Home / DOE / Charts / Pareto of Regression Coefficients / Math Details
Math Details¶
This page gives what each bar of a Pareto is, which coefficients get one, how the bars are ordered and coloured, and how many of them are drawn.
No new arithmetic happens here. Every number on this chart is already on the regression sheet. The chart takes an absolute value, sorts, and compares two p values against fixed thresholds.
Notation¶
| Term | Description |
|---|---|
| \(j\) | one coefficient of the fitted model, indexed in the order the model holds its terms |
| \(b_j\) | the coefficient of term \(j\), read from the regression table |
| \(p_j\) | the p value of term \(j\), read from the same table |
| \(\alpha\) | the threshold the colour rule uses, fixed at \(0.05\) |
| \(K\) | the number of coefficients eligible for a bar, before any limit is applied |
| \(N\) | the bar count the dialog hands the engine |
| \(B\) | the number of bars actually drawn |
Which coefficients get a bar¶
Three rules decide, applied in this order to each term of the model.
The constant is never drawn. It is skipped before anything else.
An inactive term is never drawn. The chart reads the active flag for the ticked output level out of the regression table. A term that is not marked active there gets no bar, and a term of the model structure that does not appear in the regression table at all counts as inactive for the same reason. This is stricter than treating an inactive term as a zero coefficient: it is not drawn as a zero height bar, it is absent, so the bar count is the count of active terms rather than the count of terms.
A term with levels gives one bar per level except its reference level. A quantitative term gives one bar. A categorical term, and the block, give one bar for each of their levels other than the reference level, because that is exactly how many coefficients they have. Each of those bars is named with the term followed by its level, where the regression table shows the term's levels.
So \(K\) is the number of ACTIVE coefficients on the regression table for that output level, with the constant left out. It is not the number of terms in the model, and on a reduced model the two differ.
The height of a bar¶
The absolute value is taken, so a term that lowers the response and one that raises it by the same amount give bars of the same height. The chart is about size, and the coefficient table carries the sign.
The coefficient is taken exactly as the regression table holds it, which means the chart inherits that table's coding. This decides whether the bars can be compared with each other at all:
Autocoding takes the centre and the half-range of the factor's own data:
So the lowest setting codes to \(-1\), the highest to \(+1\), and the centre to \(0\). A factor with only one distinct value has a half-range of zero, so it cannot be autocoded either, and the coding stops.
With autocoded coefficients every factor spans the same coded range, so bar heights are comparable across factors. With uncoded coefficients each one is per unit of its own factor, so a factor measured in small units shows a tall bar for a small real effect and the ranking stops meaning what it appears to mean. See Design Coding.
The order of the bars¶
Bars are sorted by height, largest first. Two comparisons break a tie, in this order:
- the smaller term index first, which is the order the model holds its terms;
- then the smaller level index first, for two bars of the same term.
Because both tiebreaks are positional rather than numerical, the same model charted twice gives the same order every time, including where several coefficients are exactly equal.
The colour of a bar¶
The colour comes from the term's p value on the same regression table, against two fixed thresholds:
with \(\alpha = 0.05\), so the two thresholds are \(0.05\) and \(0.10\). They are a constant in the engine, not a setting: no dialog control changes them, and they do not follow any alpha chosen elsewhere in the product.
A missing p value is drawn in the normal colour, not left out and not marked. A coefficient whose standard error could not be estimated is the usual case, and its bar is drawn at full height in the normal colour, so an unmarked bar means either "not significant" or "not testable" and the chart does not distinguish them. The coefficient table does.
How many bars are drawn¶
All terms hands the engine \(N = 0\). Top N: hands it the typed count, with anything below \(2\) raised to \(2\) first. The engine then decides:
Read that carefully, because it is not what the dialog appears to offer. The comparison is strictly greater than two, so \(N = 2\) takes the same branch as \(N = 0\) and draws every bar. Since the dialog raises \(0\) and \(1\) up to \(2\) before handing them over, every entry below \(3\) produces a chart of all \(K\) bars, with no message.
The consequence, stated plainly: the smallest Pareto this chart can draw has three bars. A two-bar Pareto cannot be produced by any input, although the dialog's own minimum is two.
The outer minimum is the ordinary case: a count larger than the number of coefficients available simply draws all of them.
A model with \(K = 0\) produces no chart at all. The engine stops before building anything rather than writing an empty sheet.
Shared Math Details used here¶
This chart uses shared formulas defined once in Shared Math Details. See those pages for the exact definitions.
| Shared concept | Used here for | Reference |
|---|---|---|
| Design coding | whether two bars can be compared, which depends on the coding of the table they come from | Design Coding |
See Also¶
References¶
- Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.