2. The basic ANOVA situation
Two 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
3. 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
4. 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”
5. Assumptions
• Homogeneity of variance
σ21 = σ22 = ... = σ2k
– 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
6. 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
7. 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
9. The Logic of ANOVA
• If H0 True:
– F= 0 + Chance ≈ 1
Chance
• If H0 False:
– F= Treatment Effect + Chance > 1
Chance
10. 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 ≈ 1
If 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
11. Computing ANOVA
(1) Compute SS (sums of squares)
(2) Compute df
(3) Compute MS (mean squares)
(4) Compute F
17. Example
• Does presence of offer during festival season affect sales?
IV = Number of offers present
DV = 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?
MO SO NO
10 6 1
13 8 3
5 10 4
9 4 5
8 12 2
X2 = 9 X 1= 8 X 0= 3
18. Computing ANOVA
MO SO NO
10 6 1
13 8 3
5 10 4
9 4 5
8 12 2
n 5 5 5 N = 15
Xj 9 8 3 X .. = 6.67
22. Computing ANOVA
Critical Value:
• We need two df to find our critical F value from Table (Note
E.3 α =.05; E.4 α =.01)
• “Numerator” df: dfG “Denominator” df: dfE
• df = 2,12 and α = .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.”
23. 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
