4.3 Fitted Effects for Factorial Data

4.3.1 Fitted Effects for 2-Factor Studies Factor A → I levels Factor B → J levels   Equal replication, n replicates, in each treatment group. → Balanced design.

8. Bauer, Dirks, Palkovic, and Wittmer fired tennis balls out of a “Polish cannon” inclined at an angle of 45 degree, using three different Propellants and two different Charge Sizes of propellant. They observed the distances traveled in the air by the tennis balls. Their data are given in the accompanying table. (Five trials were made for each Propellant/Charge Size combination and the values given are in feet.)

Problem 8: A = charge size with I = 2 levels of 2.5 ml and 5.0 ml B = propellant with J = 3 levels of lighter fluid, gasoline and carburetor fluid Propellant Lighter Carburetor Fluid Gasoline Fluid 58 50 76 79 90 86 2.5 53 49 84 73 79 82 Charge 59 71 86 size 5.0 65 59 96 101 107 102 61 68 94 91 91 95 67 87 97

First, plot the data! As always the first step is to plot the data. Checking for -Effects of factors Main effects Interactions -Outliers -Changes in variances   If we have only 2 factors this is relatively easy.

Notations

From the plot, Propellant ordered by travelled distance are “Carburetor fluid is better than gasoline, which in turn is better than lighter fluid” “Charge size of 5 ml is better than charge size of 2.5 ml.” The distance pattern across propellant types is similar for charge size of 5ml and charge size of 2.5ml.

Fitted Effected We use the idea of fitted effects to quantify the qualitative summaries from the plot. For factorial data, the effects of factors are described as Main effect Interaction effect

Interaction effect Interactions check the extent to which main effects are consistent at different levels of the other factor. Are the propellant effects the same for each charge? Are the charge effects the same at each propellant?

The interaction between factor A and B is denoted AB or A*B. The corresponding effect sizes for interactions are abij, analogous to ai and bj. abij measures the extent to which from a fit with parallel lines. To fit parallel lines, the fitted values depend only on main effects, no interaction terms. For parallel profiles

Fitting parallel lines: Fitted value for a=1 and b=1, charge 2.5 and propellant lighter fluid 78.53 – 6.87 – 19.63 = 52.03 Compared to the overall average, we lose 6.87 feet using charge 2.5 19.63 feet using lighter fluid

The deviation of =53.8 from the parallel lines, no interaction fit is ab11 = 53.8 – 52.03 = 1.77 Using charge 1 and propellant 1 went 1.77 feet farther than expected than predicted with a no interaction model.

Interaction Effects The fitted interactions in some sense measure how much pattern the combination means carry that is not explainable in terms of the factors A and B acting separately.

Now, the overall mean, the fitted main effects, and the fitted interactions provide a decomposition or breakdown of the combination sample means into interpretable pieces. Those pieces correspond to an overall effect, the effects of factors acting separately, and the effects of factors acting jointly.

, as before, is the fraction reduction in sum of squared errors using the model fitted values compared to using a single mean to predict all y values