ANCOVA and Repeated Measures Designs

1
ANCOVA and Repeated Measures Designs
2
Analysis of Covarinace

• Extension of Anova model by adding a control
  variable to the model method of statistical
  control
• Conceptually similar to partial correlation in
  the sense that Anova is conducted on the IVS and
  DV after removing the part of the relationship
  predicted by the control variable (Think about
  residuals)
• Green, L27
• Green has a fine definition at the beginning of
  the chapter.
• Example Revisit vitamin C example except now
  have PREDAYS, a measure of base rate of cold
  symptoms the first year.
• Vitamin C treatment program applied during second
  year, main DV is DAYS, the number of days with
  cold symptoms the second year.

3
Doing Normal Statistics
ANCOVA
X (IV)
Y (DV)
Z cov
4
Applications

• Pretest, then random assignment
• Pretest, then assignment to groups based on
  pretest
• Pretest, matching based on pretest, then random
  assignment to groups
• Studies with non-equivalent initial group
  assignment - confounding

5
Green L 27

• Vitamin C example expanded
• Effect of three levels of vitamin C dosage on
  days of cold symptoms year 2, controlling for
  days of cold symptoms during year 1.
• Treatment introduced at start of year 2.
• Note It may be that there are pre-existing
  differences in cold symptom days among the three
  groups. Covariance can control for such group
  differences.
• J\PSYCH\MARTY\4123\GreenSalkind5Dat\Lesson
  27\Lesson 27 Data File 1.sav

6
Initial Data Examination

• Are there group differences on predays? On days
  without adjustment?
• There must be a linear regression relationship
  between the DV and the covariate (else,
  controlling for the covariate will not be
  useful).
• Eventually will want the common slope.
• Assumption of homogeneity of slopes should be
  checked first.
• Assumes the regression relationship between the
  DV and covariate is the same (in the population)
  in each group.
• Final Ancova Source Table The significance of
  the covariate in the model tells whether is was
  helpful as a control in accounting for variance.
• EMMeans gives adjusted means on DV (adjusted for
  the influence of the covariate). Should be
  reported.
• How are the means adjusted?
• M(j) M(j) b (Cm(j) CGM)

7
Find in Outputs-For Results

• HOCV analysis includes an IVCov interaction term
• If this interaction is NS, then the assumption is
  met, and you can proceed with the ANCOVA.
• Where in output? -Source Table
• Green reports the F, MSE, p, partial eta-squared
  for this
• Ancova analysis source table and additional
  output. Find the following
• In Source Table Ancova F, etc. for the IV (Group
  in this example)
• In Parameter Estimates Table Common slope (B)
  for the covariate (predays in this ex.)
• Estimated marginal means and pairwise comparisons
  adjusted for the covariate

8
AnCovGL27D1.sps
data list free/group predays days. Begin
data 1 0 12 1 10 8 1 5 14 1 6 9 1 10 13 1 0 0 1 12
15 1 13 15 1 6 10 1 19 20 2 14 12 2 16 13 2 5 8 2
12 10 2 0 0 2 8 4 2 12 9 2 5 10 2 19 10 2 14 8 3
0 6 3 10 3 3 11 5 3 15 9 3 6 0 3 12 8 3 9 7 3 13 7
3 0 6 3 8 13 End data. EXAMINE
VARIABLESpredays BY group /PLOTBOXPLOT/STATISTIC
SNONE/NOTOTAL. EXAMINE VARIABLESdays BY group
/PLOTBOXPLOT/STATISTICSNONE/NOTOTAL. Anova
on the covariate. Unianova predays by group
/print descriptives /design group. Split
file produces analyses for each subgroup
separately. SORT CASES BY group . SPLIT FILE
LAYERED BY group . GRAPH /SCATTERPLOT(BIVAR)pre
days WITH days /MISSINGLISTWISE . Split file
off. Test for homogeneity of covariance
assumption. UNIANOVA days BY group WITH
predays /METHOD SSTYPE(3) /INTERCEPT
INCLUDE /CRITERIA ALPHA(.05) /DESIGN
grouppredays group predays . Assumption of
HOCOV is tenable - met, proceed with
ANCOVA. UNIANOVA days BY group WITH predays
/METHOD SSTYPE(3) /INTERCEPT INCLUDE
/EMMEANS TABLES(group) WITH(predaysMEAN)
COMPARE ADJ(SIDAK) /PRINT ETASQ HOMOGENEITY
Parameter /CRITERIA ALPHA(.05) /DESIGN
predays group . Replicate Graph in Green, p
219. GRAPH /SCATTERPLOT(BIVAR)predays WITH
days BY group /MISSINGLISTWISE .
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DATA ViCov INPUT group predays
days DATALINES 1 0 12 1 10 8 1 5 14 1 6 9 1 10 1
3 1 0 0 1 12 15 1 13 15 1 6 10 1 19 20 2 14 12 2 1
6 13 2 5 8 2 12 10 2 0 0 2 8 4 2 12 9 2 5 10 2 19
10 2 14 8 3 0 6 3 10 3 3 11 5 3 15 9 3 6 0 3 12 8
3 9 7 3 13 7 3 0 6 3 8 13 Proc print data
ViCov (obs5) Run Proc boxplot data
ViCov plot predaysgroup plot
daysgroup run Proc Univariate data ViCov
normal plot / May be more than you
want./ var days class group run Visual
Examination of homogeneity of slopes - reg line
for each group. Better sort group Proc sort
dataViCov by group run Symbol Valuedot IR
/Sm / Add SMS, where the last S is to sort
the x values./ Proc Gplot Data ViCov by
group Plot dayspredays / regeqn Run test
for homogeneity of slopes Proc GLM data ViCov
/Green L27 vit c days./ class group model
days group predays predaysgroup / ss3
means group / hovtest lsd tukey
regwq run quit Homogeneity of slopes
assumption is tenable. Proceed with Ancova. Proc
GLM data ViCov /Green L27 vit c
days./ class group model days group
predays / ss3 solution lsmeans group /
pdiff adjustsidak run quit
10

