Home / DOE / Charts / Pareto of Regression Coefficients

Pareto of Regression Coefficients¶

This chart draws one bar per model term, the height being the size of that term's coefficient regardless of sign, sorted largest first. It puts the terms in order of how much they move the response.

QXL DOE New > Charts > Analysis > Pareto Regression Coefficients

When to use it¶

The coefficient table gives each term's effect and its p value as numbers. The Pareto ranks the terms by the absolute value of their coefficients and colours each bar by its p value, at two fixed thresholds.

Because the bars are absolute values, a term that lowers the response ranks alongside one that raises it by the same amount. The chart is about magnitude; the sign stays in the coefficient table.

Where the numbers come from¶

This chart is built from a regression sheet only. Design sheets are not offered, because the chart plots a value that only a fitted model can produce.

A chart built from a regression sheet does not re-fit the model. It reads the coefficients already on that sheet and evaluates the prediction equation with them.

Two things follow. The chart always agrees with the sheet it came from, so if you edit a coefficient the next chart changes with it. And a chart is only as current as its regression sheet: change the design and the old charts do not update, because nothing re-runs the regression for them.

A term that is not in the regression table, or that is switched off for the output level being charted, contributes exactly zero to the prediction. It is not an error and there is no warning: the chart is simply drawn from the reduced model. So a chart reflects which terms are active at the moment it is created, and two charts of the same output can differ because the active terms changed between them.

A term switched off in the regression table gets no bar at all, so the Pareto shows the model as it currently stands rather than every term the design could support.

The dialog¶

The left side of the dialog is a box headed Outputs holding a tree with three levels: the worksheet, then each output on it, then each level of that output. Tick the levels you want a chart for. The dialog opens with every level on the active sheet already ticked.

The two upper levels are three-state. Ticking a worksheet or an output ticks everything under it, and clearing it clears everything under it. When only some of the children are ticked, the parent shows a partial state rather than a tick, so the tree tells you at a glance whether a selection is complete.

One worksheet at a time. Ticking anything on a second worksheet silently clears every tick on the one you had selected, and the options on the right rebuild for the new sheet. There is no warning and no way to draw charts from two source sheets in one Create.

If you press Create with nothing ticked, the dialog tells you At least one chart must be selected. and stays open.

Bars¶

The box headed Bars decides how many terms are drawn.

All terms draws every active term.

Top N: draws only the largest, with the number in the box beside it. The working range starts at 3. A value of 2 does not restrict the chart: it falls through and draws every bar, as does any value below 2. If you want two bars, this option cannot currently give them to you.

Layout¶

The box headed Layout turns the chart:

Vertical puts the terms along the bottom and the bars upright.
Horizontal puts the terms down the side and the bars across. This is usually the readable choice when term names are long, which they often are once interactions and level names are included.

This command writes its report in one step. There is no status line and no progress bar however many charts you asked for, and Cancel closes the dialog rather than stopping a run, because there is no between-chart point at which to stop.

What you get¶

A worksheet called Pareto, placed immediately to the right of the regression sheet. The heading rows name both the design sheet and the regression sheet the chart came from.

The term axis is labelled Factor and the value axis |Coeff|, the bars being absolute coefficients. Choosing Horizontal turns the chart, so the term axis becomes the vertical one; the two titles stay with their own axes either way.

Bar colour marks significance¶

Colour	Meaning
Red	p value at or below 0.05
Blue	p value above 0.05 and at or below 0.10
Normal	p value above 0.10, or no p value available

Both thresholds are fixed and cannot be changed. A term with no p value, for example one whose standard error could not be estimated, is drawn in the normal colour rather than being marked.

Read the colours as a rough flag and the coefficient table as the record: the table carries the actual p values, and two bars in the same colour can have very different ones.

Which terms get a bar¶

The constant is never drawn, since it is not an effect.

A quantitative term gets one bar. A categorical term gets one bar per level except its reference level, because that is how many coefficients it has, and each bar is named with the term followed by the level.

For a categorical factor one level is the reference. The other levels each get a coefficient measured against it, and the reference itself has none, which is why the regression table shows one fewer coefficient than the factor has levels. The reference level is not missing from the model: it is the baseline the others are expressed relative to.

Bars are sorted by absolute coefficient, largest first. Where two are equal, the order follows the term order in the model, which keeps repeated runs of the same chart in the same order.

How it is calculated¶

The bar height is the absolute value of the coefficient exactly as it appears in the regression table:

\[ \text{bar} = \left| b_j \right| \]

So the chart inherits the coding of that table, and this matters for how it should be read.

Autocoding takes the centre and the half-range of the factor's own data:

\[ m_c = \frac{x_{\max} + x_{\min}}{2} \qquad s_c = \frac{x_{\max} - x_{\min}}{2} \]

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 the bars are comparable. With uncoded coefficients they are not: each is per unit of its own factor, and a factor measured in small units will show a large coefficient for a small real effect. Full details: Design Coding.