Anova With Repeated Measures

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Introduction

The current paper provides a one-way analysis of variance (ANOVA) with repeated measures of the data supplied in the file “TutorMarks.sav,” which accompanies the task #2 provided in the book by Field (2013, p. 589); the file can be found on the web page “Datasets” (n.d.). The analysis was conducted using the SPSS software. This analysis is important because assessing whether there is a significant difference between certain sets of data may be paramount in science (Campbell & Stanley, 1963). After discussing the underlying assumptions for the ANOVA and testing them, the author of this paper describes certain ways to amend the violations of such assumptions. After that, the results of the test are provided and explained. Contents of the SPSS syntax file are also provided; in addition, the SPSS output can be found in the Appendices.

Review of Chapter 14 (Field, 2013, pp. 543-590)

In Chapter 14 of his book, Field (2013) explains what the one-way ANOVA with the repeated measures design is, and provides instructions for conducting it. First, the crux of the repeated-measures design is explained; the assumption of sphericity, which is necessary for the test, is thoroughly discussed. The effects of a violation of sphericity are described; ways to assess and amend it are provided. After that, some advanced theory of a one-way repeated measures ANOVA is elaborated. Then, the procedure of conducting the ANOVA in SPSS is described, and the interpretation of the results and post-hoc tests is discussed. Finally, the effect sizes of the test are explained, and the ways to report it are demonstrated. After this point, the factorial repeated-measures ANOVAs are discussed in a manner similar to the way in which the one-way repeated measures ANOVAs were elaborated. Lastly, some exercises are provided at the end of the chapter.

Underlying Assumptions for a One-Way ANOVA with Repeated Measures

The main assumptions for a one-way ANOVA with repeated measures are as follows (Warner, 2013; Field, 2013):

The dependent variable should be measured by using a continuous scale (interval or ratio), whereas scores on the independent variable need to have a multivariate normal distribution;
the distributions of scores should be approximately normal across the dependent variables;
there should not be extreme outliers;
the S-matrix of the sample in which the set of repeated measures is contained ought to be capable of being characterized as having a particular structure. This is often a spherical structure. Sphericity resembles the assumption of homogeneity of variance and means that the pairs of experimental conditions can be characterized as having a similar relationship, i.e., the variances in differences between tutors are similar (Field, 2013, pp. 544-545).

Testing the Assumptions and Addressing their Violations

First, the descriptive statistics for the data are as follows:

Descriptive Statistics.
	N	Mean	Std. Deviation	Variance
Dr. Field	8	68.88	5.643	31.839
Dr. Smith	8	64.25	4.713	22.214
Dr. Scrote	8	65.25	6.923	47.929
Dr. Death	8	57.38	7.909	62.554
Valid N (listwise)	8

The assumption 1 is met, for the grades are measured by using a continuous scale.

The assumptions 2 and 3 are tested by using histograms which can be found in Appendix 1. It appears that all the four distributions are close to normal, and that there are no extreme outliers.

The assumption 4 of sphericity can be tested by using the Mauchly’s test:

Mauchly’s Test of Sphericity^a
Measure: MEASURE_1
Within Subjects Effect	Mauchly’s W	Approx. Chi-Square	df	Sig.	Epsilon^b
					Greenhouse-Geisser	Huynh-Feldt	Lower-bound
tutor_marks	.131	11.628	5	.043	.558	.712	.333
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.
a. Design: Intercept Within Subjects Design: tutor_marks
b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table.

In the case of the given data Mauchly’s test yielded F(5)=.131, p=.043 (<.05). This test checks the null hypothesis that the variances of differences between the conditions in the data are equal. Thus, because p<.05, the null hypothesis is rejected, which means that the data is not spherical. Therefore, the assumption 4 of sphericity is violated.

Addressing Violation of Assumptions

Assumption 1. If the assumption 1 (the continuous measurement of the dependent variables) is violated, it does not make sense to conduct ANOVA. It might be possible to convert the variable into a continuous scale by using ranks (Warner, 2013; George & Mallery, 2016).
Assumption 2. If the assumption 2 (non-normal distribution) is violated, it is possible to apply transformations to the whole data, for instance, by raising every value of each dependent variable to the same power (Warner, 2013).
Assumption 3. Violation of the assumption 3 can be addressed by removing the extreme outliers from the data set, although it would mean that a whole case will be removed.
Assumption 4. To address the violation of the assumption of sphericity (to which a repeated measures ANOVA is rather susceptible, for the probability of a Type I error is increased greatly), it is possible to adjust the degrees of freedom in the repeated measures ANOVA by multiplying them by a certain estimate of sphericity. After the ANOVA is run, it will be needed to examine the Greenhouse-Geisser estimate (provided in the Mauchly’s test of sphericity table generated by SPSS), which varies between 1 and 1/(k – 1), where k is the number of conditions of the reported measures. If this value is lesser than.75, it is recommended to adjust the degrees of freedom by multiplying them by this Greenhouse-Geisser’s estimate; however, if this value is greater than.75, it is advised to multiply the degrees of freedom by the Huynh-Feldt estimate instead (Field, 2013).