• See Greens APA Results Section, p. 220. Final
  write-up should include a statement about the
  significance of the covariate and the value of
  the common slope something like this. This
  sentence should be added to the end of the first
  paragraph of the model Results section, p. 220.
• The covariate, predays (the number of days with
  cold symptoms the first year) showed a
  significant linear regression relationship with
  the DV, days, F(1,26)14.53, p .001, b 0.457.

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One-Way Repeated Measures Anova

• Extension of the dependent t test to more than
  two occasions of measurement.
• Experimental units (usually subjects) are
  measured three or more times.
• The main H0 is that the means of the trials are
  all equal none are different
• M1 M2 M4
• Added assumption is sphericity, sometimes called
• Compound symmetry of the variance-covariance
  matrix
• Amounts to homogeneity of variances and
  covariances of measures
• (Part of this is that the correlations between
  the levels is the same in the population.)
• Green calls this homogeneity of variances of
  differences
• Where the differences between levels are
  subtracted 1-2, 2-3, and so on.
• If assumption is violated, we can
• Apply a correction (GG or HF) involving a df
  adjustment, or
• Use a multivariate test that does not require the
  assumption.

12
GL29 ExampleDoes Desire to Express Worry Change
over Time

• Longitudinal Men take the DEW scale at 0, 5, 10,
  and 15 years of marriage.
• What is happening over time?
• One within-subjects factor
• Within-subjects designs have high power because
  each subject serves as its own control.

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One factor Repeated measures within-subjects. Pr
oc GLM data DEW /Green L29 example DEW over
time./ Title "One-factor within-subjects 4
class group" model time1--time4 / NOUNI
repeated time 4 (0 5 10 15) run quit Proc
GLM data DEW /Green L29 example DEW over
time./ Title "One-factor within-subjects 4
class group" model time1--time4 / NOUNI
/All the contrasts you want. / repeated time
4 contrast(1)/NOM summary repeated time 4
contrast(2)/NOM summary repeated time 4
contrast(3)/NOM summary run quit
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The GLM Procedure
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for
Within Subject Effects

Adj Pr gt F Source
DF Type III SS Mean Square F Value Pr gt
F G - G H - F time
3 310.733333 103.577778 7.66
0.0001 0.0009 0.0006 Error(time)
87 1175.766667 13.514559
Greenhouse-Geisser Epsilon
0.6979 Huynh-Feldt
Epsilon 0.7534
The GLM Procedure
Repeated Measures Analysis of
Variance Analysis of
Variance of Contrast Variables time_N represents
the contrast between the nth level of time and
the 1st Contrast Variable time_2
Source DF Type III SS
Mean Square F Value Pr gt F Mean
1 4.0333333
4.0333333 0.17 0.6859 Error
29 700.9666667
24.1712644 Contrast Variable time_3
Source DF Type III SS
Mean Square F Value Pr gt F Mean
1 218.7000000
218.7000000 8.41 0.0071 Error
29 754.3000000
26.0103448 Contrast Variable time_4
Source DF Type III SS
Mean Square F Value Pr gt F Mean
1 448.533333
448.533333 9.01 0.0055 Error
29 1443.466667
49.774713
No significant drop from 0 to 5 years.
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EXAMINE VARIABLEStime1 time2 time3 time4
/COMPARE VARIABLE/PLOTBOXPLOT
/STATISTICSNONE/NOTOTAL /MISSINGLISTWISE
. GLM time1 time2 time3 time4 /WSFACTOR
time 4 Polynomial /METHOD SSTYPE(3) /PLOT
PROFILE( time ) /EMMEANS TABLES(time) COMPARE
ADJ(LSD) /PRINT DESCRIPTIVE ETASQ /CRITERIA
ALPHA(.05) /WSDESIGN time .
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Spss Output
Assumption violation is mild corrections have
little effect
17

• Green has sample results based on Wilks Lambda.
  Standard univariate F with H-F p value would be
  just as good. These results suggest that men are
  more eager to express worry to their wives early
  in their marriage, and this desire decreases
  after 5 years of marriage.

