Repeated Measures Lecture - 1
2/6/2016
Repeated Measures: Field Ch 14, 15
The analyses considered here are those involving comparison of means in two or more research conditions
when either the same people or matched people have participated in the conditions.
Two major categories of design
1. Participants as their own controls design. (Also called Within Subjects designs.)
Same people participant in all conditions of the research.
Simplest: 2 conditions; Comparing performance of a group of people before treatment and after treatment.
Called a Before-After or Pre-Post design.
Longitudinal studies comparing performance of people over several time periods.
So, many repeated measures designs involve comparisons over time.
The key advantage of repeated measures designs is the similarity of participants in the two or more conditions.
For this reason, the maximum similarity is achieved by using the same people in all conditions. But there are
many designs in which you can’t use the same people. Learning designs are a good example. Once you’ve
learned something, you can’t unlearn it. The next design is a way around that.
2. Matched participants designs. (Matched people)
Different people experience the different conditions, but they’re matched with respect to one or more variables
that are correlated with the dependent variable, making them “as if” they were the same people in each
condition. Of course, you can’t match on everything, so matched participants are not as nearly identical as
participants matched with themselves.
3. Cloned participants designs!!!!!??? (Totally matched people)
Having your cake and eating it to. This is a way of having nearly perfectly matched participants without having
to use the same people in all conditions. The ultimate matching.
Repeated Measures Lecture - 2
2/6/2016
Advantages of Repeated Measures designs
1. Absence of confounds from extraneous variables. Since the people in each condition are the same or are
matched, differences found between conditions are more likely to be due to treatment differences rather than to
participant differences on variables we’re not interested in.
2. Increase in Power.
Repeated measures designs offer increased power for the same or smaller samples when
compared with between subjects designs.
Consider the relationship between the independent groups t-test and the correlated groups t-test.
The correlated-groups t-test is often expressed as
X-bar1 – X-bar2
---------------------------------------------S12 + S22 – 2rS1S2
r is the correlation between matched participants.
-----------------------N
If S1=S2=S, this can be rewritten as
X-bar1 – X-bar2
t = -----------------------2S2 (1-r)
-------------N
The key to this is the (1-r) part.
r is the correlation between the paired scores. In an independent groups design, r = 0.
In a repeated measures design, r is usually > 0.
This means that for a given difference in means (X-bar1 – X-bar2), the larger the value of r, the smaller the
denominator of the t statistic and therefore the bigger the value of t. The bigger the value of t, the more
likely it is to be a rejection t.
That is, power to detect a difference is a positive function of r - the larger the correlation between paired scores,
the greater the likelihood of detecting a difference, if there is a difference.
So designs with a positive correlation between scores in the two conditions will be more powerful than designs
with zero correlation, e.g., independent groups designs.
Of course, if there is no difference between the means of the two populations, then power is not an issue.
But if there IS a difference between the population mean, the correlated groups design will be more likely to
detect it.
Repeated Measures Lecture - 3
2/6/2016
How should data be put in the data editor for repeated measures designs?
Between Subjects Designs: Different conditions are represented in different rows of the data matrix.
C
o
n
d
i
t
i
o
n
1
C
o
n
d
i
t
i
o
n
2
Repeated Measures Designs: Different conditions are represented in different columns of the data matrix with
each row representing a person or matched persons.
C
o
n
d
i
t
i
o
n
C
o
n
d
i
t
i
o
n
1
2
Repeated Measures Lecture - 4
2/6/2016
Data Matrix of Designs with 2 Repeated Measures Factors
This example has two repeated measures factors, one with 2 levels and one with 3.
Example: Two types of material are being taught – material involving multiplication and material involving
division. Three data collection time periods are used at equally spaced intervals.
Multiplication
Multiplication
Time
1
Time
2
Levels of outer factor
Division
Time
3
Time
1
Time
2
Time
3
Levels of inner factor
Combination Designs: Between Subjects and Repeated Measures factors . . .
1B 1W: In this example, the Between-subjects factor has two levels and the Repeated Measures factor has 3.
For example, two ways of teaching statistics (with lab vs. without) measured across 3 tests.
Lab
Lab
Lab
T1
T2
T3
No
Lab
No
Lab
No
Lab
T1
T2
T3
Repeated Measures Lecture - 5
2/6/2016
The simplest type of repeated measures analysis – the paired samples t-test. – Start here on 9/23/15.
Comparing conscientiousness under honest instructions with conscientiousness under faking instructions.
The data editor
4.407 – 3.697
d = ------------------------ = 1.04
.685
This is a HUGE effect size.
Repeated Measures Lecture - 6
2/6/2016
The paired sample t as an analysis of difference scores.
We’ll get the exact same result by analyzing the single column of f-h difference scores.
Null hypothesis is that the mean of the difference scores is 0.
GET FILE='G:\MdbR\0DataFiles\Wrensen_070114.sav'.
DATASET NAME DataSet1 WINDOW=FRONT.
T-TEST /TESTVAL=0 /MISSING=ANALYSIS /VARIABLES=dcons /CRITERIA=CI(.95).
T-Test
[DataSet1] G:\MdbR\0DataFiles\Wrensen_070114.sav
One-Sample Statistics
N
dcons
Mean
166
Std. Deviation
.710
Std. Error Mean
.8431
.0654
One-Sample Test
Test Value = 0
95% Confidence Interval of the Difference
t
dcons
10.854
df
Sig. (2-tailed)
165
.000
Mean Difference
.7102
Lower
Upper
.581
.839
From above, for reference . . .
Many statistical procedures involving repeated measures make use of difference scores.
Repeated Measures Lecture - 7
2/6/2016
One-way Repeated Measures ANOVA Example
Data from Altman, D. G. Practical Statistics for Medical Research.
Data are from Table 12.2, p. 327. "Table 12.2 shows the heart rate of nine patients with congestive heart failure
before and shortly after administration of enalaprilat, an angiotensin-converting enzyme inhibitor.
Measurements were taken before and at 30, 60, and 120 minutes after drug administration.
Presumably, the drug should lower heart rate. So this is a basic longitudinal study, although the intervals
between measurements are not equal.
SUBJECT
TIME0
1
2
3
4
5
6
7
8
9
96
110
89
95
128
100
72
79
100
TIME30
92
106
86
78
124
98
68
75
106
TIME60
86
108
85
78
118
100
67
74
104
TIME120
92
114
83
83
118
94
71
74
102
Analyze -> General Linear Model -> Repeated Measures
Mike – demo this live.
Name the repeated-measures factor, and
enter the number of levels. Then click on
[Add].
Move the names of the columns representing the
levels of the repeated measures factor into the
appropriate place under [Within-Subjects
Variables]. (Note the change in terminology
from Repeated Measures to Within-Subjects.)
Repeated Measures Lecture - 8
2/6/2016
Since the 1st measurement appears to be special, I
specified a dummy variable repeated measures
contrast in which the all levels were compared with
level 1 of the RM factor.
Dummy variable coding is called Simple coding in
SPSS.
Since the times are not equally spaced – 30,60,120 – I
can’t easily use orthogonal polynomial contrasts to
look at the shape of change over time.
Check the usual set of optional statistics.
Output for the Parameter Estimates has been
erased.
Repeated Measures Lecture - 9
2/6/2016
The syntax for the analysis.
GLM
time0 time30 time60 time120
/WSFACTOR = time 4 Simple(1)
<<<<---- Specifies the RM factor and the contrast.
/METHOD = SSTYPE(3)
/PLOT = PROFILE( time )
/PRINT = DESCRIPTIVE ETASQ OPOWER PARAMETER
/CRITERIA = ALPHA(.05)
/WSDESIGN = time .
General Linear Model
Hypothesis Tested in the Analysis
The null hypothesis that is tested in the analysis can be viewed in two different and essentially equivalent ways.
Version 1 of the Null hypothesis: Equality of Means
The first is that the four means are all equal: H0: µTime0 = µTime30 = µTime60 = µTime120
Version 2 of the Null hypothesis: Means of Differences Equal 0
If the means are all equal then the differences between pairs of means are all 0.
Compute
µDiff1 = µTime0 - µTime30
µDiff2 = µTime30 - µTime60
µDiff3 = µTime60 - µTime120
(Any other 3 nonredundant differences would work.)
Then the 2nd version of the null hypothesis is: H0: µDiff1 = µDiff2 = µDiff3 = 0
Note that the = 0 at the end of the 2nd null is very important. The null is that every difference is 0.
