Chapter 10 Analysis of Variance 393
One-Way Analysis of Variance
Earlier we used the t test to compare two treatment groups, such as two
groups taught by two different methods. What if there are four treatment
groups? We might have 40 subjects split into four groups, with each group
receiving a different treatment. The treatments might be four different drugs,
four different diets, or four different advertising videos. Analysis of variance,
or ANOVA, provides a test to determine whether to accept or reject the
hypothesis that all of the group means are equal.
The model we’ll use for analysis of variance, called a means model, is
y5mi1e
Here, mi is the mean of the ith group, and e is a random error following a
normal distribution with mean 0 and variance s^2. If there are P groups, the
null and alternative hypotheses for the means model are
H 0 : m 15 m 25 c 5 mP.
Ha: Not all of the mi are equal.
Note that the assumptions for the means model are similar to those used
for regression analysis.
The errors are normally distributed.
The errors are independent.
The errors have constant variance s^2.
The similarity to regression is no accident. As you will see later in
this chapter, analysis of variance can be thought of as a special case of
regression.
To verify analysis of variance assumptions, it is helpful to make a plot that
shows the distribution of observations in each of the treatment groups. If the
plot shows large differences in the spread among the treatment groups, there
might be a problem of nonconstant variance. If the plot shows outliers, there
might be a problem with the normality assumption. Independence could
also be a problem if time is important in the data collection, in which case
consecutive observations might be correlated. However, there are usually no
problems with the independence assumption in the analysis of variance.
Regression One-Way Analysis of Variance and
Hotel Prices
Some professional associations are reluctant to hold meetings in New York
City because of high hotel prices and taxes. Are hotels in New York City
more expensive than hotels in other major cities?
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