1. Joyful mood is a meritorious deed that cheers up people around you like the showering of cool spring breeze.

2. Applied Statistics Using SAS and SPSS Topic: Two Way ANOVA By Prof Kelly Fan, Cal State Univ, East Bay

3. Two Way ANOVA Consider studying the impact of two factors on the yield (response): Here we have R = 3 rows (levels of the Row factor), C = 4 (levels of the column factor), and n = 2 replicates per cell [n ij for (i,j) th cell if not all equal] NOTE: The “1”, “2”,etc... mean Level 1, Level 2, etc..., NOT metric values NOTE: The “1”, “2”,etc... mean Level 1, Level 2, etc..., NOT metric values 1234 17.9, 18.1 17.8, 17.8 18.1, 18.2 17.8, 17.9 18.0, 18.2 18.0, 18.3 18.4, 18.1 18.1, 18.5 18.0, 17.8 17.8, 18.0 18.1, 18.3 18.1, 17.9 BRAND 123123 DEVICE

4. Statistical model: i = 1,..., R j = 1,..., C k= 1,..., n In general, n observations per cell, R C cells. Y ijk =  ij  ijk

5. 1)H o : All Row Means  i. Equal H 1 : Not all Row Means Equal 2)H o : All Col. Means . j Equal H 1 : Not All Col. Means Equal 3)H o : No Interaction between factors H 1 : There is interaction between factors 1)H o : Level of row factor has no impact on Y H 1 : Level of row factor does have impact on Y 2)H o : Level of column factor has no impact on Y H 1 : Level of column factor does have impact on Y 3)H o : The impact of row factor on Y does not depend on column H 1 : The impact of row factor on Y depends on column

6. Two basic ways to look at interaction: B L B H A L 58 A H 10? If A H B H = 13, no interaction If A H B H > 13, + interaction If A H B H < 13, - interaction -When B goes from B L  B H, yield goes up by 3 (5  8). - When A goes from A L  A H, yield goes up by 5 (5  10). -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 INTERACTION 1)

7. 2) -Holding B L, what happens as A goes from A L  A H ?+5 -Holding B H, what happens as A goes from A L  A H ?+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. B L B H A L 58 A H 1017 NOTE: - Holding A L, B L B H has impact + 3 - Holding A H, B L B H has impact + 7 (AB) = (BA) or (9-5) = (7-3).

9. 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 DF Seq SS Adj SS Adj MS F P brand 3 0.21000 0.21000 0.07000 2.80 0.085 device 2 0.28000 0.28000 0.14000 5.60 0.019 brand*device 6 0.11000 0.11000 0.01833 0.73 0.633 Error 12 0.30000 0.30000 0.02500 Total 23 0.90000 S = 0.158114 R-Sq = 66.67% R-Sq(adj) = 36.11%

10. 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

11. MFMF Interesting Example:* Frontiersman April 50 people per cell Mean Scores “Frontiersman” “April” “Frontiersman” “April” Dependentmalesmalesfemalesfemales Variables(n=50)(n=50)(n=50) (n=50) Intent-to- purchase4.443.502.044.52 (*) Decision Sciences”, Vol. 9, p. 470, 1978 Brand Name Appeal for Men & Women:

13. ANOVA Results DependentSource d.f. MS F Variable Intent-to-Sex (A)123.805.61* purchaseBrand name (B)129.646.99** (7 pt. scale)A x B1146.2134.48*** Error1964.24 *p<.05 **p<.01 ***p<.001

14. 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 tempera- ture and mode of use is not necessarily superior at other treatment combination. The batteries were being tested at 4 diffe- rent temperatures for three modes of use (I for intermittent, C for continuous, S for sporadic). Analyze the data.

15. Battery Lifetime (2 replicates) Mode of use 1234 I12, 1615, 1931, 3953, 55 C15, 1917, 1730, 3451, 49 S11, 1724, 2233, 3761, 67 Temperature