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Foundations of ANOVA – Assumptions and Hypotheses for One-Way ANOVA (12-3) - YouTube
Channel: Research By Design
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Let's look into the assumptions, the
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hypotheses, and how to determine the
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critical value for a one-way ANOVA test.
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Here are the requirements for a one-way
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ANOVA; that is an ANOVA conducted with a
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single factor, typically with three or
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more levels. The one-way ANOVA is a
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parametric procedure. We are using
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sampled data to estimate population
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parameters. Any time we look at data
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drawn from a population, we are using
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parametric procedures. A one-way ANOVA
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can compare three or more groups.
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Whenever your independent variable has
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three or more levels, use ANOVA, not a
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t-test. Although ANOVA is designed for
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three or more groups, ANOVA can be
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conducted with only two groups (i.e.when
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you would typically use an independent
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samples t-test.) In fact, you can convert
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between t and F where t equals the
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square root of F. Just remember that a
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t-test can be either positive or
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negative, unlike the f test that is
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always positive, because it is calculated
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with squared values. On the other hand,
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the sign of a t-value only tells us the
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direction of change; whether the first
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group or the second group has a higher
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mean. If you were to switch the groups,
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the sign would also switch. But if you do
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an ANOVA with just two groups and you
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want to calculate t, just be aware that
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the square root of F will always be
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positive. The independent variable groups
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must be independent. "Independent" means
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that the scores at one level do not
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influence the probability of scores in
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another level. The groups are not
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influencing one another. The samples
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should not be related. If the samples are
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related, then you would use a
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repeated-measures ANOVA. So here are the
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assumptions that must be satisfied for
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us to use the one-way ANOVA.
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The independent variable (the IV) is categorical
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or nominal, with three or more
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groups. The samples should be randomly
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selected. The samples can be randomly
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assigned, such as in an experimental
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design, but you could also use them
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naturally occurring, in what is called a
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"quasi-experimental design." The sample
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sizes for each group should be roughly
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equal. Unequal sample sizes increase the
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likelihood of Type I error rates.
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What do I mean by "roughly equal"? As a
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rule, the largest n divided by the
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smallest n of your groups should not
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exceed 1.5. The dependent variable must
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be quantitative, at the scale level, so
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that's interval or ratio. Now, some people
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will tell you that if you use Likert
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survey scales (like one = "strongly
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disagree" up to 5 = "strongly agree"),
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that those data are ordinal and you have
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to use non-parametric statistics. In fact,
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John Dawes (2008) demonstrated that when
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Likert survey scales have five or more
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item options, they function like scale
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data, and they can be used with
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parametric statistics, like this one way
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ANOVA. Check for outliers in your data
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cleaning, using the Explore command. You
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should delete or Windsorize severe
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outliers. Scores in one group should not
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influence the probability of scores in
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another group. This is called
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"independence." The participants in one
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group are in only one group; they are not
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in neither, they are not in both.
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Normality: the scores on the dependent
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variable within each group should be
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approximately normally distributed. The
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Shapiro Wilk test, and/or a QQ plot, or a
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PP plot ,or a box plot are all used to
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test for normality. See my video about
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the Central Limit Theorem for a fuller
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discussion about the implications of
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normality. If the dependent variables are
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not normally distributed, or if the
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sample sizes are not roughly equal, use
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the nonparametric Kruskal-Wallis
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one-way ANOVA test. However, the parametric
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one-way ANOVA that we are learning about
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now is "robust" as long as there is a
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minimum of 30 subjects in each sample.
