ANOVA with Repeated Measures

Running head: REPEATED MEASURES ANOVA

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REPEATED MEASURES ANOVA

Repeated Measures ANOVA

Student’s Name

Institution Affiliation

Repeated Measures ANOVA

Introduction

In this paper, we are going to discuss the Repeated Measures ANOVA. We are going to consider an experiment done by students who wanted to check the consistency of marking of the lecturers. The students submitted the same essay to four different lecturers and the grade given by each of the four lecturers. The total number of essays assigned to each lecturer was eight. Here, the independent variable is the lecturer who is marking while the dependent variable is the marks given by each lecturer.

In this paper, we are going to look at the hypothesis of the ANOVA test, and then the descriptive statistics of the variables, the posthoc tests to test the assumptions and lastly the main ANOVA test (effects of the main variables). The assumption of this test is the sphericity assumption, which will be tested by Mauchly's Test. All the data analysis is done using IBM SPSS.

Hypothesis

Ho: All the tutors are consistent in marking

Ha: The tutors are not consistent in marking

Descriptive Statistics

The mean and standard deviation of the marks given by each tutor are given in the table below.Descriptive Statistics
	Mean	Std. Deviation	N
Dr. Field	68.88	5.643	8
Dr. Smith	64.25	4.713	8
Dr. Scrote	65.25	6.923	8
Dr. Death	57.38	7.909	8

Table 1: Table of descriptive statistics

From the table above, we see that Dr. Field has the highest mean which is 68.88 with a standard deviation of 5.643. He is followed by Dr. Scrote who has a mean of 65.25 with a standard deviation of 6.923. Then we have Dr. Smith who has a mean of 64.25 and a standard deviation of 5.6 and lastly is Dr. Death with a mean of 57.38 and a standard deviation of 7.9.

Post-hoc tests

We assess the assumption of sphericity. Here, the data should have significant diversion from spericity. Below is the output of Mauchly's Test of Sphericity.

Mauchly's Test of Sphericityb
Measure:MEASURE_1
Within Subjects Effect	Mauchly's W	Approx. Chi-Square	df	Sig.	Epsilona
					Greenhouse-Geisser	Huynh-Feldt	Lower-bound
Tutor	.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. 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.
b. Design: Intercept Within Subjects Design: Tutor

Table 2: Post-Hoc test

From Table 2 above, we see that we have a p-value of 0.043. Since this p-value is less than 0.05, it means that the assumption of sphericity has been violated. Also, the table 2 above shows the level of sphericity. We have Greenhouse-Geisser value as 0.558 and Huynh-Feldt value as 0.712. It is known that if a data is perfectly spherical, then both of the estimates should be closer to 1 (Andey, 2013). For the Greenhouse-Geisser value, we always compute the lowest value as 1/(k-1), where k is the number of levels. Therefore, since we have four conditions, the lowest bound is 0.333. From the Greenhouse-Geisser value above, we see that its coefficient is closer to 0.333 (lower value) than it is to 1 which is the upper value. This implies that there is a substantial diversion from sphericity.

Main ANOVA

The table below shows the ANOVA output, where we are comparing the means of the four tutors. Tests of Within-Subjects Effects
Measure:MEASURE_1
Source	Type III Sum of Squares	df	Mean Square	F	Sig.
Tutor	Sphericity Assumed	554.125	3	184.708	3.700	.028
	Greenhouse-Geisser	554.125	1.673	331.245	3.700	.063
	Huynh-Feldt	554.125	2.137	259.329	3.700	.047
	Lower-bound	554.125	1.000	554.125	3.700	.096
Error(Tutor)	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

Table 3: Main ANOVA test

From the table above, we see that the four means are significantly different since F(3,1)= 3.7, p-value=.0.28. Since the p-value is less than 0.05, we reject the null hypothesis and conclude that the tutors are not consistent in marking

Conclusion

From the above tests, we have seen that the assumption of sphericity has been met. This gives us a go ahead to conduct the ANOVA test. The ANOVA test above shows that the lecturers are not consistent in marking. Different lecturers give different marks for the same test.

References

 

Andey Field, (2013). Discovering Statistics Using IBM SPSS Statistics. Washington: Sage Publication