One Factor Repeated Measures Design on SPSS for Windows The
Multivariate Approach The Example An experimenter wants to evaluate
the relative merits of four versions of an instrument used to
display altitude in a helicopter. Eight helicopter pilots, with
from 500 to 3000 flight hours, are available as participants.
Accuracy in reading the altimeter at low altitudes is of prime
importance, so the dependent variable is the amount of reading
error. It is anticipated that the amount of previous flying
experience may affect pilots' performance with the experimental
altimeters. So in order to isolate this nuisance variable, a
repeated measures design was used. Each participant made 100
readings under simulated flight conditions with each of the
altimeters. The sequence in which the four altimeters were
presented was randomised for each pilot. Do differences exist
between the altimeters? Altimeter 1 1 2 3 4 5 6 7 8 3 6 3 3 1 2 2 2
2 7 8 7 6 5 6 5 6 3 4 5 4 3 2 3 4 3 4 7 8 9 8 10 10 9 11
Pilot
For a repeated measures design, SPSS requires that we enter the
data for each level of the factor into a different column of its
data editor. We can do this and give each column a reasonable name
like alt1, alt2, alt3, and alt4.
To run the one way repeated measures ANOVA, we click on Analyze,
General Linear Model, Repeated Measures:
1
This opens up the Repeated Measures Define Factor(s) dialogue
box:
SPSS has provided a default factor name, factor1, which I
suggest you should replace with something a bit more meaningful,
say, altimeter. Once you have done that, you need to indicate that
there are four levels of the factor by entering 4 in the Number of
Levels box. As soon as you do this, the Add box will become
emboldened, and if you click on this, altimeter and 4 will
disappear, and altimeter(4) will appear in the box to the right of
Add:
2
You can give the dependent variable a name by clicking in the
box next to Measure Name and typing in an appropriate name. Were
measuring the amount of error with each altimeter, so type error in
the Measure Name box and click on the (now emboldened Add). The
name error appears in the bottom box.
Now you need to click on Define to open up the Repeated Measures
dialogue box:
3
Click on alt1 in the box on the left and drag down to alt4, then
click on the right arrow to paste these into the _?_(1,error),
_?_(2,error), _?_(3,error) and _?_(4,error) gaps in the
WithinSubjects Variables box.
Click on Post Hoc and confirm that the usual post hoc methods
are not available for repeated measures factors on SPSS. Now click
on Options and click on Descriptive statistics. In the top part of
the Options dialogue box click on altimeter and on the right arrow
to put it under Display Means for: When you do this the small box
beside Compare main effects becomes active.
Click in the Compare main effects box and the Confidence
interval adjustment box becomes active. Click on the down arrow and
select Bonferroni.
4
Click on Continue and then, click on OK in the Repeated Measures
dialogue box. Heres the output:
General Linear Model[DataSet1] G:\PY0701\One Way Repeated
Measures Design.savWithin-Subjects Factors Measure: error altimeter
1 2 3 4 Dependent Variable alt1 alt2 alt3 alt4
Descriptive Statistics alt1 alt2 alt3 alt4 Mean 2.7500 6.2500
3.5000 9.0000 Std. Deviation 1.48805 1.03510 .92582 1.30931 N 8 8 8
8
5
b Multivariate Tests
Effect altimeter
Pillai's Trace Wilks' Lambda Hotelling's Trace Roy's Largest
Root
Value .971 .029 33.276 33.276
F Hypothesis df 55.461a 3.000 55.461a 3.000 a 55.461 3.000
55.461a 3.000
Error df 5.000 5.000 5.000 5.000
Sig. .000 .000 .000 .000
a. Exact statistic b. Design: Intercept Within Subjects Design:
altimeter
b Mauchly's Test of Sphericity
Measure: error Epsilon Within Subjects Effect Mauchly's W
altimeter .072 Approx. Chi-Square 15.077 df 5 Sig. .011 Greenhous
e-Geisser .419a
Huynh-Feldt .468
Lower-bound .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: altimeter
Tests of Within-Subjects Effects Measure: error Source altimeter
Type III Sum of Squares 194.500 194.500 194.500 194.500 28.500
28.500 28.500 28.500 df 3 1.257 1.403 1.000 21 8.798 9.818 7.000
Mean Square 64.833 154.750 138.678 194.500 1.357 3.239 2.903 4.071
F 47.772 47.772 47.772 47.772 Sig. .000 .000 .000 .000
Error(altimeter)
Sphericity Assumed Greenhouse-Geisser Huynh-Feldt Lower-bound
Sphericity Assumed Greenhouse-Geisser Huynh-Feldt Lower-bound
Tests of Between-Subjects Effects Measure: error Transformed
Variable: Average Source Intercept Error Type III Sum of Squares
924.500 12.500 df 1 7 Mean Square 924.500 1.786 F 517.720 Sig.
.000
Estimated Marginal Means altimeter6
Estimates Measure: error altimeter 1 2 3 4 Mean 2.750 6.250
3.500 9.000 Std. Error .526 .366 .327 .463 95% Confidence Interval
Lower Bound Upper Bound 1.506 3.994 5.385 7.115 2.726 4.274 7.905
10.095
Pairwise Comparisons Measure: error Mean Difference (I-J) Std.
