PSY 307 – Statistics for the Behavioral Sciences Chapter 16 – One-Factor Analysis of Variance (ANOVA)

PSY 307 – Statistics for the Behavioral Sciences

Chapter 16 – One-Factor Analysis of Variance (ANOVA)

Fisher’s F-Test (ANOVA)

Ronald Fisher

Testing Yields in Agriculture

X1 X2 X1= =

ANOVA

Analysis of Variance (ANOVA) – a test of more than two population means.

One-Way ANOVA – only one factor or independent variable is manipulated.

ANOVA compares two sources of variability.

Two Sources of Variability

Treatment effect – the existence of at least one difference between the population means defined by IV.

Between groups variability – variability among subjects receiving different treatments (alternative hypothesis).

Within groups variability – variability among subjects who receive the same treatment (null hypothesis).

F-Test

If the null hypothesis is true, the numerator and denominator of the F-ratio will be the same. F = random error / random error

If the null hypothesis is false, the numerator will be greater than the denominator and F > 1. F = random error + treatment effect

random error

Difference vs Error

Difference on the top and the error on the bottom: Difference is the variability between the

groups, expressed as the sum of the squares for the groups.

Error is the variability within all of the subjects treated as one large group.

When the difference exceeds the variability, the F-ratio will be large.

F-Ratio

F = MSbetween

MSwithin

MS = SS df

SS is the sum of the squared differences from the mean.

F-Ratio

F = MSbetween

MSwithin

MSbetween treats the values of the group means as a data set and calculates the sum of squares for it.

MSwithin combines the groups into one large group and calculates the sum of squares for the whole group.

Testing Hypotheses

If there is a true difference between the groups, the numerator will be larger than the denominator. F will be greater than 1

Writing hypotheses:H0: 1 = 2 = 3

H1: H0 is false

Formulas for F

Description in words of what is being computed.

Definitional formula – uses the SS, described in the Witte text

Computational formula – used by Aleks and in examples in class.

Formula for SStotal

SStotal is the total Sum of the Squares It is the sum of the squared deviations

of scores around the grand mean.

SStotal = ∑(X – Xgrand)2

SStotal = ∑(X2 – G2/N) Where G is the grand total and N is its

sample size

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above

aboveSSwithin

Formula for SSbetween

SSbetween is the between Sum of the Squares It is the sum of the squared deviations

for group means around the grand mean.

SSbetween = n∑(X – Xgrand)2

SSbetween = ∑(T2/n – G2/N) Where T is each group’s total and n is

each group’s sample size

definition

computation

Formula for SSwithin

SSwithin is the within Sum of the Squares It is the sum of the squared deviations

for scores around the group mean.

SSwithin = ∑(X – Xgroup)2

SSwithin = ∑X2 – ∑T2/n) Where T is each group’s total and n is

each group’s sample size

definition

computation

Degrees of Freedom

dftotal = N-1 The number of all scores minus 1

dfbetween = k-1 The number of groups (k) minus 1

dfwithin = N-k The number of all scores minus the

number of groups (k)

Checking Your Work

The SStotal = SSbetween + SSwithin.

The same is true for the degrees of freedom:

dftotal = dfbetween + dfwithin

Calculating F (Computational)

SSbetween = T2 – G2

n N Where T is the total for each group and

G is the grand total

SSwithin = X2 - T2

N SStotal = X2 – G2/N

F-Distribution

Critical value

Common – retain null

Rare – reject null

Look up F critical value in the F table using df for numerator and denominator

ANOVA Assumptions

Assumptions for the F-test are the same as for the t-test

Underlying populations are assumed to be normal with equal variances.

Results are still valid with violations of normality if: All sample sizes are close to equal Samples are > 10 per group

Otherwise use a different test

Cautions

The ANOVA presented in the text assumes independent samples.

With matched samples or repeated measures use a different form of ANOVA.

The sample sizes shown in the text are small in order to simplify calculations. Small samples should not be used.