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Using SPSS to Compute Analysis of Variance (ANOVA)
 

In the SPSS lessons on t-tests, you compared the mean for one group to the mean for another group to determine whether the means were significantly different. Although you could use this approach to compare the means for three or more groups, you know from reading the ANOVA lesson that doing multiple t-tests increases the experimentwise error rate. In this lesson you will compare the means of three or more groups using ANOVA to determine whether they differ significantly. Upon completing the lesson, you should be able to:

  • Determine how many groups are being compared in an ANOVA based on SPSS output.
  • Identify the between-groups sums of squares, mean squares, and degrees of freedom on a sample SPSS output.
  • Identify the within-groups sums of squares, mean squares, and degrees of freedom on a sample SPSS output.
  • Calculate the mean square and F ratio from the sums of squares and degrees of freedom on a sample SPSS output.
  • Use SPSS output to determine whether to reject a specified null hypothesis.
  • Use SPSS output to determine whether the groups being compared using ANOVA are significantly different.
  • Conduct an ANOVA test using SPSS.

To practice generating ANOVA output using SPSS, download one of the following mathtest data files:

(Note: If the linked file does not begin downloading when you click the link, right-click on the link and select save target as or save link as from the menu.)


Computing ANOVA

In this lesson you will use the ANOVA statistic to compare the means of three or more groups using SPSS.

Grouping Variable

One of the things to check when preparing to conduct an ANOVA test is the number of groups. If you only have a total of two groups to compare, ANOVA would not be necessary because you could use a t-test to compare the groups. However, if you conduct a ANOVA with only two groups, the results of the ANOVA and the t-test will be the same. By including descriptive information in the ANOVA output (like the sample shown here) you can determine how many groups are being analyzed.

The grouping variable, which is also called a factor, is the independent variable that identifies what group a person is in or a score belongs to). In this example from the mathtest data, the grouping variable is curriculum (not listed on the output) and has four levels (listed in the first column of the descriptives output [shown above]). The values that correspond to these levels are 1, 2, 3, and 4, based on the type of curriculum that each person received. The name of the dependent variable that is being used to compare the groups is listed above the table of descriptive statistics.

In this example, the dependent variable is score, which represents the students' scores on a math test. The descriptive information in the above output, as well as a means plot for the groups (shown below) indicates whether the mean score for each group (31.6, 31.58, 26.8, and 27.18) is different from the other means.

You would use ANOVA to determine whether the means were significantly different from one another.

 

Sums of Squares and Mean Square

Recall from the lesson on ANOVA that analysis of variance testing involves an examination of the between-group and within-group variation (variance) by partitioning the sums of squares. The sums of squares due to between-group variation () tells you how much the group means deviate from the grand mean. The sums of squares due to within-group variation () tells you how much the individual scores deviate from the grand mean. On a SPSS ANOVA output, these values are in the sums of squares column.

The value is listed in the between groups row, and the value is in the within groups row. In some statistical packages, is labeled as model in the output and is labeled as error, which represents another way of conceptualizing the ANOVA test. The between-group variation is what you are using to examine whether your model of the groups is accurate. For example, maybe your model depicts group 1 as being very different from group 2 and very different from group 3. The model sums of squares is a measure of the variation between Group 1 and Group 2, Group 2 and Group 3, and Group 3 and Group 1. In the picture below, the difference between the groups are the shapes of the groups. The ANOVA test tells you if those shapes are significantly different.

The within-group variation is what detracts from the accuracy of the model. As you may recall from the ANOVA lesson, the within-group variation represents error. In the picture above, the within-group variation is indicated by the color of each happy face. The different colors make it more complicated to determine whether the groups are truly different (due to their shape) or whether the people in the groups (due to the people's individual characteristics) are what is causing the groups to be different.

Using the model and error logic, what are the mean square for the model and the mean square for error? Conceptually, the mean square is the weighted average of the sums of square. Computationally, you can determine the mean square by dividing the sums of squares by the corresponding degrees of freedom.

Note that if you are given a partial SPSS output containing only the sums of squares and the degrees of freedom, you could calculate the mean squares using these formulas.

F Test

When conducting ANOVA testing, you divide the mean square between by the mean square within to calculate an F ratio: . Then you determine whether the calculated F ratio is greater than the critical F ratio. You could calculate the F ratio manually or you could check the SPSS output. The F ratio is shown in the column labeled F. In our example, the F ratio is 0.762. You can find the critical value by looking up the degrees of freedom in an F distribution table. At = 3 and = 46, the critical F value is 2.81. Based on this value are the group means significantly different? Since the calculated F value is less than the critical F value, you would not reject the null hypothesis that the group means are equal. You would then conclude that the groups do not have significantly different math scores.

Does the decision match the SPSS output? To answer that question, look at the probability column (labeled sig on the output). Since the probability that the null hypothesis being true is > 0.05, you would conclude that the group means are not significantly different.

Group Comparisons

In our example, the F test indicated that none of the groups was significantly different from the other groups. Even if we had determined that the groups were significantly different in terms of their means, we would still not know whether Group 1's mean differed significantly from Group 2's mean, whether Group 1's mean differed significantly from Group 3's mean, and so on. The ANOVA results do not tell you specifically which means were significantly different; the test only tells you whether at least one of the group means was different from the others.

The descriptive information for each group (shown on the right) contains information to help us analyze how the specific groups differ. To determine whether specific group differences are significant, though, you would need to conduct multiple comparison procedures, which are covered in a more advanced statistics course.

 


Generating ANOVA Output

To statistically compare the means of three or more independent samples using ANOVA:

  1. Open the analyze menu, and select one-way ANOVA from the compare means submenu.
  2. In the One-Way ANOVA dialog box, you need to specify what the grouping variable is and which means (variables) to compare:
    • Select the grouping variable from the list of variables and add it to the factor box. In this example, curriculum is the grouping variable.
    • Select the dependent variable to be used in the F test and add it to the dependent list. In this example, score is the dependent variable.
    • Click the options button to set additional options for the ANOVA.
  3. In the One-Way ANOVA: Options dialog box:
    • Select the descriptive option to include descriptive statistics in the ANOVA output.
    • Select the means plot option to include a line graph of the group means.
    • Click the continue button to save the options and return to the One-Way ANOVA dialog box.
  4. Click the OK button in the One-Way ANOVA dialog box to run the ANOVA test and generate the output.

 


Review

In this module, you have interpreted ANOVA output from SPSS to determine whether the means of three or more groups are significantly different.

  • Given sample SPSS output you can:
    • Determine how many groups are being compared in an ANOVA.
    • Identify the between-groups sums of squares, mean squares, and degrees of freedom.
    • Identify the within-groups sums of squares, mean squares, and degrees of freedom.
    • Calculate the mean square and F ratio from the sums of squares and degrees of freedom.
    • Determine whether to reject a specified null hypothesis.
    • Determine whether the groups being compared using ANOVA are significantly different.
  • Conduct an ANOVA test using SPSS.

To practice generating ANOVA output using SPSS, download one of the following mathtest data files:

(Note: If the linked file does not begin downloading when you click the link, right-click on the link and select save target as or save link as from the menu.)

 

 

 

© 2007 by Melissa Kelly and L. K. Curda. All rights reserved. Updated on November 14, 2007