two-way anova overview & spss interpretation
DESCRIPTION
Overview for a statistics courseTRANSCRIPT
Two independent variables
• 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 twofactors as Independent Variables– Treatment type
– Age Group
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 categoriesH0: Treatment and Age Group interact in affecting Outcome
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
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)
Total Variability
Between Treatments
Factor A
Factor B Interaction
Within Treatments
N
GXSStotal
22
treatment each insidetreatmentswithin SSSS
N
G
n
TSS treatmentsbetween
22
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
effect treatment no withexpected es)(differenc variance
treatments between es)(differenc varianceF
• 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.
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
21
21
:
:
1
0
AA
AA
H
H
21
21
:
:
1
0
BB
BB
H
H
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 nosignificant effect in a Oneway ANOVA becomes significant if another effect is controlled.
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
• 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.
The equations come later!
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 twofactors as Independent Variables– Type of noise filter (Octel vs Standard)
– Size of car (Small, Midsize, Large)
Standard Filter Octel Filter
Type of Noise Filter
760
770
780
790
800
810
820
830
840
850
860
No
ise L
eve
l R
ea
din
g
Group Statistics
18 815.56 32.217
18 804.72 25.637
Type of Noise Filter
Standard Filter
Octel Filter
Noise Level Reading
N Mean Std. Deviation
First Variable: Filter Type
• Nominal – Dichotomy
• Dependent variable isnoise 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.
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.
Noise Level Reading
12 824.17 7.638
12 833.75 13.505
12 772.50 10.335
36 810.14 29.216
Small
Mid-Size
Large
Total
N Mean Std. Dev'n
Small Mid-Size Large
Size of Car
760
780
800
820
840
860
No
ise L
evel R
ead
ing
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
Small
Large
Small
Mid-Size
(I) Size of Car
Small
Mid-Size
Large
Mean
Difference
(I-J) Std. Error Sig.
The mean difference is significant at the .05 level.*.
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.
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?
Is car size related to noise level, ifeffect 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
Small Mid-Size Large
Size of Car
760
780
800
820
840
860
No
ise
Lev
el R
ea
din
g (
De
cib
els
)
Type of Noise Filter
Standard Filter
Octel Filter
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
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
Tests of Betw een-Subjects Effects
Dependent Variable: Noise Level Reading (Decibels)
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
Type III Sum
of Squares df Mean Square F Sig.
R Squared = 1.000 (Adjusted R Squared = 1.000)a.
Means of each combination of Size & Type
INTERACTION: Whenever lines not parallel
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.
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.
Total Variability
Between Treatments
Factor A
Factor B Interaction
Within Treatments
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
Notation describes procedureTables 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
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.
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
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
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
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 otherImages from Trochim’s Research Methods Knowledge Base at http://www.socialresearchmethods.net/kb/index.php
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
Total Variability
Between Treatments
Factor A
Factor B Interaction
Within Treatments
We will compute problems by hand to gain understanding, but not on a test
N
GXSStotal
22
treatment each insidetreatmentswithin SSSS
N
G
n
TSS treatmentsbetween
22
Total Variability
Between Treatments
Factor A
Factor B Interaction
Within Treatments
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
reatmentst within
reatmentst withinreatmentst within
df
SSMS
AxB
AxBAxB
B
BB
A
AA
df
SSMS
df
SSMS
df
SSMS
within
AxBAxB
within
BB
within
AA
MS
MSF
MS
MSF
MS
MSF
η2, is computed as the percentage of
variability not explained by other factors.
treatments withinA
A
AxBBtotal
AA
SSSS
SS
SSSSSS
SS
2
treatments withinB
B
AxBAtotal
BB
SSSS
SS
SSSSSS
SS
2
treatments withinAxB
AxB
BAtotal
AxBAxB
SSSS
SS
SSSSSS
SS
2
Two independent variables