One of the assumptions of ANOVA is independence of groups being compared. Sometimes measures are repeated several times, in this case variability of the data is reduced. This increases the power to detect effects but assumption of independence is violated. When two groups are compared, paired t-test is used; however, for three or more groups there is a special type of ANOVA called repeated measures ANOVA.
During application of parametric tests we should take into consideration so-called “sphericity” which indicates that the relationship between pairs of groups is equal. Violating sphericity means that the F statistic cannot be compared to the normal tables of F, and so SPSS software cannot calculate a significance value. A procedure called Mauchly's test helps to understand if the assumption of sphericity has been violated. If the Mauchly’s test statistic is nonsignificant (p >0.05), it indicates that the variances of differences are not significantly different; however, if the Mauchly’s test statistic is significant (p less or equal 0.05), this means that the condition of sphericity has not been met, and, therefore, we cannot trust the F-ratios unless applying a special correction.
In SPSS for repeated measures variables there are no proper post hoc tests. However, in this situation, the paired t-test procedure can be used in order to compare all pairs of levels of the independent variable. Then it is necessary to apply the Bonferroni correction to the probability at which we accept any of paired t-tests; this correction is calculated by dividing the probability value from t-tests by the number of tests conducted. And only the resulting probability value should be used as the criterion for statistical significance. For example, if we perform three comparisons, then standard criterion of significance (0.05) will be 0.05/3 = 0.0167, that is, we accept t-tests as being significant only if p <0.0167.
An example of data for the one-way repeated measures ANOVA is presented in the table. We studied activity of tea tree oil against 36 bacterial species by disk diffusion method. The experiment was performed in triplicates and results of measurements of inhibition zones are written as variables Tea_Tree_Diffusion, Tea_Tree_Diffusion_2 and Tea_Tree_Diffusion_3 (see Example 4).
Research question: Are there differences between results of three repeatings of experiment?
Before starting any comparison we should check distribution of variables, as it was already described for one-way ANOVA. Results of normality tests indicate that all groups being compared have normal distribution: p-values are not significant in any table:
To specify one-way repeated measures ANOVA:
1) Click the Analyze menu, point to General Linear Model, and select Repeated Measures… :
The Repeated Measures Define Factor(s) dialog box opens:
2) In the Within-Subject Factor Name box type the name for factor which will be convenient for further analysis, for example, “Diffusion” and also type the number of repeats (“3”).
3) Click the Add button; the new factor name with the number of levels is moved to the box located to the right from the button Add:
4) Click the Define button; the Repeated Measures dialog box opens:
5) Select all three “Tea_Tree_Diffusion” variables, click the transfer arrow button . The selected variables are moved to the Within-Subjects Variables(Diffusion): list box.
6) Click the Options… button; the Repeated Measures: Options dialog box opens
7) In the Factor(s) and Factor Interactions: list box select the variable “Diffusion”; click the transfer arrow button . The selected variable is moved to the Display Means For: list box.
8) Select Compare Main Effects check box.
9) For the Confidence Interval Adjustment section select the Bonferroni option by pointing to the triangle to the right from LSD(none):
10) In Display section select Descriptive Statistics check box and click the Continue button, this returns us to the Repeated Measures dialog box.
11) Click the OK button. An Output Viewer window opens with statistical results.
The results of one-way repeated measures ANOVA are presented in several tables:
1) The Within-Subject Factors and the Descriptive Statistics tables list studied factors and their descriptive statistics, such as mean, standard deviation and number of observations.
2) The Multivariate Tests table.
3) The Mauchly’s Test of Sphericity table:
4) The Tests of Within-Subjects Effects table:
5) The Tests of Within-Subjects Contrasts table:
6) The Tests of Between-Subjects Effects table:
7) The Estimated Marginal Means set of tables. It includes the table Estimates with descriptive statistics for each variables, including mean, error of mean and confidence intervals, the table Pairwise Comparisons which demonstrates results of paired comparisons of studied variables, and also the table Multivariate Tests:
From all these tables the most important are tables with Mauchly’s test, with within-subjects effects and with pairwise comparisons. The Tests of Within-Subjects Effects table is the key table in repeated measures of ANOVA, it demonstrates presence or absence of differences between variables. It contains several tests, which are selected depending on results of Mauchly’s test. If the Mauchly’s test is not significant (p>0.05), then results of ANOVA can be used without any corrections; in this case a significance value should be taken from line “Sphericity assumed” from table with within-subjects effects. However, if the Mauchly’s test is significant, then next look should be taken to the Epsilon value in the table with results of the Mauchly’s test. If the epsilon is > 0.75, then the Huynh-Feldt correction should be used, if the epsilon is less than 0.75 or nothing is known about sphericity at all, then the Greenhouse-Geisser correction should be used from table with within-subjects effects. In our example assumption of sphericity has been met, so no corrections are necessary.
As we can see from the table Tests of Within-Subjects Effects, there are no significant differences between results of three measurements (p = 0.208). Therefore, it is not necessary to look at the table with paired comparisons, all comparisons are also not significant.
Formal reporting of the one-way repeated measures ANOVA: There were no significant differences between results of different measurements of inhibition zones, F(2,70) = 1.61, p = 0.208.
In a case if assumption of sphericity has not been met, reporting of results might look, as following: “Mauchly’s test indicated that the assumption of sphericity had been violated (chi-square = 8.16, p = 0.018), therefore degrees of freedom were corrected using Huynh-Feldt estimates of sphericity (epsilon = 0.78) (these numbers are taken only as illustrative examples). Then reporting is similar to above described, but degrees of freedom and levels of p are chosen from values corresponding the Huynh-Feldt correction.