Fixed effects analysis in a Two – way ANOVA

Two-way ANOVA analysis focusing on fixed effects with crossed main effects and nested experimental units. Understand the parameters, hypotheses, expected mean squares, and ANOVA effect tests to evaluate Phosphor Type, Glass Type, and their interaction. Perform interaction plots and examine main effects to draw meaningful conclusions.

• Fixed Effects
• Experimental Design

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1. Fixed effects analysis in a Twoway ANOVA 1

2. Problem 5.6 Layout 2

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 3

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. 4

5. Linear Model = + + + + Y P G PG ( ) ij k ijk i j ij 5

6. Parameters and hypotheses 6

7. Expected mean squares- EMS 7

8. Problem 5.6 ANOVA Effect Tests Source DF Sum of Squares F Ratio Prob > F Phos. Type Glass Type 2 933.33 8.8421 0.0044* 1 14450.0 273.78 <.0001* 1.2632 0.3178 Phos. Type*Glass Type 2 133.333 8

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. 9

10. Interaction Plot 10

11. Phosphorous Type 11

12. Tukey HSD Level 2 A 1 3 Least Sq Mean 273.33333 260.00000 256.66667 B B Levels not connected by same letter are significantly different. 12

13. Glass Effect Plot 13

14. Residuals and Normality Plot 14

15. Residuals by Predicted 15

16. Problem 5.10 Layout 16

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 17

18. Linear Model = + + + + Y G T GT ( ) ij k ijk i j ij 18

19. Problem 5.10 Source DF Sum of Squares F Ratio Prob > F Glass Type 2 150864.5 Temp. 2 1970334.5 Glass *Temp. 4 290551.7 Error 18 2418330.1 206.3706 2695.259 198.7257 <.0001* <.0001* <.0001* 19

20. Interaction Plot 20

21. LS Means Table (usually put in appendix) Level Least Sq Mean Std Error 1, 100 572.6667 1, 125 1087.3333 1, 150 1386.0000 2, 100 553.0000 2, 125 1035.0000 2, 150 1313.0000 3, 100 573.3333 3, 125 1054.6667 11.038093 11.038093 11.038093 11.038093 11.038093 11.038093 11.038093 11.038093 11.038093 3, 150 886.6667 21

22. Now this is slick Level 1, 150 2, 150 B 1, 125 3, 125 2, 125 3, 150 3, 100 1, 100 2, 100 Least Sq Mean 1386.0000 1313.0000 1087.3333 1054.6667 1035.0000 886.6667 573.3333 572.6667 553.0000 A C C C D E E E Levels not connected by same letter are significantly different. 22

23. Residuals by Predicted 23

24. Residual Plot and Normality Plot 24

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