1. Two-way ANOVA problems
Fixed effects analysis in a Two–way ANOVA

2. Problem 5.6 Layout

3. Crossing and Nesting, Balanced
Phosphor Type and Glass Type (the main effects) are crossed
For example, Phosphor Type 1 means the same thing regardless of Glass Type
Since main effects are crossed we can check for interaction
Experimental units are nested within combinations of Phosphor Type and Glass Type
Experimental unit 1 for one combination of Phosphor Type and Glass Type does not mean the same thing as Experimental unit 1 for a different combination of Phosphor Type and Glass Type
Since there are three observations for each cell the design is balanced

4. Fixed and Random effects
Suppose we ran the experiment again.
Phosphor type and Glass type would mean the same thing in another run of the experiment so we call them Fixed Effects in the Model.
The experimental units are not the same, so we call them Random Effects in the Model.

5. Linear Model

6. Parameters and hypotheses

7. Expected mean squares- EMS

8. Problem 5.6 ANOVA
Effect Tests Source DF Sum of Squares F Ratio Prob > F Phos. Type * Glass Type <.0001* Phos. Type*Glass Type

9. Test and do interaction plot, then look at main effects.
F-test and plot of interaction. (F-test and plot of cell means)
If not significant, go on to test Main Effects.

10. Interaction Plot

11. Phosphorous Type

12. Tukey HSD Level Least Sq Mean 2 A 273.33333 1 B 260.00000
Levels not connected by same letter are significantly different.

13. Glass Effect Plot

14. Residuals and Normality Plot

15. Residuals by Predicted

16. Problem 5.10 Layout

17. Crossed and Nested Temperature and Glass are crossed
Can check for Interaction
Experimental units are Nested within Treatment combinations
There are three observations per cell so the design is balanced

18. Linear Model

19. Problem 5.10
Source DF Sum of Squares F Ratio Prob > F Glass Type <.0001* Temp <.0001* Glass *Temp <.0001* Error

20. Interaction Plot

21. LS Means Table (usually put in appendix)
Level Least Sq Mean Std Error 1, , , , , , , , ,

22. Now this is slick… Level Least Sq Mean 1, 150 A 1386.0000
2, B
1, C
3, C
2, C
3, D
3, E
1, E
2, E
Levels not connected by same letter are significantly different.

23. Residuals by Predicted

24. Residual Plot and Normality Plot

25. Normality test Shapiro-Wilk W Test W Prob<W 0.966954 0.5237
Note: Ho = The data is from the Normal distribution. Small p-values reject Ho.