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Statistical Package Usage Topic: Two-Way ANOVA By Prof Kelly Fan, Cal State Univ, East Bay

Two Way ANOVA Yijk = ijijk Statistical model: Consider studying the impact of two factors (row and column) on the yield (response). Statistical model: NOTE: The “1”, “2”,etc... mean Level 1, Level 2, etc..., NOT metric values Yijk = ijijk i = 1, ..., R (row level) j = 1, ..., C (column level) k= 1, ..., n (replicates) 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 mi. Equal H1: Not all Row Means Equal 2) Ho: All Col. Means m.j Equal H1: Not All Col. Means Equal 3) Ho: No Interaction between factors H1: There is interaction between factors

Example: 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.

Battery Lifetime (2 replicates) Temperature Mode of use 1 2 3 4 I 12, 16 15, 19 31, 39 53, 55 C 17, 17 30, 34 51, 49 S 11, 17 24, 22 33, 37 61, 67

Anova Table

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

BL BH AL 5 8 AH 10 17 2) - 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 BH has impact + 3 - Holding AH, BL BH has impact + 7 (AB) = (BA) or (9-5) = (7-3).

Example: Battery

Model Selection Find the final model (with or without interaction term) Assumption checking only for the final model

Remarks Only when the interaction term is not significant, we can conduct multiple comparison procedures or tests of contrasts for either row or column factors.