repeated measures/mixed-model anova:

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

33.7123.481

94710.222

.000

Box's MFdf1df2Sig.

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 itemsTime 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 53Treatment 45Female 79Male 19

01

group

12

sex

Value Label N

Multivariate Testsb

.976 1877.669a 2.000 93.000 .000

.024 1877.669a 2.000 93.000 .00040.380 1877.669a 2.000 93.000 .00040.380 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 TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy'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 itemsTime 2 Generality=Meanof all ASQ Stability andGlobality scores for badeventsTotal BDI Score=Sum ofall 21 BDI itemsTime 2 Generality=Meanof all ASQ Stability andGlobality scores for badeventsTotal BDI Score=Sum ofall 21 BDI itemsTime 2 Generality=Meanof all ASQ Stability andGlobality scores for badeventsTotal BDI Score=Sum ofall 21 BDI itemsTime 2 Generality=Meanof all ASQ Stability andGlobality scores for badeventsTotal BDI Score=Sum ofall 21 BDI itemsTime 2 Generality=Meanof all ASQ Stability andGlobality scores for badeventsTotal BDI Score=Sum ofall 21 BDI itemsTime 2 Generality=Meanof all ASQ Stability andGlobality scores for badeventsTotal BDI Score=Sum ofall 21 BDI itemsTime 2 Generality=Meanof all ASQ Stability andGlobality scores for badeventsTotal BDI Score=Sum ofall 21 BDI itemsTime 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 46Treatment 44White 80African-American 7

Asian 2Arabic 1

01

group

12

36

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 TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest Root

EffectIntercept

t1gen

t1bidtot

group

ethnicit

group * ethnicit


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.


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


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


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


.533 1 .533 2.453 .121

t1bidtot



.018 1 .018 .084 .773

group



.770 3 .257 1.182 .322

ethnicit



.967 2 .483 2.225 .115

group * ethnicit



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 .31098 98 112 112

Pearson CorrelationSig. (2-tailed)NPearson CorrelationSig. (2-tailed)

N

Pearson CorrelationSig. (2-tailed)N

Pearson CorrelationSig. (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

t1chgent2chgent3chgent4chgen

Time1234

DependentVariable

Between-Subjects Factors

Treatment 28Female 21Male 7

1group12

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 TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest Root

EffectTime

Time * group

Time * sex

Time * group * sex


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 .0233.764 2.162 1.740 3.374 .0383.764 2.456 1.532 3.374 .0323.764 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 .37229.002 56.224 .51629.002 63.861 .45429.002 26.000 1.115

Sphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-bound

SourceTime

Time * group

Time * sex

Time * group * sex

Error(Time)


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 .6606.589 26 .2535.258 26 .202

TimeLinearQuadraticCubicLinearQuadraticCubicLinearQuadraticCubicLinearQuadraticCubicLinearQuadraticCubic

SourceTime

Time * group

Time * sex

Time * group * sex

Error(Time)



Measure: MEASURE_1Transformed 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

SourceInterceptgroupsexgroup * sexError


repeated measures/mixed-model anova:

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