Home / Shared Math Details / DOE / Prediction Equation
Prediction Equation¶
Every predicted value in the DOE tools comes from one equation. The regression sheet uses it to answer the set points you type, and it is what a chart evaluates wherever the chart's source is the regression sheet: Surface, Contour and Trellis always, and Cube Plot, Interaction Plots, Main Effects Plots and Thumbnail Plot when they are drawn from the model.
Drawn from the design sheet instead, those four charts do not predict anything. They report the run data: the response value at a plotted setting is the average of the responses there, and the variation value is the average of the within-group standard deviations. A cube corner with no runs, or a setting with no replicate group of two or more, is left blank rather than filled in by the model. See Chart Source.
The form of the answer depends on the output type: a value for a quantitative output, a probability for a binary or nominal one.
Notation¶
| Term | Description |
|---|---|
| \(c_j\) | the coded value of term \(j\) at the settings being predicted |
| \(b_j\) | the coefficient of term \(j\), read from the regression table |
| \(\hat{y}\) | the predicted response |
| \(\hat{s}\) | the predicted standard deviation, from the standard deviation model |
| \(\eta_k\) | the linear sum for level \(k\) of a nominal output |
| \(p\), \(p_k\) | a predicted probability |
| \(\Phi\) | standard normal cumulative distribution function |
| \(L\) | the number of non-reference levels of a nominal output |
A quantitative output¶
A prediction is the sum, over every term in the model, of that term's coded value times its coefficient:
where \(c_j\) is the coded value of term \(j\) at the settings you asked about and \(b_j\) is its coefficient from the regression table. The constant term has \(c_j = 1\), so its coefficient enters as itself.
A term that is absent from the regression table, or switched off for the level being predicted, contributes \(b_j = 0\).
The coded values are what the coding setting produced, so a prediction is only meaningful together with the coding of the sheet it came from.
A binary output¶
For a binary output the same sum is formed first, then passed through the logistic function to give a probability:
That is the probability of the first level. The probability of the second is \(1 - p\).
A nominal output¶
For a nominal output with \(L\) non-reference levels there is one sum per level, \(\eta_k = \sum_j c_j b_{jk}\), and the probability of level \(k\) is
The \(1\) in the denominator is the reference level, whose \(e^{\eta}\) is \(1\) by construction. So the reference level's own probability is
and the probabilities across all levels sum to \(1\).
Interactions between categorical factors¶
An interaction between categorical factors has one coefficient per combination of their levels, and the combinations are numbered in mixed radix. Reading the factors in the interaction from left to right, each categorical factor's weight is the product of the level counts of the categorical factors to its right, and a quantitative factor has weight \(0\). The index is then
where \(\ell_i\) is the level factor \(i\) sits at and \(w_i\) is its weight.
For an interaction \(ADEG\) where \(A\) has 4 levels, \(D\) is quantitative, \(E\) has 3 and \(G\) has 2, the weights are \(6, 0, 2, 1\). With \(A\) at level 2, \(E\) at level 0 and \(G\) at level 1 the index is \(2(6) + 0(2) + 1(1) = 13\).
The prediction interval¶
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.
Capability from a prediction¶
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.
These figures inherit every assumption of the two models behind them: that the response is normal about its predicted mean, and that the predicted standard deviation is the right spread at those settings.
See Also¶
References¶
- Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.
- Hosmer, D. W., Lemeshow, S., and Sturdivant, R. X. (2013). Applied Logistic Regression, 3rd edition. Wiley.
- Agresti, A. (2013). Categorical Data Analysis, 3rd edition. Wiley.
- NIST/SEMATECH e-Handbook of Statistical Methods, sections on process modeling and process capability.