The ANOVA procedure is one of the most powerful applications in statistics
 
 
 
 
 
 
 

Introduction to ANOVA
 
 
 
 
 
 
 
 
 
 
 
 

What is a factor?
 
 
 
 
 
 

The 1-way 
ANOVA
 
 
 

The 2-way or 3-way ANOVA
 
 
 

Hypotheses that can be tested in an ANOVA
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

The n-way ANOVA
 

The Analysis of Variance (ANOVA)

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. 

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 after and before 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. 

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. 

• Levels may be quantitative numbers or, in many cases, simply "present" or "not present" ("0" or "1"). 

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 one-way or one factor 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 two-way  or two factor ANOVA. Furthermore, we may be interested in a third factor, the effect of time. Now we deal with a three-way or three factor ANOVA. In each of these ANOVA's we test a variety of hypotheses of equality of means (or average responses when the factors are varied). 

First consider the one-way 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 2-way ANOVA, the possible null hypotheses are: 

1 There is no difference in the means of factor A 
2 There is no difference in means of factor B 
3 There is no interaction between factors A and B 

The alternative hypothesis for cases 1 and 2 is: : the means are not equal. 

The alternative hypothesis for case 3 is: there is an interaction between A and B. 

For the 3-way ANOVA: The mains effects are factors A, B and C   The 2-factor interactions are: AB, AC, and BC.  There is also a three-factor 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. 

In addition, in all ANOVA's one can also test if the overall mean is equal to 0, with the alternative of not being equal to 0. 

In general, the number of main effects and interactions  can be found by the following expression: 

In what follows we will only discuss the 1 and 2 way ANOVA.