Repeated Measures Lecture - 10
2/6/2016
The Sequence of Tests performed by GLM
The Multivariate test that all mean differences are 0. The first tests performed by GLM are what are called
Multivariate Tests. They're called that because they are a test of the hypothesis that the multiple difference
variables (3 in our case) are all 0. So they’re multivariate tests.
Four different multivariate tests are performed. Each is based on slightly different assumptions, and in some
instances, the results for the four may be different. In this case, they are all equivalent.
The multivariate tests are the most robust tests of the null hypothesis. This means that they are less affected by
nonnormality of the distributions than are the tests that follow. The price paid for that robustness is loss of
power. The multivariate tests are less powerful than those that follow. This means that if you're worried about
power and are hoping to reject the null, rejecting the null with the multivariate tests means you're home free.
But if you fail to reject the null with the multivariate tests, then you have to hope that you meet the restrictions
for the more powerful tests that follow.
In this case, we fail to reject the null using the multivariate tests.
The test of sphericity. When conducting a one-way repeated measures ANOVA, the values of variances and
covariances should fall within specific ranges. If this is so, the matrix is said to meet the sphericity criterion. If
the sphericity criterion is met, then the most powerful test of the null can be employed. If it not met, then either
the multivariate test must be used, or one of the special tests devoted to getting around the failure to find
sphericity must be employed. Fortunately, SPSS gives you all the tests, so you can pick the one which is
appropriate. The null hypothesis in Mauchly's test is that the sphericity condition holds, so we generally hope
to not reject the null.
In this instance, the condition of sphericity holds (the test is not significant).
Whew – the spericity condition is met.
Repeated Measures Lecture - 11
2/6/2016
The Univariate Tests. Since the sphericity condition is met, we can use the test of significance printed in the
top line of the following box.
If sphericity didn't hold, then we would either report the multivariate F above or report one of the F's from the
2nd, 3rd, or 4th lines of the table below. Each of them performs an adjustment for lack of sphericity. The
specifics of the adjustment vary across tests. Note that in the top three lines, the null is rejected. The last line
reports the most conservative adjustment. But since sphericity holds, we can report the F in the top line and
conclude that there are significant differences between mean heart rates across time periods.
The difference between these results and those of the Multivariate tests shown above highlight the power
differences between the two types of tests. If you want to detect differences, use the tests shown below.
Tests of user-specified Contrasts The following tests are of the contrast specified above. If no contrasts had
been specified, it would not be printed. The individual contrasts compare the means of the 30-, 60-, and 120minute measurements with the pre-measure. The means at the 60- and 120-minute intervals were significantly
lower than the pre-measure mean as we might expect if it took time for the drug to take effect.
Tests of Between-Subjects Effects. What are between subjects factors doing in a repeated measures analysis?
The answer is that there is only a technical between subjects factor here - the difference between the mean of all
the scores and 0, known in regression parlance as the intercept. That's what's being tested in the box below.
The null hypothesis is that the intercept is 0. Don’t let its presence confuse you.
Repeated Measures Lecture - 12
The plot below is that printed by GLM.
2/6/2016
Repeated Measures Lecture - 13
2/6/2016
Repeated Measures ANOVA 1 Between Groups Factor / 1 Repeated Measures Factor
The Bridge effectiveness study.
Situation: A school system provides middle and high school students a period for whatever intellectual activity
they wish to engage in. Some choose chess; some choose reading, some choose programming, some choose to
play bridge.
One of the people involved with the bridge classes was interested in whether playing bridge during the hour
periods over a long period of time would lead to greater performance on standardized tests of school-related
achievement than the other activities. The data here provide a test of that notion.
File: BridgeData1_3Periods_RM090208For595.sav
Key variables:
group0Vs1.
Variable representing the research condition.
0=Control; 1=Bridge
ss_math_tot1 ss_math_tot2
ss_math_tot3
Standardized total scores on math achievement at time
period 1, time period 2 and time period 3.
ss_comput1
ss_comput2
ss_comput3:
Standardized “computational??” achievement
ss_lang_tot1
ss_lang_tot2
ss_lang_tot3:
Standardized language achievement
(The names of the variables are idiosyncratic because of the way the data were given to me. I’ve just been too
lazy to change them to something easier to follow.)
What outcome would be ideal here??
A. Groups start out at about the same level since the two groups should be equivalent at the start.
B. The Bridge group scores on standardized tests increase at a faster rate than the Control group scores.
C. The interaction term would be significant.
Pictorially, this is the ideal outcome of this research project . . .
Bridge Group
Control Group
Repeated Measures Lecture - 14
The data
Analyze -> General Linear Model -> Repeated Measures.
2/6/2016
Repeated Measures Lecture - 15
2/6/2016
Repeated Measures Lecture - 16
2/6/2016
The syntax – analysis of math_tot scores.
GLM ss_math_tot1 ss_math_tot2 ss_math_tot BY group0Vs1
/WSFACTOR=time 3 Polynomial
/METHOD=SSTYPE(3)
/PLOT=PROFILE(time*group0Vs1)
/PRINT=DESCRIPTIVE ETASQ OPOWER PARAMETER
/CRITERIA=ALPHA(.05)
/WSDESIGN=time
/DESIGN=group0Vs1.
The cell means displayed in a 2 way table
Group
Control
Bridge
Time1
234
233
Time2
249
250
Time3
261
261
The significant “time” effect reflects the fact that the scores for all kids changed over time – the means
increased.
Alas, the nonsignificant “time*group0Vs1” effect means that neither group increased at a higher rate than the
other.
Repeated Measures Lecture - 17
2/6/2016
There is a Between-subjects effect in this analysis – the Bridge group vs Control group factor.
The table below presents a test of the significance of difference between the overall mean of the Bridge group scores vs the overall
mean of the Control group scores. The difference is not significant.
Repeated Measures Lecture - 18
I’ve rarely seen such a striking affirmation of the null.
2/6/2016
Repeated Measures Lecture - 19
2/6/2016
Policy Capturing using
Repeated Measures
Those in the I-O program know that cognitive ability is the single best predictor of performance in a variety of
situations, including academia.
Although we know that cognitive ability is a valid predictor, it’s possible that others do not have that
knowledge. If those others are involved in selection of employees, that lack of knowledge could be a serious
problem for the organization at which they are employed.
This is a hypothetical research project to investigate whether or not HR persons involved in selection
understand these basic results. It uses a technique called policy capturing. In policy capturing, persons are
given scenarios and asked to rate each scenario on one or more dimensions. The scenarios are created so that
they differ in specific ways, although the raters are not told about these differences. After the ratings have been
gathered, analyses of the ratings are conducted to determine if the differences in the scenarios affected the
ratings. If the raters had specific, perhaps hidden, policies controlling their ratings of the scenarios, e.g.,
prejudice against Blacks or females, those policies would be revealed by differences in ratings of the different
subgroups distinguished by the characteristics impacted by the policies.
In this example, the dependent variable, the ratings, will be of suitability for employement. Let’s suppose that
we have a four-item scale of suitability, with reliability = .8.
We will have persons rate two different hypothetical applicants. The applicants will be described as having
different levels of cognitive ability – either low or high
Cognitive
Ability
High
Low
Each person rated two scenarios. The order in which each person saw the scenarios was randomized across
participants.
If the participants understood contemporary validity research and implemented that knowledge in their policies
regarding suitability for employment, we would expect
A positive relationship of suitability to cognitive ability – scenarios describing applicants with high
cognitive ability will be rated higher than scenarios describing applicants with low cognitive ability.
Whatever the results, we can say that our analysis has captured the policies of the raters.
If the policy was to ignore cognitive ability when rating suitability for employment, that policy would be
reflected by a nonsignificant difference in mean suitability ratings between the low and high cognitive ability
scenarios.
Repeated Measures Lecture - 20
2/6/2016
The Analysis
The analysis could be a simple paired-sample t-test, but I’ll do it using the GLM Repeated Measures procedure,
since I want to expand the example following this analysis.
Here are some of the data . . .
Repeated Measures Lecture - 21
2/6/2016
General Linear Model
Notes
Output Created
Comments
Input
Missing Value Handling
23-SEP-2015 10:18:01
Filter
Weight
Split File
N of Rows in Working Data File
Definition of Missing
Cases Used
Syntax
Resources
Within-Subjects Factors
Measure: MEASURE_1
ca
Dependent Variable
1
lowca
2
highca
Processor Time
Elapsed Time
<none>
<none>
<none>
80
User-defined missing values are treated as missing.