Error -3.500* .267 -.750 .313 -6.250* .881 3.500* .267 2.750* .250
-2.750* .726 .750 .313 -2.750* .250 -5.500* .707 6.250* .881 2.750*
.726 5.500* .707 95% Confidence Interval for a Difference Lower
Bound Upper Bound -4.472 -2.528 -1.889 .389 -9.454 -3.046 2.528
4.472 1.841 3.659 -5.389 -.111 -.389 1.889 -3.659 -1.841 -8.071
-2.929 3.046 9.454 .111 5.389 2.929 8.071
(I) altimeter 1
2
3
4
(J) altimeter 2 3 4 1 3 4 1 2 4 1 2 3
Sig. .000 .288 .001 .000 .000 .041 .288 .000 .001 .001 .041
.001
a
Based on estimated marginal means *. The mean difference is
significant at the .05 level. a. Adjustment for multiple
comparisons: Bonferroni.
The first table simply identifies the four levels of the
within-subjects factor ALTIMETER. The second table gives the mean,
standard deviation and sample size for each of the levels. They
would be included in a Results section. Something like this:
7
Table of means (and standard deviations) of the errors made with
four altimeters. (n=8) Altimeter 1 2.75 (1.49) The Multivariate
Approach We used to have to analyse repeated measures designs using
the univariate anova approach. Unfortunately this involved us in
having to make some rather unlikely assumptions about the data, or
in testing the assumptions and trying to allow for violations of
them by adjusting the univariate anovas degrees of freedom. Much of
SPSSs output is related to this approach. In recent years it has
become clear that it is possible to analyse repeated measures
designs using the class of statistical tests known as multivariate
analysis of variance which do not make such unrealistic
assumptions. This is the approach we now take to analysing all
repeated measures factors. The results of such tests are shown in
the third table of output. The degrees of freedom associated with
such tests are often quite small, but this is a small price to pay
for the freedom from restrictive assumptions the multivariate tests
bring with them. SPSS presents four different multivariate tests,
but they all lead to the same conclusion here. We will report
Wilkss Lambda as it seems to be the most frequently used as well as
being the one recommended in most textbook. Using this approach we
are led to the conclusion that the number of errors made did depend
significantly on the altimeter used, Wilks Lambda = .029, F(3,5) =
55.461, p < .001. Post Hoc Analysis Using Bonferroni Corrected
Repeated Measures t Tests When post hoc comparisons are carried out
following a significant independent groups anova (we used Tukey
earlier in the module), these employ an error term that is derived
from all the conditions in the study, not just the two conditions
we are comparing with a particular comparison. Things are very
different in repeated measures designs. The received wisdom is that
it is safer to use only the data in the two conditions being
compared when conducting post hoc comparisons in repeated measures
designs. The approach is to conduct repeated measures t tests
between every pair of conditions, and to control for inflation of
type one error by dividing the family-wise significance level
equally between the tests (using the Bonferroni method). In terms
of the p value for one of the tests conducted, following Bonferroni
correction, the p value will be j times bigger than the p value
that would be given if we were conducting just one t test (instead
of j t tests). The next table of output gives the mean and standard
error for each altimeter (and a confidence interval for each mean).
The table following shows the results of the Bonferroni corrected
repeated measures t tests that have been conducted to compare every
altimeter with every other one (six tests in all, though SPSS give
each one twice!). The table below shows the results in a more
user-friendly way. Altimeter 2 6.25 (1.04) Altimeter 3 3.50 (0.93)
Altimeter 4 9.00 (1.31)
8
Table showing mean differences between errors made for all pairs
of altimeters, and p values resulting from Bonferroni corrected
repeated measures t tests. Alt 1 2.75 Alt1 2.75 Alt2 6.25 Alt3 3.50
Alt 2 6.25 -3.500 p < .001 Alt3 3.50 -0.750 p = .288 2.750 p
< .001 Alt4 9.00 -6.250 p = .001 -2.750 p = .041 -5.500 p =
.001
The p values above are all six times bigger than they would have
been if we had only been conducting one repeated measures t test.
For example a repeated measures t test to compare altimeters 1 and
3 yields a p value of .048. Six times this value is .288, the
Bonferroni corrected p value in the table above for the comparison
of altimeters 1 and 3. Conclusions If you look at the right hand
side of the table, youll see that all the three tests involving
altimeter 4 are significant. We can conclude from this (and from
looking at the condition means) that, following Bonferroni
correction: Altimeter 4 (mean = 9.00) leads to significanly more
error than Altimeter 1 (mean 2.75) and Altimeter 3 (mean 3.50), p =
.001, and than Altimeter 2 (mean 6.25), p = .041. The other two
significant results both involve altimeter 2. We can conclude that,
following Bonferroni correction: Altimeter 2 (mean 6.25) leads to
significantly more error than Altimeter 1 (mean 2.75) and Altimeter
3 (mean 3.50), p < .001. Following Bonferroni correction,
Altimeter 1 (mean 2.75) and Altimeter 3 (mean 3.50) do not lead to
significantly different amounts error, p = .288. (Note that if the
only test we had conducted had been between Altimeters 1 and 3, we
would have concluded that they did lead to significantly different
amounts of error, as the uncorrected p value was .048, as mentioned
above.)
9