Home / DOE / Charts / Pareto of Regression Coefficients / Pareto How-To

Pareto How-To¶

This walkthrough builds a Pareto of regression coefficients from a replicated three factor experiment, and then uses the same chart to show one thing the Top N: box does not do.

Every chart in this family is drawn from a fitted model, so the walkthrough starts by building one. The same experiment is used on all ten chart walkthroughs, so once you have the regression sheet you can work through any of the others without setting the data up again.

The data is on this page rather than in a file to download. Press Copy for Excel, then paste it into a blank worksheet.

The data¶

A three factor experiment on a bonding process, run twice through. Temp is in degrees C, Press in psi and Time in seconds, and the response Strength is the peel strength of the finished bond. Each of the eight factor combinations was run twice, giving 16 runs.

Temp	Press	Time	Strength
180	40	10	49.5
220	40	10	65.4
180	60	10	61.8
220	60	10	66.8
180	40	20	52.3
220	40	20	70.7
180	60	20	60.7
220	60	20	70.0
180	40	10	50.6
220	40	10	66.4
180	60	10	61.2
220	60	10	67.2
180	40	20	51.4
220	40	20	67.1
180	60	20	61.9
220	60	20	71.6

Each factor takes exactly two values, so this is a full factorial of the three at two levels each, replicated once. That matters for more than one chart: a factor with exactly two observed values is the only kind that can carry a Cube Plot axis, and a design with centre points would have failed that test.

Steps¶

Put the data in Excel

Press Copy for Excel above the table. In Excel, open a blank worksheet, click cell A1, and press Ctrl+V. You should have headers in row 1 and the 16 runs in rows 2 through 17.
Make a design sheet to hold it

From the Excel ribbon: QXL DOE New > Create Design > Special > Create Custom Design.

On the first page set the number of factors to 3 and the number of runs to 16. On the second page set the number of outputs to 1. That gives you an empty design sheet of the right shape, which is how data collected outside Quantum XL gets analysed.
Fill in the design sheet

Copy the four columns of data into the three factor columns and the one output column. Name the factors Temp, Press and Time, and the output Strength, so the names on your charts match the ones quoted below.
Check the model holds the three two-factor interactions

The charts below assume a model of the three main effects plus AB, AC and BC. Add any that are missing with QXL DOE New > Modify Design > Inputs > Add/Remove Interactions.
Run the regression

QXL DOE New > Analyze Design > Run Regression. A worksheet called Regression is added after the design sheet. Every chart in this family reads that sheet.
Draw the Pareto

QXL DOE New > Charts > Regression > Pareto Regression Coefficients.

In the Outputs tree on the left, tick Strength Y-Hat. In the Bars box leave All terms selected, and in Layout leave Vertical selected. Press Create.

Check the fit first¶

Before drawing anything, check the coefficient table on the Regression sheet against this. If these numbers match, every chart number quoted further down will match too; if they do not, the design was not entered the way this walkthrough assumes and nothing below will line up.

Term	Coefficient	Standard error	t	p
Constant	62.1625	0.2912	213.477	0.000000
Temp (A)	5.9875	0.2912	20.562	0.000000
Press (B)	2.9875	0.2912	10.260	0.000003
Time (C)	1.0500	0.2912	3.606	0.005696
AB	-2.2375	0.2912	-7.684	0.000030
AC	0.6500	0.2912	2.232	0.052506
BC	-0.1500	0.2912	-0.515	0.618877

The fitted equation is

Strength = 62.1625 + 5.9875 A + 2.9875 B + 1.0500 C - 2.2375 AB + 0.6500 AC - 0.1500 BC

in coded units, where each factor runs from -1 at its low value to +1 at its high value.

Every standard error is the same 0.2912. That is not a coincidence and it is a useful sign that the design was entered correctly: on a balanced full factorial every coded column is orthogonal to every other and carries the same amount of information, so every coefficient is estimated equally precisely.

What you should see¶

A worksheet called Pareto with one chart on it, six bars, tallest first. The term axis is titled Factor and the value axis |Coeff|.

Position	Bar	Height	Colour
1	Temp (A)	5.9875	red
2	Press (B)	2.9875	red
3	AB	2.2375	red
4	Time (C)	1.0500	red
5	AC	0.6500	blue
6	BC	0.1500	normal

Six bars from a seven term model. The constant is never drawn, so the seven rows of the coefficient table give six bars.

The heights are absolute values, and two of them lost a sign. AB is 2.2375 on the chart and -2.2375 in the table, and BC is 0.1500 against -0.1500. The chart ranks by size; the table carries the direction.

The colours split four, one and one. Four bars are red, meaning p at or below 0.05. AC is blue, its p of 0.052506 falling in the band above 0.05 and at or below 0.10, and it is the only bar in that band. BC is drawn in the normal colour with a p of 0.618877.

AC is the bar worth noticing. At p = 0.052506 it misses the red threshold by about two parts in a thousand. The colour is a rough flag and the coefficient table is the record, which is exactly why the chart carries three colours rather than two.

Now show what Top N does not do¶

Draw the same chart three more times, changing only the Bars box. Everything else stays as it was.

Top N:	Bars you get	What that tells you
5	5, dropping BC	the box works
3	3: Temp (A), Press (B), AB	the box works
2	6, every bar	the box did not apply
1	6, every bar	the box did not apply

A typed 2 draws the same chart as All terms, and nothing says so. The dialog will not let you type below 2, raising a 0 or a 1 to 2 for you, and the engine only applies a limit above 2. So every value the box will accept below 3 produces the full six bar chart with no message.

The two bar Pareto the box appears to offer cannot be produced by any input. Three is the smallest count that does anything. This page describes what the software does today rather than what the control implies.

Things to try next¶

Turn the chart. Set Layout to Horizontal and draw it again. The bars run side to side, the Factor title moves to the vertical axis with its axis, and the ranking still reads from the top downwards because the plot order is reversed to keep it that way.
Reduce the model and redraw. Remove BC from the model and run the regression again. The new Regression (2) sheet gives a five bar Pareto, and the remaining coefficients change, because dropping a term changes the fit rather than just hiding a bar.
Compare against the runs. The tallest bar, Temp at 5.9875, says the fitted response moves 5.9875 for a one unit change in coded Temp, which is half its full range. Twice that, 11.975, is what Main Effects Plots How-To measures directly off the data.