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Thumbnail Plot¶

A thumbnail plot is the whole model on one sheet: a square grid with the main effect of each factor down the diagonal and the interaction between each pair off it.

QXL DOE New > Charts > Analysis > Thumbnail Plot

When to use it¶

This command puts the whole model on one sheet: every ticked factor's main effect and every pair's interaction, laid out as a square grid. Four ticked factors give sixteen small charts on one sheet.

The charts are small by design, so read them for shape and not for values. The full size versions of the same charts are Interaction Plots and Main Effects Plots, which share this dialog.

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.

The dialog¶

This dialog is shared with Interaction Plots and Main Effects Plots.

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.

The box headed Factors to plot carries a checkbox per factor, and the factors ticked here are the ones drawn. This chart has no single-pair option and no axis pickers, so the tick boxes are the only control over what appears.

The factors you tick set the size of the grid, and every cell of it is filled. Ticking four factors gives a four by four grid of sixteen charts: four main effects on the diagonal and twelve interaction cells. Each pair appears twice, mirrored about the diagonal, once with each factor on the horizontal axis.

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.

A thumbnail of many factors is a lot of charts: the count grows as the square of the number ticked, so six factors is thirty six cells. Expect it to take a moment, and use Cancel if you asked for more than you meant.

What you get¶

A worksheet named Thumbnail Plot - output level name, placed immediately to the right of the sheet it came from.

The grid is read like a correlation matrix. A cell on the diagonal is the main effect of that factor. A cell off the diagonal is the interaction between the factor of its row and the factor of its column.

Scaling is shared within a kind of chart but not across kinds. Every main effects cell uses the same value axis as every other main effects cell, so their slopes are directly comparable. Every interaction cell shares a value axis with the other interaction cells. But the main effects axis and the interaction axis are not the same, so do not compare the steepness of a diagonal cell with an off diagonal one.

A cell is drawn with dotted lines when the factor on its horizontal axis, which is its column factor, is categorical, and with solid lines when that factor is quantitative. The style follows the horizontal axis, not the factor a line belongs to, so the two mirrored cells of one pair can differ. See Interaction Plots for why.

How it is calculated¶

Each cell is computed exactly as the full size chart of the same kind would be: a diagonal cell like a main effects plot, an off diagonal cell like an interaction plot, both with the remaining factors 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 \(b_j = 0\).

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.