Chapter 19: The Two-Factor ANOVA for Independent Groups An extension of the One-Factor ANOVA experiment has more than one independent variable, or ‘factor’

Chapter 19: The Two-Factor ANOVA for Independent Groups

An extension of the One-Factor ANOVA experiment has more than one independent variable, or ‘factor’.

For example, suppose we were interested in how both caffeine and beer influence response times.

You could run two separate studies, one comparing caffeine to a control group, and another comparing beer to another control group.

However, a more interesting experiment would be a ‘two-factor’ design and put subjects into one of four categories, which includes beer only, caffeine only and beer and caffeine.

Note that for two-factor ANOVAS, the sample size in each group is always the same.

Note: we’ll be skipping sections 19.8, 19.9, 19.10 and 19.13 from the book

Chapter 19: The Two-Factor ANOVA for Independent Groups

Here’s are some summary statistics for an example data set for n=12 subjects in each group (or cell)

SSW is the sums of squared deviations from the means within each cell, just like for the 1-Factor ANOVA

No Beer Beer

No Caffeine Mean: 1.08SSW = 0.66

Mean: 1.16SSW = 0.54

Caffeine Mean: 0.80SSW = 0.67

Mean: 1.02SSW = 0.32

Grand mean: 1.02SStotal = 3.07

No Beer Beer0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Res

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(sec

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No CaffeineCaffeine

It is common to plot the results of Two-Way experiments like this, with error bars representing the standard error of the mean.

No Beer Beer


Mean: 1.16SSW = 0.54


Mean: 1.02SSW = 0.32


No Beer Beer0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Res

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(sec

)

No CaffeineCaffeine

What statistical tests can we conduct on these results?

1) Effect of Beer on response times, averaged across Caffeine levels – a ‘main effect’ for Beer2) Effect of Caffeine on response times, averaged across Beer levels – a ‘main effect for Caffeine3) Interaction between Caffeine and Beer.

A significant interaction means that the main effects do not collectively explain all of the influence of the factors on the dependent variable.

Graphically, interactions happen when the lines are not parallel.

No Beer Beer0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Res

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(sec

)

No CaffeineCaffeine

The main effect for rows the difference between the means for the rows, averaging across the columns.

No Beer Beer


Mean: 1.16SSW = 0.54


Mean: 1.02SSW = 0.32

Main effect for ROWS

In this example, it is used to test for the effect of Caffeine on response times, averaging across the No Beer and Beer groups.

Statistically, its significance is determined by a One-Factor ANOVA, ‘collapsing’ across the columns

Graphically, it is a test if to see if the middle of the blue line is different than the middle of the green line.

No Beer Beer0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Res

pons

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(sec

)

No CaffeineCaffeine

The main effect for columns the difference between the means for the columns, averaging across the rows.

No Beer Beer


Mean: 1.16SSW = 0.54


Mean: 1.02SSW = 0.32

Main effect for COLUMNS

In this example, it is used to test for the effect of Beer on response times, averaging across the No Caffeine and Caffeine groups.

Graphically, it is a test if to see if the midpoint between the blue and green lines differs across the groups (or columns).

Statistically, its significance is determined by a One-Factor ANOVA, ‘collapsing’ across the rows

No Beer Beer Row meansNo Caffeine Mean: 1.08

SSW = 0.66Mean: 1.16SSW = 0.54

Mean: 1.12


Mean: 1.02SSW = 0.32

Mean: 0.91

Column means Mean: 0.94 Mean: 1.09 Grand mean: 1.02SStotal = 3.07

No Beer Beer0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

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(sec

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No CaffeineCaffeine

Main effects for rows and columns are calculated by averaging the data across rows and columns.

