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ANOVA Analysis of Variance

- 1-way ANOVA

ANOVA

- What is Analysis of Variance
- The F-ratio
- Used for testing hypotheses among more than two

means - As with t-test, effect is measured in numerator,

error variance in the denomenator - Partitioning the Variance
- Different computational concerns for ANOVA
- Degrees Freedom for Numerator and Denominator
- No such thing as a negative value
- Using Table B.4
- The Source Table
- Hypothesis testing

M3

M1

M2

ANOVA

- Analysis of Variance
- Hypothesis testing for more than 2 groups
- For only 2 groups t2(n) F(1,n)

BASIC IDEA

Grp 1 Grp 2 Grp 3

Is the Effect Variability Large Compared to the

Random Variability

M1 1 M2 5 M3 1

Effect V

Random V

- As with the t-test, the numerator expresses the

differences among the dependent measure between

experimental groups, and the denominator is the

error. - If the effect is enough larger than random error,

we reject the null hypothesis.

BASIC IDEA

- If the differences accounted for by the

manipulation are low (or zero) then F 1 - If the effects are twice as large as the error,

then F 3, which generally indicates an effect.

Sources of Variance

Why Is It Called Analysis of Variance?Arent We

Interested In Means, Not Variance?

- Most statisticians do not know the answer to this

question? - If were interested in differences among means

why do an analysis of variance? - The misconception is that it compares ?12 to ?22.

No - The comparison is between effect variance

(differences in group means) to random variance.

Learning Under Three Temperature Conditions

T is the treatment total, G is the Grand total

M2

M1

M3

Computing the Sums of Squares

How Variance is Partitioned

- This simply disregards group membership and

computes an overall SS - Variability Between and Within Groups is

Included

How Variance is Partitioned

- Imagine there were no individual differences at

all. - The SS for all scores would measure only the

fact that there were group differences.

Grp 1 Grp 2 Grp 3

1 5 1 1 5 1 1 5 1 1 5 1 1 5 1

How Variance is Partitioned

- SS computed within a column removes the mean.
- Thus summing the SSs for each column computes

the overall variability except for the mean

differences between groups.

Grp 1 Grp 2 Grp 3

1-12-12-10-10-1

0-11-13-11-1 0-1

4-53-56-53-54-5

M1 1 M2 5 M3 1

How Variance is Partitioned

Grp 1 Grp 2 Grp 3

M1 1 M2 5 M3 1

Computing Degrees Freedom

- df between is k-1, where k is the number of

treatment groups (for the prior example, 3, since

there were 3 temperature conditions) - df within is N-k , where N is the total number of

ns across groups. Recall that for a t-test with

two independent groups, df was 2n-2? 2n was all

the subjects N and 2 was the number of groups, k.

Computing Degrees Freedom

How Degrees Freedom Are Partitioned

- N-1 (N - k) (k - 1)
- N-1 N - k k 1

Partitioning The Sums of Squares

Computing An F-Ratio

Consult Table B-4

Take a standard normal distribution, square each

value, and it looks like this

Table B-4

Two different F-curves

ANOVA Hypothesis Testing

Basic Properties of F-Curves

Property 1 The total area under an F-curve is

equal to 1. Property 2 An F-curve starts at 0

on the horizontal axis and extends indefinitely

to the right, approaching, but never touching,

the horizontal axis as it does so. Property 3

An F-curve is right skewed.

Finding the F-value having area 0.05 to its right

Assumptions for One-Way ANOVA

- 1. Independent samples The samples taken from

the populations under consideration are

independent of one another. - 2. Normal populations For each population, the

variable under consideration is normally

distributed. - Equal standard deviations The standard

deviations of the variable under consideration

are the same for all the populations.

