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Repeated Measures/Mixed-Model ANOVA:
SPSS Lab #4
MANOVA Multivariate ANOVA (MANOVA)
Both 2+ IV’s and 2+ DV’s SPSS won’t run with only 1 DV
Click “Analyze” “General Linear Model” “Multivariate…”
Same as “Univariate…” command, but lets you add 2+ DV’s
Multivariable ANOVA = Either 2+ IV’s or 2+ DV’s
Factorial ANOVA = 2+ IV’s
MANOVA Assumptions
Same as one-way and factorial ANOVA Independence of Observations Normality
Use Shapiro-Wilk’s W or z-tests of individual skewness/kurtosis
MANOVA robust to violations of this with larger n’s, unless group sizes are unequal
MANOVA Homoscedasticity
Use Box’s M and Levene’s Test
Box’s M tests for homoscedasticity in all DV’s at one (omnibus test)
MANOVA robust to violations of this unless group sizes are unequal
Correct using appropriate transformation
Box's Test of Equality of Covariance Matricesa
33.712
3.481
9
4710.222
.000
Box's M
F
df1
df2
Sig.
Tests the null hypothesis that the observed covariancematrices of the dependent variables are equal across groups.
Design: Intercept+group+sex+group * sexa.
Levene's Test of Equality of Error Variancesa
2.703 3 94 .050
2.451 3 94 .068
Total BDI Score=Sum ofall 21 BDI items
Time 2 Generality=Meanof all ASQ Stability andGlobality scores for badevents
F df1 df2 Sig.
Tests the null hypothesis that the error variance of the dependent variable isequal across groups.
Design: Intercept+group+sex+group * sexa.
MANOVA Multivariate Omnibus Tests
Univariate omnibus tests Difference somewhere between levels of IV, when
averaging across them Multivariate omnibus tests
Difference somewhere between levels of IV on 1+ DV’s, when averaging across both levels and DV’s
Even more vague than univariate omnibus test Several different tests
Pillai’s Trace most supported in research Wilks’ λ (lambda) most popular
Do you interpret univariate tests without a significant omnibus test?
MANOVABetween-Subjects Factors
Control 53
Treatment 45
Female 79
Male 19
0
1
group
1
2
sex
Value Label N
Multivariate Testsb
.976 1877.669a 2.000 93.000 .000
.024 1877.669a 2.000 93.000 .000
40.380 1877.669a 2.000 93.000 .000
40.380 1877.669a 2.000 93.000 .000
.089 4.525a 2.000 93.000 .013
.911 4.525a 2.000 93.000 .013
.097 4.525a 2.000 93.000 .013
.097 4.525a 2.000 93.000 .013
.004 .179a 2.000 93.000 .836
.996 .179a 2.000 93.000 .836
.004 .179a 2.000 93.000 .836
.004 .179a 2.000 93.000 .836
.010 .481a 2.000 93.000 .620
.990 .481a 2.000 93.000 .620
.010 .481a 2.000 93.000 .620
.010 .481a 2.000 93.000 .620
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
EffectIntercept
group
sex
group * sex
Value F Hypothesis df Error df Sig.
Exact statistica.
Design: Intercept+group+sex+group * sexb.
