![]() ![]() 05), we can reject the null hypothesis and accept the alternative hypothesis that the variances of the differences are not equal (i.e., sphericity has been violated). ![]() Thus, if Mauchly's Test of Sphericity is statistically significant ( p <. Mauchly's Test of Sphericity tests the null hypothesis that the variances of the differences are equal. However, despite these shortcomings, because it is widely used, we will explain the test in this section and how to interpret it. This is probably due to its automatic print out in SPSS for repeated measures ANOVAs and the lack of an otherwise readily available test. Although this test has been heavily criticised, often failing to detect departures from sphericity in small samples and over-detecting them in large samples, it is nonetheless a commonly used test. Testing for Sphericity: Mauchly's Test of SphericityĪs just mentioned, Mauchly's Test of Sphericity is a formal way of testing the assumption of sphericity. This data is from a fictitious study that measured aerobic capacity (units: ml/min/kg) at three time points (Time 1, Time 2, Time 3) for six subjects. To illustrate the concept of sphericity as equality of variance of the differences between each pair of values, we will analyse the fictitious data in the Table 1 below. Firstly, we will illustrate what sphericity is by way of an example. We will discuss this in more detail later in this guide. This is achieved by estimating the degree to which sphericity has been violated and applying a correction factor to the degrees of freedom of the F-distribution. ![]() Luckily, if violations of sphericity do occur, corrections have been developed to produce a more valid critical F-value (i.e., reduce the increase in Type I error rate). Therefore, determining whether sphericity has been violated is very important. The violation of sphericity is serious for the repeated measures ANOVA, with violation causing the test to become too liberal (i.e., an increase in the Type I error rate). Sphericity can be likened to homogeneity of variances in a between-subjects ANOVA. Violation of sphericity is when the variances of the differences between all combinations of related groups are not equal. Sphericity is the condition where the variances of the differences between all combinations of related groups (levels) are equal. ANOVAs with repeated measures (within-subject factors) are particularly susceptible to the violation of the assumption of sphericity. ![]()
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