This document discusses two-way analysis of variance (ANOVA). It explains that two-way ANOVA allows researchers to study the effects of two independent variables on a single dependent variable. Researchers can test for main effects of each independent variable as well as interactions between the variables. The document provides examples of how to set up a two-way ANOVA study, calculate the relevant statistics, interpret results from ANOVA tables, and draw conclusions about significant main effects and interactions.

1. Two independent variables

2. • Often, we wish to study 2 (or more) factors in a
single experiment
– Compare two or more treatment protocols
– Compare scores of people who are young, middle-aged,
and elderly
• The baseline experiment will therefore have two
factors as Independent Variables
– Treatment type
– Age Group

3.  Factorial (Two or more way) ANOVA
• One dependent variable
 interval or ratio with a normal distribution
• Two independent variables
 nominal (define groups), and independent of each other
• Three hypothesis tests:
 Test effect of each independent variable controlling for
the effects of the other independent variable
 One: H0: Treatment type has no impact on Outcome
 Two: H0: Age Group has no impact on Outcome
 Three: Test interaction effect for combinations of categories
H0: Treatment and Age Group interact in affecting Outcome

4.  First stage
• Identical to independent samples ANOVA
• Compute SSTotal, SSBetween treatments and
SSWithin treatments
 Second stage
• Partition the SSBetween treatments into three
separate components, differences attributable
to Factor A, to Factor B, and to the AxB
interaction

6.  The validity of the ANOVA presented in
this chapter depends on three
assumptions common to other hypothesis
tests
1. The observations within each sample must
be independent of each other
2. The populations from which the samples are
selected must be normally distributed
3. The populations from which the samples are
selected must have equal variances
(homogeneity of variance)

7. Total Variability
Between
Treatments
Factor
A
Factor B Interaction
Within
Treatments

8. G
N
SS X total
2
2  
2 2
G
N
T
  
n
SS between treatments
 withintreatments inside each treatment SS SS

10.  Factorial designs
• Consider more than one factor
• Joint impact of factors is considered.
 Three hypotheses tested by three F-ratios
• Each tested with same basic F-ratio structure
variance (differenc es) between treatments
variance (differenc es) expected with no treatment effect
F 

11. • Factor 1 (independent variable, e.g. type of crop)
• Always nominal or ordinal (it defines distinct groups)
• Factor 2 (independent variable, e.g., fertilizer)
• Always nominal or ordinal (it defines distinct groups)
• Outcome (dependent variable, e.g. yield)
• Always interval or ratio
• Mean Outcomes of the
groups defined by Factor 1
and Factor 2 are being
compared.

12.  Mean differences among levels of one factor
• Differences are tested for statistical significance
• Each factor is evaluated independently of the
other factor(s) in the study
 

A A
1 2
1 2
:
:
0
1
A A
H
H
 

 

B B
1 2
1 2
:
:
0
1
B B
H
H
 


13.  Not the same as experimental control.
 Statistical control: we look for the effect of
one independent variable within each group of
the other dependent variable.
 This removes the impact of the other
independent variable.
 Sometimes a variable which showed no
significant effect in a Oneway ANOVA becomes
significant if another effect is controlled.

14.  The mean differences between individuals
treatment conditions, or cells, are different
from what would be predicted from the
overall main effects of the factors
 H0: There is no interaction between
Factors A and B
 H1: There is an interaction between
Factors A and B

15. • First:
• Does Factor 1 have any
impact on the Outcome?
• Null: The groups defined by
Factor 1 will have the same
Mean Outcome.
• Second:
• Does Factor 2 have any impact on the Outcome?
• Null: The groups defined by Factor 2 will have the
same Mean Outcome.
• Third:
• Do Factor 1 and Factor 2 interact in influencing
Outcome?
• Null: No combination of Factor 1 and Factor 2
produces unusually high or unusually low mean
Outcome scores.