1RX

2CX1CX

2RX

X

We can partition the total variance into these components:SStotal

scoresall

XX 2)(

SSwithin cell

scoresall

cellXX 2)(

SSbetween

SSrows

rowsall

RR XXn 2)(

SScols

colsall

CC XXn 2)(

SSrows x cols

colsrowswithintotal SSSSSSSS

With degrees of freedom: dftotal=ntotal-1

dfwithin cell=ntotal-RxC SSbetween

dfrows=R-1 dfcols=C-1 dfrows x cols=(R-1)(C-1)

C (R) is the total number of scores for that column (row)

We can partition the total variance into these components:

Three F-tests can then be conducted by computing the following four variances:

wc

wcwc df

SSs 2 s2

wc estimates the variance within each cell, or ‘inherent variance’. This is also an estimate of the population variance s2. s2

wc is used as the denominator for all of the F-tests – just like the 1-way ANOVA

R

RR df

SSs 2 s2

R estimates the inherent variance plus the main effect for the row factor. It increases with variance across the row means.

C

CC df

SSs 2

s2C estimates the inherent variance plus the main effect for the

column factor. It increases with variance across the column means.

RxC

RxCRxC df

SSs 2 s2

RxC estimates the inherent variance plus the interaction effect. If s2

RxC is small (near s2wc) then the total variance is completely explained

by the inherent variance plus the effects of the row and column factors alone. Hence, no interaction between the two factors.

2

2

wc

R

s

sF

The four variances are used to compute the three F-ratios to make the three hypothesis tests about the row factor, column factor and the interaction:

tests for the main effect of the row factor and has dfs of (R-1) and (ntotal – RxC)

2

2

wc

C

s

sF tests for the main effect of the column factor

and has dfs of (C-1) and (ntotal – RxC)

2

2

wc

RxC

s

sF tests for the interaction between the row and column factors

and has dfs of (R-1)x(C-1) and (ntotal – RxC)

Source SS df s2 F

Rows SSR R-1 SSR/dfR s2R/s2

wc

Columns SSC C-1 SSC/dfC s2C/s2

wc

RxC SSRxC (R-1)x(C-1) SSRxC/dfRxC s2RxC/s2

wc

Within cells SSwc ntotal-RxC SSwc/dfwc

Total SStotal ntotal-1

Typically, we conduct a two-factor ANOVA by filling in a table like this:


SSW = 0.66Mean: 1.16SSW = 0.54

Mean: 1.12


Mean: 1.02SSW = 0.32

Mean: 0.91


Source SS df s2 FRows SSR 1 SSR/dfR s2

R/s2wc

Columns SSC 1 SSC/dfC s2C/s2

wc

RxC SSRxC 1 SSRxC/dfRxC s2RxC/s2

wc

Within cells SSwc 44 SSwc/dfwc

Total 3.07 47

We can start filling in the table from our example about beer and caffeine.


