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Chapter 11 1Chapter 11 Factorial Design Interaction Between Factors When a change in one factor produces a different change in the response variable at two levels of another factor, there is an interaction between the two factors. It is possible to test for the significance of an interaction only if more than one observation is taken for each experimental condition.Constructing the ANOVA for Two-FactorFactorial Experiment Terms: SSA= Sum of Squares for Factor A SSB= Sum of Squares for Factor B SSAB= Sum of Squares for Interaction of A and B SSE= Sum of Squares for Error (variability within) SST= Total Sum of Squares SST= SSA+ SSB + SSAB + SSE()∑∑∑===-=ricjnkijkXXSST11'12()∑=-=riiXXcnSSA12..'()∑=-=cijXXrnSSB12..'()∑∑==+--=ricjjiijXXXXnSSAB112.....'()∑∑∑===-=ricjnkijijkXXSSE11'12Chapter 11 2(Fixed Effects) Analysis of Variance for Factorial Design (2-Factor) Source of Variation Sum of Squares Degrees of Freedom Mean Square F0A Treatments SSAa-1 MSAMSA/MSEB Treatments SSBb-1 MSBMSB/MSEAB Interaction SSAB(a-1)(b-1) MSABMSAB/MSEError SSEab(n-1) MSETotal SSTabn-1 Example 1 (Two Factor)An experiment is conducted to study the influence of operating temperature (three levels) and three types of face-plate glass in the light output of an oscilloscope tube. Randomization There are 9 possible combinations of the factor levels. If there are 3 replicates to be run at each combination, then there are 27 observations to be collected. The order in which the 27 observations are taken must be completely randomized. The following data are collected: Temperature Glass Type 100 125 150 580 1090 1392 1 568 1087 1380 570 1085 1386 550 1070 1328 2 530 1035 1312 579 1000 1299 546 1045 867 3 575 1053 904 599 1066 889