1
One-Way Analysis of
Variance
Comparing means of more
than 2 independent
samples
KNR 445
Statistics
ANOVA (1w)
Slide 2
1
1.
2
2.
3
3.
4
4.
4
Why not multiple t-tests?
# comparisons = k(k-1)/2, where k is the # of groups. In
other words, when there are many more than 2 groups,
the number of comparisons is large
In each separate comparison, you only use information
within that pair – but information within other pairs may
increase power
If you do multiple tests, you have multiple answers, not
just one
With many tests, type 1 error rates increase radically
KNR 445
Statistics
ANOVA (1w)
Slide 3
ANalysis Of VAriance
 1-way ANOVA
 Grouping variable = factor = independent
variable
1
2
 The variable will consist of a number of levels
 If 1-way ANOVA is being used, the number will be >2.
 E.G. What type of program has the greatest
3
impact on aggression?
 Violent movies, soap operas, or “infomercials”?
 Type of program is the independent variable or
factor, while soap operas is one level of the factor, &
aggression is the dependent variable
KNR 445
Statistics
ANOVA (1w)
Slide 4
1-way ANOVA: Hypotheses
 Null:
1
H0 : 1  2    k
 Alternative (experimental):
2
3
H1 : not H0
 Note: no directional hypothesis; Null may
be false in many different ways
KNR 445
Statistics
ANOVA (1w)
Slide 5
1
2
1-way ANOVA: Logic
 Change in test statistic
 Recall with t-tests, test statistic is
difference between sample means
t
difference expected due to chance (error)
 But with ANOVA, test statistic is
variance (differenc e) between sample means
F
variance (differenc e) expected due to chance (error)
KNR 445
Statistics
ANOVA (1w)
Slide 6
1-way ANOVA: Logic
 Return to our example…
1
2
3
TV Movie
Soap Opera
Infomercial
1
6
10
3
8
13
4
10
5
5
4
9
2
12
8
X Movie= 3
X soap = 8
X sales= 9
4
KNR 445
Statistics
ANOVA (1w)
Slide 7
1-way ANOVA: Logic
 Variance between sampling means
1
2
X Movie= 3
X soap= 8
X sales= 9
 Known as “between group variance”
 Think of what could cause these means to differ
(vary) from each other
 Treatment effect: differences due to the different
ways the groups were treated (systematic variation)
 Chance: Individual differences, experimental error
(unsystematic, unexplained variation)
KNR 445
Statistics
ANOVA (1w)
Slide 8
1-way ANOVA: Logic
 Variance expected by chance (error)
1
TV Movie
Soap Opera
Infomercial
1
6
10
3
8
13
4
10
5
5
4
9
2
12
8
 Known as “within group variance”
 Think of what could cause the scores within the groups
to differ from each other
 Chance: Individual differences, experimental error
(unsystematic, unexplained variation)
KNR 445
Statistics
ANOVA (1w)
Slide 9
1-way ANOVA: Logic
 Partitioning the variance
Total variance
Between group
variance:
• Treatment
• Chance
=
1
+
Within group
variance
• Chance
3
2
KNR 445
Statistics
ANOVA (1w)
Slide 10
1-way ANOVA: Logic
 The test statistic for the 1-way ANOVA
 The F-ratio
between group variance
1
F
within group variance
 If null is true
0  chance
F
1
chance
2
 If null is false
treatment.effect  chance
F
1
chance
3
KNR 445
Statistics
ANOVA (1w)
Slide 11
1-way ANOVA: Logic
 More on 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  1
 If treatment effect, observed F will be > 1
 How large does F have to be to conclude there is a
treatment effect (to reject H0)?
1
 Compare observed F to critical values based on sampling
distribution of F
 A family of distributions, each with a pair of degrees of
freedom
KNR 445
Statistics
ANOVA (1w)
Slide 12
The F-distribution
 As with t-distribution, a
Probability
family of curves
 Shape of distribution
0
1
F values
1
changes with df
Region of
rejection
2
3
4
5
6
F.05 = 5.14
2
 If null is true, F≈1
 “p” largest at 1
 “p” tapers as F > 1
7
8
 F-values always positive
(variance can’t be
negative)
KNR 445
Statistics
ANOVA (1w)
Slide 13
1
Hypothesis Testing with ANOVA
1. Research question
 Does the type of programming affect levels of
aggression?
2. Statistical hypotheses
 H0: 1 = 2 = ... = K
 H1: At least 2 means are significantly different
3. Decision rule (critical value)
4. Compute observed F-ratio from data
5. Make decision to reject or fail to reject H0
6. If H0 rejected, conduct multiple comparisons as
needed
KNR 445
Statistics
ANOVA (1w)
Slide 14
1
2
Computing ANOVA
 For those interested (?!) we’ll cover the steps and
do an example (not necessary if you understand
the concept…but it may help)
 If you’re really confident, skip to slide 21
 Steps to completion
1. Compute SS (sums of squares)
2. Compute df
3. Compute MS (mean squares)
4. Compute F
3
KNR 445
Statistics
ANOVA (1w)
Slide 15
Computing ANOVA
 Vocabulary/Symbols:
 k = Number of groups
 nj = Sample size of the jth group (n1, n2,…nj,…nk)
 N = Total sample size
 X j = Mean of the jth group X 1 , X 2 ,... X j ,... X k
 X T = Grand (overall) mean
 SS (Sum of squares) = Sum of squared
deviations around a mean

