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ANOVA
Independent ANOVA
• Scores vary – why?
• Total variability can be divided up into 2
parts
• 1) Between treatments
• 2) Within treatments
Between design
Total variability
Between treatments
Treatment effect
Individual differences
Experimental error
Within treatments
Individual differences
Experimental error
• F = variance between treatments
variance within
F = treatment + indiv diffs + error
indiv diffs + error
INDEPENDENT ANOVA
assumptions
1) the observations within each
sample must be independent
2) the populations from which the
samples were selected must be
normal
3) the populations from which the
samples are selected must have
equal variances (homogeneity of
variance)
Same assumptions for t test
• F measures variance which is standard
deviation squared
• F = (differences between sample)2
(differences expected by chance)2
F = t2
Figure 1 Mean calories in hotdogs
Between-Subjects Factors
N
hotdog
Beef
Meat
Poultry
20
17
17
Descriptive Statistics
Dependent Variable: calories
hotdog
Beef
Meat
Poultry
Total
Mean
156.8500
158.7059
118.7647
145.4444
Std. Deviation
22.64201
25.23580
22.55141
29.38339
N
20
17
17
54
Tests of Between-Subjects Effects
Dependent Variable: calories
Type III Sum
Source
of Squares
Corrected Model
17692.195 a
Intercept
1125187.490
hotdog
17692.195
Error
28067.138
Total
1188080.000
Corrected Total
45759.333
df
2
1
2
51
54
53
Mean Square
8846.098
1125187.490
8846.098
550.336
a. R Squared = .387 (Adjusted R Squared = .363)
F
16.074
2044.546
16.074
Sig.
.000
.000
.000
Post hoc test
Multiple Comparisons
Dependent Variable: calories
LSD
(I) hotdog
Beef
Meat
Poultry
(J) hotdog
Meat
Poultry
Beef
Poultry
Beef
Meat
Mean
Difference
(I-J)
-1.8559
38.0853*
1.8559
39.9412*
-38.0853*
-39.9412*
Std. Error
7.73883
7.73883
7.73883
8.04645
7.73883
8.04645
Bas ed on observed means .
*. The mean difference is significant at the .05 level.
Sig.
.811
.000
.811
.000
.000
.000
95% Confidence Interval
Lower Bound Upper Bound
-17.3922
13.6805
22.5489
53.6216
-13.6805
17.3922
23.7872
56.0951
-53.6216
-22.5489
-56.0951
-23.7872
• Information about the purpose of the
experiment.
• The number of participants.
• Descriptive Statistics (effect size –more
later)
• Analysis result
• Post hoc differences
The difference in calories between beef, (n=20),
meat (n=17) and poultry (n=17) hotdogs was
tested. Mean calorie level was determined for
each hotdog type, beef 156.85 (22.64), meat
158.71 (25.24) and poultry 118.76 (22.55).
Confidence intervals for the means in each
group are shown in figure 1.The calorie level
varied between the hotdog types F(2,51) =
16.07, p<.05. MS = 8846. Post hoc test show
poultry had fewer calories than both meat (LSD
=-38 p<.05) and beef (LSD= -39.9 p<.05).
APA
To examine the effects of memory training on retention of words, 20
college students were randomly assigned to four training conditions
(n=5) defined by the instructions to participants: story method,
imagery method thyme method, and control (no specific
instructions). Mean recall out of a possible 20 words (and the
sample standard deviation) were for each of the conditions: story
13.2(1.3), imagery 14.4 (1.8), rhyme 13.4 (1.3) and control 10.0
(1.6). Confidence intervals for the means in each group are shown
in figure 1. Mean recall differed significantly among the four
instruction conditions, F(3,16) = 7.8, p<.05. MS = … Post hoc
test show….
Repeated Measures
ANOVA
Total variability
Between treatment variability
Treatment effect
Experimental error
Between subject variability
Individual differences
Within treatment variability
Individual differences
Experimental error
Error variability
Experimental error
F= treatment effect + experimental error
experimental error
Figure 1 Mean Anxiety by Trial Number
Mauchly's Test of Sphericityb
Meas ure: MEASURE_1
a
Eps ilon
Within Subjects Effect Mauchly's W
trial
.398
Approx.
Chi-Square
8.957
df
5
Sig.
.112
Greenhous
e-Geiss er
.622
Huynh-Feldt
.744
Lower-bound
.333
Tes ts the null hypothes is that the error covariance matrix of the orthonormalized transformed dependent variables is
proportional to an identity matrix.
a. May be us ed to adjus t the degrees of freedom for the averaged tests of significance. Corrected tes ts are displayed in
the Tests of Within-Subjects Effects table.
b.
Des ign: Intercept
Within Subjects Design: trial
Tests of Within-Subjects Effects
Meas ure: MEASURE_1
Source
trial
Error(trial)
Sphericity As s umed
Greenhouse-Geiss er
Huynh-Feldt
Lower-bound
Sphericity As s umed
Greenhouse-Geiss er
Huynh-Feldt
Lower-bound
F(3,33) = 127.56
Type III Sum
of Squares
991.500
991.500
991.500
991.500
85.500
85.500
85.500
85.500
df
3
1.865
2.231
1.000
33
20.512
24.536
11.000
Mean Square
330.500
531.709
444.504
991.500
2.591
4.168
3.485
7.773
F
127.561
127.561
127.561
127.561
Sig.
.000
.000
.000
.000
assumptions