Heiman G. Basic Statistics for the Behavioral Sciences

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Source Sum of Squares /dfⴝ Mean Square F

Between

Factor A SS

⎯⎯⎯⎯→

(volume)

Factor B SS

⎯⎯⎯⎯→

(gender)

Interaction SS

AB

⎯⎯⎯→

AB

(vol  gen)

Within SS

⎯⎯⎯

Total SS

tot

TABLE 14.4

Summary Table of

Two-Way ANOVA

328 CHAPTER 14 / The Two-Way Analysis of Variance

The summary table in Table 14.4 shows the preceding components. (Notice that if a

component is not labeled “within” or “total,” then it is one of our many between-groups

components.) To complete the ANOVA, for factor A, divide by to produce

. For factor B, divide by to produce . For the interaction, divide

by to produce

The following sections show how to compute the above components.

Computing the Sums and Means

Your first step is to organize the data in each cell. Table 14.5 shows the persuasiveness

data, as well as the various components to compute.

Step 1 Compute and in each cell. For example, in the male–soft cell,

; Also note the of the cell

(here ) Then compute the mean for each cell (for the male–soft cell, ).

These are the interaction means.

Step 2 Compute vertically in each column. Sum the from the cells in a col-

umn (for example, for soft, ). Note the in each column (here )

and compute the mean for the scores in each column (for example, ). (Or aver-

age the cell means in the column.) These are the main effect means for factor A.

Step 3 Compute horizontally in each row. Sum the from the cells in a row

(for males, ). Note the in each row (here ) Com-

pute the mean of the scores in each row (for example, ). (Or average the

cell means in the row.) These are the main effect means for factor B.

Step 4 Compute . Sum the from the levels (columns) of factor A, so

. (Or add the from the levels of factor B.)

Step 5 Compute . Sum the from each cell, so

. Note the (here )

We will use the components from the previous five steps to compute the sums of

squares, then the degrees of freedom, then the mean squares, and finally the . To keep

track of your computations and prevent brain strain, fill in the ANOVA summary table

as you go along.

N 5 18.N56 1 470 1 116 5 2075

©X

tot

5 218 1 377 1 838 1©X

©X

tot

©X©X

tot

5 36 1 69 1 68 5 173

©X©X

tot

male

5 11.89

n 5 9.n©X 5 24 1 33 1 50 5 107

©Xs©X

soft

5 6

n 5 6n©X 5 24 1 12

©Xs©X

X 5 8n 5 3.

n©X

5 4

1 9

1 11

5 218.©X 5 4 1 9 1 11 5 24

©X

A3B

.MS

A3B

Factor A: Volume

Soft Medium Loud

4818

912 17

11 13 15

Male X

苶

 8 X

苶

 11 X

苶

 16.67 X

苶

male

 11.89

©X  24 ©X  33 ©X  50 ©X  107

©X

 218 ©X

 377 ©X

 838 n  9

n  3 n  3 n  3

Factor B:

Gender

29 6

610 8

417 4

Female X

苶

 4 X

苶

 12 X

苶

 6 X

苶

fem

 7.33

©X  12 ©X  36 ©X  18 ©X  66

©X

 56 ©X

 470 ©X

 116 n  9

n  3 n  3 n  3

苶

soft

 6 X

苶

med

 11.5 X

苶

loud

 11.33 ©X

tot

 173

©X  36 ©X  69 ©X  68 ©X

tot

 2075

n  6 n  6 n  6 N  18

Computing the Two-Way ANOVA 329

TABLE 14.5

Summary of Data for

3  2 ANOVA

Computing the Sums of Squares

First, compute the sums of squares.

Step 1 Compute the total sum of squares.

This says to divide by and then subtract the answer from .

From Table 14.5, , , and Filling in the formula

gives

Note: The quantity is also used when computing other sums of squares.

We call it the correction (here the correction equals 1662.72).

Step 2 Compute the sum of squares for factor A. As in the diagrams here, always have

factor A form your columns.

1©X

tot

5 412.28

tot

5 2075 2 1662.72

tot

5 2075 2 a

11732

N 5 18.©X

tot

5 2075©X

tot

5 173

©X

tot

N1©X

tot

The formula for the total sum of squares is

tot

5 ©X

tot

2 a

1©X

tot

330 CHAPTER 14 / The Two-Way Analysis of Variance

This says to square the in each column of factor A and divide by the in the

column. Then add the answers together and subtract the correction.

