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Group Difference Methods
By Rama Krishna Kompella
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The basic ANOVA situationTwo variables: 1 Categorical (IV), 1 Continuous (DV)
Main Question: Do the (means of) the quantitative variables depend on which group (given by categorical variable) the individual is in?
If categorical variable has only 2 values: • 2-sample t-test
ANOVA allows for 3 or more groups
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ANOVA - Analysis of Variance
• Extends independent-samples t test• Compares the means of groups of
independent observations– Don’t be fooled by the name. ANOVA does not
compare variances.
• Can compare more than two groups
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ANOVA – Null and Alternative Hypotheses
Say the sample contains K independent groups
• ANOVA tests the null hypothesis
H0: μ1 = μ2 = … = μK
– That is, “the group means are all equal”
• The alternative hypothesis is
H1: μi ≠ μj for some i, j
– or, “the group means are not all equal”
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Assumptions• Homogeneity of variance
21 = 2
2 = ... = 2k
– Moderate departures are not problematic, unless sample sizes are very unbalanced
• Normality– Scores with in each group are normally distributed around their
group mean– Moderate departures are not problematic
• Independence of observations– Observations are independent of one another– Violations are very serious -- do not violate
• If assumptions violated, may need alternative statistics
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The Logic of ANOVA
t = difference between sample means
difference expected by chance (error) F = variance (differences) between sample means
variance (difference) expected by chance (error) Concerned with variance:
variance = differences between scores
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The Logic of ANOVA
Two sources of variance: Between group variance: Differences between
group means Within group variance: Differences among
people within the same group
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The Logic of ANOVA
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The Logic of ANOVA
• If H0 True: – F = 0 + Chance 1
Chance
• If H0 False: – F = Treatment Effect + Chance > 1
Chance
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The F statistic
• F is a statistic that represents ratio of two variance estimates
• Denominator of F is called “error term” • When no treatment effect, F 1If treatment effect, observed F will be > 1 • How large does F have to be to conclude there is a
treatment effect (to reject H0)? • Compare observed F to critical values based on sampling distribution of F
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Computing ANOVA
(1) Compute SS (sums of squares) (2) Compute df (3) Compute MS (mean squares) (4) Compute F
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Computing ANOVA
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Computing ANOVA
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Computing ANOVA
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Computing ANOVA
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Example• Does presence of offer during festival season affect sales? IV = Number of offers presentDV = Sales (in units)• Three conditions: No offer, Only one offer on a product,
Multiple offers on a product• Is there a significant difference among these means?
M O SO NO10 6 113 8 35 10 49 4 58 12 2
2= 9 1= 8 0= 3X X X
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Computing ANOVA
MO SO NO10 6 113 8 35 10 49 4 58 12 2
n 5 5 5 N = 159 8 3 = 6.67jX ..X
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Computing ANOVA
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Computing ANOVA
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Computing ANOVA
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Computing ANOVACritical Value:• We need two df to find our critical F value from Table (Note E.3
=.05; E.4 =.01)• “Numerator” df: dfG “Denominator” df: dfE
• df = 2,12 and = .05 Fcritical= 3.89
Decision: Reject H0 because observed F (7.38)
exceeds critical value (3.89) Interpret findings: • At least two of the means are significantly different from each other.• “The amount of sales generated is influenced by the number of
offers present on the product, F(2,12) = 7.38, p .05.”
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Types of ANOVA
• One-way ANOVA, is used to test for differences among two or more independent groups.
• Factorial ANOVA, is used in the study of the interaction effects among treatments.
• Repeated measures ANOVA, is used when the same subject is used for each treatment.
• Multivariate analysis of variance (MANOVA), is used when there is more than one response variable
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Questions?