ANOVA: Analysis of Variance
The Titanic Disaster
Were there differences in survival rates by
 gender/age groups (men, women, children) or
 economic groups (1st, 2nd, and 3rd class tickets
and crew)?
IF SO, WHY?
Survival rate = Proportion of
the group that survived
Durkheim: Suicide
Were there differences in the suicide rates of
Protestants, Catholics, and Jews?
If so, why?
The quantitative analysis sparks the theoretical
and qualitative analysis.
(Remember, this is a low absolute risk.)
What Does ANOVA Do?
It compares the means (of an interval-ratio
dependent variable) for the groups defined by the
categories of an independent variable measured at
the nominal or ordinal level.
Example: Are mean weights different for fans of
action movies, horror movies, and romantic
comedies?
The Null Hypothesis
The null hypothesis is that all of the group means
are equal to each other — that there are no
differences.
μ1 = μ2 = μ3 = μ4 etc. for however many groups
there are.
Rejecting the null hypothesis implies that, in at
least one pair of group means, the means were
different from each other.
Remember the Variance?
 Subtract to find deviation of each score from the
mean.
 Square each difference.
 Sum the squared differences.
 Find the mean squared difference by dividing by
the number of cases (if it’s a sample, use n – 1).
Can you write the formula?
So What Does the Variance
Have to Do with It?
Remember: Variances are measures of deviations
from means.
We are going to use variances to compare means.
See Garner (2010), from page 203.
3 Variances in Play!
These are
 the grand or total variance based on the
deviations between each score and the overall
mean
 the between-groups variance based on
deviations between group means and the
overall mean
 the within-groups variance based on the
deviation between each score and ITS group
mean
How Does ANOVA Do It?
Between-groups variance is “good” — it suggests
that there are differences among the groups’ DV
means that are related to the grouping variable
(the IV).
Within-groups variance is “bad” or error — it
suggests that the independent variable is not a
good predictor of differences in the DV means.
This variation remains unexplained.
 Is there more variability between the groups
than variability within the groups?
We will compute the ratio of between-groups
variance to within-groups variance.
Example
Are there significant differences among the GPA
means of 4 groups defined by “movie faves” —
action, horror, drama, and comedy?
We will use j to designate the number of groups, in
this case, 4.
The Data: GPAs of Movie Fans
Group 1 (action): 0, 1, 2, 3, 4
mean = 2.
Group 2 (drama): 0, 0, 4, 4
mean = 2.
Group 3 (horror): 3, 3, 3, 3, 3
mean = 3.
Group 4 (comedy): 4, 4, 4, 4
mean = 4.
Question: Is there a significant difference among
these means?
Null hypothesis: μ1 = μ2 = μ3 = μ4.
The F-ratio
The test-statistic is called the F-ratio and is
computed by a division:
 The numerator (top) is computed as betweengroups variation: Deviations of the group means
from the overall (grand) mean, with each
squared deviation weighted by the number of
cases in the group, and then summed and
averaged for all the groups.
 Denominator—next slide, please!
The Denominator of the F-ratio
 The denominator (bottom) is computed as
within-groups variation: Squared deviations of
the value for each case from the mean of its own
group, and then all the squared deviations
summed and averaged for all the cases.
F-ratio (continued)
HEY, LET’S TAKE THOSE CALCULATIONS ONE
STEP AT A TIME!
OK, but first, let’s see what we are going to do with
the F-ratio (the test statistic) once it is
computed.
Significance?
The null hypothesis is that all the group means are
equal to each other.
 If F is significant (exceeds critical value), it has
a low p-value (p < .05) or “Sig” in SPSS/PASW.
We can reject the null hypothesis of equal group
means.
Steps in Calculating F
 Our first step is to calculate the within-groups
variance (a mean sum of squares).
 Our second step is to calculate the betweengroups variance (a mean sum of squares)
 Our third step is to calculate the ratio: Betweengroups mean sum of squares divided by withingroups mean sum of squares). Ugh.
