Transcript TITLE

One-Way Between Subjects
ANOVA
Overview
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Purpose
How is the Variance Analyzed?
Assumptions
Effect Size
ANOVA as Regression
Comparison Methods
Purpose of the One-Way
ANOVA
• Compare the means of two or more groups
• Usually used with three or more groups
• Independent variable (factor) may or may
not be manipulated; affects interpretation
but not statistics
Why Not t-tests?
• Multiple t-tests inflate the experimentwise
alpha level.
• ANOVA controls the experimentwise alpha
level with an omnibus F-test.
Why is it One Way?
• Refers to the number of factors
• How many WAYS are individuals grouped?
• NOT the number of groups (levels)
Why is it Called ANOVA?
• Analysis of Variance
• Analyze variability of scores to determine
whether differences between groups are big
enough to reject the Null
HOW IS THE VARIANCE
ANALYZED?
• Divide the variance into parts
• Compare the parts of the variance
Dividing the Variance
• Total variance: variance of all the scores in
the study.
• Model variance: only differences between
groups.
• Residual variance: only differences within
groups.
Model Variance
• Also called Between Groups variance
• Influenced by:
– effect of the IV (systematic)
– individual differences (non-systematic)
– measurement error (non-systematic)
Residual Variance
• Also called Within Groups variance
• Influenced by:
– individual differences (non-systematic)
– measurement error (non-systematic)
Sums of Squares
• Recall that the SS is the sum of squared
deviations from the mean
• Numerator of the variance
• Variance is analyzed by dividing the SS into
parts: Model and Residual
Sums of Squares
• SS Model = for each individual, compare
the mean of the individual’s group to the
overall mean
• SS Residual = compare each individual’s
score to the mean of that individual’s group
Mean Squares
• Variance
• Numerator is SS
• Denominator is df
– Model df = number of groups -1
– Residual df = Total df – Model df
Comparing the Variance
MS Model
F=
MS Residual
non - systematic + effect of i.v.
F =
non - systematic
ASSUMPTIONS
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Interval/ratio data
Independent observations
Normal distribution or large N
Homogeneity of variance
– Robust with equal n’s
EFFECT SIZE FOR ANOVA
• Eta-squared (h2)indicates proportion of
variance in the dependent variable explained
by the independent variable
SS Model
h =
SS Total
2
Example APA Format Sentence
A one-way between subjects ANOVA
indicated that the three conditions differed
significantly in the number of words
recalled, F(2,57) = 88.55, p < .001, h2 = .76.
ANOVA AS REGRESSION
• Predict scores on Y (the DV)
• Predictors are dummy variables indicating
group membership
Dummy Variables
• Group membership is categorical
• Need one less dummy variable than the
number of groups
• If you are in the group, your score on that
dummy variable = 1
• If you are not in that group, your score on
that dummy variable = 0
Example of Dummy Variables for
Three Groups
X1
X2
Group 1
1
0
Group 2
0
1
Group 3
0
0
Regression Equation for ANOVA
Y = b0  b1X1 + b2 X2 + ....+ ei
• bo is mean of base group
• b1 and b2 indicate differences between base
group and each of the other two groups
COMPARISON METHODS
• A significant F-test tells you that the groups
differ, but not which groups
• Multiple comparison methods provide
specific comparisons of group means
Planned Contrasts
• Decide which groups (or combinations) you
wish to compare before doing the ANOVA.
• The comparisons must be orthogonal to
each other (statistically independent).
Choosing Weights
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Assign a weight to each group.
The weights have to add up to zero.
Weights for the two sides must balance.
Check for orthogonality of each pair of
comparisons.
Example of a Planned Comparison
Group
Placebo
Treatment A
Treatment B
Weight
+2
-1
-1
This compares the average of
Treatments A and B to the Placebo
mean.
Another Planned Comparison
Group
Placebo
Treatment A
Treatment B
Weight
0
-1
+1
This one leaves out the Placebo
group and compares the two treatments.
Check for Orthogonality
Group
Placebo
Treatment A
Treatment B
C1
+2
-1
-1
C2
0
-1
+1
0
+1
-1
Multiply the weights and then add up the
products. The two comparisons are
orthogonal if the sum is zero.
Non-Orthogonal Comparisons
Group
Placebo
Treatment A
Treatment B
C1
+2
-1
-1
C2
+1
0
-1
+2
0
+1
These two comparisons do not ask
independent questions
Selecting Comparisons
• Maximum number of comparisons is
number of groups minus 1
• Start with the most important comparison.
• Then find a second comparison that is
orthogonal to the first one.
• Each comparison must be orthogonal to
every other comparison.
How Planned Contrasts Work
• A Sum of Squares is computed for each
contrast, depending on the weights.
• An F-test for the contrast is then computed.
SPSS Contrasts
• Deviation: compare each group to the
overall mean
• Simple: compare a reference group to each
of the other groups
• Difference: compare the mean of each
group to the mean of all previous group
means
More SPSS Contrasts
• Helmert: compare the mean of each group
to the mean of all subsequent group means
• Repeated: compare the mean of each group
to the mean of the subsequent group
• Polynomial: compare the pattern of means
across groups to a function (e.g., linear,
cubic, quadratic)
POST HOC COMPARISONS
• Done after an ANOVA has been done
• Need not be orthogonal
• Less powerful than planned contrasts
Fisher’s LSD
• Least Significant Difference
• Pairwise comparisons only
• Liberal
Bonferroni
• Pairwise comparisons only
• Divide alpha by number of tests
• More conservative than LSD
Tukey’s HSD
• Similar to Bonferroni, but more powerful
with large number of means
• Pairwise comparisons only
• Critical value increases with number of
groups