Two-Way (Independent) ANOVA. PSYC 6130A, PROF. J. ELDER 2 Two-Way ANOVA “Two-Way” means groups...

Two-Way (Independent) ANOVA

PSYC 6130A, PROF. J. ELDER 2

Two-Way ANOVA

• “Two-Way” means groups are defined by 2 independent variables.

• These IVs are typically called factors.

• An experiment in which any combination of values for the 2 factors can occur is called a completely crossed factorial design.

• If all cells have the same n, the design is said to be balanced.

• Still have only 1 dependent variable


Example: Visual Grating Detection in Noise

200 ms

Until Response

500 ms

500 ms


2 x 3 Design

Noise Contrast

GratingFrequency

(c/deg)

0.5

1.7

4.3% 14.8% 50.0%


Balanced Design

10 10 10

10 10 10

Signal to Noiseat Threshold

.500


1.700

SpatialFrequency(cpd)

Count

.043

Count

.148

Count

.500

Noise Contrast (Michelson units)

Factor B

Factor A


Descriptive Statistics

.006 .003 .003

.004 .005 .008


.500


1.700


Std Deviation

.043

Std Deviation

.148

Std Deviation

.500


.078 .064 .065 .069

.095 .089 .098 .094

.087 .076 .082 .082


.500

1.700

Spatial Frequency(cpd)

Group Total

Mean

.043

Mean

.148

Mean

.500


Mean

GroupTotal

210.014571

1 ijT

s X XN


Interactions

• If there is no interaction between the factors (spatial frequency, noise contrast), the dependent variable (SNR) for each condition (cell) can be predicted from the independent effects of factors A and B:

– Cell mean = Grand mean + Row effect + Column effect

.078 .064 .065 .069

.095 .089 .098 .094

.087 .076 .082 .082


.500

1.700


Group Total

Mean

.043

Mean

.148

Mean

.500


Mean

GroupTotal


Interactions

• If there are no interactions, curves should be parallel (effect of noise contrast is independent of spatial frequency).


Types of Effects


Interactions

• In the general case,

Cell mean = Grand mean + Row effect + Column effect + Interaction effect

• Score deviations from cell means are considered error (unpredictable).

• Thus:

Score = Grand mean + Row effect + Column effect + Interaction effect + Error

• OR

Score - Grand mean = Row effect + Column effect + Interaction effect + Error


Sum of Squares Analysis

errorwhere WSS SS

A B AB bet

betwhere is the between-groups sum-of-squares that would

be calculated by lumping all groups into one factor in a 1-Way ANOVA.

SS SS SS SS

SS

total A B AB errorSS SS SS SS SS

AB bet A B

AB

Thus

This provides a means for calculating .

SS SS SS SS

SS

Multiple Subscript and Summation Notation


Single Subscript Notation

3X

X

1

2

12

3

14


Double Subscript Notation

ijX



• The first subscript refers to the row that the particular value is in, the second subscript refers to the column.

11 12 13

21 22 23

31 32 33

X X X

X X X

X X X



• Test your understanding by identifying in the table below.

1 43 13

23 42 33

12 11 23

32X

1 43 13

23 42 33

12 11 23



• We will follow the notation of Howell:

number of in each cell in a balanced sco desr .e ns ign

number of levels for Factor .a A

number of levels for Facto .Br b


Multi-Subscript Notation

• In two-way ANOVA, 3 indices are needed:

kijXIndex identifies the lev Factorel of (the w) roi A

Index identifies the leve Factor l of (t che )olumnj B

Index identifies the individual within cell (k score , ).i j

Together, ( , ) identify the of the data table.celi j l


Multi-Subscript Notation• Statistics are calculated by summing over scores within

cells, and thus the third subscript (k) is dropped:

1

1 n

ij ijkk

X Xn


Multi-Subscript Notation

23Thus refers to the sample mean for the cell in , of the data table

(2nd level of and 3

R

r

ow 2

d leFa vector l of

Column 3

Facto )r

X

A B

41 refers to the sample mean for the cell in , of the data table

(4th level of and 1

Colu

st l

Ro

ev

w 4

Fact el of

mn 1

Facto or ) r

X

BA


Pooled Statistics

• Multi-factor ANOVA requires the calculation of statistics that pool, or ‘collapse’ data over one or more factors.

• We indicate the factors over which the data are being pooled by substituting a ‘bullet’ • for the corresponding index.


Pooled Statistics

1 1 1

1 1 is a 'column mean'

obtained by averaging over all scores in column

(all levels A of Factor )

a n a

j ijk iji k i

X X Xan a

j

1 1 1

1 1 is a 'row mean'

obtained by averaging over all scores in row

(all levels of Factor )B

b n b

i ijk ijj k j

X X Xbn b

i

a b

j1 1 1 1 1 i=1 j=1

1 1 1 1= X

b

is a 'grand mean' obtained by averaging

over all scores in the table.

a b n a b

ijk ij ii j k i j

X X X Xabn ab a

Six Step Procedure


0.62 0.29 0.31

0.36 0.54 0.80


.500


1.700


Std Deviation

.043

Std Deviation

.148

Std Deviation

.500


Example

211.5

1 ijT

s X XN

7.8 6.4 6.5 6.9

9.5 8.9 9.8 9.4

8.7 7.6 8.2 8.2

Signal to Noiseat Threshold (%)

.500

1.700


Group Total

Mean

.043

Mean

.148

Mean

.500


Mean

GroupTotal


Step 1. State the Hypothesis

• Null hypothesis has 3 parts, e.g.,

– Mean SNR at threshold same for both spatial frequencies

– Mean SNR at threshold same for all noise levels

– No interactions


Step 2. Select Statistical Test and Significance Level

• Normally use same -level for testing all 3 F ratios.


