Introduction
Sphericity is a crucial assumption in repeated measures ANOVA, ensuring the equality of variances of the differences between conditions. This post explores the concept of sphericity, its importance, how to test for it using SPSS, and how to handle violations.
What is Sphericity?
Sphericity refers to the condition where the variances of the differences between all possible pairs of within-subject conditions are equal. It’s essential for the validity of the repeated measures ANOVA results.
Example Scenario
Consider a study examining the effects of three different teaching methods on students’ test scores. Each student is tested under each method, resulting in repeated measures data. To validly use repeated measures ANOVA, we must ensure sphericity.
Testing Sphericity in SPSS
SPSS provides Mauchly’s Test of Sphericity to assess whether the assumption holds. Here’s a step-by-step guide to performing the test:
- Data Entry: Input your data into SPSS. Each row should represent a participant, and each column should represent the scores under different conditions.
- Analyze: Navigate to
Analyze > General Linear Model > Repeated Measures. - Define Factors: Specify the within-subject factor and the number of levels.
- Run Analysis: After defining the factor, click on the
Optionsbutton and ensureMauchly's Test of Sphericityis checked.
SPSS Output Interpretation
The SPSS output for Mauchly’s Test includes:
- Mauchly’s W: The test statistic.
- Approx. Chi-Square: The Chi-Square statistic.
- df: Degrees of freedom.
- Sig.: The p-value.
Example Output Table
| Within Subjects Effect | Mauchly’s W | Approx. Chi-Square | df | Sig. |
|---|---|---|---|---|
| Condition | 0.632 | 10.482 | 2 | 0.005 |
APA-Style Results Interpretation
If the p-value (Sig.) is less than 0.05, sphericity is violated.
Mauchly’s Test of Sphericity indicated that the assumption of sphericity had been violated, χ2(2) = 10.482, p = .005.
Handling Violations of Sphericity
When sphericity is violated, use corrections to adjust the degrees of freedom:
- Greenhouse-Geisser: Conservative, used when sphericity is severely violated.
- Huynh-Feldt: Less conservative, used when sphericity is less severely violated.
SPSS automatically provides these corrections in the output. Report the corrected F-values and degrees of freedom:
A repeated measures ANOVA with a Greenhouse-Geisser correction determined that mean test scores differed significantly between conditions (F(1.38, 27.58) = 5.87, p = .011).
Practical Example with SPSS
Study Design
We have three teaching methods (A, B, and C) and test scores for 10 students under each method.
Data Entry
| Student | Method_A | Method_B | Method_C |
|---|---|---|---|
| 1 | 85 | 88 | 90 |
| 2 | 78 | 82 | 84 |
| 3 | 92 | 91 | 95 |
| … | … | … | … |
Running the Analysis
- Enter the data into SPSS.
- Navigate to
Analyze > General Linear Model > Repeated Measures. - Define the factor (Teaching_Method) with three levels.
- Run the analysis and request Mauchly’s Test of Sphericity.
Output Interpretation
| Within Subjects Effect | Mauchly’s W | Approx. Chi-Square | df | Sig. |
|---|---|---|---|---|
| Teaching_Method | 0.721 | 8.562 | 2 | 0.014 |
Since p = .014, sphericity is violated. We use the Greenhouse-Geisser correction:
| Source | Type III Sum of Squares | df | Mean Square | F | Sig. |
|---|---|---|---|---|---|
| Teaching_Method | 136.800 | 1.428 | 95.780 | 10.527 | .004 |
With the Greenhouse-Geisser correction, there was a significant effect of teaching method on test scores (F(1.428, 27.858) = 10.527, p = .004).
Incorporating Previous Posts
For detailed discussions on related topics, refer to:
- Mastering SPSS: Understanding Stratified Random Sampling
- Mastering SPSS: Understanding Measures of Spread – Standard Deviation
- Mastering SPSS: Ensuring External Validity in Research Statistics
- Mastering SPSS: Snowball Sampling Using SPSS Statistics
Conclusion
Understanding and testing for sphericity is vital in repeated measures ANOVA. Using SPSS, you can easily assess this assumption and apply necessary corrections to ensure valid results. Properly addressing sphericity enhances the reliability of your statistical analyses and the credibility of your research findings.