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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 - PowerPoint PPT Presentation
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Two-way ANOVA problemsFixed effects analysis in a Two–way ANOVA
Problem 5.6 Layout
Crossing and Nesting, BalancedPhosphor Type and Glass Type (the main effects) are
crossedFor example, Phosphor Type 1 means the same thing
regardless of Glass TypeSince main effects are crossed we can check for interactionExperimental units are nested within combinations of
Phosphor Type and Glass TypeExperimental 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
Linear Model
( )ijk i j ij ij kY P G PG
Problem 5.6 ANOVAEffect TestsSource 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
Interaction Plot
Phosphorous Type
Tukey HSD
Level Least Sq Mean2 A 273.333331 B 260.000003 B 256.66667
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 NestedTemperature and Glass are crossedCan check for InteractionExperimental units are Nested within Treatment
combinationsThere are three observations per cell so the
design is balanced
Linear Model
( )ijk i j ij ij kY G T GT
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 Error1, 100 572.6667 11.0380931, 125 1087.3333 11.0380931, 150 1386.0000 11.0380932, 100 553.0000 11.0380932, 125 1035.0000 11.0380932, 150 1313.0000 11.0380933, 100 573.3333 11.0380933, 125 1054.6667 11.0380933, 150 886.6667 11.038093
Now this is slick…
Level Least Sq Mean1, 150 A 1386.00002, 150 B 1313.00001, 125 C 1087.33333, 125 C 1054.66672, 125 C 1035.00003, 150 D 886.66673, 100 E 573.33331, 100 E 572.66672, 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
Note: Ho = The data is from the Normal distribution. Small p-values reject Ho.