Adequate Precision
Adequate or sufficient precision is a signal-to-noise ratio comparing the range of predicted values to their average standard error at the design points (Box, Hunter, and Hunter, pg. 524). Assuming no other problems with the model fit, ratios greater than 4 indicate that the model adequately estimates the response function (Draper & Smith, 1998).
Where,
\(\hat{Y}\) are the predictions at the run settings.
\(\hat{\sigma}^2\) is the residual mean square from the ANOVA table.
\(p\) is the number of terms in the model.
\(n\) is the number of runs in the design.
\(\bar{\mathrm{V}}\) is the average variance of the predictions at the run settings.
Adequate precision is a simpler but equivalent statistic to the Box-Wetz criterion which, instead, uses an estimate of the non-centrality parameter related to the regression F-statistic (Box & Wetz, 1973). Both adequate precision and the Box-Wetz criterion are attempts to address the problem that model significance alone is not sufficient to guarantee that it is useful for prediction. For a thorough and accessible discussion, including the justification of the ratio of 4, see the textbook by Draper and Smith, Chapter 11.
References
George E.P. Box, William G. Hunter, and Stuart J. Hunter. Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. John Wiley & Sons, Inc., 1978.
George E.P. Box and John Wetz. Criteria for judging adequacy of estimation by an approximating response function. Technical Report 9, University of Wisconsin Statistics Department, 1973.
Norman R. Draper and Harry Smith. Applied Regression Analysis. Wiley, third edition, 1998.
See Also
George E.P. Box and Norman R. Draper. Empirical Model-Building and Response Surfaces. Wiley, 1987.
Norman R. Draper. The box-wetz criterion versus r2. Journal of the Royal Statistical Society. Series A (General), 147(1):100–103, 1984.