From Table 14.5, the three columns produced sums of 36, 69, and 68, and n was 6.

Filling in the above formula gives

Step 3 Compute the sum of squares for factor B. In your diagram, the levels of

factor B should form the rows.

5 117.45

5 1780.17 2 1662.72

5 1216 1 793.5 1 770.6722 1662.72

5 a

1362

1692

1682

b2 a

11732

n©X

This says to square the for each level of factor B and divide by the n in the level.

Then add the answers and subtract the correction.

In Table 14.5, the two rows produced sums of 107 and 66, and n was 9. Filling in the

formula gives

Step 4 Compute the sum of squares between groups for the interaction. This requires

two substeps. First, compute something called the total sum of squares between groups,

identified as .SS

5 93.39

5 1756.11 2 1662.72

5 a

1107

166

b2 1662.72

©X

The formula for the sum of squares between groups for

factor A is

5 © a

1Sum of scores in the column2

n of scores in the column

b2 a

1©X

tot

The formula for the sum of squares between groups for

factor B is

5 © a

1Sum of scores in the row2

n of scores in the row

b2 a

1©X

tot

Computing the Two-Way ANOVA 331

Find for each cell and divide by the n of the cell. Then add the answers together

and subtract the correction.

From Table 14.5,

The equals the sum of squares for factor A plus the sum of squares for factor B

plus the sum of squares for the interaction. Therefore, we find by subtracting

the and found above (in Steps 2 and 3) from the total Thus,SS

.SS

A3B

5 313.61

5 1976.33 2 1662.72

5 a

1242

1332

1502

1122

1362

1182

b2 1662.72

1©X2

Above, , , and , so

Step 5 Compute the sum of squares within groups. The sum of squares within groups

plus the total sum of squares between groups equals the total sum of squares. There-

fore, subtract the total in Step 4 from the in Step 1 to obtain the .SS

tot

A3B

5 102.77

A3B

5 313.61 2 117.45 2 93.39

5 93.39SS

5 117.45SS

5 313.61

Above, and , so

The previous sums of squares are shown in Table 14.6.

Now determine the df.

5 98.67

5 412.28 2 313.61

5 313.61SS

tot

5 412.28

The formula for the total sum of squares between groups is

5 © a

1Sum of scores in the cell2

n of scores in the cell

b2 a

1©X

tot

The formula for the sum of squares between groups for the

interaction is

A3B

5 SS

2 SS

The formula for the sum of squares within groups is

5 SS

tot

2 SS

332 CHAPTER 14 / The Two-Way Analysis of Variance

Computing the Degrees of Freedom

1. The degrees of freedom between groups for factor A is , where is the

number of levels in factor A. (In our example, is the three levels of volume,

so )

2. The degrees of freedom between groups for factor B is , where is the num-

ber of levels in factor B. (In our example, is the two levels of gender, so )

3. The degrees of freedom between groups for the interaction is the for factor

multiplied times the for factor (In our example, and , so

)

4. The degrees of freedom within groups equals , where is the total of

the study and is the number of cells in the study. (In our example, is 18

and we have six cells, so )

5. The degrees of freedom total equals . Use this to check your previous calcu-

lations, because the sum of the above should equal (In our example

)

Place each in the ANOVA summary table as in Table 14.7. Perform the remainder

of the computations by working directly from this table. Next, we compute the mean

squares.

Computing the Mean Squares

Any mean square equals the appropriate sum of squares divided by the appropriate .

Therefore, for factor A,

tot

5 17.

tot

.dfs

N 2 1

5 18 2 6 5 12.

A3B

NNN 2 k

A3B

5 2.

5 1df

5 2B.df

Adf

5 1.k

2 1

5 2.