Step 3. Select Samples and Collect Data

• Strive for a balanced design

• Ideally, randomly sample

• More probably, random assignment


Step 4. Find Regions of Rejection

• Generally have 3 different critical values for each F test

1Bdf b

for all tests.W Tdf N ab

1Adf a

AB A Bdf df df

Denominator

Numerator

T 1Tdf N

Note that

T A B AB Wdf df df df df


Degrees of Freedom Tree

1Bdf b

W Tdf N ab

1Adf a

AB A Bdf df df

T 1Tdf N

1betdf ab


Step 5. Calculate the Test Statistics

2 2( 1)W ij ijk ij ijSS SS X X n s

2

bet ijSS n X X

2

A iSS bn X X

2

B jSS an X X

AB bet A BSS SS SS SS

2 2Sanity Check: ( 1)W A B AB T ij TSS SS SS SS SS X X N s


Step 5. Calculate the Test Statistics

AA

A

SSMS

df

BB

B

SSMS

df

ABAB

AB

SSMS

df

WW

W

SSMS

df

AA

W

MSF

MS

BB

W

MSF

MS

ABAB

W

MSF

MS


Step 6. Make the Statistical Decisions

• Note that 3 independent statistical decisions are being made.

• Thus the probability of one or more Type I errors is greater than the α value used for each test.

• It is not common to correct for this.

• You should be aware of this issue as both a producer and consumer of scientific results!


SPSS Output

Tests of Between-Subjects Effects

Dependent Variable: Signal to Noise at Threshold

.011a 5 .002 81.107 .000 .882

.399 1 .399 14644.217 .000 .996

.009 1 .009 344.657 .000 .865

.001 2 .001 18.802 .000 .411

.001 2 .000 11.637 .000 .301

.001 54 2.73E-005

.412 60

.013 59

SourceCorrected Model

Intercept

SpatialFreq

NoiseContrast

SpatialFreq *NoiseContrast

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

Partial EtaSquared

R Squared = .882 (Adjusted R Squared = .872)a.

Main effects

Interaction


SPSS Output

2nX 2nX

2nX

Tests of Between-Subjects Effects


.011a 5 .002 81.107 .000 .882

.399 1 .399 14644.217 .000 .996

.009 1 .009 344.657 .000 .865

.001 2 .001 18.802 .000 .411

.001 2 .000 11.637 .000 .301

.001 54 2.73E-005

.412 60

.013 59

SourceCorrected Model

Intercept

SpatialFreq

NoiseContrast

SpatialFreq *NoiseContrast

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

Partial EtaSquared

R Squared = .882 (Adjusted R Squared = .872)a.

betSS2nX

ASS

BSS

ABSS

WSS2iX

TSS


Assumptions of Two-Way Independent ANOVA

• Same as for One-Way

• If balanced, don’t have to worry about homogeneity of variance.


Advantages of 2-Way ANOVA with 2 Experimental Factors

• One factor may not be of interest (e.g., gender), but may affect the dependent variable.

• Explicitly partitioning the data according to this ‘nuisance’ variable can increase the power of tests on the independent variable of interest.


Simple Effects

• When significant main effects are discovered, it is common to also test for simple effects.


Simple Effects

• A main effect is an effect of one factor measured by collapsing (pooling) over all other factors.

• A simple effect is an effect of one factor measured by fixing all other factors.

• Although we found significant main effects, given the significant interaction, these main effects do not necessarily imply similarly significant simple effects.


Simple Effects

• Thus, particularly when a significant interaction is observed, a factorial ANOVA is often followed up by a series of one-way ANOVAS to test simple effects.

• For our example, there are a total of 5 possible simple effects to test.


Simple Effects

• To conduct follow-up one-way ANOVA tests of simple effects in SPSS:

– Select Split File … from the Data menu

– Click on Organize Output by Groups

– Transfer the factor to be held constant to the space labeled “Groups Based On.”

– Now proceed with one-way ANOVAS as usual.


Simple Effects

Test of Homogeneity of Variances a

Signal to Noise at Threshold

5.120 2 27 .013

Levene

Statistic df1 df2 Sig.

Spatial Frequency (cpd) = .500a.

ANOVAa


.001 2 .001 32.990 .000

.001 27 .000

.002 29

Between GroupsWithin Groups

Total

Sum ofSquares df Mean Square F Sig.

Spatial Frequency (cpd) = .500a.

