anova

```S519: Evaluation of
Information Systems
Social Statistics
Inferential Statistics
Chapter 11: ANOVA
This week




When to use F statstic
How to compute and interpret
Using FTEST and FDIST functions
How to use the ANOVA
The problem with t-tests…

We could compare three groups with multiple
ttests: M1 vs. M2, M1 vs. M3, M2 vs. M3
What is ANOVA?




“Analysis of Variance”
A hypothesis-testing procedure used to evaluate
mean differences between two or more
treatments (or populations).
Related to: t-tests using independent-measures
or repeated- measures design.


1) Can work with more than two samples.
2) Can work with more than one independent variable
What is ANOVA?





In ANOVA an independent or quasiindependent variable is called a factor.
Factor = independent (or quasi-independent)
variable.
Levels = number of values used for the
independent variable.
One factor → “single-factor design”
More than one factor → “factorial design”
What is ANOVA?

An example of a single-factor design

A example of a two-factor design
F value

Variance between treatments can have two
interpretations:


Variance is due to differences between
treatments.
Variance is due to chance alone. This may be due
to individual differences or experimental error.
Three Types of ANOVA



Independent measures design: Groups are
samples of independent measurements
(different people)
Dependent measures design: Groups are
samples of dependent measurements
(usually same people at different times; also
matched samples) “Repeated measures”
Factorial ANOVA (more than one factor)
Excel: ANOVA

Three different ANOVA:



Anova: single factor - independent
Anova: two factors with replication - factorial
Anova: two factors without replication - dependent
Example (independent)

Three groups of preschoolers and their
language scores, whether they are overall
different?
Group 1 Scores
Group 2 Scores
87
86
76
56
78
98
77
66
75
67
Group 3 Scores
87
89
85
91
99
96
85
87
79
89
81
90
82
89
78
96
85
96
91
93
F test steps

Step1: a statement of the null and research
hypothesis

One-tailed or two-tailed (there is no such thing in
ANOVA)
H 0 : 1  2  3
H1 : at least one  is different
F test steps

Step2: Setting the level of risk (or the level of
significance or Type I error) associated with
the null hypothesis

0.05
F test steps

Step3: Selection of the appropriate test
statistics


See Figure 11.1 (S-p227)
Simple ANOVA (independent)
F test steps

Between-group degree of freedom=k-1


k: number of groups
Within-group degree of freedom=N-k

N: total sample size
F test steps

Step4: determination of the value needed for
rejection of the null hypothesis using the
appropriate table of critical values for the
particular statistic



Table B3 (S-p363)
df for the denominator = n-k=30-3=27
df for the numerator = k-1=3-1=2
F test steps

Step5: comparison of the obtained value and
the critical value



If obtained value > the critical value, reject the null
hypothesis
If obtained value < the critical value, accept the
null hypothesis
8.80 and 3.36
F test steps

Step6 and 7: decision time


How do you interpret F(2, 27)=8.80, p<0.05
Example (dependent)
Five participants took a series of test on a
new drug
T1
T2
T3
T4
P1
3
4
6
7
P2
0
3
3
6
P3
2
1
4
5
P4
0
1
3
4
P5
0
1
4
3
F test steps

Step1: a statement of the null and research
hypothesis

One-tailed or two-tailed (there is no such thing in
ANOVA)
H 0 : 1  2  3  4
H1 : at least one  is different
F test steps

Step2: Setting the level of risk (or the level of
significance or Type I error) associated with
the null hypothesis

0.05
F test steps

Step3: Selection of the appropriate test
statistics


See Figure 11.1 (S-p227)
Simple ANOVA (independent)
F test steps

Between-group degree of freedom=k-1


Within-group degree of freedom=N-k


N: total sample size
Between-subject degree of freedom=n-1


k: number of groups
n: number of subjects
Error degree of freedom=(N-k)-(n-1)
F test steps

Step4: determination of the value needed for
rejection of the null hypothesis using the
appropriate table of critical values for the
particular statistic



Table B3 (S-p363)
df for the denominator = (N-k)-(n-1)=16-4=12
df for the numerator = k-1=4-1=3
F test steps

Step5: comparison of the obtained value and
the critical value



If obtained value > the critical value, reject the null
hypothesis
If obtained value < the critical value, accept the
null hypothesis
24.88 and 3.49
F test steps

Step6 and 7: decision time
