Recall that in a one‐way fixed effects ANOVA, there is only a single independent variable, and hence we can only draw conclusions about population mean differences on that single variable. However, oftentimes we wish to consider more than a single variable at a time. This will allow us to hypothesize not only main effects (i.e. the effect of a single factor on the dependent variable) but also interac- tions. What is an interaction? An interaction is the effect of one independent variable on the depend‑
ent variable but whose effect is not consistent across levels of another independent variable in the model. An example will help illustrate the nature of an interaction.
A one‐way random effects analysis of variance (ANOVA) was conducted on the achievement data to test the null hypothesis that variance due to teachers on achievement was equal to 0. It was found that approximately 83% of the variance in achievement scores can be attributed to teacher differences, either those sampled for the given experiment or in the population from which these teachers were drawn.
Suppose that instead of simply studying the effect of teacher on achievement, we wished to add a second independent variable to our study, that of textbook used. So now, our overall hypothesis is that both teacher and textbook will have an effect on achievement scores. Our data now appear as follows (Denis 2016):
Teacher 2
1
Textbook 3 4
69
70 85 95
68
67 86 94
70
65 85 89
76
75 76 94
77
76 75 93
75 73
1 1 1 2 2
2 73 91
Achievement as a Function of Teacher and Textbook
When we expand our SPSS data file, our data looks as on the left.
We run the factorial ANOVA in SPSS as follows:
ANALYZE → GENERAL LINEAR MODEL → UNIVARIATE
Under Options, we move (OVERALL), teach, text, and teach*text over under Display Means for, and we also check off Estimates of effect size and Homogeneity tests:
We can see that the data on the left corresponds exactly to the data above in the table. For instance, case 1 has an ac score of 70 and received teacher 1 and textbook 1. Case 2 has an ac score of 67 and received teacher 1 and textbook 1.
We move ac to the Dependent Variable box as usual and move teach and text to the Fixed Factor(s) box (left). Next, click on Plots so we can get a visual of the mean differences and potential interaction:
When we run the ANOVA, we obtain:
UNIANOVA ac BY teach text /METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/POSTHOC=teach(SCHEFFE BONFERRONI) /PLOT=PROFILE(teach*text)
/EMMEANS=TABLES(OVERALL) /EMMEANS=TABLES(teach) /EMMEANS=TABLES(text)
/EMMEANS=TABLES(teach*text) /PRINT=ETASQ HOMOGENEITY /CRITERIA=ALPHẶ05)
/DESIGN=teach text teach*text.
Above SPSS confirms that there are 6 observations in each teach level and 12 observations in each text group. Levene’s test on the equality of variances leads us to not reject the null hypothesis, and so we have no reason to doubt the null that variances are equal.
Next, SPSS generates the primary output from the ANOVA:
Tests of Between-Subjects Effects Dependent Variable: ac
a. R Squared = .976 (Adjusted R Squared = .965) Source Type III Sum
of Squares df Mean Square F Sig. Partial Eta Suared Corrected Model
Intercept teach text teach*text Error Total Corrected Total
2088.958 149942.042
1764.125 5.042 319.792 52.000 152083.000 2140.958
7 1 3 1 3 16 24 23
298.423 149942.042 588.042 5.042 106.597 3.250
91.822 .000 .976
1.000 .971 .088 .860 .000
.000 .231 .000 46136.013
180.936 1.551 32.799
We move teach to the Horizontal Axis box and text to the Separate Lines box. Next, click Add so that it appears as follows:
Between-Subjects Factors teach
text 1.00 2.00 3.00 4.00 1.00 2.00
6 N
6 6 6 12 12
Levene’s Test of Equality of Error Variancesa Dependent Variable: ac
Tests the null hypothesis that the error
a. Design: Intercept + teach + text + teach * text
Variance of the dependent variable is equal across groups.
F 2.037
df1 df2 Sig.
7 16 .113
We see that there is a main effect of teach (p = 0.000) but not of text (p = 0.231).
There is evidence of an interaction effect teach*text (p = 0.000).
Recall that Partial Eta‐squared is similar in spirit to Eta‐squared but is computed partialing out other sources of variance in the denominator rather than including them as Eta‐squared does in SS total. Partial Eta‐squared is calculated as
Partial SS effect SS effect SS error
2
Notice that the denominator is not SS total. It only contains SS effect and SS error. In this way, we would expect Partial Eta‐squared to be larger than Eta‐squared, since its denominator will not be as large as that used in the computation of Eta‐squared. We compute partial Eta‐squared for teach:
Partial2 1764 125
1764 125 52 000. 0 971
. . .
SPSS generates for us the plot of the interaction effect:
text 1.00 2.00 Estimated Marginal Means of ac
95.00 90.00 85.00 80.00 75.00 70.00 65.00
Estimated Marginal Means
1.00 2.00 3.00 4.00
teach
A two‐way fixed effects analysis of variance was performed on the achievement data to learn of any mean differences on teach and text and whether evidence presented itself for an interaction between these two factors. Evidence for a main effect for teach was found (p < 0.001) as well as an interaction effect of teach and text (p < 0.001), with partial eta‐squared values of 0.971 and 0.860, respec- tively. No evidence was found for a text effect (p = 0.231). An interaction plot was obtained to help visualize the teach by text interaction as evidenced from the two‐way analysis of variance. It is evident from the plot that means for text 2 were higher than means for text 1 for teachers 1 and 2, but this effect reversed itself for teacher 3. At teacher 4, means were equal.
We make the following observations regarding the plot:
● The presence of an interaction effect in the sample is evident. Across levels of teach, we notice the mean differences of text are not constant.
● At teach = 1, we can see that the mean achieve- ment is higher for text = 2 than it is for text = 1.
● At teach = 2, we see the above trend still exists, though both means rise somewhat.
● At teach = 3, we notice that text = 1 now has a much higher achievement mean than does text = 2 (the direction of the mean difference has reversed).
● At teach = 4, it appears that there is essentially no difference in means between texts.