Let's look into the assumptions, the
hypotheses, and how to determine the
critical value for a one-way ANOVA test.
Here are the requirements for a one-way
ANOVA; that is an ANOVA conducted with a
single factor, typically with three or
more levels. The one-way ANOVA is a
parametric procedure. We are using
sampled data to estimate population
parameters. Any time we look at data
drawn from a population, we are using
parametric procedures. A one-way ANOVA
can compare three or more groups.
Whenever your independent variable has
three or more levels, use ANOVA, not a
t-test. Although ANOVA is designed for
three or more groups, ANOVA can be
conducted with only two groups (i.e.when
you would typically use an independent
samples t-test.) In fact, you can convert
between t and F where t equals the
square root of F. Just remember that a
t-test can be either positive or
negative, unlike the f test that is
always positive, because it is calculated
with squared values. On the other hand,
the sign of a t-value only tells us the
direction of change; whether the first
group or the second group has a higher
mean. If you were to switch the groups,
the sign would also switch. But if you do
an ANOVA with just two groups and you
want to calculate t, just be aware that
the square root of F will always be
positive. The independent variable groups
must be independent. "Independent" means
that the scores at one level do not
influence the probability of scores in
another level. The groups are not
influencing one another. The samples
should not be related. If the samples are
related, then you would use a
repeated-measures ANOVA. So here are the
assumptions that must be satisfied for
us to use the one-way ANOVA.
The independent variable (the IV) is categorical
or nominal, with three or more
groups. The samples should be randomly
selected. The samples can be randomly
assigned, such as in an experimental
design, but you could also use them
naturally occurring, in what is called a
"quasi-experimental design." The sample
sizes for each group should be roughly
equal. Unequal sample sizes increase the
likelihood of Type I error rates.
What do I mean by "roughly equal"? As a
rule, the largest n divided by the
smallest n of your groups should not
exceed 1.5. The dependent variable must
be quantitative, at the scale level, so
that's interval or ratio. Now, some people
will tell you that if you use Likert
survey scales (like one = "strongly
disagree" up to 5 = "strongly agree"),
that those data are ordinal and you have
to use non-parametric statistics. In fact,
John Dawes (2008) demonstrated that when
Likert survey scales have five or more
item options, they function like scale
data, and they can be used with
parametric statistics, like this one way
ANOVA. Check for outliers in your data
cleaning, using the Explore command. You
should delete or Windsorize severe
outliers. Scores in one group should not
influence the probability of scores in
another group. This is called
"independence." The participants in one
group are in only one group; they are not
in neither, they are not in both.
Normality: the scores on the dependent
variable within each group should be
approximately normally distributed. The
Shapiro Wilk test, and/or a QQ plot, or a
PP plot ,or a box plot are all used to
test for normality. See my video about
the Central Limit Theorem for a fuller
discussion about the implications of
normality. If the dependent variables are
not normally distributed, or if the
sample sizes are not roughly equal, use
the nonparametric Kruskal-Wallis
one-way ANOVA test. However, the parametric
one-way ANOVA that we are learning about
now is "robust" as long as there is a
minimum of 30 subjects in each sample.