SSW = 0.66Mean: 1.16SSW = 0.54

Mean: 1.12


Mean: 1.02SSW = 0.32

Mean: 0.91


SSwithin= 18.232.054.067.066.0)( 2 scoresall

cellXX

06.0)54.028.018.2(07.3 colsrowswithintotal SSSSSSSS

Source SS df s2 F

Rows SSR 1 SSR/dfR s2R/s2

wc


wc


wc

Within cells 2.18 44 0.0495

Total 3.07 47

0495.044

18.22 wc

wcwc df

SSs

Source SS df s2 F

Rows SSR 1 SSR/dfR s2R/s2

wc


wc


wc

Within cells SSwc 44 SSwc/dfwc

Total 3.07 47


SSW = 0.66Mean: 1.16SSW = 0.54

Mean: 1.12


Mean: 1.02SSW = 0.32

Mean: 0.91


54.01

54.02 R

RR df

SSs 98.10

0495.0

54.02

2

wc

R

s

sF

Source SS df s2 FRows 0.54 1 0.54 10.98Columns SSC 1 SSC/dfC s2

C/s2wc


wc


Total 3.07 47

54.0)02.191.0(24)02.112.1(24)( 222 rowsall

RR XXnSSrows=


SSW = 0.66Mean: 1.16SSW = 0.54

Mean: 1.12


Mean: 1.02SSW = 0.32

Mean: 0.91


28.0)02.109.1(24)02.194.0(24)( 222 rowsall

CC XXnSScols=

28.01

28.02 C

CC df

SSs

Source SS df s2 F

Rows 0.54 1 0.54 10.98

Columns 0.28 1 0.28 5.62


wc


Total 3.07 47

62.50495.0

28.02

2

wc

C

s

sF


SSW = 0.66Mean: 1.16SSW = 0.54

Mean: 1.12


Mean: 1.02SSW = 0.32

Mean: 0.91


SSrows x cols = 064.0)54.028.018.2(07.3 colsrowswithintotal SSSSSSSS

Source SS df s2 F

Rows 0.54 1 0.54 10.98

Columns 0.28 1 0.28 5.62

RxC 0.064 1 0.064 1.30


Total 3.07 47

064.01

064.02 RxC

RxCRxC df

SSs 30.1

0495.0

064.02

2

wc

RxC

s

sF

Source SS df s2 F Fcrit

Rows 0.54 1 0.54 10.98 4.06Columns 0.28 1 0.28 5.62 4.06

RxC 0.064 1 0.064 1.30 4.06

Within cells 2.18 44 0.0495Total 3.07 47

We can either use our F-tables (Table E) to find the critical values of F

Source SS df s2 F P-value

Rows 0.54 1 0.54 10.98 0.0019

Columns 0.28 1 0.28 5.62 0.0214

RxC 0.064 1 0.064 1.30 0.2604


Total 3.07 47

Or, more commonly, we can use our F-calculator to calculate the corresponding p-value for our observed values of F.

Source SS df s2 F P-valueRows 0.54 1 0.54 10.98 0.0019Columns 0.28 1 0.28 5.62 0.0214

RxC 0.064 1 0.064 1.30 0.2604

Within cells 2.18 44 0.0495Total 3.07 47

No Beer Beer0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Res

pons

e T

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(sec

)

No CaffeineCaffeine

We show a significant main effect for rows (Caffeine) and for Columns (Beer), but not a significant interaction between rows and columns (Caffeine x Beer).

1 280

90

100

110

120

130

140

Columns

Sco

re

Row 1Row 2

Source SS df s2 F p-value

Rows 2629.420 1 2629.420 11.519 0.0015

Columns 6306.492 1 6306.492 27.628 0.0000

RxC 779.260 1 779.260 3.414 0.0714

Within 10043.672 44 228.265

Total 19758.844 47

Let’s play “guess that significance!”

Row 1Row 2

1 270

80

90

100

110

120

130

Columns

Sco

re


Rows 5431.309 1 5431.309 19.340 0.0001

Columns 366.798 1 366.798 1.306 0.2593

RxC 3354.865 1 3354.865 11.946 0.0012

Within 12356.377 44 280.827

Total 21509.349 47

Guess that significance!