Robust Tests of Equality of Meansb


21.413 2 16.975 .00032.990 2 17.382 .000

WelchBrown-Forsythe

Statistica df1 df2 Sig.

Asymptotically F distributed.a.

Spatial Frequency (cpd) = .500b.


Simple Effects

Test of Homogeneity of Variancesa


2.037 2 27 .150

LeveneStatistic df1 df2 Sig.

Spatial Frequency (cpd) = 1.700a.

ANOVAa


.000 2 .000 5.899 .007

.001 27 .000

.001 29

Between GroupsWithin GroupsTotal

Sum ofSquares df Mean Square F Sig.


Robust Tests of Equality of Meansb


5.527 2 16.511 .0155.899 2 19.883 .010

WelchBrown-Forsythe

Statistica df1 df2 Sig.

Asymptotically F distributed.a.

Spatial Frequency (cpd) = 1.700b.


Simple Effects

• Again note that multiple independent statistical decisions are being made.

• Conditioning the test for simple effects on a significant main effect provides protection if only 2 simple effects are being tested.

• Otherwise, the probability of one or more Type I errors is greater than the α value used for each test.

• It is not common to correct for this.

• You should be aware of this issue as both a producer and consumer of scientific results!

End of Lecture

April 8, 2009


Planned or Posthoc Pairwise Comparisons

• If significant main (and possibly simple) effects are found, it is common to follow up with one or more pairwise tests.

• It is most common to test differences between marginal means within a factor (i.e., pooling over the other factor).

• In this example, there are only 3 meaningful posthoc tests on marginal means. Why?


Pairwise Comparisons on Marginal Means

• Since there are 3 levels of noise, we can consider using Fisher’s LSD.

• However, since variances do not appear homogeneous, we should not use an LSD based on pooling the variance over all 3 conditions.

Multiple Comparisons


LSD

.010120* .004492 .028 .00112 .01912

.004800 .004492 .290 -.00420 .01380

-.010120* .004492 .028 -.01912 -.00112

-.005320 .004492 .241 -.01432 .00368

-.004800 .004492 .290 -.01380 .00420

.005320 .004492 .241 -.00368 .01432

(J) Noise Contrast(Michelson units).148

.500

.043

.500

.043

.148

(I) Noise Contrast(Michelson units).043

.148

.500

MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound

95% Confidence Interval

The mean difference is significant at the .05 level.*.

Test of Homogeneity of Variances


12.229 2 57 .000

LeveneStatistic df1 df2 Sig.


Pairwise Comparisons on Marginal Means

• Alternative when variances appear heterogeneous:

– Compute Fisher’s LSD by hand, calculating standard error separately for each test (not difficult)

– One of the unequal variance post-hoc tests offered by SPSS

Multiple Comparisons


Games-Howell

.010120* .003787 .030 .00085 .01939

.004800 .004573 .552 -.00648 .01608

-.010120* .003787 .030 -.01939 -.00085

-.005320 .005028 .546 -.01762 .00698

-.004800 .004573 .552 -.01608 .00648

.005320 .005028 .546 -.00698 .01762


.500

.043

.500

.043

.148


.148

.500

MeanDifference






• It is also possible to test differences between cell means. Note that in this design, there are 15 possible pairwise cell comparisons.

• It doesn’t make that much sense to compare 2 cells that are not in the same row or column (i.e. that differ in both factors).

• It is more likely that you would follow a significant simple effect test with a set of pairwise comparisons within a factor while holding the other factor constant. There are 9 such comparisons possible here.

• For example, within a spatial frequency condition, what noise conditions differ significantly?

• This defines a total of 6 pairwise comparisons (2 families of 3 comparisons each).



• Alternative when variances appear heterogeneous:

– Compute Fisher’s LSD by hand, calculating standard error separately for each test (not difficult)

– One of the unequal variance post-hoc tests offered by SPSS (assumes all-pairs)

Multiple Comparisonsa


Games-Howell

.005890* .002067 .030 .00055 .01123

-.003160 .002790 .513 -.01056 .00424

-.005890* .002067 .030 -.01123 -.00055

-.009050* .003066 .024 -.01697 -.00113

.003160 .002790 .513 -.00424 .01056

.009050* .003066 .024 .00113 .01697


.500

.043

.500

.043

.148


.148

.500

MeanDifference






Interaction Comparisons

• If significant interactions are found in a design that is 2x3 or larger, it may be of interest to test the significance of smaller (e.g., 2x2) interactions.

• These can be tested by ignoring specific subsets of the data for each test (e.g., by using the SPSS Select Cases function).


Unbalanced Designs for Two-Way ANOVA

• Dealing with unbalanced designs is easy for One-Way ANOVA.

• Dealing with unbalanced designs is trickier for Two-Way.


Simple Solution

• Let n = harmonic mean of sample sizes.

• Calculate marginal means as an unweighted mean of cell means (not the pooled mean).


Better Solution

• Regression approach to ANOVA (will not cover)

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Two-Way (Independent) ANOVA. PSYC 6130A, PROF. J. ELDER 2 Two-Way ANOVA “Two-Way” means groups...

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