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"Robust" means that the Type I error rate
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does not increase if the assumptions are
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violated. Also, the groups should have
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Homogeneity of Variance, which you will
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test for using Levene's test for
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equality of variances, just as we did
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with the t-test. If the dependent
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variable groups fail this assumption,
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rerun the ANOVA and report Welch's ANOVA
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instead. Here are the hypotheses for a
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one-way ANOVA. Like a t-test, the null
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hypothesis for ANOVA is that all samples
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are drawn from a population whose means
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are equal; therefore, all of the sample
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means are the same. The null
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hypothesis might be written as H0:
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mu 1 equals mu 2, equals mu 3. The
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alternative hypothesis is that the means
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of the three groups differ. And remember
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we do not know which means differ, so the
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alternative hypothesis might be written
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as H1: mu 1 does not equal mu 2,
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does not equal mu 3. Unless instructed
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otherwise, you should use the alpha of
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0.05 for your level of significance for
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the Omnibus ANOVA test. The critical
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value will be determined based upon your
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degrees of freedom and the Alpha level
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both of which are needed to use the
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ANOVA FTL I'll explain this in detail
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next you can safely assume alpha equals
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point 05 two-tailed test on the other
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hand you might protest that ANOVA is
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always a one-tailed test because the F
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distribution has only one tail true but
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the ANOVA is an omnibus test and the
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post-hoc tests are the ones that really
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matter we could use the post-hoc tests
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to hypothesize directional group
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differences but instead we're just going
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to assume that all groups are equal and
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let the post-hoc tests tell us which
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groups
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differ and in which direction if we
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wanted to do directional post-hoc tests
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we would actually be better off to use a
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different technique called planned
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contrasts more about that in another
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video finding the critical value for a
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one-way ANOVA requires knowing the
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degrees of freedom for the numerator and
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the denominator of the F ratio these are
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called the degrees of freedom between
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the degrees of freedom within and the
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degrees of freedom total the degrees of
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freedom between equals K minus 1 where k
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is the number of categories or levels
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this is the degrees of freedom for the
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numerator so in our example where we
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have 20 people who are randomly assigned
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to four groups k equals 4 so the degrees
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of freedom between would be four
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categories of diet minus one or three
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degrees of freedom within equals n minus
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K where n is the total number of
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participants and K is the total number
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of categories this is the degrees of
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freedom for the denominator the degrees
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of freedom within would be the total
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number of participants 20 minus the
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total number of categories for or 16 the
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degrees of freedom total equals n minus
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1 as it always does with n being the
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total number of participants in all
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groups there are 20 participants so the
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degrees of freedom total would be 19
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finding the critical value requires
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turning to the ANOVA f the degrees of
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freedom between are contained in the
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columns at the top of the table the
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values for the degrees of freedom within
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are in the rows the intersection of the
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column and the row is the critical value
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so for three degrees of freedom between
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and 16 degrees of freedom within the
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critical value is 3.2 for
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and that is written cv 3 comma 16 both
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of those in parentheses equals 3.24
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let's revisit that assumption of
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homogeneity of variance the one way
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ANOVA test assumes that there is equal
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variance of scores across groups IE
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homogeneity of variance homogeneity
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means of the same nature or the same
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kind of variability Levene's test for
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equality of variances is a test of
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whether the variances of the samples or
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groups are approximately equal or
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homogeneous now this is the same test
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that we learned about with T tests we
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will ask for this when we run the ANOVA
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on the computer however as long as your
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sample sizes are greater than 30 and the
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groups have essentially the same number
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of people in each one in other words n 1
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equals n 2 equals and 3 then anova is
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robust to violations of homogeneity of
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variance robust means that the type 1
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error rate does not increase if the
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assumptions are violated another unique
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feature of the ANOVA test is something
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called the ANOVA summary table because
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there is so much information to know
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about in the ANOVA model and the sources
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of variance it is much clearer to put
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all of this information into a table the
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ANOVA summary table tells us seven
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things number one the source of
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variability between within total that
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goes in column one second is the degrees
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of freedom also between within and total
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degrees of freedom goes in column two
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column 3 is the sum of squares
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associated with each between within and
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total column 4 is variance also called
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the mean square so mean square think of
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average of the squares the mean square
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is the sum of squares divided by the
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degrees of freedom for each row
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you only need the mean square for the
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first two rows between and within fifth
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is the F ratio the F ratio is the mean
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square between / the mean square within
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the F ratio tells us that there is 17.6
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19 times more variability between then
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there is within groups p less than 0 0 1
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as you can see if you have degrees of
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freedom and the sum of squares you could
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calculate the rest of the table from
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these values a sixth is a column for
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probability values and another column
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for a 2 squared the effect size and just
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so that you know those numbers in blue
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one through seven above each of the
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columns were added for ease of
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identifying the columns you should not
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include those numbers in your actual and
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ova summary table when you use spss to
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conduct an ANOVA the results will be
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presented to you in an ANOVA table like
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this you would include the ANOVA table
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in your write-up for your research paper
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I have also created this table titled
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elements of the one-way ANOVA summary
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table as a reference for learning about
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the ANOVA summary table this table is a
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useful guide for what information goes
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in each part of the table for example
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under degrees of freedom on the road
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between you see k minus 1 that is the
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formula for degrees of freedom between
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this table is not meant to be explained
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in a video but rather to serve as a
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reference as you are constructing your
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own anova summary table
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