Two-Way Independent Samples ANOVA with SPSS

Two-Way Independent Samples ANOVA with SPSSã
Obtain the file ANOVA2.SAV on my SPSS Data page. The data are those that appear in Table 17-3 of Howell’s Fundamental statistics for the behavioral sciences (6th ed.) and in Table 13-2 of Howell’s Statistical methods for psychology (6th ed.). The independent variables are age of participant (young or old) and depth of cognitive processing (manipulated by the instructions given to participants prior to presentation of a list of words). The dependent variable is number of words correctly recalled later.
Bring the data file, ANOVA2.SAV, into SPSS. To conduct the factorial analysis, click Analyze, General Linear Model, Univariate. Scoot Items into the Dependent Variable box and Age and Condition into the Fixed Factors box. Click Plots and scoot Conditon into the Horizontal Axis box and Age into the Separate Lines box. Click Add, Continue. Click Post Hoc and scoot Conditon into the "Post Hoc Tests for" box. Check REGWQ. Click Continue. Click options, check Descriptive Statistics and Estimates of Effect Size, click Continue. Click OK.
Look at the plot. The plot makes it pretty clear that there is an interaction here. The difference between the oldsters and the youngsters is quite small when the experimental condition is one with little depth of cognitive processing (counting or rhyming), but much greater with higher levels of depth of cognitive processing. With the youngsters, recall performance increases with each increase in depth of processing. With the oldsters, there is an interesting dip in performance in the intentional condition. Perhaps that is a matter of motivation, with oldsters just refusing to follow instructions that ask them to memorize a silly list of words.
Do note that the means plotted here are least squares means (SPSS calls them estimated means). For our data, these are the same as the observed means. We had the same number of scores in each cell of our design. If we had unequal numbers of scores in our cells, then our independent variables would be correlated with one another, and the observed means would be 'contaminated' by the correlations between independent variables. The estimated means represent an attempt to estimate what the cells means would be if the independent variables were not correlated with one another. These estimated means are also available in the Options dialog box.
Look at the output from the omnibus ANOVA. We generally ignore the F for the "Corrected Model” -- that is the F that would be obtained if we were to do a one-way ANOVA, where the groups are our cells. Here it simply tells us that our cell means differ significantly from one another. The two-way factorial ANOVA is really just an orthogonal partitioning of the treatment variance from such a one-way ANOVA -- that variance is partitioned into three components: The two main effects and the one interaction. We also ignore the test of the intercept, which tests the null hypothesis that the mean of all the scores is zero. If you divide each effect's SS by the total SS, you see that the condition effect accounts for a whopping 57% of the total variance, with the age effect only accounting for 9% and the interaction only accounting for 7%. Despite the fact that all three of these effects are statistically significant, one really should keep that in mind, and point out to the readers of the research report that the age and interaction effects are much less in magnitude than is the effect of recall condition (depth of processing).
Look at the within-cell standard deviations. In the text book, Howell says "it is important to note that the data themselves are approximately normally distributed with acceptably equal variances." I beg to differ. Fmax is 4.52 / 1.42 > 10 -- but I am going to ignore that here.
The interpretation of the effect of age is straightforward -- the youngsters recalled significantly more items than did the oldsters, 3.1 items on average. The pooled within-age standard deviation is computed by taking the square root of the mean of the two groups’ variances -- . The standardized difference, d, is then 3.1/4.977 = .62. Using Cohen's guidelines, that is a medium to large sized effect. In terms of percentage of variance explained,
The interpretation of the recall condition means is also pretty simple. With greater dept of processing, recall is better, but the difference between the intentional condition and the imagery condition is too small to be significant, as is the difference between the rhyming condition and the counting condition. The pooled standard deviation within the intentional recall and the counting conditions is . Standardized effect size, d, is then , an enormous effect. In terms of percentage of variance explained by recall condition,
Although the significant interaction effect is small (h2 = .07) compared to the main effect of recall condition, we shall investigate it by examining simple main effects. For pedagogical purposes, we shall obtain the simple main effects of age at each level of recall condition as well as the simple main effects of recall condition for each age.
Notice that SPSS gives you values of partial eta-squared. Also note that they sum to more than 100% of the variance. If you want to place confidence intervals on the obtained values of eta-squared, you must compute an adjusted F for each effect, as I have shown you elsewhere. To place confidence intervals on partial eta-squared you need only the F and df values that SPSS reports. Using the NoncF script, here are the confidence intervals:

