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 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

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.

Linear Model

Parameters and hypotheses

Expected mean squares- EMS

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

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.

Interaction Plot

Phosphorous Type

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

Glass Effect Plot

Residuals and Normality Plot

Residuals by Predicted

Problem 5.10 Layout

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

Linear Model

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

Interaction Plot

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

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.

Residuals by Predicted

Residual Plot and Normality Plot

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