Statistics are based on all cases with valid data for all variables in
the model.
GLM lowca highca
/WSFACTOR=ca 2 Polynomial
/METHOD=SSTYPE(3)
/PLOT=PROFILE(ca)
/PRINT=DESCRIPTIVE ETASQ OPOWER
/CRITERIA=ALPHA(.05)
/WSDESIGN=ca.
00:00:00.14
00:00:00.14
Repeated Measures Lecture - 22
2/6/2016
Descriptive Statistics
lowca
highca
Mean
.1536
.4583
Std. Deviation
.91630
.86124
N
80
80
Multivariate Testsa
Effect
ca
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
a. Design: Intercept
Within Subjects Design: ca
b. Exact statistic
c. Computed using alpha = .05
Value
.176
.824
.214
.214
F
16.872b
16.872b
16.872b
16.872b
Hypothesis df
1.000
1.000
1.000
1.000
Error df
79.000
79.000
79.000
79.000
Partial Eta
Squared
.176
.176
.176
.176
Sig.
.000
.000
.000
.000
Noncent.
Parameter
16.872
16.872
16.872
16.872
Observed
Powerc
.982
.982
.982
.982
Mauchly's Test of Sphericitya
Measure: MEASURE_1
Epsilonb
Approx. ChiGreenhouseWithin Subjects Effect
Mauchly's W
Square
df
Sig.
Geisser
Huynh-Feldt
Lower-bound
ca
1.000
.000
0
.
1.000
1.000
1.000
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: ca
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.
Tests of Within-Subjects Effects
Measure: MEASURE_1
Source
ca
Error(ca)
Sphericity Assumed
GreenhouseGeisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Type III Sum
of Squares
3.714
GreenhouseGeisser
Huynh-Feldt
Lower-bound
1
Mean
Square
3.714
F
16.872
Sig.
.000
Partial Eta
Squared
.176
Noncent.
Parameter
16.872
Observed
Powera
.982
3.714
1.000
3.714
16.872
.000
.176
16.872
.982
3.714
3.714
1.000
1.000
3.714
3.714
16.872
16.872
.000
.000
.176
.176
16.872
16.872
.982
.982
17.390
79
.220
17.390
79.000
.220
17.390
79.000
.220
17.390
79.000
.220
df
a. Computed using alpha = .05
Tests of Within-Subjects Contrasts
Measure: MEASURE_1
Type III Sum of
Source
ca
Squares
ca
Linear
3.714
Error(ca)
Linear
17.390
df
1
Mean Square
3.714
79
.220
F
16.872
Sig.
.000
Partial Eta
Squared
.176
Noncent.
Parameter
16.872
Observed
Powera
.982
a. Computed using alpha = .05
Tests of Between-Subjects Effects
Measure: MEASURE_1
Transformed Variable: Average
Type III Sum of
Source
Squares
Intercept
14.978
Error
107.535
a. Computed using alpha = .05
df
1
Mean Square
14.978
79
1.361
F
11.004
Sig.
.001
Partial Eta
Squared
.122
Noncent.
Parameter
11.004
Observed
Powera
.906
Repeated Measures Lecture - 23
Profile Plots
2/6/2016
Repeated Measures Lecture - 24
2/6/2016
Policy Capturing using
Factorial Repeated Measures
(Pardon the repetition from the previous example.)
Those in the I-O program know that cognitive ability is the single best predictor of performance in a variety of
situations, including academia. Many also know that conscientiousness is also a valid, though less efficacious
predictor of performance, also including performance in academia.
Although we know that these two characteristics are valid predictors, it’s possible that others do not have that
knowledge. If those others are involved in selection of employees, that could be a serious problem for the
organization at which they are employed.
This is a hypothetical research project to investigate whether or not HR persons involved in selection
understand these basic results. It uses a technique called policy capturing. In policy capturing, persons are
given scenarios and asked to rate each scenario on one or more dimensions. The scenarios are created so that
they differ in specific ways, although the raters are not told about these differences. After the ratings have been
gathered, analyses of the ratings are conducted to determine if the differences in the scenarios affected the
ratings. If the raters had specific policies regarding the differences in the scenarios, e.g., prejudice against
Blacks or females, those policies would be revealed by differences in ratings of the different subgroups
distinguished by those policies.
In this example, the dependent variable, the ratings, will be of suitability for employement. Let’s suppose that
we have a four-item scale of suitability, with reliability = .8.
We will have persons rate four different hypothetical applicants. The applicants will be described as having
different combinations of cognitive ability and conscientiousness – four combinations arranged factorially as
shown in the following table . . .
Conscientiousness
High
Low
Cognitive
Ability
High
Low
H-L
L-L
H-H
L-H
Each person rated four scenarios. The order in which each person saw the scenarios was randomized across
participants.
If the participants understood contemporary validity research, we would expect
1. A positive relationship of suitability to cognitive ability – scenarios describing applicants with high cognitive
ability will be rated higher than scenarios describing applicants with low cognitive ability.
2. A positive reltionship of suitability to conscientiousness – scenarios describing applicants with high
conscientiousness will be rated higher than scenarios describing applicants with low conscientiousness.
3. No interaction between the effects of cognitive ability and conscientiousness on ratings – the effect of
cognitive ability will be the same across conscientiousness levels and the effect of conscientiousness will be the
same across levels of cognitive ability.
Repeated Measures Lecture - 25
2/6/2016
The data would be analyzed using a repeated measures factorial design.
The data would be conceptualized in the data editor as 4 columns
ID
1
2
3
4
5
Etc
Low C
Low CA
rating
rating
.
.
.
etc
Low C
High CA
rating
rating
.
.
.
etc
High C
Low CA
rating
rating
.
.
.
etc
1
Here’s part of the actual hypothetical data . . .
High C
High CA
rating
rating
.
.
.
etc
Repeated Measures Lecture - 26
2/6/2016
Repeated Measures Lecture - 27
2/6/2016
General Linear Model
Notes
Output Created
Comments
Input
Missing Value Handling
23-SEP-2015 10:27:13
Filter
Weight
Split File
N of Rows in Working Data File
Definition of Missing
Cases Used
<none>
<none>
<none>
80
User-defined missing values are treated as missing.
Statistics are based on all cases with valid data for all variables in
the model.
GLM lowC_lowCA lowC_highCA highC_lowCA highC_highCA
/WSFACTOR=C 2 Polynomial CA 2 Polynomial
/METHOD=SSTYPE(3)
/PLOT=PROFILE(CA*C)
/EMMEANS=TABLES(C*CA)
/PRINT=DESCRIPTIVE ETASQ OPOWER
/CRITERIA=ALPHA(.05)
/WSDESIGN=C CA C*CA.
00:00:00.17
00:00:00.16
Syntax
Resources
Processor Time
Elapsed Time
Within-Subjects Factors
Measure: MEASURE_1
C
CA
Dependent Variable
1
1
lowC_lowCA
2
lowC_highCA
2
1
highC_lowCA
2
highC_highCA
Descriptive Statistics
lowC_lowCA
lowC_highCA
highC_lowCA
highC_highCA
Mean
.0213
.3186
.2859
.5980
Std. Deviation
.98327
1.00520
1.10178
.95945
N
80
80
80
80
Multivariate Testsa
Effect
C
Value
.135
.865
.156
.156
.176
.824
.214
.214
.000
1.000
.000
.000
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
CA
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
C * CA Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
a. Design: Intercept
Within Subjects Design: C + CA + C * CA
b. Exact statistic
c. Computed using alpha = .05
F
12.288b
12.288b
12.288b
12.288b
16.872b
16.872b
16.872b
16.872b
.009b
.009b
.009b
.009b
Hypothesis df
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Error df
79.000
79.000
79.000
79.000
79.000
79.000
79.000
79.000
79.000
79.000
79.000
79.000
Sig.
.001
.001
.001
.001
.000
.000
.000
.000
.923
.923
.923
.923
Partial Eta
Squared
.135
.135
.135
.135
.176
.176
.176
.176
.000
.000
.000
.000
Noncent.
Parameter
12.288
12.288
12.288
12.288
16.872
16.872
16.872
16.872
.009
.009
.009
.009
Observed
Powerc
.934
.934
.934
.934
.982
.982
.982
.982
.051
.051
.051
.051
Mauchly's Test of Sphericitya
Measure: MEASURE_1
Epsilonb
Approx. ChiGreenhouseWithin Subjects Effect
Mauchly's W
Square
df
Sig.
