OBJECTIVE
Verify
whether two or more groups are significantly different.
DESCRIPTION
While with
a t-test we can only test two groups, with ANOVA we can test several groups and
decide whether the means of these samples are significantly different. The
simplest analysis is one-way ANOVA, in which the variance depends on
one factor. Suppose that we want to know whether the age of people buying three
different products is significantly different to focus the promotional
campaigns better. In this case we have just one factor (type of product); therefore,
we run a one-way ANOVA after checking the normality assumption (see 36. INTRODUCTION TO REGRESSIONS).
Output of a One-way ANOVA Analysis
Once the
test is concluded, we check the p-value (< 0.05) and F-value (larger than
the F-critical value) to reject the null hypothesis and infer that the populations
are not equal. In the proposed example, there is a significant age difference
among products. However, we should perform a t-test of each pair of groups to determine
where the difference lies.
In a two-way ANOVA we have two factors to be tested. For example,
we are selling products A, B, and C in three countries (1, 2, and 3). In the
proposed example, we use a two-way ANOVA “without replication,” since only one
observation is recorded for each factor’s combination (we will use an ANOVA
“with replication” if there are more observations recorded for each
combination). In a two-way ANOVA, there are two null hypotheses to be tested,
one for each factor, and it is possible that the hypothesis will be rejected
for one factor but not for the other one.
In our
example we reject the null hypothesis for the factor “type of product” (rows), since
its p-value is smaller than 0.05, but we cannot reject the null hypothesis for
the factor “country” (columns).
Output of a Two-Way ANOVA Analysis
We can also
perform an ANOVA with repeated measures when we have repeated measures within the same
group. In our example a company decided to start a four-week training program for
five employees to diminish the number of errors made at work. In this case the
repeated measures are the errors of each employee in the same week of training.
Results of a Single-Factor ANOVA with Repeated Measures
The
template contains the calculations for a single-factor repeated-measures ANOVA.
In our example, since the p-value is lower than 0.05 (our chosen alpha), we
reject the null hypothesis of no difference among the week’s means and infer
that the training has had an impact on the number of errors.
We can use the
Excel Data Analysis complement to perform a two-factor repeated-measures ANOVA,
choosing the test “Anova: Two-Factor With Replication.” In the proposed example,
we are selling different product versions in different markets, and we want to
test whether either the product or the market (or both) have an impact on the
number of products sold. The results show that, while the kind of product (rows)
affects the sales (p-value < 0.05), the market (columns) does not.
Results of a Two-Factor ANOVA with Repeated Measures
An
extension of ANOVA is MANOVA, which allows the test to be run with more than
one dependent variable. For example, it is possible to run a MANOVA using “level
of education” as the categorical independent variable and “test score” and “yearly
income” as the continuous dependent variables.
TEMPLATE
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