16. The equations come later!

17. From one-way to two-way designs:
• Often, we wish to study 2 (or more) factors in a
single experiment
– Compare a new and standard style of noise filter (inside
a muffler) on a car
– The size of the car might also be an important factor in
noise level.
• The baseline experiment will therefore have two
factors as Independent Variables
– Type of noise filter (Octel vs Standard)
– Size of car (Small, Midsize, Large)

18. Standard Filter Octel Filter
Type of Noise Filter
860
850
840
830
820
810
800
790
780
770
760
Noise Level Reading
Group Statistics
18 815.56 32.217
18 804.72 25.637
Type of Noise Fil ter
Standard Fi l ter
Octel Fi l ter
Noise Level Reading
N Mean Std. Deviation
 First Variable: Filter Type
• Nominal – Dichotomy
• Dependent variable is
noise level (ratio level)
 Test: Two-Sample t
• Compare means (above)
• View boxplot (at right)
• t (34)=1.116, p = .272
 RETAIN H0
 Type of filter does not cause a
significant difference in noise.

19. N Mean Std. Dev'n
12 824.17 7.638
12 833.75 13.505
12 772.50 10.335
36 810.14 29.216
Smal l
Mid-Size
Large
Total
 Second Variable: Car Size
• Nominal – 3 groups
• Dep.Var: noise level (ratio)
 Test: Oneway ANOVA
• Compare means (above)
• View boxplot (at left)
• F (2,33) =112.44, p < .0005
 REJECT H0
 Size of car is related to a significant
difference in noise.
Small Mid-Size Large
Size of Car
860
840
820
800
780
760
Noise Level Reading

20. ANOVA is significant, so we need Post-hoc Tests.
Groups: Same Size so Test: Tukey HSD
- Small vs Large = Sig.
- Midsize vs Large = Sig.
- Small vs Midsize = n.s.
Multiple Comparisons
Dependent Variable: Noise Level Reading
Tukey HSD
-9.583 4.394 .089
51.667* 4.394 .000
9.583 4.394 .089
61.250* 4.394 .000
-51.667* 4.394 .000
-61.250* 4.394 .000
(J) Size of Car
Mid-Size
Large
Smal l
Large
Smal l
Mid-Size
(I) Size of Car
Smal l
Mid-Size
Large
Mean
Di fference
(I-J) Std. Error Sig.
The mean di fference is significant *. at the .05 level.

21.  Filters – Octel vs Standard
• Independent sample t-test
• No significant differences
 Size of Car – Small, Midsize, Large
• ANOVA
• Significant differences
• Large cars are significantly more quiet
 BUT – is it possible that the Octel filter might
work better with just one of the types of
cars?

22.  Is car size related to noise level, if
effect of filter type is controlled?
 Is filter type related to noise level, if
effect of size of car is controlled?
 Is there a combination of Size of Car and
Noise Filter Type that is especially loud, or
especially soft?
• called an INTERACTION effect.
 Multiple comparison tests

23. Small Mid-Size Large
Size of Car
860
840
820
800
780
760
Noise Level Reading (Decibels)
Type of Noise Filter
Standard Filter
Octel Filter

24.  Factorial (Two or more way) ANOVA
• One dependent variable
 interval or ratio
 normal distribution
• Two independent variables
 nominal (define groups)
 independent of each other
• Test effect of each I.V. controlling for the effects
of the other I.V.
• Test interaction effect for combinations of
categories

25. Tests of Between-Subjects Effects
Dependent Variable: Noise Level Reading (Decibels)
Type III Sum
of Squares df Mean Square F Sig.
23655612.5a 6 3942602.083 60269.076 .000
26051.389 2 13025.694 199.119 .000
1056.250 1 1056.250 16.146 .000
804.167 2 402.083 6.146 .006
1962.500 30 65.417
23657575.0 36
Source
Model
size
type
size * type
Error
Total
R Squared = 1.000 (Adjusted a. R Squared = 1.000)
 SIZE effect is still significant
 TYPE effect is significant when size is controlled
 INTERACTION effect is significant
• There is a combination which shows more than the combined
impact of SIZE and TYPE

26. Means of each combination of Size & Type
INTERACTION: Whenever lines not parallel

27. Manufacturers of the new Octel noise filter claim that it
reduces noise levels in cars of all sizes. In a Two-Way
ANOVA, this claim proved to be true. The Size of Car
effect was significant (F(2,36) = 199.119, p < .001). When
the impact of size was controlled, the Filter Type effect
was also significant (F(1,36) = 16.146, p < .001), with the
Octel Filter having lower noise levels than standard filters.
The Interaction effect was also significant (F(2,30) =
6.146, p = .006). For Small cars, the noise difference
between filter types was 3.33; for Large cars it was 5.000,
but Midsize cars with the Octel filter averaged 24.166
points lower on the Noise Level scale.
A complete report would include the Mean and SD of each cell where a
significant difference occurred, either in a table or in narrative. It
would include effect size (η2) for significant effects.

