{y}\pm t_{(1-\frac{\alpha}{2},n-p)}\cdot SE_{pred}\)

where the prediction error for one future response measurement at point \(x_0\) is,

\(SE_{pred} = s \cdot \sqrt{1 + x_0(X^T X)^{-1}x_0^T}\)

These are the prediction intervals available on the graphs. See the Confirmation node entry for an example of how this formula changes for multiple confirmation observations at \(x_0\).

(1-α)*100% Tolerance Interval for P * 100% of the population

\(\hat{y}\pm s \cdot (\mathrm{TI~Multiplier})\)

where the multiplier is,

\(\mathrm{TI~Multiplier} = t_{(1-\alpha, n-p)} \cdot \sqrt{x_0 (X^TX)^{-1}x_0^T} + \Phi^{-1}\left(0.5 + P/2\right) \cdot\sqrt{\frac{n-p}{\chi^2_{(\alpha, n-p)}}}\)

The TI uses only \(\scriptstyle \alpha\) rather than \(\scriptstyle \alpha/2\) to compute the two-tailed interval.

Symbols:

*

The expanded model matrix (X) has one row for each run in the design and one column for each term in the model. The values in the X matrix are assumed to be coded values. The first column is typically all 1’s to represent the intercept term of the model. For mixture designs there is no intercept term in the Scheffé polynomial thus this column is not present.

†

The expanded point vector is a way to represent the settings of the factors for a particular location for purposes of prediction. Think of it as a matrix with one row. It has a similar structure to a row of the expanded model matrix; one element for each term in the model. The order of the terms represented by the model matrix’s columns and the point vector’s elements must match.

‡

The residual (error) degrees of freedom for a split-plot design are estimated using the procedure outlined by Kenward and Roger. This value is shown on the point prediction output.

References