Post on 13-Dec-2015
One-Way ANOVA
Two ways to run a one-way ANOVA1. Analyze Compare Means One-Way
ANOVA Use if you have multiple DV’s, but only one IV
2. Analyze General Linear Model Univariate
Use if you have only one DV bc/ can provide effect size statistics
More on this later (factorial ANOVA section)
Method #1: Compare Means
First we have to test if we meet the assumptions of ANOVA: Independence of Observations
Cannot be tested statistically, is determined by research methodology only
Normally Distributed DataShapiro-Wilk’s W statistic, if significant, indicates
significant non-normality in dataAnalyze Descriptive Statistics Explore
Click on “Plots”, make sure “Normality Plots w/Tests” is checked
Testing Assumptions
Tests of Normality
.511 111 .000 .371 111 .000
.386 86 .000 .685 86 .000
PESSGRPOptimistic
Pessimistic
BDI214Statistic df Sig. Statistic df Sig.
Kolmogorov-Smirnova
Shapiro-Wilk
Lilliefors Significance Correctiona.
Testing Assumptions
Homogeneity of Variances (Homoscedasticity)
Tested at the same time you test ANOVAAnalyze Compare Means One-Way
ANOVA Click on “Options” and make sure “Homogeneity of
variance test” is checked If violated, use Brown-Forsythe or Welch statistics,
which do not assume homoscedasticity
Method #1: Compare Means
One-Way ANOVA Analyze Compare Means One-Way ANOVA “Dependent List” = DV’s; “Factor” = IV Options
Descriptive Fixed and random effects Homogeneity of variance test
Levene’s Test: Significant result Non-homogenous variances
Brown-Forsythe Welch Means plot
Method #1: Compare Means
Descriptives
BDI2TOT
201 11.46 9.921 .700 10.08 12.84 0 49
25 8.88 8.472 1.694 5.38 12.38 0 28
2 8.00 7.071 5.000 -55.53 71.53 3 13
2 4.50 3.536 2.500 -27.27 36.27 2 7
5 19.20 10.232 4.576 6.49 31.91 10 36
235 11.26 9.799 .639 10.00 12.52 0 49
9.756 .636 10.01 12.51
1.682 6.59 15.93 3.262
Caucasian
African American
Asian American
Hispanic
Other
Total
Fixed Effects
Random Effects
Model
N Mean Std. Deviation Std. Error Lower Bound Upper Bound
95% Confidence Interval forMean
Minimum Maximum
Between-Component
Variance
Test of Homogeneity of Variances
BDI2TOT
.403 4 230 .806
LeveneStatistic df1 df2 Sig.
Robust Tests of Equality of Means
BDI2TOT
1.992 4 3.765 .268
2.378 4 10.901 .116
Welch
Brown-Forsythe
Statistica
df1 df2 Sig.
Asymptotically F distributed.a.
Method #1: Compare Means
RACE
OtherHispanicAsian AmericanAfrican AmericanCaucasian
Me
an
of
BD
I2T
OT
30
20
10
0
Method #1: Compare Means
One-Way ANOVAPost-Hoc
Can only be done if your IV has 3+ levels Pointless if only 2 levels, just look @ the means
Click the test you want, either with equal variances assumed or not assumed
DON’T just click all of them and see which one gives what you want (that’s cheating), select the test you want priori
Method #1: Compare Means
ContrastsClick “Polynomial”, Leave “Degree” at default
(“Linear”)
Enter in your coefficients # of coefficients should equal # of levels of your IV
Doesn’t count missing cells, so if you have 3 levels, but no one in one of the levels, you should have 2 coefficients
Coefficients need to sum to 0
Method #1: Compare Means
ContrastsEnter in your coefficients
IV = Race – 1=Caucasian, 2=African American, 3=Asian American, 4=Hispanic, 5=Native American, 6=Other, BUT there were no Native Americans in the sample
If you want to compare Caucasians to “Other”, coefficients = 1, 0, 0, 0, -1
Caucasians vs. everyone else = -1, .25, .25, .25, .25
Method #1: Compare Means
ANOVA
BDI2TOT
577.335 4 144.334 1.517 .198
