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Math Details¶
This page gives the values a surface plot draws and how each one is formed: where the grid points sit, what number each one carries, how the height axis is chosen, and which cells of a grid report are drawn at all.
One engine builds the Surface Plot and the Contour Plot. The two compute the same numbers from the same inputs: only the chart Excel is asked to draw differs, and the rules that decide which factor pairs are allowed. Everything below applies to both unless it says otherwise.
Notation¶
| Term | Description |
|---|---|
| \(A\) | the factor on the horizontal axis, which the product calls Factor1 |
| \(B\) | the factor on the series axis, Factor2 |
| \(A_{\text{lo}}\), \(A_{\text{hi}}\) | the From and to values of \(A\)'s plotted range, used only when \(A\) is quantitative |
| \(B_{\text{lo}}\), \(B_{\text{hi}}\) | the same two values for \(B\) |
| \(M_A\), \(M_B\) | the number of positions on each axis |
| \(u_i\) | the \(i\)th position along \(A\), for \(i = 0, 1, \dots, M_A - 1\) |
| \(v_m\) | the \(m\)th position along \(B\), for \(m = 0, 1, \dots, M_B - 1\). The index letter \(m\) is used because \(j\) below belongs to the model term |
| \(t_k\) | the set point typed or picked for factor \(k\) |
| \(\mathbf{z}\), \(z_k\) | the vector of factor settings handed to the prediction equation, and its entry for factor \(k\) |
| \(\hat{y}_{im}\) | the height plotted at \((u_i, v_m)\): a fitted response on the response level, a fitted standard deviation on the standard deviation level, a fitted probability on a binary or nominal level |
| \(c_j\), \(b_j\) | the coded value and the coefficient of model term \(j\) |
| \(\hat{y}_{\min}\), \(\hat{y}_{\max}\) | the smallest and the largest height on one chart |
| \(\Delta\) | the tick spacing of the height axis |
| \(r\) | the rounded range the height axis is built from |
| \(\hat{y}_{\text{low}}\), \(\hat{y}_{\text{high}}\) | the two ends of the height axis |
| \(n\) | the number of ticked factors in a grid report |
Where the grid points sit¶
Every position is computed, not read from the runs. This is the deepest difference between this chart and the interaction family, which puts its points only at settings the experiment actually visited. A surface sweeps its two axes across a range and predicts at every intersection, so most of its points sit at factor settings no run ever used.
A quantitative axis carries 41 positions. The count is fixed in the engine and no dialog control changes it. The positions are evenly spaced from the From value to the to value inclusive, which is 40 steps between them:
The last position is assigned rather than computed, so the top of the range is exactly the number you typed, with no rounding drift. Every other position is the low value plus a whole multiple of the step, formed by multiplication rather than by adding the step repeatedly, so the positions can be reproduced one at a time without accumulating error. The series axis is built the same way from \(B_{\text{lo}}\) and \(B_{\text{hi}}\).
A categorical axis carries one position per level, and the position is the level's own index:
The block counts as a categorical factor here, with one position per block.
So the number of predictions in one chart is \(M_A \times M_B\). Two quantitative axes give \(41 \times 41 = 1681\) of them. A quantitative axis against a three level categorical factor gives \(41 \times 3 = 123\).
The height at a grid point¶
The engine takes the set points as they stand, overwrites the two entries belonging to the plotted factors with that point's own position, and hands the result to the prediction equation:
The set points of \(A\) and \(B\) are overwritten before every prediction, so they are never read on this chart. In the single-pair layout the dialog hides those two rows for exactly that reason. In a grid report the rows stay visible and stay editable, because a factor that is an axis in one cell is held in every cell that does not plot it.
The vector is copied afresh for each chart, and only that chart's own two entries are replaced. In a grid report that means one set-point box can move most of the sheet and leave a few cells alone: the factor is swept on the cells that plot it and held at its set point everywhere else.
Each entry is then coded, and the coded values are combined with the coefficients already on the regression sheet:
Every quantitative factor is put on a common scale before the regression runs. One expression does it, whichever coding is in force, using a centre \(m_c\) and a scale \(s_c\) that the coding chooses:
A categorical factor is never coded this way. Its coded value is \(1\), and the level it sits at selects which coefficient is used instead.
An interaction's coded value is the product of the coded values of the factors in it.
A prediction is the sum, over every term in the model, of that term's coded value times its coefficient:
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 output the same sum is formed and passed through the logistic function:
For a binary output the same sum is formed first, then passed through the logistic function to give a probability:
That is the probability of the first level. The probability of the second is \(1 - p\).
For a nominal output there is one sum per level:
For a nominal output with \(L\) non-reference levels there is one sum per level, \(\eta_k = \sum_j c_j b_{jk}\), and the probability of level \(k\) is
The \(1\) in the denominator is the reference level, whose \(e^{\eta}\) is \(1\) by construction. So the reference level's own probability is
and the probabilities across all levels sum to \(1\).
Which coefficients are used follows the level you ticked. On a quantitative output the Y-Hat level reads the response model's coefficients and the S-Hat level reads the standard deviation model's, so on that level the surface is a fitted standard deviation over the two factors. The set-point vector, the coding and the sum are the same either way.
A held factor left at its default set point codes to zero under Autocode. The set-point box opens at the midpoint of the factor's own low and high, and Autocode centres its coding on that same midpoint, so the default value makes the factor's coded entry exactly \(0\) and drops its term, and every interaction it appears in, out of the sum. Under Standardize the coding is centred on the mean of the observed values instead, which is the midpoint only on a symmetric design, so the same box left alone does not generally code to zero. See Design Coding.