Geisser
Huynh-Feldt
Lower-bound
C
1.000
.000
0
.
1.000
1.000
1.000
CA
1.000
.000
0
.
1.000
1.000
1.000
C * CA
1.000
.000
0
.
1.000
1.000
1.000
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: C + CA + C * CA
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.
Repeated Measures Lecture - 28
2/6/2016
Tests of Within-Subjects Effects
Measure: MEASURE_1
Source
C
Error(C)
Sphericity Assumed
GreenhouseGeisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Type III Sum
of Squares
5.916
GreenhouseGeisser
Huynh-Feldt
Lower-bound
CA
Error(CA)
Sphericity Assumed
GreenhouseGeisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
GreenhouseGeisser
Huynh-Feldt
Lower-bound
C * CA
Error(C*CA
)
Sphericity Assumed
GreenhouseGeisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
GreenhouseGeisser
Huynh-Feldt
Lower-bound
1
Mean
Square
5.916
5.916
1.000
5.916
12.288
.001
.135
12.288
.934
5.916
5.916
1.000
1.000
5.916
5.916
12.288
12.288
.001
.001
.135
.135
12.288
12.288
.934
.934
38.034
79
.481
38.034
79.000
.481
38.034
79.000
.481
38.034
79.000
.481
7.428
1
7.428
16.872
.000
.176
16.872
.982
7.428
1.000
7.428
16.872
.000
.176
16.872
.982
7.428
7.428
1.000
1.000
7.428
7.428
16.872
16.872
.000
.000
.176
.176
16.872
16.872
.982
.982
34.780
79
.440
34.780
79.000
.440
34.780
79.000
.440
34.780
79.000
.440
.004
1
.004
.009
.923
.000
.009
.051
.004
1.000
.004
.009
.923
.000
.009
.051
.004
.004
1.000
1.000
.004
.004
.009
.009
.923
.923
.000
.000
.009
.009
.051
.051
36.940
79
.468
36.940
79.000
.468
36.940
79.000
.468
36.940
79.000
.468
df
F
12.288
Sig.
.001
Partial Eta
Squared
.135
Noncent.
Parameter
12.288
Observed
Powera
.934
a. Computed using alpha = .05
Tests of Within-Subjects Contrasts
Measure: MEASURE_1
Source
C
C
Linear
Error(C)
Linear
CA
Type III Sum
of Squares
df
Mean Square
5.916
1
5.916
38.034
79
.481
CA
Linear
7.428
1
7.428
Error(CA)
Linear
34.780
79
.440
C * CA
Linear
Error(C*CA) Linear
Linear
Linear
.004
1
.004
36.940
79
.468
F
Sig.
Partial Eta
Squared
Noncent.
Parameter
Observed
Powera
12.288
.001
.135
12.288
.934
16.872
.000
.176
16.872
.982
.009
.923
.000
.009
.051
a. Computed using alpha = .05
Tests of Between-Subjects Effects
Measure: MEASURE_1
Transformed Variable: Average
Type III Sum of
Source
Squares
Intercept
29.956
Error
215.070
a. Computed using alpha = .05
df
1
Mean Square
29.956
79
2.722
F
11.004
Sig.
.001
Partial Eta
Squared
.122
Noncent.
Parameter
11.004
Observed
Powera
.906
Repeated Measures Lecture - 29
2/6/2016
Estimated Marginal Means
C * CA
Measure: MEASURE_1
C
1
2
CA
1
2
1
2
Mean
.021
.319
.286
.598
Std. Error
.110
.112
.123
.107
95% Confidence Interval
Lower Bound
Upper Bound
-.197
.240
.095
.542
.041
.531
.384
.812
Profile Plots
The CA difference indicates that the hypothetical raters value persons with high CA more than those with low
CA.
The C difference indicates that the hypothetical raters value persons high in C more than those with low C.
There is no interaction of the effectsof CA and C on ratings.
Repeated Measures Lecture - 30
2/6/2016
Repeated Measures
1 Quantitative Between Subject Factor
1 Repeated Measures Factor
Nhung Nguyen’s dissertation data.
Nhung’s dissertation
1. Gave the Big 5 to 200 persons under two instruction conditions- once to respond honestly (Honest
instruction) and again to respond in a “way that would best guarantee that you would get a customer service
representative job.” (Instructed Faking).
2. Gave the Wonderlic test – a test of cognitive ability that correlates very highly with standard IQ tests.
3. Computed a faking ability score for each Big 5 dimension. This score was computed as the difference
between the score under the Faking instruction and the score under the Honest instruction standardized so that
faking ability scores on the Big Five dimensions were comparable. Called faking ability because participants
were instructed to fake in the Instructed Faking condition.
Questions:
1. The Repeated Measures Main Effect:
Are there differences in the mean faking amounts of the different Big 5 Personality dimensions? That
is, are the Big Five dimensions equally fakable?
2. The Between Subjects Main Effect:
Are there differences in overall average faking between the various levels of cognitive ability? Did
people who scored high on cognitive ability fake more than those who score low? Put another way: is
there a relationship between faking ability and cognitive ability?
3. The Interaction of the Between Subjects and Repeated Measures Effects:
Are differences in faking between the Big 5 measures the same across levels of cognitive ability?
or
Is the relationship of faking ability to cognitive ability the same for different Big 5 dimensions.
The analysis is a repeated measures ANOVA with a Quantitative Between subjects factor (the Wonderlic
scores).
Because the Between Subjects Factor is quantitative, it will be specified differently than a nominal factor and
the effects associated with it will be interpreted and presented differently than those in the previous example,
which involved a nominal (group0vs1) between subjects factor.
GET FILE='E:\MdbR\Nhung\SJTPaper
\ThesisPaperFile030907.sav'.
GLM dsurg dagree dcons des dopen WITH gscore
/WSFACTOR = persdim 5 Polynomial
/METHOD = SSTYPE(3)
/PLOT = PROFILE( persdim )
/EMMEANS = TABLES(persdim) WITH(gscore=MEAN)
/PRINT = DESCRIPTIVE ETASQ OPOWER
/CRITERIA = ALPHA(.05)
/WSDESIGN = persdim
/DESIGN = gscore .
The syntax generated by the pull down menus shown below.
Repeated Measures Lecture - 31
2/6/2016
Specification of Factors: Analyze -> General Linear Model -> Repeated Measures
Quantitative between subjects factors must be put in the covariates box in GLM.
Repeated Measures Lecture - 32
2/6/2016
Hypotheses Tested
The Repeated Measures Main Effect (Surgency is extraversion)
µdsurg
µdagree
µdcons
µdes
µdopen
H0: Equality of means of population of faking scores on the 5 personality dimensions.
Repeated Measures Lecture - 33
2/6/2016
The Between Subjects Main Effect
Average Diff scores
-.10 = Av Diff Score 1
-.11 = Av Diff Score 2
0 = Av Diff Score 3
1.14 = Av Diff Score 4
1.15 = Av Diff Score 5
0.86 = Av Diff Score 6
-0.04 = Av Diff Score 7
0.06 = Av Diff Score 8
And so on . .
Faking
Ability
H0: The correlation of gscore with the average difference scores in population is 0.
gscore
Note that this hypothesis is NOT stated as equality of means. If we had observed, say, 50 people with CA = 20,
50 with CA=25, and 50 with CA=30, and 50 with CA=35, then it could have been stated as a hypothesis that the
means of the faking scores at the 4 levels of CA were equal. But since we have multiple CA values with just a
few scores at each, perhaps only 1, it is more efficient to examine the relationship of faking to CA, rather than
to examine the differences in means.
The Interaction of Persdim and gscore
H0 :
The correlation of gscore with individual dimension faking scores will be the same for each
individual dimension.
or
Differences between dimension faking score means will be the same for each level of gscore.
Repeated Measures Lecture - 34
2/6/2016
General Linear Model
The multivariate tests below indicate that
there is NOT a main effect of Persdim
(Personality dimension). This means that
there are not overall significant differences
in the amount of faking between the 5
dimensions.
Within-S ubj ec ts Fa ctors
Me asure : MEA SURE_1
PE RSDI
M
1
De pende nt
Va riable
DS URG
2
DA GREE
3
DCONS
4
DE S
5
DO PEN
The significant interaction suggests,
however, that there are gscore-specific
differences in the amount of faking between
the 5 dimensions, i.e., differences that vary
with gscore..
