11.2 - Coding for Carry-over Covariates

Earlier in the course I mentioned that coding in ANOVA can be tricky business, and here is a prime example. Steve Arnold, formerly teaching this course and now retired, devised this coding strategy to code for carry-over effects. The best way to see how it works is going through an example.

The example we will work through appears in the textbook Design of Experiments, by Kuehl, as example 16.1. Investigators want to evaluate the effect of 3 diets on Neutral Detergent Fiber (NDF) levels in steer. The three diets are administered to each steer in a sequence over 3 periods. A total of 6 sequences were used and two steers were assigned to each sequence of treatments.

The cross-over design can be summarized as:

Sequence 1 2 3
1 A B C
2 B C A
3 C A B
4 A C B
5 B A C
6 C B A

If we look at the the first part of the dataset (Steer Data) for this example in Excel , we can see the following:

steer data example screen shot

We need now to add two columns to use an effect-type coding for the 3 treatment levels. We can use:

  \(x_1\) \(x_2\)
A 1 0
B 0 1
C -1 -1

Where \(x_1\) and \(x_2\) will be columns we create in the data to input:

steer data example screen shot

for all of the rows of data.

Notice there no entries for the first period because on the first application of each treatment there has been none that have preceded it. Therefore a 0 is used for both \(x_1\) and \(x_2\). Looking at Period 2, sequence 1, treatment B we can refer back to the Sequence chart and see that it was preceded by treatment level A, so we assign \(x_1 = 1\), and \(x_2 = 0\), indicating that it was treatment A that could produce a carry-over effect here.

steer data example screen shot

We carefully do this for each entry in the dataset. The coded variables \(x_1\) and \(x_2\) then are entered into the general linear model as continuous covariates and LSmeans for treatments are adjusted for carry-over effects.