We have previously discussed methods of testing whether
- Two populations have equal means (independent two-sample tests).
- The means from two measurements on one population are the same
(paired tests).
Analysis of Variance (ANOVA) allows us to extend this to more than two
populations or measurements (treatments/). That is, we can test
the following:
- Are all the means from more than two populations equal?
- Are all the means from more than two treatments on one population
equal? (This is equivalent to asking whether the treatments have any
overall effect.)
To set our notation, let I be the number of populations or
treatments being compared and let 
be the I means. Then the hypotheses for testing are
To test these hypotheses, we require a random sample from each
population or treatment.
NOTE:\ For computational
purposes, the ANOVA equations for the multiple population case and the
multiple treatment on one population case are the same. However, the
interpretation of hypotheses and results is slightly different. Thus,
- Multiple populations:
is the true mean of population i.
- Multiple treatments:
is the true average response when treatment i is applied.
 | One-way
ANOVA
Recall that we have I populations or I treatments on
one population and we wish to test whether the means for all I
groups are equal. The notation we will use for our random variables and
observed data values is:
 | 
: The random variable denoting the jth measurement from the ith
population or treatment.
| 
: The observed value of
when the experiment is performed. |
|
|
| The
Model
|
The mathematical model describing analysis of variance can be
expressed in two different ways. First,
where 
is an overall mean, 
is the effect due to treatment i, and 
. Note that the treatment effects satisfy the condition 
. If we define the ith treatment or population mean as 
, our model can be expressed as
| The
ANOVA Table
|
In this class, we will not be concerned with formulas for
computing ANOVA quantities. Instead, we will let Stataquest do that
for us. However, we will need to learn how to interpret and test
using Stataquest output.
The traditional way to present ANOVA results is in an ANOVA table
such as the one given below.
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| Computer
Lab for Week 10
|
Applicable StataQuest Commands:
Labs 
Calculating One-Way ANOVA
Statistics
ANOVA
One-way
| Concept
Lab for Week 10
|
| Ch 17: Between and Within Variation |
%
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