18
Two-Factor Designs with a Repeated Measure on One
Factor

• Green, L27 Revisited
• Vitamin C example simplified to a 2 x (2)
• Effect of two levels of vitamin C dosage
  (placebo, low dose) on days of cold symptoms year
  2, with pretest data for days of cold symptoms
  during year 1.
• Treatment introduced at start of year 2.
• Note This design may be analyzed three ways
  Independent t test on pre-post change scores
  two-factor repeated measures and Ancova treating
  the pretest as the covariate.
• J\PSYCH\MARTY\4123\GreenSalkind5Dat\Lesson
  27\Lesson 27 Data File 1.sav

19
Effects in Source Table

• Between Subjects
• Groups
• Error1
• Within Subjects
• Prepost
• Prepost x Groups
• Error2

20
GET FILE'\\JADE\PSYCHOLOGY\COMMON\PSYCH\MARTY\
4123\GreenSalkind5Dat\Lesson 2' '7\Lesson 27
Data File 1.sav'. Title '2-Factor Design with
Repeated Measures on One Factor -Comparative
Analyses'. Note Data for group 3 hi vitamin C
were dropped. data list free/group predays
days. Begin data 1 0 12 1 10 8 1 5 14 1 6 9 1 10 1
3 1 0 0 1 12 15 1 13 15 1 6 10 1 19 20 2 14 12 2 1
6 13 2 5 8 2 12 10 2 0 0 2 8 4 2 12 9 2 5 10 2 19
10 2 14 8 End data. GLM predays days BY group
/WSFACTOR prepost 2 Polynomial /METHOD
SSTYPE(3) /PLOT PROFILE( prepostgroup )
/EMMEANS TABLES(group) /EMMEANS
TABLES(prepost) /EMMEANS TABLES(groupprepost)
/PRINT ETASQ HOMOGENEITY /CRITERIA
ALPHA(.05) /WSDESIGN prepost /DESIGN
group . Get Change from pre to post indicating
reduction in symptom days. Compute SymDif
predays - days. Execute. Run t test on
difference scores. T-TEST GROUPS group(1 2)
/MISSING ANALYSIS /VARIABLES SymDif
/CRITERIA CI(.95) . Compare t value with
interaction term in previous analysis. UNIANOVA
days BY group WITH predays /METHOD
SSTYPE(3) /INTERCEPT INCLUDE /EMMEANS
TABLES(group) WITH(predaysMEAN) /PRINT ETASQ
PARAMETER HOMOGENEITY /CRITERIA ALPHA(.05)
/DESIGN predays group .
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DATA ViCov INPUT group predays
days DATALINES 1 0 12 1 10 8 1 5 14 1 6 9 1 10 1
3 1 0 0 1 12 15 1 13 15 1 6 10 1 19 20 2 14 12 2 1
6 13 2 5 8 2 12 10 2 0 0 2 8 4 2 12 9 2 5 10 2 19
10 2 14 8 Proc print data ViCov
(obs5) Run Proc boxplot data ViCov plot
predaysgroup plot daysgroup run Proc
Univariate data ViCov normal plot / May be
more than you want./ var days class
group run Visual Examination of homogeneity
of slopes - reg line for each group. Better
sort group Proc sort dataViCov by
group run Symbol Valuedot IR /Sm / Add
SMS, where the last S is to sort the x
values./ Proc Gplot Data ViCov by
group Plot dayspredays / regeqn Run test
for homogeneity of slopes Proc GLM data
ViCov /Green L27 vit c days./ class
group model days group predays
predaysgroup / ss3 means group / hovtest
lsd tukey regwq run quit Here's the 2 x (2)
with predays as premeasure. Proc GLM data
ViCov /Green L27 vit c days group 3
dropped./ Title1 "Two-way Anova with a Repeated
Measure on One Factor" class group model
predays days group / NOUNI ss3 Solution
Repeated Prepost 2 (1 2) means
group lsmeans group / pdiff adjustsidak run q
uit Homogeneity of slopes assumption is
tenable. Proceed with Ancova for
comparison. Proc GLM data ViCov /Green L27
vit c days./ class group model days group
predays / ss3 Solution lsmeans group / pdiff
adjustsidak run quit