Two-Way ANOVA

16 Pages·2012·0.17 MB·English

by  Xuanyao He

Two-Way ANOVA
Chapter 10
3/25/11 Lecture 24 1
Quick Questions about one-way
ANOVA and RCBD
•  Is there a difference in the average
competition scores for the population with
black, white, and other shoe colors?

•  Is there a difference in the 4 detergents, if
stain is another factor in our study but not
of interest?

3/25/11 Lecture 24 2
Brief look at Two-way ANOVA
•  Self-Reading: Section 10.1
•  Now we have two full factors of interest
–  Factor A effect
–  Factor B effect
•  Interaction between factors is possible
–  Interaction AB effect
•  Always test the interaction first, if the
interaction is significant, must discuss the
results as an interaction effect!
3/25/11 Lecture 24 3
ANOVA formulas

•  Suffice it to say that the formulas aren’t
anything different from what we are used to,
they just get more complex as summations
•  For example: a
SSA = br∑ ( A − x )2
i
i=1
•  Read Page 445-446 in the text.
•  Still have SST = SSA + SSB + SSAB + SSE
•  MS’s still SS/df
•  Still testing over the MSE.
3/25/11 Lecture 24 4
Two-way ANOVA Table
Source DF SS MS
Factor A a – 1 SSA MSA
Factor B b – 1 SSB MSB
AB interaction (a – 1)(b – 1) SSAB MSAB
Error ab(r – 1) SSE MSE
Total abr – 1 SST
3/25/11 Lecture 24 5
Three F tests— 1. Interaction effect?
•  H : There is no interaction effect
0
H : There is a significant interaction effect
a
MSAB
df = (a – 1)(b – 1), ab(r – 1)
F =
MSE
•  If we reject the null hypothesis, it shows that there
is a significant interaction
–  The next two tests aren’t as useful!!!
•  MUST interpret the results as an interaction effect.
–  Look at one Factor while fixing the other (“Slicing”)
3/25/11 Lecture 24 6
Three F tests— 2. Factor A effect?
•  H : There is no Factor A main effect
0
H : There is a significant Factor A main effect
a
MSA
F =
df = a – 1, ab(r – 1)
MSE
•  If we reject the null hypothesis, it shows that
Factor A is significant
–  IF there is an interaction effect, still must slice!
•  Result for this test don’t mean much
–  IF no interaction then:
•  Results and interpretations are similar to One-Way ANOVA
•  Can follow up with multiple comparisons to see exactly which
groups are significantly different.
3/25/11 Lecture 24 7
Three F tests— 3. Factor B effect?
•  H : There is no Factor B main effect
0
H : There is a significant Factor B main effect
a
MSB
F =
df = b – 1, ab(r – 1)
MSE
•  If we reject the null hypothesis, it shows that
Factor B is significant
–  IF there is an interaction effect, still must slice!
•  Result for this test don’t mean much
–  IF no interaction then:
•  Results and interpretations are similar to One-Way ANOVA
•  Can follow up with multiple comparisons to see exactly which
groups are significantly different.
3/25/11 Lecture 24 8
Example
•  We are interested in
Casing Casing Casing
what is causing a
1 2 3
vibration on an
13.1 15.0 14.0
Brand 1
13.2 14.8 14.3
electric motor. Brand
and Casing are both 16.3 15.7 17.2
Brand 2
15.8 16.4 16.7
potential causes, the
13.7 13.9 12.4
response is the
Brand 3
14.3 14.3 12.3
amount of vibration
15.7 13.7 14.4
in microns. Brand 4
15.8 14.2 13.9
13.5 13.4 13.2
Brand 5
12.5 13.8 13.1
3/25/11 Lecture 24 9
Example (cont) using SAS
•  SAS code
proc glm data=emotor;
class brand casing;
model vibration = brand|casing;
run;
3/25/11 Lecture 24 10