De scriptiv e Statis tics
Me an
.47 97
DS URG
Std . Deviatio n
1.0 1526
N
20 3
DA GRE
E
.35 56
1.0 0172
20 3
DCONS
.58 54
.97 258
20 3
DE S
.74 74
1.0 6989
20 3
DO PEN
.39 21
.98 581
20 3
Equivalently, the interaction means that the
correlation of gscore with faking of each
dimensions varies from one dimension to
the next.
Multiv a riate Tests c
Eff ect
PE RSDI M
PE RSDI M *
GS CORE
Pil lai's T race
Va lue
.04 0
F
Hy pothe sis df
2.0 41 b
4.0 00
Error df
19 8.000
Sig .
.09 0
Pa rtial E ta
Sq uared
.04 0
No ncen t.
Pa rame ter
8.1 63
a
Ob serve d Po wer
.60 3
Wi lks' La mbd a
.96 0
2.0 41 b
4.0 00
19 8.000
.09 0
.04 0
8.1 63
.60 3
Ho tellin g's Trace
.04 1
2.0 41 b
4.0 00
19 8.000
.09 0
.04 0
8.1 63
.60 3
Ro y's La rgest
Ro ot
.04 1
2.0 41
4.0 00
19 8.000
.09 0
.04 0
8.1 63
.60 3
Pil lai's T race
.05 1
2.6 35 b
4.0 00
19 8.000
.03 5
.05 1
10 .538
.73 0
Wi lks' La mbd a
.94 9
2.6 35 b
4.0 00
19 8.000
.03 5
.05 1
10 .538
.73 0
Ho tellin g's Trace
.05 3
2.6 35 b
4.0 00
19 8.000
.03 5
.05 1
10 .538
.73 0
Ro y's La rgest
Ro ot
.05 3
2.6 35
4.0 00
19 8.000
.03 5
.05 1
10 .538
.73 0
b
b
a. Co mput ed using a lpha = .05
b. Ex act st atistic
c.
De sign: Interc ept+ GSCORE
Wi thin S ubje cts Design : PERSDIM
Ma uchly's Te st of Sphe ricity b
Me asure : ME ASURE_1
Ep silon
Wit hin S ubje cts
Eff ect
PE RSDI M
Ma uchly's W
.88 8
Ap prox.
Ch i-Squ are
23. 663
df
9
Sig .
.00 5
Gre enho useGe isser
.94 6
a
Hu ynh-Feldt
.97 1
Lower-b ound
.25 0
Te sts the null hypo thesi s that the e rror covari ance matrix of t he orthono rmal ized transf orme d dep ende nt va riable s is p roportiona l
to a n ide ntity matrix.
a. Ma y be used to ad just th e de grees of fre edom for the a verag ed te sts of signi fican ce. Correct ed te sts are disp laye d in
the Test s of Within -Subj ects E ffect s tabl e.
b.
De sign: Intercept+G SCO RE
Wit hin S ubje cts De sign: PERSDIM
The failure to meet the sphericity criterion means that we cannot use the “Sphericity Assumed” F printed below.
Repeated Measures Lecture - 35
2/6/2016
Tes ts of Within-Subj ec ts Effects
Me asure : ME ASURE_1
So urce
PE RSDI M
Ty pe III Sum
of Squa res
5.4 37
Sp heric ity Assume d
PE RSDI M *
GS CORE
Error(PE RSDIM)
4
Me an S quare
1.3 59
F
2.2 17
Sig .
.06 6
Pa rtial E ta
Sq uared
.01 1
No ncen t.
Pa rame ter
8.8 66
df
a
Ob serve d Po wer
.65 3
Gre enho use-Geisse
r
5.4 37
3.7 85
1.4 37
2.2 17
.06 9
.01 1
8.3 89
.63 5
Hu ynh-Feldt
5.4 37
3.8 85
1.3 99
2.2 17
.06 8
.01 1
8.6 13
.64 4
Lo wer-b ound
5.4 37
1.0 00
5.4 37
2.2 17
.13 8
.01 1
2.2 17
.31 7
Sp heric ity Assume d
6.0 67
4
1.5 17
2.4 74
.04 3
.01 2
9.8 94
.70 8
Gre enho use-Geisse
r
6.0 67
3.7 85
1.6 03
2.4 74
.04 6
.01 2
9.3 61
.69 0
Hu ynh-Feldt
6.0 67
3.8 85
1.5 61
2.4 74
.04 5
.01 2
9.6 11
.69 8
Lo wer-b ound
6.0 67
1.0 00
6.0 67
2.4 74
.11 7
.01 2
2.4 74
.34 7
Sp heric ity Assume d
49 2.981
80 4
.61 3
Gre enho use-Geisse
r
49 2.981
76 0.700
.64 8
Hu ynh-Feldt
49 2.981
78 0.974
.63 1
Lo wer-b ound
49 2.981
20 1.000
2.4 53
a. Co mput ed using a lpha = .05
Tes ts of Within-Subj ec ts Contras ts
Me asure : ME ASURE_1
PE RSDI
M
Lin ear
So urce
PE RSDI M
Ty pe III Sum
of Squa res
3.7 15
Qu adrat i
c
Cu bic
PE RSDI M *
GS CORE
F
5.4 52
Sig .
.02 1
Pa rtial E ta
Sq uared
.02 6
No ncen t.
Pa rame ter
5.4 52
.05 2
1
.05 2
.08 8
.76 8
.00 0
.08 8
.06 0
a
Ob serve d Po wer
.64 2
.05 8
1
.05 8
.07 9
.77 9
.00 0
.07 9
.05 9
Ord er 4
1.6 12
1
1.6 12
3.5 97
.05 9
.01 8
3.5 97
.47 1
Lin ear
2.9 96
1
2.9 96
4.3 97
.03 7
.02 1
4.3 97
.55 1
.62 2
1
.62 2
1.0 48
.30 7
.00 5
1.0 48
.17 5
Qu adrat i
c
Cu bic
.68 1
1
.68 1
.93 4
.33 5
.00 5
.93 4
.16 1
1.7 67
1
1.7 67
3.9 43
.04 8
.01 9
3.9 43
.50 7
Lin ear
13 6.969
20 1
.68 1
Qu adrat i
c
11 9.312
20 1
.59 4
Cu bic
14 6.621
20 1
.72 9
90 .079
20 1
.44 8
Ord er 4
Error(PE RSDIM)
1
Me an S quare
3.7 15
df
Ord er 4
a. Co mput ed using a lpha = .05
The Tests of Within-subjects Contrasts are automatically printed, but make no sense since the personality
dimensions are not on a quantitative dimension.
Tes ts of Betw een-Subj ects Effec ts
Me asure : ME ASURE_1
Tra nsformed Varia ble: A vera ge
So urce
Inte rcep t
GS COR
E
Error
Typ e III Sum
of S qua res
6.2 92
1
Me an S quare
6.2 92
F
2.6 32
Sig .
.10 6
Pa rtial E ta
Sq uared
.01 3
No ncent .
Pa rame ter
2.6 32
49. 924
1
49. 924
20. 882
.00 0
.09 4
20. 882
480 .540
201
2.3 91
df
a
Ob serve d Power
.36 5
.99 5
a. Co mput ed using a lpha = .05
The significant main effect of gscore means that there is a correlation of overall faking with cognitive ability.
That is, there are differences in amount of faking at different levels of cognitive ability.
Inspection of the graph below shows that it’s positive – smarter people faked more.
Repeated Measures Lecture - 36
2/6/2016
Estimated Marginal Means
PE RSDI M
Me asure : ME ASURE_1
95 % Co nfide nce I nterva l
PE RSDI
M
1
Me an
.48 0 a
Std . Erro r
.06 9
Lo wer B ound
.34 4
Up per B ound
.61 5
2
.35 6 a
.06 8
.22 2
.48 9
3
.58 5 a
.06 7
.45 3
.71 8
4
.74 7 a
.07 3
.60 4
.89 1
5
.39 2 a
.06 9
.25 6
.52 8
a. Co varia tes ap pearing in the mode l are evalu ated at th e
fol lowin g valu es: G SCO RE wond erlic test score = 24. 61.
Profile Plots
These are identical to the observed
means since there is only one group of
respondents.
Dimensions are listed as E A C S O.
