Download BHS 307 – Statistics for the Behavioral Sciences

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: m1 = m2 = m3
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
Hours of Sleep Deprivation
SSbetween
0
24
48
0
3
6
4
6
8
2
6
10
Grand Total
6
15
24
G = 45
 (6) 2 (15) 2 (24) 2  (45) 2  36 225 576  2025



 


  9  54
3
3
3
9
3
3
3




SS within  (0) 2  (4) 2  (2) 2  (3) 2  (6) 2  (6) 2  (8) 2  (10) 2  above 
0  16  4  9  36  36  64  100  above  22
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)

definition
computation
Where T is each group’s total and n is
each group’s sample size
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)

definition
computation
Where T is each group’s total and n is
each group’s sample size
Degrees of Freedom

dftotal = N-1


dfbetween = k-1


The number of all scores minus 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 =



S
T2 – G2
n
N
Where T is the total for each group and
G is the grand total
SSwithin = S X2 - S T2
N
SStotal = S X2 – G2/N
F-Distribution
Common – retain
null
Critical value
Look up F critical value in the F table using df for
numerator and denominator
Rare –
reject
null
Effect Sizes



Effect size (h2) for ANOVA is the
squared curvilinear correlation.
The effect size (h2) is the amount of
variance in the dependent variable
explained by the independent
variable.
To calculate effect size, divide the
SSbetween by SStotal
Interpreting h2

Cohen’s rule of thumb:




If h2 approximates .01, effect size is
small.
If h2 approximates .06, effect size is
medium.
If h2 approximates .14 or more, effect
size is large.
Effect size is especially important
when large samples sizes are used.
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.