The Solution to the Factorial Analysis of Variance

As shown in the Excel file, Howell 13-2, the ANOVA analysis (in the ToolPac) yielded the following table:

 

All effects—Sample (Group), Columns (Condition), and the Group x Condition interaction—were statistically significant at p < .05. (I provide a sample write-up later.)  

For the next analysis I used VassarStats  > Two-Way Factorial ANOVA for Independent Samples. Following the directions given there, I obtained the following. (Note that the ANOVA summary table is the same as that for the Excel example.)  

Data Entered

 

Col 1

Col 2

Col 3

Col 4

Row 1

9
8
6
8
10
4
6
5
7
7

7
9
6
6
6
11
6
3
8
7

11
13
8
6
14
11
13
13
10
11

12
11
16
11
9
23
12
10
19
11

Row 2

8
6
4
6
7
6
5
7
9
7

10
7
8
10
4
7
10
6
7
7

14
11
18
14
13
22
17
16
12
11

20
16
16
15
18
16
20
22
14
19

Row 3

---

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---

---

Row 4

---

---

---

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Summary Data

Within each box:
  Item 1 = N     Item 2 =  X     Item 3 = Mean
  Item 4 =  X2     Item 5 = Variance
  Item 6 = Std. Dev.     Item 7 = Std. Err.

 

C1

C2

C3

C4

Tot.

R1

10
70
7
520
3.33
1.83
0.58

10
69
6.9
517
4.54
2.13
0.67

10
110
11
1266
6.22
2.49
0.79

10
134
13.4
1978
20.27
4.5
1.42

40
383
9.575
4281
15.74
3.97
0.63

R2

10
65
6.5
441
2.06
1.43
0.45

10
76
7.6
612
3.82
1.96
0.62

10
148
14.8
2300
12.18
3.49
1.1

10
176
17.6
3158
6.71
2.59
0.82

40
465
11.625
6511
28.34
5.32
0.84

R3

---

---

---

---

---

R4

---

---

---

---

---

Tot.

20
135
6.75
961
2.62
1.62
0.36

20
145
7.25
1129
4.09
2.02
0.45

20
258
12.9
3566
12.52
3.54
0.79

20
310
15.5
5136
17.42
4.17
0.93

80
848
10.6
10792
22.83
4.78
0.53

ANOVA Summary

Source

SS

df

MS

F

P

Rows

84.05

1

84.05

11.37

0.0012

Columns

1106.9

3

368.97

49.92

<.0001

r x c

80.05

3

26.68

3.61

0.0173

Error

532.2

72

7.39

   

Total

1803.2

79

     

Critical Values for the Tukey HSD Test

 

HSD[.05]

HSD[.01]

HSD=the absolute [unsigned] difference between any two means (row means, column means, or cell means) required for significance at the designated level: HSD[.05] for the .05 level; HSD[.01] for the .01 level. The HSD test between row means can be meaningfully performed only if the row effect is significant; between column means, only if the column effect is significant; and between cell means, only if the interaction effect is significant.

Rows [2]

1.21

1.61

Columns [4]

2.26

2.78

Cells [8]

3.8

4.48

 Since there are only two groups, the HSD test for Rows is unimportant. The F test for the row effect was already shown to be significant in the ANOVA Summary table. The significant F  for the columns effect, however, does require the post hoc HSD test.  

From the third row of the Tot subsection of the Summary Data table, the means of the four conditions are 6.75, 7.25, 12.9, and 15.5. From the Tukey HSD table we see that to be statistically significant at p <.05 we need a difference between any two means to be at least 2.26 units. The difference between means for Condition 1 and Condition 2 does not satisfy this criterion. Hence we find no evidence that rhyming was any more effective than counting. On the other hand the differences between Condition 3 and Condition 2 (an, of course, Condition 1) was statistically significant. The same can be said about Condition 4 (vs Conditions 1 & 2). Also, the difference between Condition 4 and Condition 3 is statistically significant.

There is still a problem that needs to be addressed, however. The presences of a significant interaction tells us that there is a differential effect of Conditions depending upon which group, young or old, is examined. I will treat this in the next analysis.

 ______________________________________________________

In the final analysis, I used SPSS to analyze the data in Howell Table 13-2. The syntax I used for this analysis, assuming the data have been obtained from the Excel worksheet, List, was  

SPSS Syntax for Analyzing Howell’s Table 13-2  

*The commands for this first analysis specify an overall 2x3 ANOVA.  

DATASET ACTIVATE DataSet1.
UNIANOVA Score BY Group Condition
  /METHOD=SSTYPE(3)
  /INTERCEPT=INCLUDE
  /PLOT=PROFILE(Condition*Group)
  /CRITERIA=ALPHA(0.05)
  /DESIGN=Group Condition Group*Condition.  

*Since the first analysis yielded a significant Group (or Age) by Condition Interaction, we should perform simple effects analyses within groups. This command splits the data file by Group.  

SPLIT FILE LAYERED BY Group.  

*The next set of commands re-run the earlier analysis, only this time the analysis is computed once for each group. Additionally, for each analysis, SPSS is instructed to generate plots of means.  

UNIANOVA Score BY Group Condition
  /METHOD=SSTYPE(3)
  /INTERCEPT=INCLUDE
  /POSTHOC=Condition(TUKEY)
  /PLOT=PROFILE(Condition*Group)
  /CRITERIA=ALPHA(0.05)
  /DESIGN=Group Condition Group*Condition.

 

 You can view the results for this analysis by copying the syntax above, pasting it into a new SPSS syntax file, then running the syntax.

 In the Output Navigator (the pane to the left) you can scroll down to a heading labeled, Tests of Between-Subjects Effects to display the following table.

 

As is the case with many general statists packages, this table has more information than what is typically given in an ANOVA summary table. For instance, we typically are not interested in the [Mean] Corrected Model or Intercept sources of variance since these typically are always highly significant. Moreover, we usually DO want to report the (Mean) Corrected Total Variance rather than the Total Variance. By double clicking on the table in SPSS the table can be edited so that it looks like the following:

 

As mentioned earlier, since the test for the Group x Condition was significant, the significant effect due to Conditions is different for each Group. Because of this, we would normally be motivated to compute a simple effects analysis for each Group. Computing simple effect analyses is tantamount to computing a separate ONEWAY ANOVA on Condition for each Group. The simple effects syntax, when run, yields the following ANOVA summary table (Note that there are two Anovas given in the table).

 

Again, there is more information given in the table than we need. In addition to the Corrected Model, Intercept, and Total sources of variance, we do not need the Group and Group * Condition source of variance since, in simple effects, these are not factors. By double clicking on this table in SPSS, the table can be edited to look like the following.

 

The Condition effect is significant within both groups.  

The syntax also instructed SPSS to compute post hoc comparisons, using Tukey’s HSD test. Since the data file was split on Group, these comparisons are performed separately for each group. The post hoc analyses are summarized in a homogeneous subsets table:  

For both age groups, the Adjective and Imagery conditions resulted in higher levels of recall than did the Rhyming and Counting conditions. Furthermore, the difference was greater in the Young Group. This was evident in the plots from the over all analysis: