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Trellis¶

A trellis draws the same chart many times over, once for each slice of one or two other factors, and lays the small charts out in a grid. Where a single surface shows one pair of factors at one set of held values, a trellis shows that pair repeatedly as a third factor moves.

QXL DOE New > Charts > Analysis > Trellis

When to use it¶

A trellis repeats a surface, contour or interaction chart across slices of one or two other factors, one panel per slice value, and lays the panels out as a grid on one worksheet. Every panel shares the same axis ranges and the same response scale, so they can be compared directly.

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.

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.

Chart type¶

A trellis can be built from any of three chart types, and you choose which inside this dialog rather than from the ribbon:

Surface, a three dimensional surface in each panel.
Contour, a contour map in each panel.
Interaction, an interaction plot in each panel.

This is why the ribbon has one Trellis entry rather than three.

Primary factors¶

The box headed Primary factors sets the chart that is repeated. Horizontal axis (X) and Series axis (Y) each take a factor.

Whether they also take a From and to range depends on the chart type. On a Surface or Contour trellis a quantitative primary axis carries a From and to range, so the panels all cover the same region and can be compared directly. On an Interaction trellis there are no From and to boxes on either primary axis: the chart draws the factor at its own levels instead, so there is no range to type.

Panel variables¶

The box headed Panel variables sets what changes between panels.

First panel variable starts with a Factor picker, and what appears below it depends on the factor:

A quantitative factor takes a From and to range and a number of Slices. The range is divided into that many values and one panel is drawn at each.
A categorical factor, or the block, takes a checkbox per level instead, and you get one panel per ticked level. Ticking none is refused with Tick at least one level of the panel variable name.

Ticking Add a second panel variable adds Second panel variable with its own factor, range and slice count. With two panel variables the grid becomes two dimensional: the first variable's slices run one way and the second's run the other.

The panel count is the product of the two variables' panel counts, whether each of those comes from a slice count or from a number of ticked levels. Two variables at four each is sixteen panels.

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.

On a trellis the panel variables drop out of this list too, not just the two primary axes. A factor chosen as the first or second panel variable is given a value by each panel, so there is nothing to hold it constant at and its row disappears.

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¶

Above 16 panels the software warns that the trellis may be slow to draw and hard to read, reports how many panels it is about to create, and asks whether to continue. The count is the product of the slice counts, so it grows quickly once a second panel variable is added.

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¶

One worksheet per ticked output level, each holding that level's full grid of panels.

The sheet is not named "Trellis". It takes the name of the chart type inside it, followed by the output name:

Trellis of	Worksheet
Surface	Surface Plot - output name
Contour	Contour Plot - output name
Interaction	Interactions Plot - output name

Those are the same names a single chart or a grid gets, so the sheet name does not tell you a trellis apart from an ordinary Surface, Contour or Interactions Plot. See DOE Charts for the naming rule across the family.

The sheet is placed immediately to the right of the regression sheet it came from.

Each panel is labelled with the value of the panel variable it was drawn at, and every panel shares the axis ranges you set, so the panels are directly comparable.

How it is calculated¶

Every point in every panel is a prediction from the fitted model, with the panel variable held at that panel's slice value and the remaining factors held at their set points.

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 panels plot a predicted probability instead. Full details: Prediction Equation.

Where you typed a From and to range, the factor axis runs exactly from one to the other. On an Interaction trellis, which has no such boxes, the factor axis runs over the factor's own levels instead. It is the response axis that is put on round tick values, and on a trellis that response scale is shared by every panel, which is what makes the panels comparable.