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Joyful mood is a meritorious deed that cheers up people around you like the showering of cool spring breeze. Applied Statistics Using SAS and SPSS Topic: Two Way ANOVA By Prof Kelly Fan, Cal State Univ, East Bay Two Way ANOVA Consider studying the impact of two factors on the yield (response): BRAND 1 2 DEVICE 3 4 1 17.9, 18.1 17.8, 17.8 18.1, 18.2 17.8, 17.9 2 18.0, 18.2 18.0, 18.3 18.4, 18.1 18.1, 18.5 3 18.0, 17.8 17.8, 18.0 18.1, 18.3 18.1, 17.9 NOTE: The “1”, “2”,etc... mean Level 1, Level 2, etc..., NOT metric values Here we have R = 3 rows (levels of the Row factor), C = 4 (levels of the column factor), and n = 2 replicates per cell [nij for (i,j)th cell if not all equal] Statistical model: Yijk = ijijk i = 1, ..., R j = 1, ..., C k= 1, ..., n In general, n observations per cell, R • C cells. 1) Ho: Level of row factor has no impact on Y H1: Level of row factor does have impact on Y 2) Ho: Level of column factor has no impact on Y H1: Level of column factor does have impact on Y 3) Ho: The impact of row factor on Y does not depend on column H1: The impact of row factor on Y depends on column 1) Ho: All Row Means i. Equal H1: Not all Row Means Equal 2) Ho: All Col. Means .j Equal H1: Not All Col. Means Equal 3) Ho: No Interaction between factors H1: There is interaction between factors INTERACTION Two basic ways to look at interaction: 1) AL AH BL BH 5 10 8 ? If AHBH = 13, no interaction If AHBH > 13, + interaction If AHBH < 13, - interaction - When B goes from BLBH, yield goes up by 3 (58). - When A goes from AL AH, yield goes up by 5 (510). - When both changes of level occur, does yield go up by the sum, 3 + 5 = 8? Interaction = degree of difference from sum of separate effects BL 2) AL AH 5 10 BH 8 17 - Holding BL, what happens as A goes from AL AH? +5 - Holding BH, what happens as A goes from AL AH? +9 If the effect of one factor (i.e., the impact of changing its level) is DIFFERENT for different levels of another factor, then INTERACTION exists between the two factors. NOTE: - Holding AL, BL - Holding AH, BL BH has impact + 3 BH has impact + 7 (AB) = (BA) or (9-5) = (7-3). ANOVA Table for Battery Lifetime General Linear Model: time versus brand, device Factor Type Levels Values brand fixed 4 1, 2, 3, 4 device fixed 3 1, 2, 3 Analysis of Variance for time, using Adjusted SS for Tests Source brand device brand*device Error Total DF 3 2 6 Seq SS 0.21000 0.28000 0.11000 12 23 Adj SS 0.21000 0.28000 0.11000 0.30000 0.90000 Adj MS F 0.07000 2.80 0.14000 5.60 0.01833 0.73 0.30000 0.02500 S = 0.158114 R-Sq = 66.67% R-Sq(adj) = 36.11% P 0.085 0.019 0.633 Model Selection Backward model selection: 1. Fit the full model: Y=A+B+A*B 2. Remove A*B if not significant; otherwise, stop 3. Remove the most insignificant main effect until all effects left are significant Assumption checking for the final model Brand Name Appeal for Men & Women: M F Interesting Example:* Frontiersman 50 people per cell April Mean Scores “Frontiersman” Dependent males Variables (n=50) Intent-topurchase 4.44 “April” males (n=50) “Frontiersman” females (n=50) 3.50 (*) Decision Sciences”, Vol. 9, p. 470, 1978 2.04 “April” females (n=50) 4.52 Interaction Plot - Data Means for y brand 1 2 Mean 4 3 2 1 2 gender Y 12 ANOVA Results Dependent Variable Source Intent-topurchase (7 pt. scale) Sex (A) 1 Brand name (B) 1 AxB 1 Error 196 *p<.05 **p<.01 ***p<.001 d.f. MS 23.80 29.64 146.21 4.24 F 5.61* 6.99** 34.48*** Exercise: Lifetime of a Special-purpose Battery It is important in battery testing to consider different temperatures and modes of use; a battery that is superior at one temperature and mode of use is not necessarily superior at other treatment combination. The batteries were being tested at 4 different temperatures for three modes of use (I for intermittent, C for continuous, S for sporadic). Analyze the data. Battery Lifetime (2 replicates) Temperature Mode of use 1 2 3 4 I 12, 16 15, 19 31, 39 53, 55 C 15, 19 17, 17 30, 34 51, 49 S 11, 17 24, 22 33, 37 61, 67