Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 11 Multifactor Analysis of Variance

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc Two-Factor ANOVA with K ij = 1

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Notation The Grand Mean Average when factor A is held at level i Average when factor B is held at level j

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. A Model Assume the existence of I parameters and J parameters such that so that (i = 1,…,I j = 1,…,J ) This is an additive model.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Mean response Levels of A Mean Responses AdditiveNonadditive

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. A Model to Eliminate Nonuniqueness where Are assumed independent, normally distributed, with mean 0 and common variance

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Hypotheses

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Test Procedures df = IJ – 1 df = I – 1 df = J – 1 df = (I – 1)(J – 1)

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. SSE SST = SSA + SSB + SSE The following allows SSE to be determined by subtraction.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Hypoth.Test StatisticRej. Region Level Tests

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Expected Mean Squares (when model additive)

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Identifying Significantly Differences 1.Compute 2.List the sample means in increasing order, underline those that differ by less than w. Any pair not underscored by the same line corresponds to a pair that are significantly different. Factor A Factor B

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Two-Factor Random Effects Model When K ij = 1 all independent, normally distributed rv’s with mean = 0. (i = 1,…,I j = 1,…,J )

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Mixed Model For the case when factor A is fixed and factor B is random, the mixed model is (i = 1,…,I j = 1,…,J )

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc Two-Factor ANOVA with K ij > 1

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Parameters for the Fixed Effects Model with Interaction Let

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Notation Define effect of factor A at level i effect of factor B at level j from which

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Hypotheses

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Notation For the case when factor A is at level i and factor B is at level j the model is (i = 1,…,I j = 1,…,J k = 1,…,K )

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sum of Squares Formulas df = IJK – 1 df = I – 1 df = J – 1 df = (I – 1)(J – 1)

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Formulas SST = SSA + SSB + SSAB + SSE The following allows SSAB to be determined by subtraction. df = IJ(K – 1)

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Expected Mean Squares

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. F Ratio Tests Hypoth.Test StatisticRej. Region

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Identifying Significantly Differences 1.Compute JK is the number of observations averaged to obtain each of the 2.List the sample means in increasing order, underline those that differ by less than w. Any pair not underscored corresponds to significantly different levels of factor A.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Mixed Model For the case when factor A is fixed and factor B is random, the mixed model is (i = 1,…,I j = 1,…,J k = 1,…,K)

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Hypotheses

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. F Ratio Testing Hypoth.Test StatisticRej. Region

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc Three-Factor ANOVA

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Three-Factor ANOVA With L = L ijk (i = 1,…,I j = 1,…,J k = 1,…,K l = 1,…,L) are normally distributed with mean 0 and variance and

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sums of Squares df = IJKL – 1 df = I – 1 df = (I – 1)(J – 1)

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sum of Squares df = IJK(L – 1) df = (I – 1)(J – 1)(K – 1)

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Expected Mean Squares

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. F Ratio Testing Null Hypoth.Test StatisticRej. Region

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Latin Squares are independent and normally distributed with mean (i,j,k = 1,…, N )

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sum of Squares Formulas df = N 2 – 1 df = N – 1

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sum of Squares SST = SSA + SSB + SSC + SSE df = (N – 1) (N – 2)

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc P Factorial Experiments

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. are independent and normally distributed with mean (i,j,k = 1, 2; l = 1,…, n ) 2 3 Experiments

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Examples of the Sum of Squares Formulas

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Signs for Computing Effect Contrasts condcellABCABACBCABC (1)x ax bx abx cx acx bcx abcx

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Confounding When the 2 p experimental conditions are placed in 2 r blocks, 2 r – 1 of the factor effects cannot be estimated. This is because 2 r – 1 of the factor effects are mixed up or confounded with the block effects.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Confounding (1), ab, ac, bca, b, c, abc Confounding ABC in a 2 3 experiment.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Confounding Using More Than Two Blocks In the case r = 2 (four blocks), three effects are confounded with blocks. The experimenter first chooses two defining effects to be confounded. The third effect is then the generalized interaction of the two, obtained by writing the two chosen effects side by side and then canceling any letters common to both.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Fractional Replication Including only a portion of the possible conditions in an experiment. For example, including 2 p-1 of the 2 p conditions is called a half-replicate. lose information about a single effect Confounding occurs