7.
Product and Process Comparisons
7.4. Comparisons based on data from more than two processes


Test equality of means  The procedure known as the Analysis of Variance or ANOVA is used to test hypotheses concerning means when we have several populations.  
The Analysis of Variance (ANOVA)  
The ANOVA procedure is one of the most powerful statistical techniques 
ANOVA is a general technique that can be used to test the hypothesis
that the means among two or more groups are equal, under the
assumption that the sampled populations are normally distributed.
A couple of questions come immediately to mind: what means? and why analyze variances in order to derive conclusions about the means? Both questions will be answered as we delve further into the subject. 

Introduction to ANOVA 
To begin, let us study the effect of temperature on a passive
component such as a resistor. We select three different
temperatures and observe their effect on the resistors. This
experiment can be conducted by measuring all the participating
resistors before placing \(n\)
resistors each in three different
ovens.
Each oven is heated to a selected temperature. Then we measure the resistors again after, say, 24 hours and analyze the responses, which are the differences between before and after being subjected to the temperatures. The temperature is called a factor. The different temperature settings are called levels. In this example there are three levels or settings of the factor Temperature. 

What is a factor? 
A factor is an independent treatment variable whose settings
(values) are controlled and varied by the experimenter. The
intensity setting of a factor is the level.


The oneway ANOVA  In the experiment above, there is only one factor, temperature, and the analysis of variance that we will be using to analyze the effect of temperature is called a oneway or onefactor ANOVA.  
The twoway or threeway ANOVA  We could have opted to also study the effect of positions in the oven. In this case there would be two factors, temperature and oven position. Here we speak of a twoway or twofactor ANOVA. Furthermore, we may be interested in a third factor, the effect of time. Now we deal with a threeway or threefactor ANOVA. In each of these ANOVA techniques we test a variety of hypotheses of equality of means (or average responses when the factors are varied).  
Hypotheses that can be tested in an ANOVA 
First consider the oneway ANOVA. The null hypothesis is: there
is no difference in the population means of the different levels of
factor \(A\) (the only factor).
The alternative hypothesis is: the means are not the same. For the twoway ANOVA, the possible null hypotheses are:
The alternative hypothesis for case 3 is: there is an interaction between \(A\) and \(B\). For the threeway ANOVA, the main effects are factors \(A\), \(B\), and \(C\), and the twofactor interactions are \(AB\), \(AC\), and \(BC\). There is also a threefactor interaction, \(ABC\). For each of the seven cases the null hypothesis is the same: there is no difference in means, and the alternative hypothesis is the means are not equal. 

The \(n\)way ANOVA 
In general, the number of main effects and interactions can be
found by the following expression:
$$ N = \left( \begin{array}{c} n \\ 0 \end{array} \right) +
\left( \begin{array}{c} n \\ 1 \end{array} \right) +
\left( \begin{array}{c} n \\ 2 \end{array} \right) + \ldots +
\left( \begin{array}{c} n \\ n \end{array} \right) \, . $$
The first term is for the overall mean, and is always 1. The second
term is for the number of main effects. The third term is for the
number of twofactor interactions, and so on. The last term is for
the \(n\)factor
interaction and is always 1.
In what follows, we will discuss only the oneway and twoway ANOVA. 