45.416 1 45.416 .477 .490
531.919 3 177.306 1.863 .137
21889.831 230 95.173
22467.166 234
(Combined)
Weighted
Deviation
Linear Term
BetweenGroups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
Contrast Coefficients
-1 0 0 0 1Contrast1
CaucasianAfrican
AmericanAsian
American Hispanic Other
RACE
Contrast Tests
7.74 4.417 1.753 230 .081
7.74 4.629 1.672 4.189 .167
Contrast1
1
Assume equal variances
Does not assume equalvariances
BDI2TOT
Value ofContrast Std. Error t df Sig. (2-tailed)
Method #2: Univariate
Univariate works for both one-way (1 IV) and factorial ANOVA’s (2+ IV’s)
Allows for specification of both fixed and random factors (IV’s)
Assumptions Independence of ObservationsNormally Distributed Data
Both same as one-way ANOVA
Factorial ANOVA
Assumptions:Homoscedasticity
Tested at the same time you test ANOVAClick on Analyze General Linear Model
Univariate Click on “Options” and make sure “Homogeneity tests”
is checked
Factorial ANOVA
Options Estimated Marginal Means
Displays means, SD’s, & CI’s for each level of each IV selected
If “Compare main effects” is checked, works as one-way ANOVA on each IV selected
“Confidence interval adjustments” allows you to correct for inflation of alpha using Bonferroni or Sidak method
Descriptive statistics Estimates of effect size Observed power
Pointless, adds nothing to interpretation of p-value and e.s. Homogeneity tests
Levene’s test
Factorial ANOVA
Between-Subjects Factors
Female 135
Male 98
Caucasian 199
AfricanAmerican
25
AsianAmerican
2
Hispanic 2
Other 5
0
1
gender
1
2
3
4
6
race
Value Label N Descriptive Statistics
Dependent Variable: genbad
3.9771 1.05540 112
3.9417 .96621 20
4.6667 . 1
4.1250 1.82669 2
3.9792 1.04116 135
4.2400 .99602 87
3.1833 1.16726 5
3.0000 . 1
4.5833 .11785 2
4.8611 .48829 3
4.1995 1.01337 98
4.0921 1.03558 199
3.7900 1.03053 25
3.8333 1.17851 2
4.5833 .11785 2
4.5667 1.05640 5
4.0718 1.03312 233
raceCaucasian
African American
Asian American
Other
Total
Caucasian
African American
Asian American
Hispanic
Other
Total
Caucasian
African American
Asian American
Hispanic
Other
Total
genderFemale
Male
Total
Mean Std. Deviation N
Levene's Test of Equality of Error Variancesa
Dependent Variable: genbad
1.117 8 224 .353F df1 df2 Sig.
Tests the null hypothesis that the error variance of thedependent variable is equal across groups.
Design: Intercept+gender+race+gender * racea.
Pairwise Comparisons
Dependent Variable: genbad
.546 .267 .419 -.211 1.303
.275 .729 1.000 -1.793 2.343
-.475b .729 1.000 -2.543 1.593
-.384 .474 1.000 -1.729 .960
-.546 .267 .419 -1.303 .211
-.271 .770 1.000 -2.453 1.912
-1.021b .770 1.000 -3.203 1.162
-.931 .534 .829 -2.445 .584
-.275 .729 1.000 -2.343 1.793
.271 .770 1.000 -1.912 2.453
-.750b 1.026 1.000 -3.660 2.160
-.660 .864 1.000 -3.109 1.789
.475c .729 1.000 -1.593 2.543
1.021c .770 1.000 -1.162 3.203
.750c 1.026 1.000 -2.160 3.660
.090c .864 1.000 -2.359 2.539
.384 .474 1.000 -.960 1.729
.931 .534 .829 -.584 2.445
.660 .864 1.000 -1.789 3.109
-.090b .864 1.000 -2.539 2.359
(J) raceAfrican American
Asian American
Hispanic
Other
Caucasian
Asian American
Hispanic
Other
Caucasian
African American
Hispanic
Other
Caucasian
African American
Asian American
Other
Caucasian
African American
Asian American
Hispanic
(I) raceCaucasian
African American
Asian American
Hispanic
Other
MeanDifference
(I-J) Std. Error Sig.a
Lower Bound Upper Bound
95% Confidence Interval forDifference
a
Based on estimated marginal means
Adjustment for multiple comparisons: Bonferroni.a.
