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Analysis of Variance (ANOVA) in SPSS.
The ANOVA family of analysis are used for testing whether or not a
significant difference exists between more than two groups. There are many forms
of ANOVA which allows it to be used in a variety of situations. The simplest is
the oneway ANOVA which is used for testing multiple groups of one independent
variable's effect on one continuous or nearly continuous dependent variable. The
oneway name implies one independent variable.
All examples below will utilize the
ExampleData001.sav data file.
(1) Oneway ANOVA
First, click on Analyze, Compare Means, One-Way ANOVA...
Next, highlight / select the Recalled (Time1) variable and use the top arrow
button to move it to the Dependent List: box. Then, highlight the Stimuli
Presented variable and use the bottom arrow button to move it to the Factor: box.
Next, click on the Post Hoc... button and specify which tests you want for
equal variances assumed and equal variances not assumed. Then click the Continue
button.
Next, click on options and select Descriptive (provides descriptive
statistics for each group), Homogeneity of variance test (a Levene's test for
testing the assumption of equal variances), Welch (a robust F test), and
Means plot (provides a line graph showing each group's mean). Then click the
Continue button. Then click the OK button.
The output should look like that displayed below.
The means plot is a graphical representation of the differences between the
means, but a much better graphical representation can be produced using the
Boxplot function.
First, click on Graphs, Legacy Dialogs, Boxplot...
Next, click the Define button (the default Simple and Summaries for groups of
cases are appropriate for this example). Then, highlight the Recalled (Time1)
variable and use the top arrow button to move it to the Variable: box. Then,
highlight the Stimuli Presented variable and use the second arrow to move it to the
Category Axis: box.
Then, click the OK button. The boxplot should be similar to the one displayed
below.
The boxplot above really highlights what the ANOVA does for us. It uses
variance to test for mean differences. The ANOVA does this by comparing the
between groups variance to the within groups variance. If the between groups
variance is greater than the within groups variance, then we tend to have a
significant effect. Looking at the boxplot, if we focus on the height of the
Printed group's box and whiskers, then we see a representation of that
group's variance (i.e. within that group variance). If we visually sum
each group's variance, then we have our within group variance for comparison to
the between group variance. The between group variance can be seen by comparing
the horizontal line of each group's box (their means). Going further, the
Levene's test for homogeneity of variance is specifically concerned with whether
or not the variance of each group is significantly different (from one another).
If the Levene's test is significant (e.g. p < .05), then the assumption
of homogeneity of variance is violated and we cannot have confidence in the
omnibus F
test results. To be more specific, we could not be confident that the
observed mean differences were attributable to the conditions of each group or
if those differences were attributable to individual by treatment interaction
effects (also called a subject by treatment interaction). Since the Levene's test was
significant, we would interpret the Welch's Robust test table rather than the
ANOVA summary table when interpreting the F statistic. We would also then
interpret the Games-Howell post hoc results rather than the REGW-Q. As mentioned
in the t test tutorial, consult
this article
for a more thorough discussion of the Levene's test and the homogeneity of
variance assumption. We can
interpret the Welch's Robust ANOVA as indicating a significant mean difference among the the
participants of three groups in terms of their number of words recalled, F(2,
32.80) = 13.60, p < .001. Furthermore, the Games-Howell post hoc testing
reveals a significant difference between the Printed and Spoken group and the
Printed group, as well as a significant difference between the Printed and
Spoken group and the Spoken group indicating that the group which received both
types of stimuli recalled significantly more words than each group receiving
only one type of stimuli.
(2) Oneway ANCOVA
The Oneway ANCOVA is an extension of the oneway ANOVA. The oneway Analysis of
Covariance (ANCOVA) simply allows us to test for mean differences among more
than two groups of one independent variable while controlling for one or more
continuous or nearly continuous covariates.
First, click on Analyze, General Linear Model, Univariate...
Next, highlight the Recall (Time1) variable and use the top arrow button to
move it to the Dependent Variable: box. Then, highlight the Stimuli Presented variable
and use the second arrow button to move it to the Fixed Factor(s): box. Then,
highlight the Age variable and use the fourth arrow button to move it
to the Covariate(s): box.
Next, click on the Plots... button. Highlight the Stimuli variable and move
it to the Horizontal Axis: box. Then click on the Add button to move it to the
Plots: box. Then click the Continue button.