Tests of Between-Subjects Effects
77.657a
3 25.886 .541 .656
3.424b
3 1.141 3.700 .014
3543.298 1 3543.298 74.027 .000
1170.647 1 1170.647 3795.270 .000
62.816 1 62.816 1.312 .255
2.636 1 2.636 8.547 .004
6.660 1 6.660 .139 .710
.082 1 .082 .267 .606
22.361 1 22.361 .467 .496
.194 1 .194 .629 .430
4499.322 94 47.865
28.994 94 .308
10690.000 98
1978.369 98
4576.980 97
32.418 97
Dependent VariableTotal BDI Score=Sum ofall 21 BDI items
Time 2 Generality=Meanof all ASQ Stability andGlobality scores for badevents
Total BDI Score=Sum ofall 21 BDI items
Time 2 Generality=Meanof all ASQ Stability andGlobality scores for badevents
Total BDI Score=Sum ofall 21 BDI items
Time 2 Generality=Meanof all ASQ Stability andGlobality scores for badevents
Total BDI Score=Sum ofall 21 BDI items
Time 2 Generality=Meanof all ASQ Stability andGlobality scores for badevents
Total BDI Score=Sum ofall 21 BDI items
Time 2 Generality=Meanof all ASQ Stability andGlobality scores for badevents
Total BDI Score=Sum ofall 21 BDI items
Time 2 Generality=Meanof all ASQ Stability andGlobality scores for badevents
Total BDI Score=Sum ofall 21 BDI items
Time 2 Generality=Meanof all ASQ Stability andGlobality scores for badevents
Total BDI Score=Sum ofall 21 BDI items
Time 2 Generality=Meanof all ASQ Stability andGlobality scores for badevents
SourceCorrected Model
Intercept
group
sex
group * sex
Error
Total
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
R Squared = .017 (Adjusted R Squared = -.014)a.
R Squared = .106 (Adjusted R Squared = .077)b.
MANOVA Follow-up inspection of univariate tests
with multiple comparison procedures Just like with “Univariate…” command
Analysis of Covariance (ANCOVA) Same as ANOVA, but allows removal of
variance attributable to a covariate Used frequently if group differences are
found on some IV IV = treatment, Levels = treatment and control
groups Ideally, both groups differ ONLY on presence
of treatment If differ on something else, mean differences may be
due to that instead of treatment
ANCOVA IV = treatment, Levels = treatment and control
groups Ideally, both groups differ ONLY on presence
of treatment If differ on something else (i.e. gender ratio), mean
differences may be due to that instead of treatment Use “something else” as covariate to remove the
effects of that variable
ANCOVA Use same Analyze General Linear Model
Univariate… (if only 1 DV) or Multivariate… (if 2+ DV’s) commands
Specify a “Covariate”
Between-Subjects Factors
Control 46
Treatment 44
White 80
African-American
7
Asian 2
Arabic 1
0
1
group
1
2
3
6
Ethnicity
Value Label N
Multivariate Testsc
.075 3.257a 2.000 80.000 .044
.925 3.257a 2.000 80.000 .044
.081 3.257a 2.000 80.000 .044
.081 3.257a 2.000 80.000 .044
.276 15.221a 2.000 80.000 .000
.724 15.221a 2.000 80.000 .000
.381 15.221a 2.000 80.000 .000
.381 15.221a 2.000 80.000 .000
.431 30.270a 2.000 80.000 .000
.569 30.270a 2.000 80.000 .000
.757 30.270a 2.000 80.000 .000
.757 30.270a 2.000 80.000 .000
.055 2.336a 2.000 80.000 .103
.945 2.336a 2.000 80.000 .103
.058 2.336a 2.000 80.000 .103
.058 2.336a 2.000 80.000 .103
.127 1.824 6.000 162.000 .097
.874 1.854a 6.000 160.000 .092
.143 1.883 6.000 158.000 .087
.136 3.685b 3.000 81.000 .015
.105 2.250 4.000 162.000 .066
.897 2.238a 4.000 160.000 .067
.113 2.225 4.000 158.000 .069
.085 3.428b 2.000 81.000 .037
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
EffectIntercept
t1gen
t1bidtot
group
ethnicit
group * ethnicit
Value F Hypothesis df Error df Sig.
Exact statistica.
The statistic is an upper bound on F that yields a lower bound on the significance level.b.
Design: Intercept+t1gen+t1bidtot+group+ethnicit+group * ethnicitc.
Tests of Between-Subjects Effects
Source Dependent Variable Type III Sum of Squares df Mean Square F Sig.
