Interpreting the Categoric Model

Stat-Ease, Inc. does not recommend interpreting the model directly. Higher-order terms, especially interactions, can be confusing and difficult to picture without the aid of the model graphs produced by Stat-Ease.

We get the most questions about how to interpret the terms of the coded model when categoric factors are in the model.

Let’s assume that A is a three-level factor, and B is a four-level factor. The main effect (ME) terms take the form of A[1], A[2], and, B[1], B[2], B[3]. The interactions are A[1]B[1], A[2]B[1], A[1]B[2], A[2]B[2], A[1]B[3], and A[2]B[3].

There is one less term than there are levels in the factor.

The number of interaction terms is the product of the number of terms in the main effects.

The terms look the same using either Nominal or Ordinal coding, but the estimated coefficients are different and are interpreted differently.

Nominal Case:

The coefficients are the difference between the average response data at a particular factor setting and the grand average. At this point the usual question is something to the effect of, “Isn’t there a level missing?”

While there is no term for the “last” (L) level, we can use a little bit of math to figure things out. We know the grand mean of the response; it is the average of all the data. We can also compute the average at each level of a factor. Once we have computed the grand average and every average except Ls, the Lth level’s average is the grand average less the differences between the grand and the average at each level.

Numeric Example for a three-level A:

Grand Average = 100, A[1] average = 110, A[2] average = 85. The coefficient differences from grand average are +10 and -15 respectively.

A[3] average = 100 - 10 - (-15) = 105; A[3]’s coefficient would have been +5 had it been arbitrarily assigned to the first or second level of the A factor.

To use these models outside of Stat-Ease, to make a prediction at the first level of A, substitute +1 for A[1], and 0 for A[2]. For the second level, substitute 0 for A[1] and +1 for A[2]. For the third level, substitute -1 for A[1], and -1 for A[2].

The coded equation from the ANOVA is

+100

+10*A[1]

-15*A[2]

Ordinal Case:

The terms are now based upon polynomial contrasts. A[1] is the linear effect or the change in the response when the factor changes from the low level of A up to the high level of A. A[2] is the quadratic effect or how much different the average at the middle level is from the prediction from the linear model.

Most statistics text books show how orthogonal polynomial contrasts are arranged when the numeric levels are evenly spaced. The exact interpretation of the coefficients depends on the number of levels and the exact numeric settings used. If they are not evenly spaced numeric levels then the underlying contrast will be different than the standard.

Assuming that the levels of A are 1, 2, and 3 (evenly spaced) and using the values above where the prediction at A[1] is 110, A[2] is 85, and A[3] is 105…

The model is

+100

-2.5 * A[1]

+7.5 * A[2]

The linear coefficient for A[1] is half the difference between the average at the high and low level of A (105 - 110 = -5 / 2 = -2.5). The quadratic coefficient for A[2] is half the difference between the average at the middle level (85) and the prediction that would have come from the linear coefficient alone (100).

To get predictions from the ordinal coded models outside of the software requires knowing the polynomial contrasts used when the design was built. Substitute the value in the cell corresponding the factor level on the left, and the terms across the top.

For evenly spaced numbers, the ordinal contrast for a three-level factor is.

For more complex designs, right-click on the factor column header and choose Edit Info. The contrast table is displayed there.