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Math Details¶

A custom design is supplied by you rather than generated, so there is no construction to describe. What the product does compute is the coded value of every cell, because the fit and every chart work in coded units. This page gives that coding, which is derived from the data in the column and from nothing else.

Notation¶

Term	Description
\(v\)	the value in a cell of the supplied design
\(v_{\min}\), \(v_{\max}\)	the smallest and largest value in that cell's column
\(c\)	the column's center, \((v_{\max} + v_{\min}) / 2\)
\(h\)	the column's half range, \((v_{\max} - v_{\min}) / 2\)
\(x\)	the coded value written for that cell

Each column is classified first¶

Every column of the supplied design is examined and treated as either text or numeric. The classification is made from the column's contents, and the two are coded by different rules, so the rule that applies to a factor is decided by what is in its column rather than by anything declared about it.

Numeric columns¶

The column's own smallest and largest values are found, and every cell is then coded against them:

\[ c = \frac{v_{\max} + v_{\min}}{2}, \qquad h = \frac{v_{\max} - v_{\min}}{2}, \qquad x = \frac{v - c}{h} \]

so the smallest value in the column codes to exactly \(-1\), the largest to exactly \(+1\), and a value at the midpoint to \(0\). Values are placed proportionally between.

The range comes from the data, not from a low and high you declare. This is the difference between a custom design and a generated one. In a generated design the coding is anchored to the factor's stated low and high, so a run at the low setting codes to \(-1\) by construction. Here the anchors are whatever the column happens to contain, so adding a single run at a more extreme setting moves \(v_{\min}\) or \(v_{\max}\) and re-codes every other row of that column.

A constant column is coded as all zeros. When \(v_{\min} = v_{\max}\) the half range is zero and the formula above would divide by it, so the case is taken separately and the whole column is set to \(0\).

Text columns¶

A text column is coded by order of first appearance. The distinct values are collected as the column is read from the top, and each cell receives the position of its value in that list:

\[ x = 0, 1, 2, \dots \]

with \(0\) going to the value that appears in the first row, \(1\) to the first value that differs from it, and so on.

These codes are not centered on zero and they are not symmetric. A two level text column codes to \(0\) and \(1\), not to \(-1\) and \(+1\), which is what a generated design would give the same factor. The codes are also order dependent: the same two levels swap codes if the rows are reordered so that the other one appears first. Nothing in the coding treats the values as ordered or as equally spaced; the numbers are labels standing for the levels.

What follows from the coding¶

Because the coding is read off the supplied data, two properties of a generated design are not guaranteed here.

The design need not be balanced or orthogonal. Nothing checks or enforces either. Whatever the supplied rows imply is what the fit carries.
The coded values need not be the small set of levels a generated design uses. A numeric column holding many distinct values codes to many distinct numbers spread over \([-1, +1]\), not to \(\pm 1\) and \(0\).

For how coded values are used once they exist, see the prediction equation and the coding page below.

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	how coded values are used by the fit and the charts	Design Coding
Prediction equation	what the coded columns are fed into	Prediction Equation

References¶

Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.