Home / DOE / Charts / Main Effects Plots

Main Effects Plots¶

A main effects plot shows the response against one factor at a time, with every other factor held fixed. One small chart per factor, side by side.

QXL DOE New > Charts > Analysis > Main Effects Plot

When to use it¶

This command draws one panel per ticked factor: the response against that factor, with every other factor held at its set point. The panels sit on one sheet, drawn from the same model.

Each panel shows one factor alone. If two factors interact, each one's line depends on where the other is held, and that dependence is what Interaction Plots draws instead.

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.

Design sheets are offered for quantitative outputs, where the plotted value is the average of the runs at each setting. For a binary or nominal output only a regression sheet will do.

The dialog¶

This dialog is shared with Interaction Plots and the Thumbnail Plot.

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.

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.

Set points matter more here than they look as though they should. A main effects plot is a slice of the model at the held values, so when a factor interacts with something, its line changes as the set point of that other factor changes. Two main effects plots of the same model at different set points can disagree, and both are correct.

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¶

A worksheet named Main Effects Plot - source sheet name, placed immediately to the right of the sheet it came from. This is the one chart in the family whose sheet name carries the source sheet rather than the output level.

Everything you selected goes on one combined sheet, unlike the interaction and thumbnail plots which take a sheet each. Each output gets its own row of charts, and the rows follow the order the responses appear on the design sheet, with the response model before its standard deviation model. So the sheet reads in the same order as the design.

A panel whose factor is categorical is drawn with a dotted line, and one whose factor is quantitative with a solid line. The style follows the factor on the horizontal axis, which on this chart is the panel's own factor. See Interaction Plots for why.

How it is calculated¶

From a regression sheet each point is a prediction with the plotted factor at that point's value 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 \(b_j = 0\).

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.

From a design sheet the plot is sometimes called a marginal means plot, because each point is the mean of the runs at that setting of the factor, averaged over whatever the other factors were doing in those runs.