Return to the Data Editor. Click Data, Split File. Tell SPSS to organize the output by groups based on the Conditon variable. OK. Click Analyze, Compare Means, One-Way ANOVA. Scoot Items into the Dependent List and Age into the Factor box. OK.
The results show that the youngsters recalled significantly more items than did the oldsters at the higher levels of processing (adjective, imagery, and intentional), but not at the lower levels (counting and rhyming). The tests we have obtained here employ individual error terms – that is, each test is based on error variance from only the two groups being compared. Given that there is a problem with heterogeneity of variance among our cells, that is actually a good procedure. If we did not have that problem, we might want to get a little more power by using a pooled error term. What we would have to do is take the treatment MS for each of these tests, divide it by the error MS from the overall factorial analysis, and evaluate each resulting F with the same error df used in the overall ANOVA. Our error df would then be 90 instead of 18, which would give us a little more power.
Return to the Data Editor. Click Data, Split File. Tell SPSS to organize the output by groups based on the Age variable. OK. Click Analyze, Compare Means, One-Way ANOVA. Leave Items in the Dependent List and replace Age with Conditon in the Factor box. OK.
Note that the effect of condition is significant for both age groups, but is larger in magnitude for the youngsters (h2 = .83) than for the oldsters (h2 = .45). I don't think that the pairwise comparisons here add much to our understanding, but lets look at them briefly. Among the oldsters, mean recall in the adjective, intentional, and imagery conditions was significantly greater than in the rhyming and counting conditions. Among the youngsters, mean recall in the adjective conditions was significantly greater than that in the counting and rhyming conditions and significantly less than that in the imagery and intentional conditions.

Writing up the Results – Here is an Example
A 2 x 5 factorial ANOVA was employed to determine the effects of age group and recall condition on participants’ recall of the items. A .05 criterion of statistical significance was employed for all tests. The main effects of age, F(1, 90) = 29.94, p < .001, hp2 = .25, CI.95 = .11, .38, and recall condition, F(4, 90) = 47.19, p < .001, hp2 = .68, CI.95 = .55, .74, were statistically significant, as was their interaction, F(4, 90) = 5.93, p < .001, hp2 = .21, CI.95 = .05, .32; MSE = 8.03 for each effect. Overall, younger participants recalled more items (M = 13.16) than did older participants (M = 10.06). The REGWQ procedure was employed to conduct pairwise comparisons on the marginal means for recall condition. As shown in the table below, recall was better for the conditions which involved greater depth of processing than for the conditions that involved less cognitive processing.

Table 1. The Main Effect of Recall Condition

Recall Condition

Counting
Rhyming
Adjective
Imagery
Intentional
Mean
6.75 A
7.25A
12.90B
15.50C
15.65C
Note. Means sharing a letter in their superscript are not significantly different from one another according to REGWQ tests.

The interaction is displayed in the following figure. Recall condition had a significant simple main effect in both the younger participants, F(4, 45) = 53.06, MSE = 6.38, p < .001, h2 = .83, CI.95 = .70, .87) and the older participants, F(4, 45) = 9.08, MSE = 9.68, p < .001, h2 = .45, CI.95 = .18, .57), but the effect was clearly stronger in the younger participants than in the older participants. The younger participants recalled significantly more items than did the older participants in the adjective condition, F(1, 18) = 7.85, MSE = 9.2, p = .012, h2 = .30, CI.95 = .02, .55 the imagery condition, F(1, 18) = 6.54, MSE = 13.49, p = .020, h2 = .27, CI.95 = .005, .52, and the intentional condition, F(1, 18) = 25.23, MSE = 10.56, p < .001, h2 = .58, CI.95 = .23, .74, but the effect of age fell well short of significance in the counting condition, F(1, 18) = 0.46, MSE = 2.69, p = .50, h2 = .03, CI.95 = .00, .25, and in the rhyming condition, F(1, 18) = 0.59, MSE = 4.18, p = .45, h2 = .03, CI.95 = .00, .27.


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Copyright 2007, Karl L. Wuensch - All rights reserved.
ã Copyright 2007, Karl L. Wuensch - All rights reserved.