Repeated Measures Anova

1. Analysis of Variance with
Repeated Measures

2. Repeated Measurements: Within Subjects Factors
Repeated measurements on a subject are called within subjects
factors.
Example: Strength was measured pre, 8 weeks, 16 weeks and 24
weeks. Strength is a within subjects factor with 4 levels (pre, 8,
16, 24).
Subjects rode for 15 minutes, divided
into five 3-minute periods for the
purpose of collecting data. Data were
collected on the number of balance
errors during the last minute of each 3-
minute period, and resistance was
increased at the end of each 3-minute
period. In this design, the dependent
variable is balance errors and the
independent variable is increase in
resistance (fatigue).

3. Advantages of Repeated Measures over
Independent Groups ANOVA
In repeated measures subjects serve as their
own controls.
Differences in means must be due to:
the treatment
variations within subjects
error (unexplained variation)
Repeated measures designs are more powerful
than independent groups designs.

4. Roller Ergometer Data. Within Subjects Factor with
5 levels (3, 6, 9, 12, 15 min)

5. Repeated Measures ANOVA Summary Table
How is the F ratio of 18.36 computed?
How are the Mean Squares
computed?
Does fatigue effect balance? If so,
which means are different?

6. Post hoc comparisons

7. SPSS Output
Within-Subjects Factors
Measure: MEASURE_1
Minute_3
Minute_6
Minute_9
Minute_12
Minute_15
treatmnt
1
2
3
4
5
Dependent
Variable
Descriptive Statistics
8.5000 4.50309 10
11.4000 7.96102 10
16.4000 10.80329 10
31.1000 12.55610 10
36.5000 21.13055 10
Minute_3
Minute_6
Minute_9
Minute_12
Minute_15
Mean Std. Deviation N
General Linear Model
Multivariate Tests
c
.866 9.694b 4.000 6.000 .009 .866 38.777 .934
.134 9.694b 4.000 6.000 .009 .866 38.777 .934
6.463 9.694b 4.000 6.000 .009 .866 38.777 .934
6.463 9.694b 4.000 6.000 .009 .866 38.777 .934
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Effect
treatmnt
Value F Hypothesis df Error df Sig.
Partial Eta
Squared
Noncent.
Parameter
Observed
Power
a
Computed using alpha = .05
a.
Exact statistic
b.
Design: Intercept
Within Subjects Design: treatmnt
c.

8. Repeated Measure ANOVA Assumptions: Sphericity?
Mauchly's Test of Sphericity
b
Measure: MEASURE_1
.024 27.594 9 .001 .371 .428 .250
Within Subjects Effect
treatmnt
Mauchly's W
Approx.
Chi-Square df Sig.
Greenhous
e-Geisser Huynh-Feldt Lower-bound
Epsilon
a
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is
proportional to an identity matrix.
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.
a.
Design: Intercept
Within Subjects Design: treatmnt
b.
Mauchly’s Test of Sphericity indicated that sphericity was
violated [ W(9) =27.59, p = .001
You don’t want this to be significant. Since Sphericity is
violated, we must
use either the G-G
or H-F adjusted
ANOVAs

9. SPSS Output: Within Subjects Factors
Tests of Within-Subjects Effects
Measure: MEASURE_1
6115.880 4 1528.970 18.359 .000 .671 73.437 1.000
6115.880 1.485 4117.754 18.359 .000 .671 27.268 .995
6115.880 1.710 3575.916 18.359 .000 .671 31.400 .998
6115.880 1.000 6115.880 18.359 .002 .671 18.359 .967
2998.120 36 83.281
2998.120 13.367 224.289
2998.120 15.393 194.776
2998.120 9.000 333.124
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Source
treatmnt
Error(treatmnt)
Type III Sum
of Squares df Mean Square F Sig.
Partial Eta
Squared
Noncent.
Parameter
Observed
Power
a
Computed using alpha = .05
a.
If Sphericity was okay then the statistics would be F(4,36) = 18.36, p =
.000, power = 1.000
But since Sphericity was violated we use the adjusted values:
F(1.485,13.367) = 18.36, p = .000, power = .995, effect size or partial η2
= .67 Which means are significantly different?
What is the difference between this power (post hoc) and an a priori
power analysis?

10. SPSS Output: Between Subjects Effects
Tests of Between-Subjects Effects
Measure: MEASURE_1
Transformed Variable: Average
21590.420 1 21590.420 45.801 .000 .836 45.801 1.000
4242.580 9 471.398
Source
Intercept
Error
Type III Sum
of Squares df Mean Square F Sig.
Partial Eta
Squared
Noncent.
Parameter
Observed
Power
a
Computed using alpha = .05
a.
If we had a between subjects factor like
Gender, the ANOVA results would be printed
here.

11. SPSS Output: Effect Size & Confidence Intervals
1. Grand Mean
Measure: MEASURE_1
20.780 3.070 13.834 27.726
Mean Std. Error Lower Bound Upper Bound
95% Confidence Interval
Estimates
Measure: MEASURE_1
8.500 1.424 5.279 11.721
11.400 2.517 5.705 17.095
16.400 3.416 8.672 24.128
31.100 3.971 22.118 40.082
36.500 6.682 21.384 51.616
treatmnt
1
2
3
4
5
Mean Std. Error Lower Bound Upper Bound
95% Confidence Interval

12. Post hoc Tests for Main Effects (Treatment means)
4 (12 min)
is diff from:
1,2,3 (3,6,9
min)
5 (15 min)
is diff from:
1,2,3 (3,6,9
min)
Pairwise Comparisons
Measure: MEASURE_1
-2.900 1.656 .114 -6.647 .847
-7.900* 2.718 .017 -14.049 -1.751
-22.600* 3.194 .000 -29.826 -15.374
-28.000* 6.354 .002 -42.375 -13.625
2.900 1.656 .114 -.847 6.647
-5.000 2.380 .065 -10.385 .385
-19.700* 2.848 .000 -26.143 -13.257
-25.100* 6.457 .004 -39.708 -10.492
7.900* 2.718 .017 1.751 14.049
5.000 2.380 .065 -.385 10.385
-14.700* 2.633 .000 -20.657 -8.743
-20.100* 4.792 .002 -30.941 -9.259
22.600* 3.194 .000 15.374 29.826
19.700* 2.848 .000 13.257 26.143
14.700* 2.633 .000 8.743 20.657
-5.400 4.525 .263 -15.635 4.835
28.000* 6.354 .002 13.625 42.375
25.100* 6.457 .004 10.492 39.708
20.100* 4.792 .002 9.259 30.941
5.400 4.525 .263 -4.835 15.635
(J) treatmnt
2
3
4
5
1
3
4
5
1
2
4
5
1
2
3
5
1
2
3
4
(I) treatmnt
1
2
3
4
5
Mean
Difference
(I-J) Std. Error Sig.
a
Lower Bound Upper Bound
95% Confidence Interval for
Difference
a
Based on estimated marginal means
The mean difference is significant at the .05 level.
*.
Adjustment for multiple comparisons: Least Significant Difference (equivalent to no
adjustments).
a.

13. Excel Spreadsheet of Means & sds
Minutes of
Exercise
Balance
Errors sd
3 8.5 4.5
6 11.4 7.96
9 16.4 10.8
12 31.1 12.56
15 36.5 21.13
0
10
20
30
40
50
60
70
0 5 10 15 20
Minutes of Exercise
Balance
Errors
Post hoc: 3, 6 & 9 minutes are significantly different from 12 minutes;
3, 6 & 9 minutes are significantly different from 15 minutes.