2/15/2007 219323 Probability and Statistics for Software Statistics for Software and Knowledge Engineers Lecture 12: Multifactor Experimental Multifactor Experimental Design and Analysis Monchai Sopitkamon, Ph.D. Outline � Experiments with Two Factors (14.1) � Experiments with Three or More Factors (14.2) 1
2/15/2007 Experiments with Two Factors (14.1) � Two-Factor Experimental Designs (14.1.1) � Models for Two-Factor Experiments (14 1 2) � Models for Two-Factor Experiments (14.1.2) � Analysis of Variance Table (14.1.3) � Pairwise Comparisons of the Factor Level Means (14.1.4) Two-Factor Experimental Designs I (14.1.1) � ANOVA in Chapter 11 concerns relationship between a response variable of interest and between a response variable of interest and various levels of a single factor of interest � Objectives: – To simultaneously investigate the relationship between a response variable and various levels of two or more factors of interest – To extend the ANOVA technique to analyze the To extend the ANOVA technique to analyze the structure of the relationship between the response variable and the two factors of interest – To assess any interaction between the two factors where the response variable in influenced 2
2/15/2007 Two-Factor Experimental Designs II (14.1.1) � Investigate how a response variable depends on two factors of interest depends on two factors of interest � Ex: washing quality (response var) of a detergent may depend on two factors: water temperature and detergent type . � Ex: a student’s grade (response var) may depend on two factors: in-class concentration and diligence . t ti d dili Two-Factor Experimental Designs III (14.1.1) A tw o factor experim ent w ith a × b experim ental configurations 3
2/15/2007 Two-Factor Experimental Designs IV: Two-Factor Experiments � Consider an experiment w/ two factors, A and B . If factor A has a levels and factor B and B . If factor A has a levels and factor B has b levels, then there are ab experimental configurations or cells. � In a complete balanced experiment, n replicate measurements are taken at each configuration, resulting in a total sample size of n T = abn data observations. of n = abn data observations � The data observation x ijk represents the value of the k th measurement obtained at the i th level of factor A and the j th level of factor B . Two-Factor Experimental Designs V: Two-Factor Experiments Data observations and sam ple averages for a tw o factor experim ent 4
2/15/2007 al Designs tor Experimenta s Experiments Two-Factor V: Two-Fac E V E T Data set for the car body assem bly line exam ple Models for Two-Factor Experiments I (14.1.2) � An observation x ijk for the k th observation of level i of factor A and level j of factor B can j be written as: x ijk = µ ij + ε ijk where µ ij are unknown parameters of interest in a two-factor experiment or cell means , and ε ijk is the error term with N (0, σ 2 ) � Consider three aspects of the relationship p p between the expected value of the response variable and the two factors of interest. – the interaction effect between factors A and B – the main effect of factor A – the main effect of factor B 5
2/15/2007 Models for Two-Factor Experiments II (14.1.2) Cell m eans for a tw o factor experim ent Models for Two-Factor Experiments III (14.1.2) I nterpretation of m odels for tw o factor experim ents 6
2/15/2007 Models for Two-Factor Experiments IV (14.1.2) I nterpretation of m odels for tw o factor experim ents Models for Two-Factor Experiments V (14.1.2) I nterpretation of m odels for tw o factor experim ents 7
2/15/2007 Models for Two-Factor Experiments VI (14.1.2) I nterpretation of m odels for tw o factor experim ents Models for Two-Factor Experiments VII (14.1.2) I nterpretation of m odels for tw o factor experim ents 8
