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

2. Applied Statistics Using 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):
17.9, , , , 17.9
18.0, , , , 18.5
18.0, , , , 17.9
BRAND 1 2 3
DEVICE
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]

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

5. 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

6. 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, = 8?
Interaction = degree of difference from sum of separate effects

7. 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).

9. ANOVA Table for Battery Lifetime
General Linear Model: time versus brand, device
Factor Type Levels Values
brand fixed , 2, 3, 4
device fixed , 2, 3
Analysis of Variance for time, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
brand
device
brand*device
Error
Total
S = R-Sq = 66.67% R-Sq(adj) = 36.11%

10. Model Selection Backward model selection:
Fit the full model: Y=A+B+A*B
Remove A*B if not significant; otherwise, stop
Remove the most insignificant main effect until all effects left are significant
Assumption checking for the final model

11. Brand Name Appeal for Men & Women: M F Interesting Example:*
Frontiersman
April
50 people
per cell
Mean Scores
“Frontiersman” “April” “Frontiersman” “April”
Dependent males males females females
Variables (n=50) (n=50) (n=50) (n=50)
Intent-to-
purchase
(*) Decision Sciences”, Vol. 9, p. 470, 1978

13. ANOVA Results Dependent Source d.f. MS F Variable
Intent-to- Sex (A) *
purchase Brand name (B) **
(7 pt. scale) A x B ***
Error
*p <.05
**p <.01
***p <.001

14. Multiple Comparisons
If A*B is not significant (so not included in the final model), conduct multiple comparison procedures as in one-way ANOVA.
If A*B is significant, create one factor, called C, which contains all combinations of A and B, then conduct one-way ANOVA and multiple comparisons on the factor C.

15. Example: Child Activity Level
placebo
ritalin
normal
hyperactive
One group of children is considered as normal and the other as hyperactive.
Each group is randomly divided, with one half receiving a placebo and the other a drug called ritalin.
A measure of activity is determined for each of the children.

16. 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.

17. 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