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Cube Plot¶

A cube plot labels the corners of a square or a cube with the response at those corners. Two factors give a square with four corners, three give a cube drawn in projection with eight.

QXL DOE New > Charts > Analysis > Cube Plot

When to use it¶

A cube plot puts the whole factorial region on one small picture with real numbers on it, so it answers "what happens at each combination of the extremes" directly. It is the natural companion to a two level design, where the corners are exactly the runs that were made.

Where the numbers come from¶

Some charts can be built from either kind of sheet, and the two answer different questions.

A regression sheet gives the fitted model's prediction: a smooth value at any combination of settings, including combinations no run used. A design sheet gives the raw data: the average of the responses actually measured at each setting, and where a setting has replicates, their standard deviation. The design sheet can only speak about settings that were run.

Where both are offered, the dialog lists every eligible worksheet in the workbook and you pick one.

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.

There is one restriction on the design source: raw run data is only available for a quantitative output.

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.

Cube axes¶

The box headed Cube axes takes the factors that form the shape:

Horizontal axis (1)
Vertical axis (2)
Depth axis, optional (3)

Depth axis, optional (3) defaults to (None), which gives a square with four corners. Choosing a factor for it gives a cube drawn in projection with eight.

The lists only offer factors with exactly two positions, which means two distinct observed values for a quantitative factor, or two levels for a categorical factor or the block. Anything else cannot form a corner pair. Two consequences follow:

A source with fewer than two eligible factors reports This source has fewer than two factors with exactly two levels or two observed values, so a cube cannot be drawn. Use a surface or contour plot instead.
A design containing a nested factor is refused outright, with Cube Plot is not available for a design with nested factors.

Set points¶

A chart can only vary the factors on its axes. Every other factor in the model has to be held at some value while the chart is drawn, and the box headed Set points (factors held constant) is where you choose those values.

The box is not always on screen. It belongs to a prediction, so:

With a design sheet as the source it disappears entirely. A design-source chart averages the runs at each setting instead of predicting, so nothing is held constant, and the extrapolation prompt cannot appear either.
On the Surface, Contour, Trellis and Cube dialogs it also disappears when no factor is being held, which happens when every factor in the model is on an axis.

Which factors get a row differs by dialog. On Surface, Contour, Trellis and Cube a factor's row is hidden while that factor is on an axis, except in grid mode, where every factor keeps a row because each one is held constant in the cells where it is not an axis. On the Interaction, Main Effects and Thumbnail dialog every factor keeps a row, and the rows for the plotted factors have no effect.

The set points are part of the prediction, not decoration. Changing one moves the whole surface, because the fitted model includes that factor and any interactions it appears in. Two charts of the same pair of factors at different set points are two different slices of the same model, and they can look nothing alike when the held factor interacts with an axis factor.

For a quantitative factor the set point is a value. For a categorical factor it is one of its levels.

A set point outside the range the factor took in the data makes the chart an extrapolation, and the software asks before drawing it. Keeping set points inside the observed range keeps the chart inside the region the experiment actually covered.

Before it draws¶

If anything you typed reaches outside the range the factor actually took in the data, the software asks before drawing. A set point is one source of that. On the dialogs that carry axis range boxes, which are Surface Plot, Contour Plot and Trellis, a From or to value counts too, as does a trellis slice range, so widening an axis past the experimental range raises the prompt on its own with every set point left alone. The prompt lists each offending entry with the factor's experimental range beside it, warns that the chart will extrapolate beyond the data, and asks whether to create it anyway. Answering no returns you to the dialog with your options intact.

This is a warning, not a refusal. A prediction outside the range of the data is an extrapolation of the fitted model, and the model was never tested there.

While charts are being written, a status line reports what is happening and a progress bar appears only when more than one chart is being drawn, since a bar for a single chart would sit at 100 percent from the moment it appeared.

Cancel stops the run cleanly between charts, not part way through one. A chart that has already been written stays; the sheet that was mid-write is deleted rather than left half finished. So cancelling never leaves a partial chart behind, but it can leave fewer sheets than you asked for.

What you get¶

All the cubes you selected land on one worksheet called Cube Plot, arranged in a grid, placed immediately to the right of the sheet they came from. The heading names the source sheet.

Each corner carries the response value at that combination of axis settings. Corners are laid out so that each axis runs from its low value to its high value in the direction its label indicates.

A cube built from raw run data can have blank corners. A corner is only filled when the design actually has runs at that combination, so a fractional design leaves some corners empty. A cube built from a regression sheet never has a blank corner, because the model can predict anywhere.

If you are charting the standard deviation from raw data, a corner needs at least two runs at that combination before it can report one. A corner with a single run is left blank rather than shown as zero.

How it is calculated¶

From a regression sheet, each corner is a prediction with the two or three axis factors at their low or high values and every other factor at its set point:

A prediction is the sum, over every term in the model, of that term's coded value times its coefficient:

\[ \hat{y} = \sum_{j} c_j \, b_j \]

where \(c_j\) is the coded value of term \(j\) at the settings you asked about and \(b_j\) is its coefficient from the regression table. The constant term has \(c_j = 1\), so its coefficient enters as itself.

A term that is absent from the regression table, or switched off for the level being predicted, contributes \(b_j = 0\).

For a binary or nominal output the corner carries the predicted probability of the level you selected. Full details: Prediction Equation.

From a design sheet the two plotted values come straight from the run data. The response value is the average of the responses at each distinct setting of the charted factors. The variation value is the average, over that same setting, of the standard deviations within each replicate group, using the sample standard deviation.

A setting with no replicate group of two or more runs has no variation to report and is left blank rather than shown as zero.