Home / Shared Math Details / DOE / Power and Sample Size
Power and Sample Size¶
Power is the chance a design detects an effect of the size you name. Quantum XL computes it from the noncentral \(F\) distribution, with two forms: one for a two level design, one for a general factorial.
This page also gives the axial value of a central composite design, since it is set by the same kind of design arithmetic and appears in the same dialogs.
Notation¶
| Term | Description |
|---|---|
| \(\alpha\) | the type I error rate, the risk of calling an effect real when it is not |
| \(\delta\) | the effect to be detected |
| \(\sigma\) | the assumed standard deviation |
| \(\lambda\) | the noncentrality parameter |
| \(df_1\), \(df_2\) | numerator and denominator degrees of freedom |
| \(r\) | number of replicates |
| \(k\) | number of runs at the corners, or the number of factors where noted |
| \(N\), \(p\) | total observations, and parameters including the constant |
| \(L_{\max}\), \(n_L\) | the largest level count on any factor, and observations at each of its levels |
| \(F\) | number of factorial runs, in the axial value formulas |
| \(c\) | number of centre points |
How power is computed¶
Power is computed from the noncentral \(F\) distribution. The critical value is the ordinary \(F\) value that cuts off \(\alpha\) in the upper tail, and the power is the probability that a noncentral \(F\) with the same degrees of freedom exceeds it:
Everything specific to the design is in the numerator degrees of freedom \(df_1\) and the noncentrality parameter \(\lambda\).
A two level design¶
For a two level design the test is on a single effect, so \(df_1 = 1\), and
where \(r\) is the number of replicates, \(k\) the number of runs at the corners, \(\delta\) the effect to be detected, \(\sigma\) the assumed standard deviation, \(N\) the total observations and \(p\) the number of parameters including the constant.
A general factorial¶
For a general factorial the test is on the factor with the most levels, so \(df_1\) is one less than that count, and the noncentrality uses the number of observations at each of its levels:
where \(L_{\max}\) is the largest number of levels on any factor and \(n_L\) is the number of observations at each level of that factor, replicates included. The error degrees of freedom is supplied rather than derived.
Note the two forms divide by \(4\) and by \(2\) respectively. They are not the same expression rearranged.
The limit on the noncentrality parameter¶
The noncentrality parameter must lie between \(0\) and \(1000\). A design with a large effect relative to \(\sigma\) and a large number of runs can produce a \(\lambda\) above that ceiling, and the calculation then returns no value rather than a power just short of \(1\).
When \(\lambda\) is smaller than \(10^{-10}\) the ordinary \(F\) distribution is used instead, since the noncentral form adds nothing at that scale.
The central composite axial value¶
The axial distance of a central composite design, in coded units, is written \(\alpha_{\text{axial}}\) here and grouped under Alpha value in the dialog; it is subscripted here because \(\alpha\) on its own is the type I error rate. It is set one of four ways. Writing \(F\) for the number of factorial runs, \(k\) for the number of factors and \(c\) for the number of centre points:
The fourth way is to type the value in.
A face centred design puts the axial points level with the factorial faces, so no factor is ever run outside its stated low and high. The rotatable value makes the prediction variance depend only on distance from the centre. The orthogonal value makes the quadratic terms orthogonal to the rest of the model.
For the recognised combinations of factors and factorial runs, the dialog opens with a standard axial value and a standard centre-point count already filled in, and the alpha type set to Manual. That is a starting value rather than an alternative to the formulas: as soon as Rotatable, Optimal orthogonality or Face CCD Alpha = 1 is selected, the value used is what that formula computes, rounded to four decimal places, for a recognised design as much as for any other. The two computed captions are recalculated as the centre-point count changes.
The prefilled values assume an unblocked design and are not revised when you choose a block count. They are filled in on the base design step, before the replicates and blocks step is reached, and are looked up as though the design had one block. The stored table does hold different standard values for the same factors and factorial runs at different block counts, so on a blocked CCD the prefilled axial value and centre-point count are the unblocked entry rather than the one for the design you end up with. Type the values you want, or pick one of the computed alpha types, which are unaffected.
The standard split of centre points between the factorial and axial portions is used only when the total you ask for equals the standard total. Otherwise the centre points are distributed across the blocks instead.
See Also¶
- Design Coding, for what the axial value means in coded units
- Sums of Squares
- Shared Math Details
References¶
- Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.
- Myers, R. H., Montgomery, D. C., and Anderson-Cook, C. M. (2016). Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 4th edition. Wiley.
- Box, G. E. P., and Wilson, K. B. (1951). On the Experimental Attainment of Optimum Conditions. Journal of the Royal Statistical Society, Series B, 13(1), 1 to 45.
- Patnaik, P. B. (1949). The Non-Central Chi-Square and F Distributions and their Applications. Biometrika, 36(1 and 2), 202 to 232.
- NIST/SEMATECH e-Handbook of Statistical Methods, sections on response surface designs and power.