In the given case, the Greenhouse-Geisser’s estimate of sphericity is.558, so it will be needed to multiply the degrees of freedom by this estimate.

Also, a completely different option when the assumption of sphericity is violated is to use the MANOVA test instead of ANOVA, because MANOVA does not require sphericity (Field, 2013).

Null and Research Hypotheses for the Test

The null hypothesis for the test: that there are no significant differences between the means of the grades given by the four different tutors to the students for their essays.

The alternative (research) hypothesis: there are significant differences between the means of the grades that were given by the four different tutors to the students.

Syntax File

The contents of the syntax file can be found in Appendix 2

SPSS Output

The SPSS output can be found in Appendix 3.

Results Tables

Tests of Within-Subjects Effects.
Measure: MEASURE_1
Source		Type III Sum of Squares	df	Mean Square	F	Sig.	Partial Eta Squared	Noncent. Parameter	Observed Power^a
tutor_marks	Sphericity Assumed	554.125	3	184.708	3.700	.028	.346	11.100	.722
	Greenhouse-Geisser	554.125	1.673	331.245	3.700	.063	.346	6.189	.522
	Huynh-Feldt	554.125	2.137	259.329	3.700	.047	.346	7.906	.603
	Lower-bound	554.125	1.000	554.125	3.700	.096	.346	3.700	.383
Error(tutor_marks)	Sphericity Assumed	1048.375	21	49.923
	Greenhouse-Geisser	1048.375	11.710	89.528
	Huynh-Feldt	1048.375	14.957	70.091
	Lower-bound	1048.375	7.000	149.768
a. Computed using alpha =.05

Pairwise Comparisons.
Measure: MEASURE_1
(I) tutor_marks	(J) tutor_marks	Mean Difference (I-J)	Std. Error	Sig.^b	95% Confidence Interval for Difference^b
(I) tutor_marks	(J) tutor_marks	Mean Difference (I-J)	Std. Error	Sig.^b	Lower Bound	Upper Bound
1	2	4.625^*	1.085	.022	.682	8.568
	3	3.625	2.841	1.000	-6.703	13.953
	4	11.500	4.675	.261	-5.498	28.498
2	1	-4.625^*	1.085	.022	-8.568	-.682
	3	-1.000	2.563	1.000	-10.320	8.320
	4	6.875	4.377	.961	-9.039	22.789
3	1	-3.625	2.841	1.000	-13.953	6.703
	2	1.000	2.563	1.000	-8.320	10.320
	4	7.875	4.249	.637	-7.572	23.322
4	1	-11.500	4.675	.261	-28.498	5.498
	2	-6.875	4.377	.961	-22.789	9.039
	3	-7.875	4.249	.637	-23.322	7.572
Based on estimated marginal means
*. The mean difference is significant at the.05 level.
b. Adjustment for multiple comparisons: Bonferroni.

Results

Therefore, a one-way repeated measures ANOVA was conducted in order to identify whether there are significant differences in the means of grades given by the four instructors to their students. The preliminary screening revealed a violation of the sphericity assumption by using the Mauchly’s test, which yielded F(5)=.131, p=.043 (<.05). The Greenhouse-Geisser estimate was.558; because it is lower than.75, the degrees of freedom for the ANOVA were adjusted by multiplying them by this estimate.

The ANOVA test with the degrees of freedom adjusted by the Greenhouse-Geisser estimate yielded marginally significant results; F(1.673)=3.7, p=0.63. The effect size as measured by the partial η² was.346, which is a large effect.

It is noteworthy, however, that if the degrees of freedom are adjusted by the Huynh-Feldt estimate, it gives results at p=.047; Field (2013) recommends to also take these results into account. In this case, F(2.137)=3.7, and the partial η²=.346, which is a large effect.

The post-hoc tests (pairwise comparisons) revealed that there was a significant difference between the first and the second tutors (Dr. Field and Dr. Smith, respectively): mean difference was 4.625 (95% confidence interval: from.682 to 8.568), p=.022. The descriptive statistics showed that for Dr. Field, mean=68.88, SD=5.643; for Dr. Smith, mean=64.25, SD=4.713. No other significant differences were found in the grades given by the rest of the other of instructors.

All in all, it may be possible to conclude that at least marginal evidence was found to support the alternative hypothesis that the means of the grades given by the four instructors to their students in some cases differ significantly.

References

Campbell, D. T., & Stanley, J. C. (1963). Experimental and quasi-experimental designs for research. Boston: Houghton Mifflin.

Datasets. (n.d.). Web.

Field, A. (2013). Discovering statistics using IBM SPSS statistics: And sex and drugs and rock’n’roll (4th ed.). Thousand Oaks, CA: Sage Publications.

George, D., & Mallery, P. (2016). IBM SPSS Statistics 23 step by step: A simple guide and reference (14th ed.). New York, NY: Routledge.

Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd ed.). Thousand Oaks, CA: SAGE Publications.