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Residual Plots¶

A residual is what the model failed to explain for one run: the measured response minus the fitted value. Residual plots put those leftovers on a chart, because their pattern is where a model's problems show up.

QXL DOE New > Charts > Regression Diagnostics > Residual Plots

When to use it¶

This is the command that draws the per-run diagnostics: the residual, both studentized forms, the leverage and Cook's D, each against a choice of horizontal quantity. It can also write those values out as numbers.

The only automatic reading of residuals in the product is the DOE Advisor's, which applies two fixed thresholds to the studentized residual and runs a normality test. Everything else the charts show is left to you. See Understanding Least Square Residuals.

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.

Residuals are differences from fitted values, so there is nothing to plot without a model.

The dialog¶

Residual Plots uses the older chart dialog rather than the one the other nine charts share, so it looks different. Its two Create buttons read Create and Exit and Create and don't exit, which do what the other dialogs call Create and Exit and Create and Keep Open.

Source¶

The control labelled Source for the plots: looks like a plain dropdown but opens the same three-level tick tree as the other nine chart dialogs: worksheet, then output, then level. It chooses the regression table, and for a quantitative output which of its two models to diagnose: the response model or the standard deviation model. Each has its own residuals.

You can tick more than one source, and two rules govern what stays ticked:

One worksheet at a time. Ticking anything on a second worksheet clears every tick on the first, the same rule the other nine chart dialogs follow.
One output type at a time, on this dialog only. Ticking a quantitative output clears any ticked binary output and the other way round, so the two can never be charted together. Two quantitative outputs, or two levels of one output, can be ticked together.

Each ticked source produces its own set of worksheets, so one Create can write several.

Nominal outputs are not offered here. A nominal output does not appear in the source list at all, because the residual plots do not support it.

The axes¶

There are two lists, labelled Plot on X-Axis and Plot on Y-Axis, and you may tick more than one item in each, which produces one chart per combination.

The two lists are not the same list. Most quantities are available on one axis only.

For a quantitative response:

Plot on X-Axis	Plot on Y-Axis
block first when the design is blocked, then one row per factor, each read as its name with its coded letter after it, for example Temperature (A)	Actual Y, the measured response
Run #, the run number from the design's Run column	Predicted Y, the fitted value
Actual Y	Leverage, how unusual that run's factor settings are
Predicted Y	Residual, measured minus fitted
	Studentized Residual, the residual scaled by its own standard error
	R-Studentized Residual, the same with that run left out of the fit
	Cook's D, how much the whole fit would move if that run were dropped

For a binary logistic response the split is sharper still:

Plot on X-Axis	Plot on Y-Axis
Predicted, the fitted probability	Delta Chi-Squared, the change in the chi-square statistic if that run were dropped
Leverage	Delta Deviance, the change in deviance if that run were dropped
	Delta Beta, the change in the coefficients if that run were dropped
	Delta Beta Standardized, the same on a standardized scale

Run # is the value in the design's Run column, which is the order the runs were meant to be performed in. It is not the row position in the regression, so a randomized design's Run # will not match the row order.

The text report¶

The checkbox labelled Create textual report writes the residual values out as numbers alongside the charts. It is off unless you tick it.

What you get¶

Up to two worksheets per ticked source, placed to the right of the regression sheet:

Residual Plot - output name, the charts.
Residual Report - output name, the values as numbers, when Create textual report was ticked. For a quantitative output the heading written inside that sheet reads Quantum XLResiduals Report, with no space between the two words. The missing space is reproduced here because it is what the sheet prints. For a binary output the same heading is spaced correctly, as Quantum XL Residuals Report.

Residual Plots is the only chart command that writes two kinds of sheet.

How to read the scaled residuals¶

A raw residual is in the units of the response, so whether one is large depends on the response. The studentized forms divide by an estimate of the residual's own standard error, which puts them on a common scale where the numbers mean the same thing for any response.

On that scale, and assuming the residuals are roughly normal, about 95 percent should fall within plus or minus 2 and about 99 percent within plus or minus 3.

Those are the two thresholds the DOE Advisor applies, and the grade it gives them differs:

Studentized residual	What the Advisor does
at or beyond 2	records Large Standardized Residuals in its Notes, and marks the run red on the design sheet
at or beyond 3	records Extreme Standardized Residuals in its Warnings, and marks the run red

High leverage also goes to the Notes and also marks the run red, so a red cell is not always about a residual. See The DOE Advisor.

R-Studentized Residual is computed with the run in question left out of the standard error estimate, so a run large enough to inflate its own denominator cannot mask itself that way.

Cook's D answers a different question from the residual plots: not how far a run is from the model, but how much the model depends on it. A run can have a small residual and a large Cook's D, which means the fit moved to accommodate it.