Two-Way ANOVA

When a researcher is interested in examining the effects of two independent variables on a single dependent variable, a two-way ANOVA is the appropriate statistical analysis.

 Illustrated Example of a Statistically Significant Main Effect

A researcher wanted to know if she could reduce the amount of social desirability (i.e., participant responding in the most socially desirable manner) in survey responses if the survey was administered on the WWW verses paper pencil. In addition, she believes that gender may have an effect on social desirability. She randomly assigns participants into groups that are administered the survey via the WWW or administered the survey via paper pencil. The results are below.

(3 tests)

    Survey Administration
    PP WWW  
Gender Female MF_PP MF_www MF
Male MM_PP MM_www MM
  Mpp Mwww  

 [Download Data]

WWW

 

Pencil/Paper

Female

Male

 

Female

Male

10

15

 

18

19

12

14

 

17

20

14

12

 

16

21

13

11

 

19

19

15

12

 

20

22

12

13

 

19

19

15

11

 

18

18

 

 

 

 

 Write Up

Prior to the data analysis, the data were screened for outliers and normality of distribution. There were no outliers and both the kurtosis and skewness test indicated no serious departures for normality (all coefficients resulted in absolute values of less than 1). Levene's test for homogeneity of group variance was nonsignificant.

A 2 X 2 ANOVA was used to examine the social desirability level by gender and type of survey administration (WWW and Pencil/Paper). The means and standard deviations for social desirability scores are reported in Table 1. There was not a statistically significant interaction effect (F=3.00, p>.05) or main effect for gender (F>.98, p>.05). There was a statistically significant main effect for type of survey administration (F=113.20, p<.05). Participants who were administered the survey via the WWW had a much lower mean (M=12.79) than those administered the survey via paper/pencil (M=18.93), d=3.98.

Table 1

Social Desirability Means and Standard Deviations for Treatment Groups and Gender

 

Treatment

Gender

M

SD

N

WWW

female

13.00

1.83

7

 

male

12.57

1.51

7

 

Total

12.79

1.63

14

 

 

 

 

 

Pencil/Paper

female

18.14

1.35

7

 

male

19.71

1.38

7

 

Total

18.93

1.54

14

 

 

 

 

 

Total

female

15.57

3.08

14

 

male

16.14

3.96

14

 

Total

15.86

3.49

28

 

Illustrated Example of a Statistically Significant Main Effect

This example uses the same research questions as previously presented. The data have been changed.

[Download Data] 

WWW

 

Pencil/Paper

Female

Male

 

Female

Male

10

15

 

18

19

12

14

 

17

20

14

12

 

16

21

13

11

 

19

19

15

12

 

20

22

12

13

 

19

19

15

11

 

18

18

 

 

 

 

Write Up

Prior to the data analysis, the data were screened for outliers and normality of distribution. There were no outliers and both the kurtosis and skewness test indicated no serious departures for normality (all coefficients resulted in absolute values of less than 1). Levene's test for homogeneity of group variance was nonsignificant.

A 2 X 2 ANOVA was used to examine the social desirability level by gender and type of survey administration (WWW and Pencil/Paper). The means and standard deviations for social desirability scores are reported in Table 1. There was a statistically significant interaction effect (F=27.08, p<.05). An illustration of the interaction is displayed in Figure 1. Males who were administered the survey via the WWW had a statistically significantly (p<.05) lower mean social desirability score (M=12.27) than (a) males administered the survey via paper/pencil (M=19.71), (b) females administered the survey via WWW (M=18.00) or (c) females in the pencil/paper group (M=18.14), with effect sizes (g) of 4.93, 2.83, and 3.89 respectively.

 

Table 1

Social Desirability Means and Standard Deviations for Treatment Groups and Gender

Treatment

Gender

M

SD

N

WWW

female

18.00

2.58

7

 

male

12.57

1.51

7

 

Total

15.29

3.47

14

 

 

 

 

 

Pencil/Paper

female

18.14

1.35

7

 

male

19.71

1.38

7

 

Total

18.93

1.54

14

 

 

 

 

 

Total

female

18.07

1.98

14

 

male

16.14

3.96

14

 

Total

17.11

3.22

28

 

 

Figure 1.