Row 1Row 2


Rows 3946.596 1 3946.596 21.685 0.0000

Columns 68.646 1 68.646 0.377 0.5423

RxC 442.586 1 442.586 2.432 0.1261

Within 8007.838 44 181.996

Total 12465.666 47

1 290

95

100

105

110

115

120

125

Columns

Sco

re


Row 1Row 2

1 285

90

95

100

105

110

115

Columns

Sco

re


Rows 7.143 1 7.143 0.027 0.8707

Columns 8.773 1 8.773 0.033 0.8568

RxC 2547.690 1 2547.690 9.565 0.0034

Within 11719.076 44 266.343

Total 14282.682 47


Row 1Row 2

1 295

100

105

110

115

120

125

130

Columns

Sco

re


Rows 402.806 1 402.806 1.906 0.1743

Columns 4266.756 1 4266.756 20.193 0.0001

RxC 13.740 1 13.740 0.065 0.7999

Within 9296.999 44 211.295

Total 13980.299 47


Row 1Row 2


Rows 4677.560 1 4677.560 20.283 0.0000

Columns 0.653 1 0.653 0.003 0.9578

RxC 6777.132 1 6777.132 29.387 0.0000

Within 10147.208 44 230.618

Total 21602.554 47

1 270

80

90

100

110

120

130

Columns

Sco

re


Row 1Row 2

1 2 380

85

90

95

100

105

Columns

Sco

re


Rows 1717.957 1 1717.957 7.282 0.0088

Columns 4.773 2 2.386 0.010 0.9899

RxC 508.478 2 254.239 1.078 0.3463

Within 15569.688 66 235.904

Total 17800.896 71


Row 1Row 2

1 2 380

90

100

110

120

130

Columns

Sco

re


Rows 439.149 1 439.149 2.011 0.1609

Columns 9877.896 2 4938.948 22.619 0.0000

RxC 324.247 2 162.123 0.742 0.4799

Within 14411.484 66 218.356

Total 25052.776 71


Row 1Row 2


Rows 608.976 1 608.976 2.710 0.1045

Columns 818.724 2 409.362 1.822 0.1698

RxC 4498.794 2 2249.397 10.010 0.0002

Within 14830.738 66 224.708

Total 20757.233 71


1 2 370

80

90

100

110

120

Columns

Sco

re

n = 12 ExerciseNone A little A lot

Diet A Mean: 20.73SSW = 20.01

Mean: 20.12SSW = 37.86

Mean: 19.21SSW = 29.84

Diet B Mean: 20.83SSW = 13.14

Mean: 20.57SSW = 44.59

Mean: 18.08SSW = 20.27


Suppose you wanted to test the effects of diet and exercise on body mass index. You choose two diets (A and B) and three levels of exercise (none, a little, a lot). You then find 12 subjects for each group and obtain the following descriptive statistics:

Conduct a two factor ANOVA to determine if there is a main effect for diet, exercise and if there is an interaction between diet and exercise.

none A little A lot17.5

18

18.5

19

19.5

20

20.5

21

21.5

Exercise

BM

I

Diet ADiet B

First, let’s plot the data with error bars as the standard error of the means.



Mean: 20.12SSW = 37.86

Mean: 19.21SSW = 29.84

Mean: 20.02


Mean: 20.57SSW = 44.59

Mean: 18.08SSW = 20.27

Mean: 19.83

Mean: 20.78 Mean: 20.35 Mean: 18.64Grand mean: 19.92

SStotal = 235.90


Rows SSR R-1 SSR/dfR s2R/s2

wc

Columns SSC C-1 SSC/dfC s2C/s2

wc

RxC SSRxC (R-1)x(C-1) SSRxC/dfRxC s2RxC/s2

wc

Within cells SSwc ntotal-RxC SSwc/dfwc

Total SStotal ntotal-1



Mean: 20.12SSW = 37.86

Mean: 19.21SSW = 29.84

Mean: 20.02


Mean: 20.57SSW = 44.59

Mean: 18.08SSW = 20.27

Mean: 19.83

Mean: 20.78 Mean: 20.35 Mean: 18.64Grand mean: 19.92

SStotal = 235.90

Source SS df s2 F p-valueRows 0.70 1 0.70 0.28 0.5999Columns 61.23 2 30.61 12.19 0.0000RxC 8.26 2 4.13 1.65 0.2007Within cells 165.71 66 2.51Total 235.90 71

No main effect for Rows (Diet)Main effect for Columns (Exercise)

No interaction between rows and columns (Diet and Exercise)


Rows 0.70 1 0.70 0.28 0.5999Columns 61.23 2 30.61 12.19 0.0000RxC 8.26 2 4.13 1.65 0.2007Within cells 165.71 66 2.51Total 235.90 71

none A little A lot17.5

18

18.5

19

19.5

20

20.5

21

21.5

Exercise

BM

I

Diet ADiet B

Documents

Chapter 19: The Two-Factor ANOVA for Independent Groups An extension of the One-Factor ANOVA experiment has more than one independent variable, or ‘factor’