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Math Details¶

This page gives how a three level full factorial is enumerated and coded, and the run count that follows.

The same engine builds this design and the general factorial. A three level full factorial is the case where every factor is given exactly three levels, so the general page's enumeration and coding apply here without change, and this page states what they come to when every \(L_j = 3\).

Notation¶

Term	Description
\(k\)	the number of factors
\(d_j\)	the level index of factor \(j\) in a given run, \(0\), \(1\) or \(2\)
\(R\)	the number of replicates
\(N\)	the run count of one replicate

The run count¶

Every factor has three levels and the design visits every combination once, so

\[ N = 3^{k} \]

and the finished sheet holds \(3^{k} \cdot R\) rows. The design is always full. There is no fractional three level design in this engine, so the run count is fixed by the factor count and cannot be chosen. It grows quickly: 9 runs at 2 factors, 27 at 3, 81 at 4, 243 at 5 and 729 at 6.

The description block written at the top of the sheet names the design as a 3 level full factorial and reports the factor count and the run count. Unlike the general factorial it prints no Levels: line, because every factor has the same three levels and the design name already says so.

The order the runs are generated in¶

The runs are enumerated as a base three count, last factor varying fastest:

\[ d_j = \left\lfloor \frac{r}{3^{\,k-j}} \right\rfloor \bmod 3, \qquad r = 0, 1, \dots, 3^{k}-1 \]

so a three factor design opens with \((0,0,0)\), \((0,0,1)\), \((0,0,2)\), \((0,1,0)\) and continues to \((2,2,2)\).

The run order setting does not reach this design. The two level designs, the central composite design and the Box Behnken design read a per-user setting that can reverse the enumeration so the first factor varies fastest. This engine passes the unreversed order regardless of that setting, so it has no effect here.

Coding¶

With three levels the coded values are symmetric about zero and evenly spaced:

\[ d_j = 0 \mapsto -1, \qquad d_j = 1 \mapsto 0, \qquad d_j = 2 \mapsto +1 \]

Three is an odd level count, so zero is used and the spacing is uniform. That is not true of every general factorial: a factor with an even number of levels skips zero and its coded values are unevenly spaced. The general factorial math details gives the rule and the cases.

Every quantitative factor is put on a common scale before the regression runs. One expression does it, whichever coding is in force, using a centre \(m_c\) and a scale \(s_c\) that the coding chooses:

\[ x_{\text{coded}} = \frac{x - m_c}{s_c} \]

A categorical factor is never coded this way. Its coded value is \(1\), and the level it sits at selects which coefficient is used instead.

An interaction's coded value is the product of the coded values of the factors in it.

Replicates¶

Replicating joins whole copies of the design end to end, so replicate 2 repeats the same \(3^{k}\) rows in the same order. No center points are added: the middle level is already a level every run can take, and the center point machinery the two level designs use is not part of this engine.

Shared Math Details used here¶

This design uses shared formulas defined once in Shared Math Details. See those pages for the exact definitions.

Shared concept	Used here for	Reference
Design coding	turning \(-1\), \(0\) and \(+1\) into the values written on the design sheet	Design Coding

References¶

Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.
Box, G. E. P., Hunter, J. S., and Hunter, W. G. (2005). Statistics for Experimenters: Design, Innovation, and Discovery, 2nd edition. Wiley.