Group Difference Methods

By Rama Krishna Kompella

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

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

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”

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

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

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

The Logic of ANOVA

The Logic of ANOVA

• If H0 True: – F = 0 + Chance   1

Chance

• If H0 False: – F = Treatment Effect + Chance >  1

Chance

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

Computing ANOVA

(1) Compute SS (sums of squares) (2) Compute df (3) Compute MS (mean squares) (4) Compute F

Computing ANOVA

Computing ANOVA

Computing ANOVA

Computing ANOVA

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

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

Computing ANOVA

Computing ANOVA

Computing ANOVA

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.”

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

Questions?