"Robust" means that the Type I error rate
does not increase if the assumptions are
violated. Also, the groups should have
Homogeneity of Variance, which you will
test for using Levene's test for
equality of variances, just as we did
with the t-test. If the dependent
variable groups fail this assumption,
rerun the ANOVA and report Welch's ANOVA
instead. Here are the hypotheses for a
one-way ANOVA. Like a t-test, the null
hypothesis for ANOVA is that all samples
are drawn from a population whose means
are equal; therefore, all of the sample
means are the same. The null
hypothesis might be written as H0:
mu 1 equals mu 2, equals mu 3. The
alternative hypothesis is that the means
of the three groups differ. And remember
we do not know which means differ, so the
alternative hypothesis might be written
as H1: mu 1 does not equal mu 2,
does not equal mu 3. Unless instructed
otherwise, you should use the alpha of
0.05 for your level of significance for
the Omnibus ANOVA test. The critical
value will be determined based upon your
degrees of freedom and the Alpha level
both of which are needed to use the
ANOVA FTL I'll explain this in detail
next you can safely assume alpha equals
point 05 two-tailed test on the other
hand you might protest that ANOVA is
always a one-tailed test because the F
distribution has only one tail true but
the ANOVA is an omnibus test and the
post-hoc tests are the ones that really
matter we could use the post-hoc tests
to hypothesize directional group
differences but instead we're just going
to assume that all groups are equal and
let the post-hoc tests tell us which
groups
differ and in which direction if we
wanted to do directional post-hoc tests
we would actually be better off to use a
different technique called planned
contrasts more about that in another
video finding the critical value for a
one-way ANOVA requires knowing the
degrees of freedom for the numerator and
the denominator of the F ratio these are
called the degrees of freedom between
the degrees of freedom within and the
degrees of freedom total the degrees of
freedom between equals K minus 1 where k
is the number of categories or levels
this is the degrees of freedom for the
numerator so in our example where we
have 20 people who are randomly assigned
to four groups k equals 4 so the degrees
of freedom between would be four
categories of diet minus one or three
degrees of freedom within equals n minus
K where n is the total number of
participants and K is the total number
of categories this is the degrees of
freedom for the denominator the degrees
of freedom within would be the total
number of participants 20 minus the
total number of categories for or 16 the
degrees of freedom total equals n minus
1 as it always does with n being the
total number of participants in all
groups there are 20 participants so the
degrees of freedom total would be 19
finding the critical value requires
turning to the ANOVA f the degrees of
freedom between are contained in the
columns at the top of the table the
values for the degrees of freedom within
are in the rows the intersection of the
column and the row is the critical value
so for three degrees of freedom between
and 16 degrees of freedom within the
critical value is 3.2 for
and that is written cv 3 comma 16 both
of those in parentheses equals 3.24
let's revisit that assumption of
homogeneity of variance the one way
ANOVA test assumes that there is equal
variance of scores across groups IE
homogeneity of variance homogeneity
means of the same nature or the same
kind of variability Levene's test for
equality of variances is a test of
whether the variances of the samples or
groups are approximately equal or
homogeneous now this is the same test
that we learned about with T tests we
will ask for this when we run the ANOVA
on the computer however as long as your
sample sizes are greater than 30 and the
groups have essentially the same number
of people in each one in other words n 1
equals n 2 equals and 3 then anova is
robust to violations of homogeneity of
variance robust means that the type 1
error rate does not increase if the
assumptions are violated another unique
feature of the ANOVA test is something
called the ANOVA summary table because
there is so much information to know
about in the ANOVA model and the sources
of variance it is much clearer to put
all of this information into a table the
ANOVA summary table tells us seven
things number one the source of
variability between within total that
goes in column one second is the degrees
of freedom also between within and total
degrees of freedom goes in column two
column 3 is the sum of squares
associated with each between within and
total column 4 is variance also called
the mean square so mean square think of
average of the squares the mean square
is the sum of squares divided by the
degrees of freedom for each row
you only need the mean square for the
first two rows between and within fifth
is the F ratio the F ratio is the mean
square between / the mean square within
the F ratio tells us that there is 17.6
19 times more variability between then
there is within groups p less than 0 0 1
as you can see if you have degrees of
freedom and the sum of squares you could
calculate the rest of the table from
these values a sixth is a column for
probability values and another column
for a 2 squared the effect size and just
so that you know those numbers in blue
one through seven above each of the
columns were added for ease of
identifying the columns you should not
include those numbers in your actual and
ova summary table when you use spss to
conduct an ANOVA the results will be
presented to you in an ANOVA table like
this you would include the ANOVA table
in your write-up for your research paper
I have also created this table titled
elements of the one-way ANOVA summary
table as a reference for learning about
the ANOVA summary table this table is a
useful guide for what information goes
in each part of the table for example
under degrees of freedom on the road
between you see k minus 1 that is the
formula for degrees of freedom between
this table is not meant to be explained
in a video but rather to serve as a
reference as you are constructing your
own anova summary table