Example Problem—Reporting Multiple ANOVAs

 An art to writing up results in research is to present the most information in the least amount of space. We are still examining the effect of type of method of administering surveys (WWW or Pencil/Paper) on the respondents. In addition to examining the social desirability, the research is also interested in examining participants’ motivation level and feelings of confidentiality. In other word, the research now has three dependent variables, and will be calculating three two-way ANOVAs. It should be noted that the researcher should consider running MANOVA, but for illustration purposes univariate ANOVAs will be conducted. Below are the data.

 [Download Data] 

WWW Administration

 

Pencil/Paper

Social

Desirability

Motivation

Confidentiality

Gender

1=Female

2=Male

 

Social

Desirability

Motivation

Confidentiality

Gender

1=Female

2=Male

10

5

2

1

 

18

5

6

1

12

4

4

1

 

17

7

7

1

14

8

3

1

 

16

8

6

1

13

6

6

1

 

19

7

8

1

15

7

2

1

 

20

5

9

1

12

5

3

1

 

19

6

7

1

15

7

4

1

 

14

7

5

1

15

2

8

2

 

19

2

6

2

14

4

9

2

 

20

3

7

2

12

5

7

2

 

21

4

8

2

9

3

8

2

 

19

5

6

2

12

4

7

2

 

22

2

6

2

13

2

6

2

 

19

3

5

2

11

6

8

2

 

18

5

8

2

 

 

You have to run the analysis three times changing the dependent variable each time. The factors will remain the same.

 Writing Up

 The means, standard deviations for all the dependent variables for gender and survey administered type are reported in Table 1. The results of the two-way ANOVAs are reported in Table 2. For social desirability there was a statistically significant main effect for survey type. Participants in the WWW administered group had on average a lower level of social desirability than those participants in the pencil/paper group (g = 3.09). There was a statistically significant main effect for gender on motivation level. Females tended to have a higher level of motivation than males (g = 2.04). The interaction effect was statistically significant for confidentiality. An illustration of this interaction is displayed in Figure 1. Follow-up analyses (Scheffe) indicated that females who were administered the survey via the WWW had on average a much lower perception of confidentiality than (a) females administered the survey via pencil/paper (g = 2.49), (b) males administered survey via WWW (g = 3.42), and (c) males administered survey via pencil/paper (g = 2.47).

 Table 1

Sample Sizes, Means, and Standard Deviations for Dependent Variables by Group  

 

 

 

Social Desirability

 

Motivation

 

Confidentiality

GROUP

GENDER

N

M

SD

 

M

SD

 

M

SD

WWW

female

7

13.00

1.83

 

6.00

1.41

 

3.43

1.40

 

male

7

12.29

1.98

 

3.71

1.50

 

7.57

0.98

 

Total

14

12.64

1.86

 

4.86

1.83

 

5.50

2.44

 

 

 

 

 

 

 

 

 

 

 

Pen/Paper

female

7

17.57

2.07

 

6.43

1.13

 

6.86

1.35

 

male

7

19.71

1.38

 

3.43

1.27

 

6.57

1.13

 

Total

14

18.64

2.02

 

4.93

1.94

 

6.71

1.20

 

 

 

 

 

 

 

 

 

 

 

Total

female

14

15.29

3.02

 

6.21

1.25

 

5.14

2.21

 

male

14

16.00

4.19

 

3.57

1.34

 

7.07

1.14

 

Total

28

15.64

3.60

 

4.89

1.85

 

6.11

1.99

 

 Table 2

Results of Two-way ANOVAs 

 

Social Desirability

 

Motivation

 

Confidentiality

Effects

F-Ratio

 

hp2

 

F-Ratio

 

hp2

 

F-Ratio

 

hp2

Survey Type

75.06

**

.76

 

.02

 

.00

 

6.88

*

.22

Gender

1.06

 

.04

 

27.38

**

.53

 

17.36

**

.42

Interaction

4.26

 

.15

 

.50

 

.02

 

22.88

**

.49

Note. dfs=1, 24 for all analyses.

 

 

Figure 1.