4.3 Fitted Effects for Factorial Data - PowerPoint Presentation

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4.3 Fitted Effects for Factorial DataSlide2

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.Slide3

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.)Slide4

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 97Slide5

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.Slide6

NotationsSlide7
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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. Slide11

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 effectSlide12
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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?Slide16
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The interaction between factor A and B is denoted AB or A*B.

The corresponding effect sizes for interactions are

ab

ij

, analogous to

a

i

and

b

j

.

ab

ij

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

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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 fluidSlide19

The deviation of

=53.8 from the parallel lines, no interaction fit is

ab

11

= 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.Slide20
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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.Slide23

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.Slide24
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, 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 valuesSlide27