ANALYSIS OF VARIANCE

  I.  Introduction.
        A.  In independent measures t-tests, the means of two
            independent samples (groups) on some interval variable
            were compared with the t-test indicating whether the
            means of the groups were different. This is fine when you
            only have two groups, such as with gender. However, when
            you have more than two groups, a different type of test
            is required.
        B.  Analysis of variance (ANOVA) is a statistical test in
            which the means of 3 or more groups on some interval
            level variable can be compared simultaneously.  It
            indicates whether at least one of the group means is
            significantly different from the others.
        C.  In analysis of variance, the total variance in the
            combined sample (all groups taken together) is
            partitioned (divided) into two parts.
              1.  The first part represents the amount of variation
                  that exists within each of the the groups being
                  compared.
              2.  The second part represents the amount of variation
                  between the groups being compared.
              3.  The ratio of the variation between the groups to
                  the variation within the groups is called an
                  F-ratio.  The F-ratio is used in Anova to determine
                  if the means of the groups being compared can be
                  considered equal.

 II.  Doing analysis of variance.
        A.  State the null hypothesis.
        B.  Choose a statistical test.
        C.  Check assumptions.
              1.  Three or more independent random probability
                  samples.
              2.  Interval-ratio variable being compared across three
                  or more categories defined by a categorical
                  variable.
              3.  Populations are normally distributed.
              4.  Population variances are equal.
                    a.  Sample size of each group is relatively large
                        (30 or more).
                    b.  Sample sizes of the groups approximately
                        equal.
        D.  Select an alpha level.
        E.  Calculating the F-ratio and making a decision about the
            null hypothesis.
              1.  Calculate the statistic.
              2.  Determine the probability associated with the
                  statistic.
              3.  Compare probability to alpha level.
              4.  Decide about the null.

III.  Limitations and extensions of the one-way analysis of variance.
        A.  The analysis of variance test just performed is called a
            one-way analysis of variance because means are being
            compared across a single set of groups or categories.
            Extensions anova also allow for comparisons across
            several different sets of groups or categories, called
            multiple analysis of variance or MANOVA, and for the
            effects of interval variables (besides the dependent
            variable) to be controlled, called analysis of
            covariance or ANCOVA.
        B. Analysis variance only tells you if at least one of the
           groups being compared differs from the others.  It does not
           tell you which groups differ from each other.  However, there
           are additional tests that can be performed that do indicate
           such differences.  These are called multiple comparison
           tests.