Learning Under Three Temperature Conditions

M1 1 M2 5 M3 1

Learning Under Three Temperature Conditions

Learning Under Three Temperature Conditions

Learning Under Three Temperature Conditions

Learning Under Three Temperature Conditions

Learning Under Three Temperature Conditions

Learning Under Three Temperature Conditions

M2

M1

M3

Learning Under Three Temperature Conditions

SX2 106

16936916

144

191

M2

M1

M3

Learning Under Three Temperature Conditions

M2

M1

M3

Learning Under Three Temperature Conditions

M2

M1

M3

Calculating the F statistic

Sstotal X2-G2/N 46 SSbetween

SSbetween 30 SStotal Ssbetween

SSwithin Sswithin 16

Distribution of the F-Statistic for One-Way ANOVA

Suppose the variable under consideration is

normally distributed on each of k populations and

that the population standard deviations are

equal. Then, for independent samples from the k

populations, the variable has the

F-distribution with df (k 1, n k) if the

null hypothesis of equal population means is

true. Here n denotes the total number of

observations.

ANOVA Source Table for a one-way analysis of

variance

The one-way ANOVA test for k population means

(1 of 3)

Step 1 The null and alternative hypotheses

are Ho ?1 ?2 ?3 ?k Ha Not all the

means are equal Step 2 Decide On the significance

level, ? Step 3 The critical value of F?, with df

(k - 1, N - k), where N is the total number of

observations.

The one-way ANOVA test for k population means

(2 of 3)

The one-way ANOVA test for k population means

(3 of 3)

Step 4 Obtain the three sums of squares, STT,

STTR, and SSE Step 5 Construct a one-way ANOVA

table Step 6 If the value of the

F-statistic falls in the rejection region, reject

H0

Post Hocs

- H0 ?1 ?2 ?3 ?k
- Rejecting H0 means that not all means are equal.
- Pairwise tests are required to determine which of

the means are different. - One problem is for large k. For example with k

7, 21 means must be compared. Post-Hoc tests are

designed to reduce the likelihood of groupwise

type I error.

Criterion for deciding whether or not to reject

the null hypothesis

One-Way ANOVA

A researcher wants to test the effects of St.

Johns Wort, an over the counter, herbal

anti-depressant. The measure is a scale of

self-worth. The subjects are clinically

depressed patients. Use a 0.01

One-Way ANOVA

Compute the treatment totals, T, and the grand

total, G

One-Way ANOVA

Count n for each treatment, the total N, and k

One-Way ANOVA

Compute the treatment means

One-Way ANOVA

(0-1)21 (1-1)20 (3-1)24 (0-1)21 (1-1)20

sum

Compute the treatment SSs

One-Way ANOVA

Compute all X2s and sum them

One-Way ANOVA

Compute SSTotal SSTotal ?X2 G2/N

One-Way ANOVA

Compute SSWithin SSWithin ?SSi

One-Way ANOVA

Determine d.f.s d.f. WithinN-k d.f.

Betweenk-1 d.f. TotalN-1 Note that

(N-k)(k-1)N-1

One-Way ANOVA

Ready to move it to a source table

One-Way ANOVA

- Compute the missing values

One-Way ANOVA

- Compute the missing values

One-Way ANOVA

- Compute the missing values

One-Way ANOVA

- Compare your F of 17.5 with the critical value at

2,12 degrees of freedom, ? 0.01 6.93 - reject H0

One-Way ANOVA

Students want to know if studying has an impact

on a 10-point statistics quiz, so they divided

into 3 groups low studying (0-5hrs./wk), medium

studying (6-15 hrs./wk) and high studying (16

hours/week). At a0.01, does the amount of

studying impact quiz scores?

One-Way ANOVA

Compute the treatment totals, T, and the grand

total, G

One-Way ANOVA

Count n for each treatment, the total N, and k

One-Way ANOVA

Compute the treatment means

One-Way ANOVA

(2-2)20 (4-2)24 (3-2)21 (0-2)24 (2-2)20 (1-2)

21 sum

Compute the treatment SSs

One-Way ANOVA

Compute all X2s and sum them

One-Way ANOVA

Compute SSTotal SSTotal ?X2 G2/N

One-Way ANOVA

Compute SSWithin SSWithin ?SSi

One-Way ANOVA

Determine d.f.s d.f. WithinN-k d.f.

Betweenk-1 d.f. TotalN-1 Note that

(N-k)(k-1)N-1

One-Way ANOVA

- Fill in the values you have

One-Way ANOVA

- Compute the missing values

One-Way ANOVA

- Compare your F of 37.97 with the critical value

at 2,15 degrees of freedom, ? 0.01 6.36 - reject H0

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