Latin and Graeco-Latin Squares What we give up to do Exploratory Designs

Hicks Tire Wear Example data

Linear Model

ANOVA with Main Effects

It’s an orthogonal design so… The Type III tests on top match the Type I tests below. Main Effects Are not confounded with each other.

We are primarily interested in Brand, but what about interactions? How does this work for 2X2X2? If we put in even one interaction, then there are no df for error and this Interaction is completely confounded with Brand.

Notice One cannot estimate and test Interaction terms since we do not have enough d.f. Interaction terms are confounded with error and other terms. As we shall see later with Fractional Factorials, they are likely confounded with each other too.

Brand is the only Fixed Effect for Inference

Tukey HSD on Tire Wear LS Means

Residuals vs. Predicted

Normal Plot of Residuals

Normality Test

Hicks Graeco-Latin Square Example

Basic ANOVA with Main Effects

Only Time is close to significance so…

Since this is a screening design….. Which variables might we investigate further? How might we collect more data? What about diagnostics on the model we fit?

Residuals Vs. Predicted Plot

Normality Plot

Normality Test

What happened with our Diagnostics? With Diagnostics we use Residuals as surrogates for Experimental Error in our Model The Diagnostics are based on the assumption that our Residuals are independently distributed This assumption was never true in an absolute sense However, if the df for Error is “large” relative to the Model df, it is close enough to “true” so that our Diagnostics make sense Remember, these designs are meant to screen factors for further study