The height axis¶
The axis is computed from the chart's own grid and no other. The engine takes the smallest and the largest of that chart's \(M_A \times M_B\) heights and rounds them out to nice numbers. A grid report therefore gives every cell its own height axis, computed from that cell alone, and two cells on one sheet are not to be read against each other by eye. This is the opposite of the interaction family, whose charts share one value axis across the sheet, and it is the opposite of the Trellis, which deliberately forces one shared scale onto all of its panels.
The ends are chosen by rounding to a nice number. Write
where \(e\) is the base ten exponent of \(r\), \(\phi\) is its leading fraction, and \(f\) is a nice fraction read from \(\phi\) by one of two tables. Rounding to the nearest takes \(f = 1, 2, 5, 10\) as \(\phi\) falls below \(1.5\), below \(3\), below \(7\), or higher; rounding up takes \(f = 1, 2, 5, 10\) as \(\phi\) is at most \(1\), at most \(2\), at most \(5\), or higher. The range is rounded up and the tick spacing to the nearest:
Three guards follow. A range below \(10^{-5}\) is replaced by the largest power of ten not exceeding the larger of \(\lvert \hat{y}_{\min} \rvert\) and \(\lvert \hat{y}_{\max} \rvert\), and failing that by \(10^{-5}\), so a flat surface still gets an axis. An end that rounding moved inside the data is pulled back to the data, so the axis can never cut off a plotted height. And if the two ends come out equal, each of them is moved by \(r/2\), the low end down and the high end up.
\(\hat{y}_{\text{low}}\) and \(\hat{y}_{\text{high}}\) are then written onto the chart as the value axis minimum and maximum, and the axis is made to cross at \(\hat{y}_{\text{low}}\).
The tick spacing that reaches the chart is not \(\Delta\). \(\Delta\) is used to place the two ends and then dropped. What is written as the major unit is the finished span divided by ten:
so a surface or contour always carries ten intervals on its height axis whatever the data.
One case is left to Excel. A surface whose series axis is categorical is drawn as a 3D line chart rather than as a surface, and the major unit rule above is skipped for that chart type alone, so its height axis takes whatever interval Excel chooses. The two ends are still forced to \(\hat{y}_{\text{low}}\) and \(\hat{y}_{\text{high}}\).
The axis labels¶
The two factor axes are labelled from the positions themselves: a quantitative position is written with the product's nice-number formatting, and a categorical position is written as the level name. Both factor axes and the height axis carry the General number format, so the number of decimals shown is Excel's.
The engine does compute a decimal count for each of the three axes, but nothing in the drawing code reads them. They reach the report structure and stop there.
Which chart Excel draws¶
The compute is identical for all three cases below. Only the chart type differs.
| Chart and axes | Drawn as |
|---|---|
| Surface, both axes quantitative | a 3D surface |
| Surface, series axis categorical | a 3D line, one line per level of \(B\) |
| Contour, both axes quantitative | a surface seen from above |
A contour is the same grid of heights as its surface, so a contour line is a path through grid values rather than an exact solution of the fitted equation. It follows the surface closely where the fit changes gently and becomes visibly angular only where the fit changes fast relative to the spacing of the 41 positions.
Cells that are not drawn¶
In a grid report the ticked factors are laid out as an \(n\) by \(n\) square. The diagonal carries factor names rather than charts, and each off-diagonal cell is the ordered pair with the column's factor on the horizontal axis and the row's factor on the series axis, so \(n\) ticked factors give \(n(n-1)\) charts.
A cell is written as NA rather than a chart when its pair fails the axis rule:
- On a surface grid the horizontal factor must be quantitative. The series factor may be categorical.
- On a contour grid both factors must be quantitative.
- The block is categorical for this purpose, always, so it can never be the horizontal factor of a drawn cell.
The grid does not swap a pair the way the single-pair dialog does. Given a categorical factor and a quantitative one, the single-pair layout of a surface moves the quantitative factor to the horizontal axis and draws the chart. The grid instead leaves that cell as NA and draws the transposed cell, the one that already has the quantitative factor on the horizontal axis. The chart is still on the sheet; it is in the other half of the grid.
Every drawn cell is the same compute as the single chart for that pair. The grid calls the per-pair engine once per valid cell with the same set points and the same ranges, so a cell and a single chart of the same pair hold the same numbers.
What the heading block reports¶
Above the chart the engine writes the chart type, the name of the regression sheet the coefficients came from, the output level, and a Constant values: line listing each held factor as name and value, separated by semicolons. A quantitative value is written with the nice-number formatting; a categorical one is written as the level's own value. The block's entry is named block, spelled lower case as the product spells it, and its value is the block number, which is the level index plus one.
A grid report lists only the factors that are not on the grid at all. A ticked factor is held constant in the cells that do not plot it, but it is not on the constants line, because it is not constant across the sheet.
Shared Math Details used here¶
This chart uses shared formulas defined once in Shared Math Details. See those pages for the exact definitions.
| Shared concept | Used here for | Reference |
|---|---|---|
| Prediction equation | the height at every grid point | Prediction Equation |
| Design coding | turning an axis position or a set point into the coded value the prediction uses | Design Coding |
See Also¶
- Surface Plot
- Options
- Contour Plot Math Details, the same engine
- Trellis, the same surface repeated across slices of another factor
- Chart Source
References¶
- Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.
- Heckbert, P. S. (1990). Nice Numbers for Graph Labels. In A. S. Glassner (Ed.), Graphics Gems. Academic Press.