Much as a t-test allows us to compare means for two groups, an ANalysis Of VAriance test allows us to compare means for 3 or more groups, There are three types of basic ANOVA tests: a one-way ANOVA, a factorial ANOVA, and a repeated measures ANOVA.
A one-way ANOVA allows us to test whether several means (for different conditions or groups) are equal across one variable.
1) Normality. Assume that the population distributions are normal. ANOVA is quite robust over moderate violations of this assumptions. Check for normality by creating a histogram.
2) Independent Observations. The observations within each treatment condition must be independent.
3) Equal Variances. Assume that the population distributions have the same variances. This assumption is quite important (if it is violated, it makes the test's averaging of the variances meaningless). As long as the largest variance is no more than 4 or 5 times the size of the smallest variance and the sample sizes are equal, then the test is robust to violations. Standard deviation is the square root of variance, so you can test this by hand using the SPSS output. Square the largest and smallest standard deviation to get the variances, and then divide the larger by the smaller.
Here, our one-way ANOVA will test whether how many hours of work Reed students do per night is different based on what year they are - first year, sophomore, junior, or senior. This data comes from the Kathy Oleson's Social Psych class survey from 2005.
1) Select "General Linear Model" in the "Analyze" menu.
2) Select "Univariate..." from this menu.
3) As seen in Figure 5.1, select a dependent variable.
4) Next, select the factor (independent variable) upon which to base your mean comparison and place it in the "Fixed Factor(s)" box.
Figure 5.1 One-Way ANOVA
5) To display the group means, click the Options button (as shown in Figure 5.2) and then add your independent variable (here, "year") to the "Display Means for" list. You may also want to select the "Descriptive Statistics" option to obtain more information about each group such as standard deviation and count in addition to the means.
Figure 5.2 Options Window for One-Way ANOVA
6) You may wish to run post-hoc tests on your ANOVA to examine individual mean differences. To do so, click "Post Hoc..." to bring up a screen similar to Figure 5.3. Add the variables you wish to test (here, "year"). Most often, you will use the Tukey test.
Figure 5.3 Post-Hoc Test Selection Screen
If you've chosen to get descriptive statistics and post-hoc tests, SPSS will give you many tables. You will see a table with the information on how many people (or whatever you're analyzing) are in each group. The next table (Figure 5.4) tells you the mean, standard deviation, and total count (N) in each group.
Figure 5.4 Descriptive Statistics Table from One-Way ANOVA
You will then see the actual ANOVA table (Figure 5.5). You are interested in the third and fourth row. The third row has the name of your independent variable in capitals; use the F-value and p-value from there. Your degrees of freedom "between" is also in that row. Use the degrees of freedom in the next row, labeled "Error," to get your degrees of freedom "within."
Figure 5.5 Output from One-Way ANOVA
You will see another means table, and then your post-hoc tests. If you used a Tukey test, it will look like the table below (Figure 5.6). Here, it lists the four levels of "year" in the first column, and then compares each level to every other level to see if they are significantly different. It lists the mean difference between levels; here, we see that first year students do an average of .99 hours less work per night than sophomores. If a two means differ significantly, SPSS will star the pair; you can also find the p-value of each comparison in the "Sig." column. SPSS also gives you another table about the Tukey test that gives you similar information.
Figure 5.6 Tukey HSD output from One-Way ANOVA
Be sure to include the type of test you used, what you are evaluating, the direction of the effect, the group means, the F-value and degrees of freedom (between, within), and the p-value.
Sometimes you may want to compare the means for a variable across two or more variables, rather than one. This is known as a factorial design. For example, if you wanted to know how gender and sleep patterns affect GPA, you could run two separate one-way ANOVAs, or you could run a factorial ANOVA. The advantage of the factorial ANOVA is that it also allows you to examine interaction effects between your independent variables. For instance, perhaps boys who sleep during the day have a higher GPA than boys who sleep during the night, but for girls the pattern is reversed. The best way to observe such a pattern is with a factorial ANOVA.
1) Normality. Assume that the population distributions for each of your cells are normal. ANOVA is quite robust over moderate violations of this assumption. Check for normality by creating a histogram.
2) Independent Observations. The observations within each cell must be independent.
3) Equal Variances. Assume that the population distributions for each cell have the same variances.
Here, we are testing whether gender, which college one attends (Reed, PSU, or University of Oregon), and the combination of the two predicts someone's reported neuroticism (also called emotional volatility). This data comes from the senior thesis of Leigh Wensman '05.
1) Click "General Linear Model" under the "Analyze" menu.
2) Select "Univariate" from the list.
3) As demonstrated in Figure 5.7, add your dependent variable and your fixed factors. Fixed factors can be numeric or string variables, but the dependent variable must be numeric.
Figure 5.7 Factorial ANOVA
4) You can also arrange for any post-hoc tests by clicking the "Post Hoc..." button on the left of the window. To get descriptive statistics about each group, click on the "Options..." button, and as seen in Figure 5.8, put each variable in the left box into the "Display Means for:" box, and click "Descriptive Statistics."
