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Two-way ANOVA problems

Two-way ANOVA problems. Fixed effects analysis in a Two–way ANOVA. Problem 5.6 Layout. 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

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Two-way ANOVA problems

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  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. Linear Model

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

  6. Interaction Plot

  7. Phosphorous Type

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

  9. Glass Effect Plot

  10. Residuals and Normality Plot

  11. Residuals by Predicted

  12. Problem 5.10 Layout

  13. 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

  14. Linear Model

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

  16. Interaction Plot

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

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

  19. Residuals by Predicted

  20. Residual Plot and Normality Plot

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

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