Next, click on the Options button and make sure OVERALL has been moved to the
Display Means for: box. Also select Descriptive statistics, Estimates of effect
size (provides Partial Eta squared), and Homogeneity tests (Levene's). You may
be tempted to select Observed power, but recall this is virtually irrelevant. As
a good researcher, you will have calculated the appropriate sample size based on
the level of power and effect size desired prior to collecting your data. Next,
click the Continue button, then click the OK button.
The output should be similar to what is displayed below.
We can see from the ANOVA summary table, Age does not have a significant
effect on Recall (Time1). However, the Levene's test was significant (p <
.05) which means we have violated our assumption of homogeneity of variances.
So, although we appear to have a significant effect for Stimuli Presented. Also
of note is our effect size; the Partial Eta squared (partial
η² = .328) indicates that only 32.8% of the variance in Recall (Time1) is
accounted for by our independent variable Stimuli Presented after we partial
out the influence of Age. Furthermore, Eta squared tends to be an overestimate
of the relationship in the population so, we can safely assume and even weaker
relationship than what we have here in this sample. A better estimate of effect
size in the ANOVA situation is Omega squared:
Omega squared offers a less biased estimate of the amount of variance
accounted for in our dependent variable by the independent variable effect(s).
(3) Factorial ANOVA
The Factorial ANOVA is an extension of the Oneway situation where the design
is composed of more than one independent variable, each with two or more groups
(sometimes called multi-way ANOVA). The major benefit of factorial ANOVA is the
ability to investigate interactions among the independent variables. The
Factorial ANOVA is still considered a univariate analysis (as opposed to a
multivariate analysis) because, it deals with only one dependent variable (where
the multivariate ANOVA deals with multiple dependent variables).
Start by clicking Analyze, General Linear Model, Univariate...
Next, highlight the Recalled (Time1) variable and use the top arrow button to
move it to the Dependent Variable: box. Then, highlight the Candy variable and
move it to the Fixed Factor(s): box. Then highlight the Stimuli Presented
variable and move it to the Fixed Factor(s): box also. Notice we could specify
one or more covariates and make this analysis a Factorial ANCOVA. Next, click on
Plots...
Now move Stimuli to the Horizontal Axis: box and move Candy to the Separate
Lines: box. Generally it is preferable to have few lines and the variable with
more groups listed along the x-axis. Next, click on the Add button. Then click
the Continue button.
Next, click on the Post Hoc... button.
Now, because we only have one (of two) independent variable with more than
two groups, we will need to specify post hoc testing for that variable. Here we
have specified the REGW-Q. Click the Continue button, then click on the
Options... button.
Here we specify which variables we want a means chart displayed for, as well
as the usual descriptive statistics, estimates of effect size, and homogeneity
tests. It may seem silly to ask for displayed means and also the descriptive
statistics; however, if there are un-even cell sizes, they will be different.
Therefore, it is good practice to always ask for a display of means for each
variable and the descriptive statistics. Next, click the Continue button and
then click the OK button to complete the analysis.
The output should look similar to that displayed below.
We can see from the Levene's test table, the assumption of homogeneity of
variances was not violated. According to the between subjects effects table, it
appears we do have significant main effects for both Stimuli Presented and
Candy; as well as a significant interaction between the two (p < .05).
Simple effects analysis would be necessary to tease out where the interaction
effect is actually significant (among all 6 conditional cells). Although, the
Partial Eta squared value associated with our interaction effect is rather
paltry.
Simple Effects Analysis
Unfortunately, SPSS does not allow for specification of simple effects
analysis through point and click options. One must use syntax to get the tests
of simple main effects. So, if we return to the Data Window, click on Analyze,
General Linear Model, Univariate... once again.
We notice the previous run and all its options are still specified. This time
however, instead of clicking the OK button, we need to click the Paste button;
which opens a new syntax window with the syntax written as specified through the
use of the point and click dialog and options.
Next, in the syntax window, we need to insert a line (or lines) specifying
the simple main effect test we want. For the current example, we will test for
differences in Stimuli at each level of Candy. The necessary line has a red
ellipse around it: /EMMEANS = TABLES(Stimuli*Candy)compare(Stimuli)
If we then highlight the entire syntax and click the green triangle (run
selection) button, we should get an additional table in the output which
provides us with the desired tests.
So, we see we have a significant effect for Stimuli Presented at each of the
two levels of Candy (p < .001).
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