So the largest amount of faking was in
the Emotional Stability dimension (4th
in the list). The least was in
Agreeableness and Openness. But
remember that the differences in means
were not officially significant.
Estimated Marginal Means of MEASURE_1
.8
.7
Note – Persdim main
effect was not
significant, so this
graph shouldn’t be
overinterpreted.
.6
.5
.4
.3
1
2
Ext
PERSDIM
3
Agr
4
Con
5
Sta
Opn
Graphing the Main Effect of gscore.
I computed an average faking score (those values illustrated on the right on p. 22 above.)
graph /scatterplot = gscore with fakescor.
Graph
4
3
2
1
FAKESCOR
0
-1
-2
Rsq = 0.0941
0
10
20
30
40
50
w onderlic test score
This is a display of the main effect of cognitive ability. The average difference scores are called FAKESCOR
in this plot. On the right, persons with similar gscore values have been grouped together. This shows the
positive relationship of faking to Gscore more strikingly.
Repeated Measures Lecture - 37
2/6/2016
The interactions.
Recall what the interaction is: The relationship of average faking to gscore differed across dimensions.
Below are displays of the relationship of faking for each dimension to cognitive ability. These displays are
analogous to the “interaction” plots when the between subjects factor is a categorical factor.
I’m not sure of the explanation of the interaction. Why is the relationship of faking to cognitive ability stronger
for Extraversion than it is for Openness, for example? That is, smart people faked Extraversion a lot more than
less smart people did. But smart people didn’t fake Openness much more than less smart people. ??
graph /scatterplot = gscore with dsurg.
Graph r = +.26
Graph r = +.27
5
5
4
4
3
3
2
2
1
1
0
0
DES
DSURG
-1
-1
-2
Rsq = 0.0685
0
-2
10
20
30
40
50
Rsq = 0.0744
0
10
20
30
40
50
w onderlic test score
w onderlic test score
graph /scatterplot = gscore with dopen.
graph /scatterplot = gscore with dagree.
Graph
Graph r = +.09
r = +.28
4
4
3
3
2
2
1
1
0
0
-1
-1
DOPEN
DAGREE
-2
-3
-2
-3
Rsq = 0.0076
0
-4
10
20
30
40
50
Rsq = 0.0802
0
10
20
30
40
50
w onderlic test score
w onderlic test score
graph /scatterplot = gscore with dcons.
Graph r = +.19
It may be that for some items, there may be
disagreement on what constitutes a “good”
response. That is, some people might have
thought that agreeing was the appropriate way
to fake while others thought that disagreeing
was the appropriate way to fake. The result
was that the relationships of faking to
cognitive ability was suppressed for those
items.
6
4
2
0
DCONS
-2
-4
Rsq = 0.0362
0
10
20
30
40
50
w onderlic test score
graph /scatterplot = gscore with des.
Repeated Measures Lecture - 38
2/6/2016
Using a correlation matrix to examine the individual dimension correlations.
correlation gscore dsurg to dopen fakescore.
Correlations
Wa rnings
Te xt: FA KES CORE
A variab le na me i s more tha n 8 characters l ong. Only the first 8 characters will be used.
Correlations
GS CORE
wo nderl ic test
sco re
GS CORE wo nderlic te st
sco re
Pe arson
Co rrelat ion
Sig . (2-t ailed )
DS URG
DA GREE
DCONS
DE S
DO PEN
FA KESCOR
DS URG
DA GREE
DCONS
DE S
DO PEN
FA KESCOR
1
.27 3
.28 3
.19 0
.26 2
.08 7
.30 7
.
.00 0
.00 0
.00 7
.00 0
.21 5
.00 0
N
203
203
203
203
203
203
203
Pe arson
Co rrelat ion
.27 3
1
.41 6
.32 9
.45 2
.37 5
.71 9
Sig . (2-t ailed )
.00 0
.
.00 0
.00 0
.00 0
.00 0
.00 0
N
203
203
203
203
203
203
203
Pe arson
Co rrelat ion
.28 3
.41 6
1
.39 7
.26 0
.33 1
.66 6
Sig . (2-t ailed )
.00 0
.00 0
.
.00 0
.00 0
.00 0
.00 0
N
203
203
203
203
203
203
203
Pe arson
Co rrelat ion
.19 0
.32 9
.39 7
1
.49 8
.43 5
.73 6
Sig . (2-t ailed )
.00 7
.00 0
.00 0
.
.00 0
.00 0
.00 0
N
203
203
203
203
203
203
203
Pe arson
Co rrelat ion
.26 2
.45 2
.26 0
.49 8
1
.45 1
.75 0
Sig . (2-t ailed )
.00 0
.00 0
.00 0
.00 0
.
.00 0
.00 0
N
203
203
203
203
203
203
203
Pe arson
Co rrelat ion
.08 7
.37 5
.33 1
.43 5
.45 1
1
.71 8
Sig . (2-t ailed )
.21 5
.00 0
.00 0
.00 0
.00 0
.
.00 0
N
203
203
203
203
203
203
203
Pe arson
Co rrelat ion
.30 7
.71 9
.66 6
.73 6
.75 0
.71 8
1
Sig . (2-t ailed )
.00 0
.00 0
.00 0
.00 0
.00 0
.00 0
.
N
203
203
203
203
203
203
203
Repeated Measures Lecture - 38 02/06/16
Repeated Measures Lecture - 39
2/6/2016
Repeated Measures ANOVA 1 Between Groups Factor
2 Repeated Measures Factors
The data are from Myers & Well, p 313 although the story describing the data is different from
theirs. Memory for two types of event, one of some interest to the persons (C1), the other of less
interest to them (C2) is tested at three time periods (B1, B2, and B3). The tests are performed under
two conditions of distraction, much distraction (A1) and little distraction (A2). The interest here is
on the effects of interest on memory, the effects of distraction on memory, and the the interaction of
interest and distraction – whether the effect of distraction is the same for memory of interesting and
uninteresting material. The data matrix looks as follows . .
Interesting
material
Uninteresting
material
C1B1 C1B2 C1B3 C2B1 C2B2 C2B3
Mike – show the pull
down menus in more
detail.
A
80
46
51
72
68
65
48
37
49
57
40
44
45
34
36
50
33
36
76
42
45
66
58
56
45
34
38
51
38
37
41
33
30
42
30
28
1
1
1
1
1
1
Much
distraction
70
88
58
63
78
84
55
69
60
57
81
82
52
66
54
52
75
80
68
91
50
61
79
80
57
74
41
58
78
73
56
70
38
56
74
76
2
2
2
2
2
2
Little
distraction
Specify the factor that varies slowest across
columns first
Then specif the factor whose levels vary
fastest next, in this case the B factor . . .
Repeated Measures Lecture - 39 02/06/16
Repeated Measures Lecture - 40
Click on the “Define” button, then . . .
Repeated Measures Lecture - 40 02/06/16
2/6/2016
Repeated Measures Lecture - 41
2/6/2016
Click on the name of each variable in the left-hand field and click on the right-pointing arrow to
move the name into the “Withing-Subjects Variables” field.
GLM
c1b1 c1b2 c1b3 c2b1 c2b2 c2b3 BY a
/WSFACTOR = c 2 Polynomial b 3 Polynomial
/METHOD = SSTYPE(3)
/PLOT = PROFILE( b*a*c )
/PRINT = DESCRIPTIVE ETASQ OPOWER HOMOGENEITY
/CRITERIA = ALPHA(.05)
/WSDESIGN = c b c*b
/DESIGN = a .
W arni ngs
Box's Test of Equality of Covarianc e
Matric es is not comput ed because
there are fewer than two
nonsingular cell covariance
matric es.
I believe it’s given because the number of persons in each cell is not larger than the number of
measures on each person.
