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# Test for non-additivity; 3 factor designs.pdf

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School
University of Alberta
Department
Statistics
Course
STAT368
Professor
Douglas Wiens
Semester
Winter

Description
117 16. Test for non-additivity; 3 factor designs Sometimes we can make only one observation per cell ( = 1). Then all 1 ¯ = 0, so = 0 on ( 1) = 0 d.f. The interaction SS, which for = 1 is X ³ ´2 ¯ ¯ + ¯ (*) is what we should be using to estimate experimental error.There is still however a way to test for inter- actions, if we assume that they take a simple form: ( ) = We carry out Tukeys one d.f. test for interaction, which is an application of the usual reduction in SS hypothesis testing principle. Our full model is = + + + + Under the null hypothesi0: = 0 of no interac- tions, the reduced model is = + + + 118 in which the minimum SS (i.e. ) is (*) above. One computes 0 = ( 1)( 1) 1 ( ) The di erence = is called the SS for non-additivity, and uses 1 d.f. to estimate the one parameter . The ANOVA becomes Source SS df MS A 1 = 1 B 1 = 1 N 1 = ( 1)( 1) 1 Error 1 = ( ) Total 1 The error SS is . To obtain it one has to minimize ³ h i´ X 2 + + + 119 After a calculation it turns out that n P ³ ´o 2 ¯ ¯ ¯ + + ¯2 = · Then is obtained by subtraction: = . An R function to calculate this, and carry out the F- test, is at R commands for Tukeys 1 df test on the course web site. Example. For the experiment at Example 5.2 of the text there are= 3 levels of temperature and= 5 of pressure; response i= impurities in a chemical product. > h
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