two-way anova problems

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.