The Within-Groups Sum of Squares
 Find the difference between each score and its
own group mean by subtracting the group
mean from the score.
 Square this difference.
 Sum the squares for the entire data set. There
will be a square for each score (case) in the
data set.
Mean Square Within
 Divide the within-groups sum of squares by the
total number of cases minus the number of
groups (n – j).
SSwithin / (n – j)
This is the denominator (bottom) of the F-ratio
and is “bad” — variability we cannot predict
from our independent variable.
Between-Groups Sum of Squares
 Subtract each group mean from the grand
mean.
 Square the difference.
 Multiply this square by the number of cases in
the group to weight the squared difference.
 Sum these weighted squares. (There will be a
weighted square for each group of the IV.)
Mean Square Between
 Divide the between-groups sum of squares
obtained at the previous step by its degrees of
freedom:
(j – 1)
where j is the number of categories of the IV.
This is “good” variability, based on differences
between (or among) the groups.
The F-ratio
F = meansqbetween / meansqwithin
The “between” term is good variability (predicted
by the IV).
The “within” term is bad or error variability (not
predicted by the IV).
Degrees of Freedom
The F-ratio has a separate df calculation for the
numerator and for the denominator.
 df for the numerator is (j – 1).
 df between
Look across the top row of the chart for the critical
values of F.
 df for the denominator is (n – j)
 df within
Look down the left column of the chart.
What’s the Next Step
First, we do the division:
meansqbetween-groups / meansqwithin-groups
So we have the value of a test statistic called F.
Is F “Big Enough” to Have a Low pValue?
We check out F in the chart.
See Garner (2010, pp. 326–29).
If F is BIG relative to its degrees of freedom (and
there are 2 of these in play — see previous
slides), it is significant — i.e., there is a lot of
between-groups variability relative to withingroups variability. (P is low.)
Drat, Not Yet Done! Post-Hoc
Multiple Comparison Procedures
The problem with finding that F is significant is that
we still need to figure out which group means are
significantly different from each other. (A significant
F just tells us SOME of them are, but we don’t yet
know which.)
For this, we need to look at the post-hoc results:
Bonferroni is one I like to look at, but it assumes
equal group variances; others prefer Tukey’s HSD.
Be Sure to Get the Descriptives!
When you run ANOVA in SPSS/PASW, be sure to
check “Descriptives” in the “One Way ANOVA”
dialog box entitled “Options” to see what the group
means actually are.
ANOVA as Part of Regression Analysis
In addition to a data analysis technique in its own
right, ANOVA is part of linear regression analysis,
so we need to be able to interpret it in that context.
It is used in figuring out what proportion of
variation in the dependent variable can be
predicted from the independent variable, a value
called the coefficient of determination, R2.
ANOVA is actually easier to understand and
calculate as part of a regression analysis than as a
compare-means procedure.
ANOVA and Independent-Samples tTests [1]
When there are only TWO groups — randomly
selected rather than paired, matched, or retested
— an independent-samples t-test is often used
instead of ANOVA.
The independent-samples t-test is usually reported
in research but the result (p-value) is IDENTICAL
to the ANOVA result.
t-Test results are a little harder to set up and read
in SPSS/PASW.
Independent-Samples t-Tests [2]
Just like ANOVA, independent-samples t-tests are
set up under the SPSS/PASW commands
Analyze–Compare Means.
 Specify the test variable (interval-ratio DV) and
the grouping variable (categoric IV).
 Enter the groups’ values for the grouping
variable.
 Output first shows Levene test for equal
variances and then the result of the t-test.
ANOVA Summary
Remember the logic of ANOVA:
 It involves the F test statistic.
 F is computed by the ratio of two kinds of
variation (between- and within-groups
variation).
ANOVA Summary (continued)
 When there is a lot of between-groups variance
and relatively little within-groups variance, the
result is significant (we can safely reject the null
hypothesis of no differences among the
means).
 In SPSS/PASW and other software, ANOVA is
often located under Analyze–Compare Means
in the menu.