Repeated Measures Lecture - 41 02/06/16
Repeated Measures Lecture - 42
2/6/2016
Hypotheses tested
The C (Interest) - Repeated Measures Main Effect
C1B1 C1B2 C1B3 C2B1 C2B2 C2B3
80
46
51
72
68
65
70
88
58
63
78
84
48
37
49
57
40
44
55
69
60
57
81
82
45
34
36
50
33
36
52
66
54
52
75
80
76
42
45
66
58
56
68
91
50
61
79
80
µC1
45
34
38
51
38
37
57
74
41
58
78
73
A
41
33
30
42
30
28
56
70
38
56
74
76
1
1
1
1
1
1
2
2
2
2
2
2
Sample is treated as
one giant group for the
repeated measures
comparisons
Much
distraction
Little
distraction
µC2
The B (Time) - Repeated Measures Main Effect
C1B1 C1B2 C1B3 C2B1 C2B2 C2B3
80
46
51
72
68
65
70
88
58
63
78
84
48
37
49
57
40
44
55
69
60
57
81
82
45
34
36
50
33
36
52
66
54
52
75
80
µB1
76
42
45
66
58
56
68
91
50
61
79
80
45
34
38
51
38
37
57
74
41
58
78
73
µB2
A
41
33
30
42
30
28
56
70
38
56
74
76
1
1
1
1
1
1
2
2
2
2
2
2
Sample is treated as
one giant group for the
repeated measures
comparisons
Little
distraction
µB3
The A – Distraction - Between-Subjects Main Effect
C1B1 C1B2 C1B3 C2B1 C2B2 C2B3
Much
distraction
A
80
46
51
72
68
65
48
37
49
57
40
44
45
34
36
50
33
36
76
42
45
66
58
56
45
34
38
51
38
37
41
33
30
42
30
28
µA1
Much
distraction
70
88
58
63
78
84
55
69
60
57
81
82
52
66
54
52
75
80
68
91
50
61
79
80
57
74
41
58
78
73
56
70
38
56
74
76
µA2
Little
distraction
Repeated Measures Lecture - 42 02/06/16
Repeated Measures Lecture - 43
General Linear Model
Wi thin-S ubj e cts Factors
Me asure: ME ASURE_ 1
C
1
2
B
1
De pend ent
Va riabl e
C1 B1
2
C1 B2
3
C1 B3
1
C2 B1
2
C2 B2
3
C2 B3
Be tw ee n-Subj ects Fac tors
Va lue L abel
A
N
1
6
2
6
De scriptiv e Statis tics
C1 B1
C1 B2
C1 B3
C2 B1
C2 B2
C2 B3
A
1
Me an Std . Deviatio n
63 .67
12 .88
N
6
2
73 .50
11 .86
6
To tal
68 .58
12 .87
12
1
45 .83
7.1 4
6
2
67 .33
11 .98
6
To tal
56 .58
14 .64
12
1
39 .00
6.8 7
6
2
63 .17
12 .37
6
To tal
51 .08
15 .82
12
1
57 .17
12 .75
6
2
71 .50
14 .79
6
To tal
64 .33
15 .14
12
1
40 .50
6.2 8
6
2
63 .50
14 .07
6
To tal
52 .00
15 .88
12
1
34 .00
6.0 3
6
2
61 .67
14 .50
6
To tal
47 .83
17 .91
12
Repeated Measures Lecture - 43 02/06/16
2/6/2016
Repeated Measures Lecture - 44
2/6/2016
The multivariate tests. The less powerful but more robust multivariate tests are always printed first.
These tests indicate that there is a significant effect associated with Factor C (Interest), with Factor B
(Time), and with the B * A (Time by Distraction) interaction.
Multiv a riate Tests c
Eff ect
C
C* A
B
B* A
C* B
Pil lai's T race
Va lue
.44 2
F
Hyp othe sis df
7.9 35 b
1.0 00
Error df
10. 000
No ncent .
Sig .
Eta Squ ared Pa rame ter
.01 8
.44 2
7.9 35
Ob serve d
a
Po wer
.71 9
Wil ks' La mbd a
.55 8
7.9 35 b
1.0 00
10. 000
.01 8
.44 2
7.9 35
.71 9
Ho tellin g's Trace
.79 3
7.9 35 b
1.0 00
10. 000
.01 8
.44 2
7.9 35
.71 9
Ro y's La rgest Root
.79 3
7.9 35 b
1.0 00
10. 000
.01 8
.44 2
7.9 35
.71 9
Pil lai's T race
.10 9
1.2 26 b
1.0 00
10. 000
.29 4
.10 9
1.2 26
.17 1
Wil ks' La mbd a
.89 1
1.2 26 b
1.0 00
10. 000
.29 4
.10 9
1.2 26
.17 1
Ho tellin g's Trace
.12 3
1.2 26 b
1.0 00
10. 000
.29 4
.10 9
1.2 26
.17 1
Ro y's La rgest Root
.12 3
1.2 26 b
1.0 00
10. 000
.29 4
.10 9
1.2 26
.17 1
Pil lai's T race
.89 3
37. 678 b
2.0 00
9.0 00
.00 0
.89 3
75. 356
1.0 00
Wil ks' La mbd a
.10 7
37. 678 b
2.0 00
9.0 00
.00 0
.89 3
75. 356
1.0 00
Ho tellin g's Trace
8.3 73
37. 678 b
2.0 00
9.0 00
.00 0
.89 3
75. 356
1.0 00
Ro y's La rgest Root
8.3 73
37. 678 b
2.0 00
9.0 00
.00 0
.89 3
75. 356
1.0 00
Pil lai's T race
.56 7
5.8 83 b
2.0 00
9.0 00
.02 3
.56 7
11. 766
.73 4
Wil ks' La mbd a
.43 3
5.8 83 b
2.0 00
9.0 00
.02 3
.56 7
11. 766
.73 4
Ho tellin g's Trace
1.3 07
5.8 83 b
2.0 00
9.0 00
.02 3
.56 7
11. 766
.73 4
Ro y's La rgest Root
1.3 07
5.8 83 b
2.0 00
9.0 00
.02 3
.56 7
11. 766
.73 4
Pil lai's T race
.32 3
2.1 50 b
2.0 00
9.0 00
.17 3
.32 3
4.2 99
.32 9
Wil ks' La mbd a
.67 7
2.1 50 b
2.0 00
9.0 00
.17 3
.32 3
4.2 99
.32 9
Ho tellin g's Trace
.47 8
2.1 50 b
2.0 00
9.0 00
.17 3
.32 3
4.2 99
.32 9
Ro y's La rgest Root
.47 8
2.1 50 b
2.0 00
9.0 00
.17 3
.32 3
4.2 99
.32 9
.18 6
1.0 28 b
2.0 00
9.0 00
.39 6
.18 6
2.0 55
.17 7
Wil ks' La mbd a
.81 4
1.0 28 b
2.0 00
9.0 00
.39 6
.18 6
2.0 55
.17 7
Ho tellin g's Trace
.22 8
1.0 28 b
2.0 00
9.0 00
.39 6
.18 6
2.0 55
.17 7
Ro y's La rgest Root
.22 8
1.0 28 b
2.0 00
9.0 00
.39 6
.18 6
2.0 55
.17 7
C * B * A Pil lai's T race
a. Co mput ed using a lpha = .05
b. Exa ct sta tistic
c.
De sign: Intercept+ A
Wit hin S ubje cts De sign: C+B +C*B
The significant C Main effect indicates that the mean amount recalled depends on the Interest.
The significant B main effect indicates that the mean amount recalled depends on the time at
which recall occurred.
The significant B*A interaction indicates that the change in mean amount over time is different
for persons under much distraction than it is for persons under little distraction.
Repeated Measures Lecture - 44 02/06/16
Repeated Measures Lecture - 45
2/6/2016
The sphericity test. Mauchly's test indicates that the sphericity condition is NOT met. So we must
either go with the multivariate tests or use one of the tests which adjusts for lack of sphericity.
Fortunately, they all give the same result with respect to significance and they all agree with the
multivariate tests with respect to significance, so the point is moot.
Ma uchly's Te st of Sphe ricity b
Me asure : ME ASURE_1
Ep silon
Ap prox.
Wit hin S ubje cts Ef fect Ma uchly's W Ch i-Squ are
C
1.0 00
.00 0
df
Sig .
0
a
Gre enho u
se-Geisser Hu ynh-Feldt Lower-b ound
.
1.0 00
1.0 00
1.0 00
B
.26 3
12. 031
2
.00 2
.57 6
.66 8
.50 0
C* B
.49 1
6.4 02
2
.04 1
.66 3
.80 1
.50 0
Te sts the null hypo thesi s that the e rror covari ance matrix of t he orthono rmal ized transf orme d dep ende nt
variable s is p roportional to an iden tity m atrix.
a. Ma y be used to ad just th e de grees of fre edom for the a verag ed te sts of sign ifican ce. Corrected
tests are displ ayed in th e Tests of Within -Sub jects Effects tab le.
b.