An estimate of the modified population marginal mean (J).b.
An estimate of the modified population marginal mean (I).c.
Estimates
Dependent Variable: genbad
4.109 .073 3.964 4.253
3.563 .257 3.057 4.068
3.833 .726 2.403 5.264
4.583a .726 3.153 6.014
4.493 .468 3.570 5.416
raceCaucasian
African American
Asian American
Hispanic
Other
Mean Std. Error Lower Bound Upper Bound
95% Confidence Interval
Based on modified population marginal mean.a.
Univariate Tests
Dependent Variable: genbad
5.892 4 1.473 1.398 .235 .024 5.593 .432
235.971 224 1.053
Contrast
Error
Sum ofSquares df Mean Square F Sig.
Partial EtaSquared
Noncent.Parameter
ObservedPower
a
The F tests the effect of race. This test is based on the linearly independent pairwise comparisons among the estimatedmarginal means.
Computed using alpha = .05a.
Factorial ANOVA
SaveDon’t worry about this for now
Post HocSelect the IV for which you wish to compare all
levels against all other levels (i.e. that you don’t plan to do planned comparisons on)
Click on the right arrow button so the IV is in the box labeled “Post Hoc Tests for”
Check the post hoc tests you want done, either with equal variances assumed or not assumed
Click “Continue”
Multiple Comparisons
Dependent Variable: genbad
.3021 .21779 .637 -.2969 .9010
.2587 .72939 .997 -1.7472 2.2646
-.4913 .72939 .962 -2.4972 1.5146
-.4746 .46474 .845 -1.7527 .8035
-.3021 .21779 .637 -.9010 .2969
-.0433 .75423 1.000 -2.1176 2.0309
-.7933 .75423 .831 -2.8676 1.2809
-.7767 .50282 .535 -2.1595 .6061
-.2587 .72939 .997 -2.2646 1.7472
.0433 .75423 1.000 -2.0309 2.1176
-.7500 1.02637 .949 -3.5727 2.0727
-.7333 .85873 .913 -3.0949 1.6283
.4913 .72939 .962 -1.5146 2.4972
.7933 .75423 .831 -1.2809 2.8676
.7500 1.02637 .949 -2.0727 3.5727
.0167 .85873 1.000 -2.3449 2.3783
.4746 .46474 .845 -.8035 1.7527
.7767 .50282 .535 -.6061 2.1595
.7333 .85873 .913 -1.6283 3.0949
-.0167 .85873 1.000 -2.3783 2.3449
.3021 .21779 .839 -.3137 .9179
.2587 .72939 1.000 -1.8036 2.3211
-.4913 .72939 .999 -2.5536 1.5711
-.4746 .46474 .975 -1.7887 .8394
-.3021 .21779 .839 -.9179 .3137
-.0433 .75423 1.000 -2.1759 2.0892
-.7933 .75423 .969 -2.9259 1.3392
-.7767 .50282 .733 -2.1984 .6450
-.2587 .72939 1.000 -2.3211 1.8036
.0433 .75423 1.000 -2.0892 2.1759
-.7500 1.02637 .998 -3.6521 2.1521
-.7333 .85873 .993 -3.1614 1.6947
.4913 .72939 .999 -1.5711 2.5536
.7933 .75423 .969 -1.3392 2.9259
.7500 1.02637 .998 -2.1521 3.6521
.0167 .85873 1.000 -2.4114 2.4447
.4746 .46474 .975 -.8394 1.7887
.7767 .50282 .733 -.6450 2.1984
.7333 .85873 .993 -1.6947 3.1614
-.0167 .85873 1.000 -2.4447 2.4114
(J) raceAfrican American
Asian American
Hispanic
Other
Caucasian
Asian American
Hispanic
Other
Caucasian
African American
Hispanic
Other
Caucasian
African American
Asian American
Other
Caucasian
African American
Asian American
Hispanic
African American
Asian American
Hispanic
Other
Caucasian
Asian American
Hispanic
Other
Caucasian
African American
Hispanic
Other
Caucasian
African American
Asian American
Other
Caucasian
African American
Asian American
Hispanic
(I) raceCaucasian
African American
Asian American
Hispanic
Other
Caucasian
African American
Asian American
Hispanic
Other
Tukey HSD
Sidak
MeanDifference
(I-J) Std. Error Sig. Lower Bound Upper Bound
95% Confidence Interval
Based on observed means.