2 1

Source Sum of Squares df Mean Square F

Between

Factor A 117.45 df

(volume)

Factor B 93.39 df

(gender)

Interaction 102.77 df

AB

(vol  gen)

Within 98.67 df

Total 412.28 df

tot

TABLE 14.6

Summary Table of Two-

Way ANOVA showing the

Sums of Squares

From Table 14.7 we find

117.45

5 58.73

The formula for the mean square between groups for

factor A is

Computing the Two-Way ANOVA 333

Source Sum of Squares df Mean Square F

Between

Factor A 117.45 2 MS

(volume)

Factor B 93.39 1 MS

(gender)

Interaction 102.77 2 MS

AB

(vol  gen)

Within 98.67 12 MS

Total 412.28 17

TABLE 14.7

Summary Table of

Two-Way ANOVA

with df and Sums

of Squares

From Table 14.7

93.39

5 93.39

Using Table 14.7

A3B

102.77

5 51.39

Table 14.7 gives

98.67

5 8.22

The formula for the mean square between groups for

factor B is

The formula for the mean square between groups for the

interaction is

A3B

The formula for the mean square within groups is

334 CHAPTER 14 / The Two-Way Analysis of Variance

Putting the above values into the summary table gives Table 14.8.

Now, finally, compute the .

Computing F

Any equals the divided by the Therefore,MS

.MS

In our example, from Table 14.8 we have

58.73

8.22

5 7.14

Source Sum of Squares df Mean Square F

Between

Factor A 117.45 2 58.73 F

(volume)

Factor B 93.39 1 93.39 F

(gender)

Interaction 102.77 2 51.39 F

AB

(vol  gen)

Within 98.67 12 8.22

Total 412.28 17

TABLE 14.8

Summary Table of Two-

Way ANOVA Showing

the Mean Squares, df,

and Sums of Squares

Thus, we have

93.39

8.22

5 11.36

The formula for F

for the main effect of factor A is

The formula for F

for the main effect of factor B is

The formula for F

AⴛB

for the interaction effect is

A3B

Computing the Two-Way ANOVA 335

And now the finished summary table is in Table 14.9.

Interpreting Each F

Each is tested in the same way as in the previous chapter: The may be larger

than 1 because (1) is true but we have sampling error or (2) is false and at least

two means represent a relationship in the population. The larger an , the less likely

that is true. If is larger than , then we reject

To find the for a particular , in the -tables (Table 5 in Appendix ), use the

that you used in computing that and your Thus,

1. To find for testing , use as the between groups and In our

example, and So, for , the is 3.88.

2. To find for testing , use as the between groups and In our

example, and So, at , the is 4.75.

3. To find for the interaction, use as the between groups and In

our example, and Thus, at , the is 3.88.

Notice that because factors A and B have different between groups, they have dif-

ferent critical values.

Thus, we end up comparing the following:

crit

 5 .05df

5 12.df

A3B

5 2

.dfdf

A3B

crit

 5 .05df

5 12.df

5 1

.dfdf

crit

 5 .05df

5 12.df

5 2

.dfdf

crit

obt

CFF

obt

crit

obt

A3B

51.39

8.22

5 6.25

Source Sum of Squares df Mean Square F

Between

Factor A 117.45 2 58.73 7.14

(volume)

Factor B 93.39 1 93.39 11.36

(gender)

Interaction 102.77 2 51.39 6.25

(vol  gen)

Within 98.67 12 8.22

Total 412.28 17

TABLE 14.9

Completed Summary

Table of Two-Way

ANOVA

obt

crit

Main effect of volume (A) 7.14 3.88

Main effect of gender (B) 11.36 4.75

Interaction (A  B) 6.25 3.88

By now you can do this with your eyes closed: Imagine a sampling distribution with

a region of rejection and in the positive tail. (If you can’t imagine this, look back

in Chapter 13 at Figure 13.1.) First, our of 7.14 is larger than the , so it lies in

the region of rejection. Therefore, we conclude that changing the volume of a message

produced significant differences in persuasiveness scores.

Likewise, the of 11.36 is significant, so we conclude that the males and females

in this study represent different populations of scores.

crit

336 CHAPTER 14 / The Two-Way Analysis of Variance

Finally, the of 6.25 is significant, so we conclude that the effect that changing

volume has in the population depends on whether it is a population of males or a popula-

tion of females. Or we can say that the difference between the male and female popula-

tions we’d see depends on whether a message is played at soft, medium, or loud volume.

Note: It is just a coincidence of your particular data which will be significant: any

combination of the main effects and/or the interaction may or may not be significant.

