Chapter 111Chapter 11Factorial DesignInteraction Between FactorsWhen a change in one factor produces a different change in the responsevariable at two levels of another factor, there is an interaction betweenthe two factors.It is possible to test for the significance of an interaction only if morethan one observation is taken for each experimental condition.Constructing the ANOVA forTwo-FactorFactorial ExperimentTerms:SSA= Sum of Squares for Factor ASSB= Sum of Squares for Factor BSSAB= Sum of Squares for Interaction of A and BSSE= Sum of Squares for Error (variability within)SST= Total Sum of SquaresSST=SSA+SSB + SSAB + SSE()∑∑∑===-=ricjnkijkXXSST11'12()∑=-=riiXXcnSSA12..'()∑=-=cijXXrnSSB12..'()∑∑==+--=ricjjiijXXXXnSSAB112.....'()∑∑∑===-=ricjnkijijkXXSSE11'12Chapter 112(Fixed Effects) Analysis of Variance for Factorial Design (2-Factor)Source of VariationSum of SquaresDegrees of FreedomMean SquareF0A TreatmentsSSAa-1MSAMSA/MSEB TreatmentsSSBb-1MSBMSB/MSEAB InteractionSSAB(a-1)(b-1)MSABMSAB/MSEErrorSSEab(n-1)MSETotalSSTabn-1Example 1 (Two Factor)An experiment is conducted to study the influence of operatingtemperature (three levels) and three types of face-plate glass in the lightoutput of an oscilloscope tube.RandomizationThere are 9 possible combinations of the factor levels.If there are 3replicates to be run at each combination, then there are 27 observationsto be collected.The order in which the 27 observations are taken mustbe completely randomized.The following data are collected:TemperatureGlass Type100125150580109013921568108713805701085138655010701328253010351312579100012995461045867357510539045991066889