Figure 5.8 Options Window for Factorial ANOVA
SPSS allows you to graph the resulting interaction in the following way:
1) Select "Line" from the "Graphs" menu.
2) Click on the "Multiple" box (Figure 5.9) before clicking "Define."
Figure 5.9 Line Charts
3) As seen in Figure 5.10, click on "Other summary function" and select your dependent variable from the list on the left.
4) Indicate which of your two factors you wish to be an axis on the graph and which one you want to be represented by colored lines. When finished, click "OK."
Figure 5.10 Setting Your Interaction Graph
You will first see some tables that list your factors and give you descriptive statistics about your data. Some of the information in the descriptive statistics box is repeated later in the marginal means tables (shown in Figure 5.12), but count (N) and standard deviation is not shown there.
You will then see the output box (Figure 5.11), which runs three separate ANOVAs: the main effects for each of your two variables, and the interaction effect between the variables. Each of these is shown on a different row, with the variable name(s) in capitals; it gives you the df-between, F-value and p-value. The next line, labeled "Error," will give you the df-within. Here, the analysis is not significant for either main effect, but is significant for the interaction.
Figure 5.11 Output from Factorial ANOVA
You will then see the marginal means tables, as shown in Figure 5.12. They show the means of the groups compared in your three ANOVAs (main effect 1, main effect 2, and interaction). You might find this more visually helpful than your descriptive statistics table.
Figure 5.12 Marginal Means Tables Output from Factorial ANOVA
Be sure to include the type of test you used and its specific groups/factors, all results regardless of whether they are significant (indicating if they are), the differences you are evaluating, the group means, the F-value and degrees of freedom (between, within), and the p-value.
Similar to a paired t-test, repeated measures ANOVA tests allow us to examine the means for two groups that are related to each other. For example, it would be appropriate to use a repeated measures ANOVA if you had only one group of participants who were measured at three or more intervals. The repeated measures ANOVA test is one of the more complex tests we will discuss because it requires assigning your data correctly.
Data for repeated measures ANOVA is initially entered just like any other variable for each case. Make sure you name your repeated measures variables something that you will recognize when running the analysis. For example, in Figure 5.13, we have entered four variables marked as "t1", "t2", "t3" and "t4". For each of the 20 participants, "t" was measured 4 times.
Figure 5.13 Entering data for repeated measures ANOVA
There is no need to enter data for repeated measures ANOVA separately from the rest of your data, just be sure to enter it in a way that will help you perform the analysis correctly later on.
1) Normality. Assume that the population distributions are normal. ANOVA is quite robust over moderate violations of this assumptions. Check for normality by creating a histogram.
2) Independent Observations. The observations within each treatment condition must be independent.
3) Equal Variances. Assume that the population distributions have the same variances.
Here, we are examining how many hours of work Reed students report doing per night at four points in the semester - during (t1) week 2 beginning of semester, (t2) week 7 midterms week, (t3) week 12 second-to-last week of classes, and (t4) finals week. This data is fictional.
1) Click on "General Linear Model" under the "Analyze" menu.
2) Select "Repeated measures" from this list.
3) You will be prompted to create a "Within-Subject Factor Name" (Figure 5.14). This name can be anything that wil help you remember what variable you are testing. Most importantly, indicate the number of levels in this factor. In our example above, we had four levels, "t1", "t2", "t3" and "t4."
Figure 5.14 Defining your repeated measures factor
4) When both pieces of information are entered, click "Add" and the "Define" button will become active. Click "Define."
5) There will be several lines in the "Within-Subjects Variables box (with your new factor name in parenthesis above it). The number of lines is dictated by the number of levels which you assigned on the last screen. In our example, we had four levels, so there are four lines. Select the variables which make up your new factor from the list of variables on the left and move them over to the box on the right (see Figure 5.15).
Figure 5.15 A Basic Repeated Measures ANOVA
6) To get descriptive statistics for the variables, click on the "Options" bubble and then on "Descriptive Statistics."
7) Click "OK" to begin examining your repeated measures ANOVA results.
SPSS gives you many exciting tables for repeated measures ANOVA, most of which you can ignore in whole or at least in part. If you have asked for descriptive statistics, you will see them near the top of the output, as in Figure 5.16.
Figure 5.16 Descriptive Statistics Table from Repeated-Measures ANOVA
You will then see many tables that do different tests and correct for different things. However, the only one you will probably be concerned with is "Tests of Within-Subjects Effects."In that table, you should use the "Sphericity Assumed" rows to get your results, as shown in Figure 5.17. The first row (with the name of your variable) will tell you df-between, F-value, and p-value (here, p < .001); you can get your df-error from the block that says Error(variablename).
Figure 5.17 Output from Repeated-Measures ANOVA
Be sure to include all information for the F-ratios and the follow-up test: the type of ANOVA used, F-values, degrees of freedom (between, error), p-values, all of the means and whether or not the differences were significant.