1. Two-level Factorial Designs Bacteria Example: Bacteria Example: –Response: Bill length –Factors:  B: Bacteria (Myco, Control)  T: Room Temp (Warm, Cold)  I: Inoculation (Eggs, Chicks) Yandell, B. (2002) Practical Data Analysis for Designed Experiments, Chapman & Hall, London

2. Two-level Factorial Designs BacteriaTemp.EggChick ControlCold39.7740.23 MycoCold39.1938.95 ControlWarm40.3741.71 MycoWarm40.2140.78

3. Cube Plot + Temp Inoculation W C C E C 39.77 40.78 40.21 41.71 38.9540.23 39.19 40.37 Bacteria M

4. Estimated Effects For a k-factor design with n replicates, the cell means are estimated as For a k-factor design with n replicates, the cell means are estimated as We can write any effect as a contrast; interaction contrasts are obtained by element-wise multiplication of main effect contrast coefficients. We can write any effect as a contrast; interaction contrasts are obtained by element-wise multiplication of main effect contrast coefficients.

5. Estimated Effects The resulting contrasts are mutually orthogonal. The resulting contrasts are mutually orthogonal. The contrasts (up to a scaling constant) can be summarized as a table of ±1’s. The contrasts (up to a scaling constant) can be summarized as a table of ±1’s.

6. Orthogonal Contrast Coefficients

7. Estimated Effects If we code contrast coefficients as ±1, the estimated effects are: If we code contrast coefficients as ±1, the estimated effects are: These effects are twice the size of our usual ANOVA effects. These effects are twice the size of our usual ANOVA effects.

8. Estimated Effects The sum of squares for the estimated effect can be computed using the sum of squares formula we learned for contrasts The sum of squares for the estimated effect can be computed using the sum of squares formula we learned for contrasts

9. Estimated Effects Bacteria Example Bacteria Example B effect=(39.19+38.95+40.21+40.78-39.77- 40.23-40.37-41.71)/4 =-.7375 =-.7375 SSB=(-.7375) 2 x2=1.088 The entire ANOVA table for this example can be constructed in this way The entire ANOVA table for this example can be constructed in this way

11. Testing Effects With replication (n>1) With replication (n>1) Without replication (k large) Without replication (k large) –Claim higher-order interactions are negligible and pool them –For k=6, if 3-way (and higher) interactions are negligible, 42 d.f. would be available for error

12. Testing Effects Without replication--Normal Probability Plots Without replication--Normal Probability Plots –If none of the effects is significant, the effects are orthogonal normal random variables with mean 0 and variance

13. Testing Effects Because the effects are normal, they are also independent Because the effects are normal, they are also independent IID normal effects can be “tested” using a normal probability plot (Minitab Example) IID normal effects can be “tested” using a normal probability plot (Minitab Example) Yandell uses a half-normal plot Yandell uses a half-normal plot You can pool values on the line as error and construct an ANOVA table You can pool values on the line as error and construct an ANOVA table

14. Testing Effects Lenth (1989) developed a more formal test of effects. Lenth (1989) developed a more formal test of effects. Denote the effects by e i, i=1,…,m. Denote the effects by e i, i=1,…,m. We say that the e i ’s are iid N(0,   ), where  is their common standard error. We say that the e i ’s are iid N(0,   ), where  is their common standard error.

15. Testing Effects Lenth develops two estimates of the common standard error, , of the c i ’s: Lenth develops two estimates of the common standard error, , of the c i ’s:

16. Testing Effects Though both are consistent estimates, PSE is more robust Though both are consistent estimates, PSE is more robust The following terms are used to test effects The following terms are used to test effects

17. Testing Effects The df term was developed from a study of the empirical distribution of PSE 2 The df term was developed from a study of the empirical distribution of PSE 2 ME is a 1-  confidence bound for the absolute value of a single effect ME is a 1-  confidence bound for the absolute value of a single effect SME is an exact (since the effects are independent) simultaneous 1-  confidence bound for all m effects SME is an exact (since the effects are independent) simultaneous 1-  confidence bound for all m effects