Time 2 Generality=Mean of all ASQ Stability and Globality scores for bad events
13.415(a) 8 1.677 7.717 .000
Corrected Model
Total BDI Score=Sum of all 21 BDI items 2166.539(b) 8 270.817 11.319 .000
Time 2 Generality=Mean of all ASQ Stability and Globality scores for bad events
1.395 1 1.395 6.420 .013
Intercept
Total BDI Score=Sum of all 21 BDI items 4.610 1 4.610 .193 .662
Time 2 Generality=Mean of all ASQ Stability and Globality scores for bad events
6.451 1 6.451 29.689 .000
t1gen
Total BDI Score=Sum of all 21 BDI items 25.021 1 25.021 1.046 .310
Time 2 Generality=Mean of all ASQ Stability and Globality scores for bad events
.533 1 .533 2.453 .121
t1bidtot
Total BDI Score=Sum of all 21 BDI items 1403.388 1 1403.388 58.653 .000
Time 2 Generality=Mean of all ASQ Stability and Globality scores for bad events
.018 1 .018 .084 .773
group
Total BDI Score=Sum of all 21 BDI items 111.417 1 111.417 4.657 .034
Time 2 Generality=Mean of all ASQ Stability and Globality scores for bad events
.770 3 .257 1.182 .322
ethnicit
Total BDI Score=Sum of all 21 BDI items 190.597 3 63.532 2.655 .054
Time 2 Generality=Mean of all ASQ Stability and Globality scores for bad events
.967 2 .483 2.225 .115
group * ethnicit
Total BDI Score=Sum of all 21 BDI items 112.689 2 56.345 2.355 .101
Time 2 Generality=Mean of all ASQ Stability and Globality scores for bad events
17.600 81 .217
Error
Total BDI Score=Sum of all 21 BDI items 1938.084 81 23.927
ANCOVA Assumptions
Independence of Observations Normality Homoscedasticity
Same as (M)ANOVA
ANCOVA Assumptions
Relationship between covariate and DV Analyze Correlate Bivariate Click covariate(s) and DV(s) into right box
Correlations
1 .160 .692** .131
.116 .000 .198
98 98 98 98
.160 1 .098 .507**
.116 .335 .000
98 98 98 98
.692** .098 1 .097
.000 .335 .310
98 98 112 112
.131 .507** .097 1
.198 .000 .310
98 98 112 112
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Total BDI Score=Sum ofall 21 BDI items
Time 2 Generality=Meanof all ASQ Stability andGlobality scores for badevents
Total BDI Score=Sum ofall 21 BDI items
Time 1 Generality=Meanof all ASQ Stability andGlobality scores for badevents
Total BDIScore=Sumof all 21 BDI
items
Time 2Generality=Mean of all
ASQ Stabilityand Globalityscores for bad
events
Total BDIScore=Sumof all 21 BDI
items
Time 1Generality=Mean of all
ASQ Stabilityand Globalityscores for bad
events
Correlation is significant at the 0.01 level (2-tailed).**.
ANCOVA Assumptions
Relationship between covariate and DV If no significant relationship is found, don’t use
covariate If multiple covariates are used, run 2 separate
ANCOVA’s with related covariates and DV’s together Relationship between IV and covariate is equal
across levels of IV If covariate x IV interaction is significant, than this
assumption in violated If violated, don’t use covariate
ANCOVA Assumptions
Relationship between IV and covariate is linear Examine best-fit line in scatterplots of DV and covariate
within levels of IV
Repeated-Measures/Mixed-Model ANOVA Repeated-Measures/Mixed-Model ANOVA
Click “Analyze” “General Linear Model” “Repeated Measures…”
“Within-Subject Factor” = IV for which same participants are included in all levels
I.e. IV = Time, Levels = Time 1, Time 2, etc. Click “Add”, after all within-subjects factors are
added click “Define” Multivariate tests
Same as MANOVA
Repeated-Measures/Mixed-Model ANOVA
Within-Subjects Factors
Measure: MEASURE_1
t1chgen
t2chgen
t3chgen
t4chgen
Time1
2
3
4
DependentVariable
Between-Subjects Factors
Treatment 28
Female 21
Male 7
1group
1
2
sex
Value Label N
Multivariate Testsb
.214 2.173a 3.000 24.000 .117
.786 2.173a 3.000 24.000 .117
.272 2.173a 3.000 24.000 .117
.272 2.173a 3.000 24.000 .117
.000 .a .000 .000 .