2/15/2007 nce Table I sis of Varian 3) (14.1.3 Analys The sum of squares decom position for a tw o factor experim ent Analysis of Variance Table II (14.1.3) : SST � Total Sum of Squares (SST) – a measure of the total variability in the data set the total variability in the data set 2 ( ) a b n a b n ∑∑∑ ∑∑∑ = − = − 2 2 SST x x x abn x ijk ... ijk ... = = = = = = 1 1 1 1 1 1 i j k i j k where a b n ∑∑∑ x ijk = = = = i i 1 1 j j 1 1 k k 1 1 overall or grand mean. x : ... n T k -th observation in level i .of factor A and x : ijk level j of factor B n T : total number of observations: abn 9
2/15/2007 The sum of squares decom position for a tw o factor experim ent Analysis of Variance Table II (14.1.3) : SSA � Sum of Squares for Factor A (SSA) – a measure of the variability due to factor A . measure of the variability due to factor A . a a ( ) ∑ ∑ = − = − 2 2 2 SSA bn x x bn x abn x ⋅ ⋅ ⋅ ⋅ i ... i ... = = i 1 i 1 where 1 ∑∑ b n ∑∑ ⋅ = x x : sample averages at each level l h l l ⋅ i ijk bn = = of factor A . j 1 k 1 b : number of levels of factor B n : number of observations at level i of factor A and level j of factor B 10
2/15/2007 Partitioning the Total Sum of Squares VIII (11.1.2) The sum of squares decom position for a tw o factor experim ent Analysis of Variance Table III (14.1.3) : SSB � Sum of Squares for Factor B (SSB) – a measure of the variability due to factor B . measure of the variability due to factor B . ( ) b b ∑ ∑ = − = − 2 2 2 SSB an x x an x abn x ⋅ ⋅ ⋅ ⋅ j ... j ... = = j 1 j 1 where 1 ∑∑ a n 1 ∑∑ ⋅ = sample averages at l x x : ⋅ j ijk an each level of factor B . = = i 1 k 1 a : number of levels of factor A n : number of observations at level i of factor A and level j of factor B 11
2/15/2007 The sum of squares decom position for a tw o factor experim ent Analysis of Variance Table IV (14.1.3) : SSAB � Sum of Squares for Interaction (SSAB) – a measure of the variability due to interaction measure of the variability due to interaction effect between factors A and B . ( ) a b ∑∑ = − − − 2 SSAB n x x x x ⋅ ⋅ ⋅ ⋅ ⋅ ij i j ... = = 1 1 i j where n 1 ∑ ⋅ = Cell averages at level i of factor x x : ij ijk A and level j of factor B . n = k 1 n : number of observations at level i of factor A and level j of factor B 12
2/15/2007 The sum of squares decom position for a tw o factor experim ent Analysis of Variance Table V (14.1.3) : SSE � Sum of Squares for Error (SSE) – a measure of the variability within each of the measure of the variability within each of the ab experimental configuration. ( ) a b n a b n a b ∑∑∑ ∑∑∑ ∑∑ = − = − 2 2 2 SSE x x x n x ⋅ ijk ij . ijk ij = = = = = = = = 1 1 1 1 1 1 1 1 i j k i j k i j where n 1 ∑ ⋅ = Cell averages at level i of factor x x : ij ijk A and level j of factor B . n = k 1 n : number of observations at level i of factor A and level j of factor B 13
2/15/2007 The Analysis of Variance Table VI (14.1.3) Analysis of variance table for a tw o factor experim ent � Reject H 0 if F -statistic > F -critical (from Table IV or Excel’s FINV function) � or Reject H 0 if p -value < α (as in previous chapter) The Analysis of Variance Table VII (14.1.3) : An Example � Ex.9 pg.654: Car Body Assembly Line The analysis of variance table for the car body assem bly line exam ple the car body assem bly line exam ple � No machine main effect (accept H 0 ) � There is solder main effect (reject H 0 ) � No interaction effect between machine and solder (accept H 0 ) Excel sheet 14
2/15/2007 The Analysis of Variance Table VIII (14.1.3) : Another Example � Ex.74 pg.655: Company Transportation Costs The analysis of variance table for the com pany transportation costs exam ple the com pany transportation costs exam ple � There is route main effect (reject H 0 ) � There is period main effect (reject H 0 ) � No interaction effect between the route taken and the period of the day (accept H 0 ) Excel sheet The Analysis of Variance Table IX (14.1.3) : Another Example A plot of the cell sam ple averages for the com pany transportation transportation costs exam ple 15
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