A3B

■

A two-way ANOVA produces an for each main

effect and for the interaction.

■

Each equals the corresponding divided by

the .

■

Each equals the appropriate between

groups divided by the

MORE EXAMPLES

In a new study, our factor A tests the effect of three

doses of a smart pill (one, two, or three pills), and fac-

tor B tests two ages (10- or 20-year olds). We obtain

the and shown here.

Source SS df MS F

A(dose) 48 2 24 6.00

B (age) 8 1 8 2.00

Interaction 38 2 19 4.75

Within 72 18 4

Total 166

Then, is ; is ; is

; is . Then, is

; is ; is .

The for A is 3.55, so is significant: ChangingF

crit

19>4 5 4.75F

A3B

8>4 5 2.00F

24>4 5 6.00

72>18 5 4MS

38>2 5 19

A3B

8>1 5 8MS

48>2 5 24MS

dfsSSs

MSF

obt

SSMS

obt

dose influenced IQ. The for B is 4.41, so is not

significant. The for is 3.55, so it is signifi-

cant: The influence that changing dose had on IQ

depends on participants’ age.

For Practice

1. For this study, complete the summary table:

Source SS df MS F

A 108 3

____ ____

B 104 2 ____ ____

Interaction 252 6 ____ ____

Within 672 48 ____ ____

Total 536 59 ____ ____

2. The for is ____, the for is ____,

and the for is ____.

3. Which effects are significant?

Answers

1. , , , and ;

, , and

2. 2.80; 3.19; 2.30

3. The main effect of and the interaction effectB

A3B

5 3.00F

5 3.71F

5 2.57

5 14MS

A3B

5 42MS

5 52MS

5 36

A3B

crit

4 3 3

A3B

crit

A QUICK REVIEW

INTERPRETING THE TWO-WAY EXPERIMENT

To understand and interpret the results of a two-way ANOVA, you should examine the

means from each significant main effect and interaction by graphing them and perform-

ing post hoc comparisons.

Graphing and Post Hoc Comparisons with Main Effects

We graph each main effect separately, plotting all main effect means, even those that

the post hoc tests may indicate are not significant. As usual, label the axis as the mean

of the dependent scores and place the levels of the factor on the axis. For our volumeX

Interpreting the Two-Way Experiment 337

factor, the main effect means were 6.00 for low, 11.50 for medium, and 11.33 for loud.

For gender, the means were 11.89 for males and 7.33 for females. Figure 14.1. shows

the resulting graphs. Volume is a continuous, ratio variable, so we create a line graph.

Gender is a discrete, nominal variable, so we create a bar graph.

To determine which main effect means differ significantly, we usually perform

Tukey’s HSD procedure because we usually have equal . (With unequal , perform

Fisher’s t-test, as in Chapter 13.) Recall that the formula for the HSD is

where is from the ANOVA, is from Table 6 in Appendix C for and

(where is the number of levels in the factor), and is the number of scores in a level.

But be careful here: Recall that the and were different for each main effect. In partic-

ular, the is always the number of scores used to compute each mean you are compar-

ing right now! Also, because depends on , when factors have a different , they have

different values of . Therefore, you will have a different HSD for each main effect

when or is different.

Our volume factor has three means, and each is 6. The . With ,

, and , the Thus,

The HSD for factor A is 4.41. The mean for soft (6.00) differs from the means for

medium (11.50) and loud (11.33) by more than 4.41. Thus, soft produces a significant

difference from the other volumes: increasing volume from soft to medium and from

soft to loud produced significant differences (increases) in scores. However, the means

for medium and loud differ by less than 4.41, so these conditions do not differ signifi-

cantly: Increasing volume from medium to loud did not produce a significant change in

scores.

After comparing the main effect means of factor A, we move on the main effect

means of factor B. When a factor contains only two levels (like our gender factor),

HSD 5 1q

b5 13.772a

8.22

b5 4.41

5 3.77.df

5 12k 5 3

 5 .05MS

5 8.22n

kkq

kdf

HSD 5 1q

nsns

Volume

Male

Gender

Mean persuasiveness

Soft Medium Loud Female

Mean persuasiveness as

a function of volume

Mean persuasiveness as

a function of gender

FIGURE 14.1

Graphs showing main

effects of volume and

gender