1.000 .a .000 25.000 .
.000 .a .000 2.000 .
.000 .000a 3.000 23.000 1.000
.045 .378a 3.000 24.000 .769
.955 .378a 3.000 24.000 .769
.047 .378a 3.000 24.000 .769
.047 .378a 3.000 24.000 .769
.000 .a .000 .000 .
1.000 .a .000 25.000 .
.000 .a .000 2.000 .
.000 .000a 3.000 23.000 1.000
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
EffectTime
Time * group
Time * sex
Time * group * sex
Value F Hypothesis df Error df Sig.
Exact statistica.
Design: Intercept+group+sex+group * sex Within Subjects Design: Time
b.
Repeated-Measures/Mixed-Model ANOVA Mauchly’s W
Tests for sphericity or multivariate homogeneity of variances assumption
If significant, indicates violations of sphericity However, very dependent on sample size – With few
subjects, fails to detect violations (Type II Error) and with too many subjects detects violations too often (Type I Error)
Tests of Within-Subjects Effects
Measure: MEASURE_1
3.764 3 1.255 3.374 .023
3.764 2.162 1.740 3.374 .038
3.764 2.456 1.532 3.374 .032
3.764 1.000 3.764 3.374 .078
.000 0 . . .
.000 .000 . . .
.000 .000 . . .
.000 .000 . . .
.550 3 .183 .493 .688
.550 2.162 .255 .493 .628
.550 2.456 .224 .493 .651
.550 1.000 .550 .493 .489
.000 0 . . .
.000 .000 . . .
.000 .000 . . .
.000 .000 . . .
29.002 78 .372
29.002 56.224 .516
29.002 63.861 .454
29.002 26.000 1.115
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
SourceTime
Time * group
Time * sex
Time * group * sex
Error(Time)
Type III Sumof Squares df Mean Square F Sig.
Mauchly's Test of Sphericityb
Measure: MEASURE_1
.573 13.772 5 .017 .721 .819 .333Within Subjects EffectTime
Mauchly's WApprox.
Chi-Square df Sig.Greenhouse-Geisser Huynh-Feldt Lower-bound
Epsilona
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables isproportional to an identity matrix.
May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed inthe Tests of Within-Subjects Effects table.
a.
Design: Intercept+group+sex+group * sex Within Subjects Design: Time
b.
Tests of Within-Subjects Contrasts
Measure: MEASURE_1
3.066 1 3.066 4.646 .041
.610 1 .610 2.408 .133
.088 1 .088 .433 .516
.000 0 . . .
.000 0 . . .
.000 0 . . .
.287 1 .287 .435 .515
.252 1 .252 .993 .328
.012 1 .012 .058 .811
.000 0 . . .
.000 0 . . .
.000 0 . . .
17.155 26 .660
6.589 26 .253
5.258 26 .202
TimeLinear
Quadratic
Cubic
Linear
Quadratic
Cubic
Linear
Quadratic
Cubic
Linear
Quadratic
Cubic
Linear
Quadratic
Cubic
SourceTime
Time * group
Time * sex
Time * group * sex
Error(Time)
Type III Sumof Squares df Mean Square F Sig.
Tests of Between-Subjects Effects
Measure: MEASURE_1
Transformed Variable: Average
1558.970 1 1558.970 442.725 .000
.000 0 . . .
1.215 1 1.215 .345 .562
.000 0 . . .
91.554 26 3.521
SourceIntercept
group
sex
group * sex
Error
Type III Sumof Squares df Mean Square F Sig.