De sign: Intercept+ A
Wit hin S ubje cts De sign: C+B +C*B
Repeated Measures Lecture - 45 02/06/16
Repeated Measures Lecture - 46
2/6/2016
Tes ts of Within-Subj ec ts Effects
Me asure : ME ASURE_1
So urce
C
C* A
Error(C)
B
B* A
Error(B)
C* B
Sp hericity Assume d
Type III Sum
of Squa res
29 2.014
Gre enho use-Geisser
29 2.014
1.0 00
29 2.014
7.9 35
.01 8
.44 2
7.9 35
.71 9
Hu ynh-Feldt
29 2.014
1.0 00
29 2.014
7.9 35
.01 8
.44 2
7.9 35
.71 9
Lo wer-b ound
29 2.014
1.0 00
29 2.014
7.9 35
.01 8
.44 2
7.9 35
.71 9
Sp hericity Assume d
45 .125
1
45 .125
1.2 26
.29 4
.10 9
1.2 26
.17 1
Gre enho use-Geisser
45 .125
1.0 00
45 .125
1.2 26
.29 4
.10 9
1.2 26
.17 1
Hu ynh-Feldt
45 .125
1.0 00
45 .125
1.2 26
.29 4
.10 9
1.2 26
.17 1
Lo wer-b ound
45 .125
1.0 00
45 .125
1.2 26
.29 4
.10 9
1.2 26
.17 1
Sp hericity Assume d
36 8.028
10
36 .803
Gre enho use-Geisser
36 8.028
10 .000
36 .803
Hu ynh-Feldt
36 8.028
10 .000
36 .803
Lo wer-b ound
36 8.028
10 .000
36 .803
Sp hericity Assume d
36 83.11 1
2
18 41.55 6
36 .410
.00 0
.78 5
72 .821
1.0 00
Gre enho use-Geisser
36 83.11 1
1.1 51
31 99.37 8
36 .410
.00 0
.78 5
41 .915
1.0 00
Hu ynh-Feldt
36 83.11 1
1.3 35
27 58.60 5
36 .410
.00 0
.78 5
48 .613
1.0 00
Lo wer-b ound
36 83.11 1
1.0 00
36 83.11 1
36 .410
.00 0
.78 5
36 .410
1.0 00
Sp hericity Assume d
61 6.333
2
30 8.167
6.0 93
.00 9
.37 9
12 .186
.83 3
Gre enho use-Geisser
61 6.333
1.1 51
53 5.385
6.0 93
.02 7
.37 9
7.0 14
.65 2
Hu ynh-Feldt
61 6.333
1.3 35
46 1.626
6.0 93
.02 1
.37 9
8.1 35
.70 1
Lo wer-b ound
61 6.333
1.0 00
61 6.333
6.0 93
.03 3
.37 9
6.0 93
.60 6
Sp hericity Assume d
10 11.55 6
20
50 .578
Gre enho use-Geisser
10 11.55 6
11 .512
87 .870
Hu ynh-Feldt
10 11.55 6
13 .351
75 .764
Lo wer-b ound
10 11.55 6
10 .000
10 1.156
df
Me an S quare
1
29 2.014
F
7.9 35
No ncen t.
Sig .
Eta Squ ared Pa rame ter
.01 8
.44 2
7.9 35
Ob serve d
a
Po wer
.71 9
Sp hericity Assume d
5.7 78
2
2.8 89
.69 1
.51 2
.06 5
1.3 83
.15 0
Gre enho use-Geisser
5.7 78
1.3 25
4.3 59
.69 1
.46 0
.06 5
.91 6
.12 9
Hu ynh-Feldt
5.7 78
1.6 03
3.6 05
.69 1
.48 4
.06 5
1.1 08
.13 8
Lo wer-b ound
5.7 78
1.0 00
5.7 78
.69 1
.42 5
.06 5
.69 1
.11 7
Sp hericity Assume d
7.0 00
2
3.5 00
.83 8
.44 7
.07 7
1.6 76
.17 3
Gre enho use-Geisser
7.0 00
1.3 25
5.2 82
.83 8
.40 9
.07 7
1.1 10
.14 6
Hu ynh-Feldt
7.0 00
1.6 03
4.3 67
.83 8
.42 7
.07 7
1.3 43
.15 8
Lo wer-b ound
7.0 00
1.0 00
7.0 00
.83 8
.38 2
.07 7
.83 8
.13 2
Error(C* B) Sp hericity Assume d
83 .556
20
4.1 78
Gre enho use-Geisser
83 .556
13 .253
6.3 04
Hu ynh-Feldt
83 .556
16 .029
5.2 13
Lo wer-b ound
83 .556
10 .000
8.3 56
C* B*A
a. Co mput ed using a lpha = .05
Repeated Measures Lecture - 46 02/06/16
Repeated Measures Lecture - 47
2/6/2016
Test of equality of variances across groups. Levene's test is for the between-subjects effects. It
indicates generally that the variances are not significantly different.
Le v ene 's Te st of Equa lity of Error Va rianc es a
C1 B1
F
.00 7
C1 B2
C1 B3
df1
1
df2
10
Sig .
.93 6
3.1 86
1
10
.10 5
4.7 78
1
10
.05 4
C2 B1
.31 1
1
10
.58 9
C2 B2
5.2 05
1
10
.04 6
C2 B3
5.0 00
1
10
.04 9
Te sts th e nul l hyp othesis tha t the error varia nce of
the dep ende nt va riable is eq ual a cross grou ps.
a.
De sign: Intercept+ A
Wi thin S ubje cts Design : C+B +C*B
Test of between-subjects main effect. The comparison between the two distracter conditions
indicates that there is a significant difference in mean recall between the two.
Tes ts of Betw een-Subj ects E ffec ts
Me asure : ME ASURE_1
Tra nsformed Varia ble: A vera ge
So urce
Inte rcep t
Typ e III Sum
of S qua res
231 767. 014
df
Me an S quare
F
1 231 767. 014 363 .878
A
726 0.12 5
1
726 0.12 5
Error
636 9.36 1
10
636 .936
11. 399
No ncent .
Sig .
Eta Squ ared Pa rame ter
.00 0
.97 3
363 .878
.00 7
.53 3
11. 399
a. Co mput ed using a lpha = .05
Repeated Measures Lecture - 47 02/06/16
Ob serve d
a
Po wer
1.0 00
.86 0
Repeated Measures Lecture - 48
2/6/2016
Profile Plots
Estimated Marginal Means of MEASURE_1
70
60
Distraction.
Main Effect of A: Mean
recall was better in the low
distraction condition.
50
40
High distraction con
Low distraction cond
A
Estimated Marginal Means of MEASURE_1
70
60
Time.
Main Effect of B: Mean recall decreased
over time periods.
50
40
1
2
3
B
Repeated Measures Lecture - 48 02/06/16
Repeated Measures Lecture - 49
2/6/2016
Estimated Marginal Means of MEASURE_1
59
58
Main Effect of Factor C: Mean recall was
greater for more interesting material.
57
56
55
54
1
2
Less
Interesting
More
Interesting
C
Estimated Marginal Means of MEASURE_1
80
70
Distraction by Time.
Interaction of A and B: The effect of
distraction was greater at longer recall
times. (This was the only significant
interaction.)
60
A
50
High distraction con
dition
40
Low distraction cond
30
ition
1
2
3
B
Repeated Measures Lecture - 49 02/06/16
Repeated Measures Lecture - 50
2/6/2016
B*A*C
Estimated Marginal Means of MEASURE_1
Even though the three way
interaction, B*A*C was not
significant, it is of value to see how it
would be visualized. A significant
three way interaction means that a
two-way interaction was not the same
across levels of a third variable.
At C = 1
80
70
60
A
50
High distraction con
dition
40
Low distraction cond
30
ition
1
2
3
B
Estimated Marginal Means of MEASURE_1
At C = 2
80
70
In this case, if the B*A*C interaction
were significant, that would mean
that the B*A interaction varied across
levels of C. The way to visualize that
is to plot the B*A interaction for each
level of C. Those plots are on the
left. The fact that they're nearly
identical supports the fact that there
was NO B*A*C interaction. If there
had been, the two plots (one for C=1
and the other for C=2) would have
been different.
So . . .
60
A
50
High distraction con
dition
40
A two-way interaction is a difference
of effect of one variable across levels
of a second variable.
Low distraction cond
30
ition
1
B
2
3
A three-way interaction is a
difference of INTERACTION of two
variables across levels of a third
variable.
Repeated Measures Lecture - 50 02/06/16