# Anova

Introduction

Anova (Analysis of variance) procedures are particular cases of more general procedures, called Factorial Experiments.

The experiments usually involve one (Anova)  or more (Manova) response variables, and one (One-Factor Anova) or two (Two-Factor Anova), or p ( 2P-Factorial Experiments) independent variables called factors
These factors may have different levels or treatments. Note that in the case of 2p-factorial experiments, there are 2 treatments for each factor.
The units assigned to each treatment are called replicates
When equal number of units are assigned to each treatment, the design is said to be balanced and will be unbalanced otherwise.

Examples of Anovas.

1. Do four brands  of gasoline have different effects on automobile fuel efficiency ?
 Factor Levels or  Treatments Response Gasoline Brand1, Brand2, Brand3, Brand4 Fuel Efficiency

2. Is there any difference in crop yields  when five different fertilizers are used?
 Factor Levels or  Treatments Response Fertilizer Fertilizer 1, Fertilizer 2, Fertilizer 3, Fertilizer , Fertilizer 5 Crop Yield

3. Is there any difference in patients response  for different doses of a given drug?
 Factor Levels or  Treatments Response Dose Dose 1, Dose 2, ....., Dose n Patient Response

4. Do four brands  of gasoline have different effects  on the  fuel efficiency  for three different car makes?
 Factor Levels or  Treatments Response Gasoline Brand1, Brand2, Brand3, Brand4 Fuel Efficiency Car Car 1, Car 2, Car 3

The goal of Anova procedures  is to compare the treatments means.

One-Factor Anova (or One-way Anova)

The One-factor Anova assesses if there is a statistical significance difference in the treatments of one  factor on one response variable.
The hypotheses to be tested are:
Assumptions for One-Factor Anova

1. The treatments populations must be normal.
2. The treatments populations must all have the same variance.

Application (One-Factor Anova)

 R SAS Minitab>
Two-Factor Anova (or Two-way Anova)

Mathematical Setting:
The Two-factor Anova assesses if there is a statistical significance difference in the treatments of two  factors on one response variable.
We shall call one factor the row factor with n levels  and the other the column factor with m levels.

For any level i of the row factor, let the average of all treatments means be

For any level j of the column factor, let the average of all treatments means be

Let the average of all treatments mean be
Define the j-th column effect as

Define the  ij-interaction effect effect as

Hypotheses to be tested:

1. Test if the additive model hold, that is, we test if the null hypothesis that all the interactions are equal to zero:
2. Test if the row effects are equal to zero:

3. Test if the column effect are equal to zero:

Assumptions for two-Factor Anova

1. The design must be complete.
2. The design must be balanced.
3. The number of replicates per treatment must be at least 2.
4. Within any treatment, the observations are simple random samples from a normal population.
5. The population variance is the same for each all treatments

Application (Two-Factor Anova)

 R SAS Minitab