28.  A table or graph of group means
 A report of the three hypothesis tests:
• One for Factor A
• One for Factor B
• One for the interaction of A with B
 Asterisks often used to report hypothesis test results
* = significant with alpha = .05
** = significant with alpha = .01
*** = significant with alpha = .001
 If there are more than two factors, there will be more
hypothesis tests for factors, and more interactions.

29. Total Variability
Between
Treatments
Factor
A
Factor B Interaction
Within
Treatments

30.  Three distinct tests
• Main effect of Factor A
• Main effect of Factor B
• Interaction of A and B
 A separate F test is conducted for each

31. Notation describes procedure
Tables usually used to present
the results
Group means (cell)
Row means
Column means
Each factor is operationalized by one
or more variables (measures)
Images from Trochim’s Research Methods Knowledge Base at
http://www.socialresearchmethods.net/kb/index.php

32.  Plot the means of each group (defined as a
combination of Factor 1 and Factor 2)
 If all the null hypotheses are true, all the points
will have about the same Mean Outcome level.

33.  The two row means
are the same
 The two column
means are the
same
 All groups have the
same mean score
 Neither factor had
any effect
Images from Trochim’s Research Methods Knowledge Base at
http://www.socialresearchmethods.net/kb/index.php

34.  Row means: the same
 Column means: differ
 No score especially high
or especially low
 Row means: differ
 Column means: the same
 No score especially high
or especially low
Images from Trochim’s Research Methods Knowledge Base at
http://www.socialresearchmethods.net/kb/index.php

35.  The mean differences between individuals
treatment conditions, or cells, are different
from what would be predicted from the
overall main effects of the factors
 H0: There is no interaction between
Factors A and B
 H1: There is an interaction between
Factors A and B

36.  Row means differ
 Column means differ
 One group is different
 Others are the same
 Row means the same
 Column means the same
 Graph shows that pattern
in one factor depends on
the
status of the other Images from Trochim’s Research Methods Knowledge Base at
http://www.socialresearchmethods.net/kb/index.php

37.  Dependence of factors
• The effect of one factor depends on the level
or value of the other
 Non-parallel lines (cross or converge) in a
graph
• Indicate interaction is occurring
 Typically called the A x B interaction

38. Total Variability
Between
Treatments
Factor
A
Factor B Interaction
Within
Treatments

39. We will compute problems by hand to gain
understanding, but not on a test

40. G
N
SS X total
2
2  
2 2
G
N
T
  
n
SS between treatments
 withintreatments inside each treatment SS SS

41. Total Variability
Between
Treatments
Factor
A
Factor B Interaction
Within
Treatments

42. dftotal = N – 1
dfwithin treatments = Σdfinside each treatment
dfbetween treatments = k – 1
dfA = number of rows – 1
dfB = number of columns– 1
dfAxB = dfbetween treatments – dfA – dfB

43. B
SS
SS
MS   
A df
within treatments
SS
within treatments
A
MS 
within treatments df
AxB
AxB
AxB
B
B
A
SS
MS
df
MS
df
AxB
within
MS
MS
F   
A AxB
MS
B
within
B
A
within
MS
F
MS
F
MS

44.  η2, is computed as the percentage of
variability not explained by other factors.
A
SS
A SS SS
A within treatments
A
total B AxB
SS
SS SS SS


 
 2 
B
SS
B SS SS
B within treatments
B
total A AxB
SS
SS SS SS


 
 2 
AxB
SS
AxB SS SS
AxB within treatments
AxB
total A B
SS
SS SS SS


 
 2 

45. Two independent variables