1

KNR 445
Statistics
ANOVA (1w)
Slide 16
1
Computing ANOVA
 Step 1: Compute Sums of Squares (SS)
 Need total, group, and error sums of squares
 These are combined with appropriate df to give
variance calculations, & then generate F-ratio
2
KNR 445
Statistics
ANOVA (1w)
Slide 17
Computing ANOVA
 Step 1: Compute Sums of Squares (SS)
 Total sum of squares (looks a little daunting,
2
but it’s really not)
4
3
1
SS total 
i 1
i N
x  X 
2
i
1. This is just the sum of the
squared deviations of each
observation from the overall
mean
T
( X )
 X 
N
2
2
2. The second part of the
formula is the one we’ll use in
the calculations, as it’s easier to
work with. The first one is easier
to conceptualize.
KNR 445
Statistics
ANOVA (1w)
Slide 18
Computing ANOVA
 Step 1: Compute Sums of Squares (SS)
 Group sum of squares
1
SS group  
j 1
n
(
X

X
)
j
j
T
j k
1. This is the sum of the squared
deviations of each group mean
from the overall mean (multiplied
by each sample’s size)
2
KNR 445
Statistics
ANOVA (1w)
Slide 19
Computing ANOVA
 Step 1: Compute Sums of Squares (SS)
 Error sum of squares
1
 Total variance is composed of SSgroup & SSerror
 SStotal = SSgroup + SSerror
 Rearrange this formula to get:
2
 SSerror= SStotal – SSgroup
1. To calculate, take the sum of squares
of each observation within a group from
its group mean, for all groups
3
KNR 445
Statistics
ANOVA (1w)
Slide 20
Computing ANOVA
 Step 2: Compute degrees of freedom
 (Used to adjust SS to variance estimates)
 df group: df
 k 1
1
group
 df total:
df total  N  1
2
 df error (or “what’s left over”):
df error  N  k
3
KNR 445
Statistics
ANOVA (1w)
Slide 21
Computing ANOVA
 Step 3: Compute Mean Squares (MS) & F-
ratio
1
 Mean Square = variance
SS group
MSgroup 
df group
SSerror
MSerror 
df error
2
F
3
MS group
MS error
MS between

MS within
KNR 445
Statistics
ANOVA (1w)
Slide 22
Computing ANOVA
 The ANOVA summary table (may help you
understand where it comes from in SPSS)
 all the processes are alluded to above
Source
Sum of
DF
Squares
MS
F
sig.
Between
SSgroup
Groups
dfgroup SSgp/dfgp=MSB MSB/MSW p-value
Within
Groups
SSerror
dferror
Total
SStotal
dftotal
Sse/dfe=MSW
1