genbad
25 3.7900
2 3.8333
199 4.0921
5 4.5667
2 4.5833
.809
25 3.7900
2 3.8333
199 4.0921
5 4.5667
2 4.5833
.753
raceAfrican American
Asian American
Caucasian
Other
Hispanic
Sig.
African American
Asian American
Caucasian
Other
Hispanic
Sig.
Tukey HSDa,b,c
Ryan-Einot-Gabriel-Welsch Range
c
N 1
Subset
Means for groups in homogeneous subsets are displayed.Based on Type III Sum of SquaresThe error term is Mean Square(Error) = 1.053.
Uses Harmonic Mean Sample Size = 4.016.a.
The group sizes are unequal. The harmonic mean of thegroup sizes is used. Type I error levels are not guaranteed.
b.
Alpha = .05.c.
Factorial ANOVA
The following graph has the IV “Race” on the horizontal axis and separate lines by the IV “Gender”
Factorial ANOVA
ModelAllows you to:
Denote which main effects and interactions you are interested in testing (default is to test ALL of them)
Specify which type of sum of squares to use
Usually you won’t be tinkering with this
Factorial ANOVA
ContrastsTests all levels within one IVConcern yourself with Simple only for now “Reference category” = What level all others are
compared to (either first or last, with this referring to how they were numbered)
Can test specific levels within one IV with specific levels in another IV, but requires knowledge of syntax
Contrast Results (K Matrix)
-.546
0
-.546
.267
.042
-1.072
-.020
-.275
0
-.275
.729
.706
-1.713
1.162
.107
0
.107
.867
.902
-1.602
1.815
.384
0
.384
.474
.418
-.550
1.319
Contrast Estimate
Hypothesized Value
Difference (Estimate - Hypothesized)
Std. Error
Sig.
Lower Bound
Upper Bound
95% Confidence Intervalfor Difference
Contrast Estimate
Hypothesized Value
Difference (Estimate - Hypothesized)
Std. Error
Sig.
Lower Bound
Upper Bound
95% Confidence Intervalfor Difference
Contrast Estimate
Hypothesized Value
Difference (Estimate - Hypothesized)
Std. Error
Sig.
Lower Bound
Upper Bound
95% Confidence Intervalfor Difference
Contrast Estimate
Hypothesized Value
Difference (Estimate - Hypothesized)
Std. Error
Sig.
Lower Bound
Upper Bound
95% Confidence Intervalfor Difference
race Simple Contrasta
Level 2 vs. Level 1
Level 3 vs. Level 1
Level 4 vs. Level 1
Level 5 vs. Level 1
genbad
Dependent
Variable
Reference category = 1a.
Test Results
Dependent Variable: genbad
5.384 4 1.346 1.278 .280
235.971 224 1.053
SourceContrast
Error
Sum ofSquares df Mean Square F Sig.
Factorial ANOVA
Tests of Between-Subjects Effects
Dependent Variable: genbad
11.653a 8 1.457 1.383 .205
358.168 1 358.168 339.997 .000
.655 1 .655 .622 .431
6.010 4 1.502 1.426 .226
5.935 3 1.978 1.878 .134
235.971 224 1.053
4110.717 233
247.624 232
SourceCorrected Model
Intercept
gender
race
gender * race
Error
Total
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
R Squared = .047 (Adjusted R Squared = .013)a.