Introduction

Different types of ANOVA

Assumption of ANOVA

Hypothesis of ANOVA

F-statistics and F-test

Techniques of analyzing variance

Application of ANOVA

References

ANOVA is the abbreviation for the full name of the method:

Analysis of variance. Invented by Ronald Fischer.

Analysis of variance (ANOVA) is a collection of statistical

models and their associated estimation procedures (such as the

“variation” among and between groups) used to analyze the

differences among group means in a sample.

Example:

A group of psychiatric patients are trying three different

therapies: counseling, medication and biofeedback. You want

to see if one therapy is better than the others.

A one way ANOVA is used to compare two means from two

independent (unrelated) groups using the F-distribution.

When to use a one way ANOVA

Situation : You have a group of individuals randomly split into

smaller groups and completing different tasks. For example, you

might be studying the effects of tea on weight loss and form

three groups: green tea, black tea, and no tea.

For example, adventurous researchers studying a

population of walruses might ask “Do the walruses

weigh more in early or late mating season?

Here, the independent variable or factor (the two

terms mean the same thing) is “month of mating

season”.

In an ANOVA, our independent variables are

organised in categorical groups.

For example, if the researchers looked at walrus

weight in December, January, February and March,

there would be four months analyzed, and therefore

four groups to the analysis.

In a one-way ANOVA there are two possible

hypotheses.

The null hypothesis (H0) is that there is no

difference between the groups and equality

between means. (Walruses weigh the same in

different months)

The alternative hypothesis (H1) is that there is a

difference between the means and groups.

(Walruses have different weights in different

months)

Normality – That each sample is taken from a

normally distributed population

Sample independence – that each sample has been

drawn independently of the other samples

Variance Equality – That the variance of data in the

different groups should be the same

Your dependent variable – here, “weight”, should be

continuous – that is, measured on a scale which can

be subdivided using increments (i.e. grams,

milligrams)

A one way ANOVA will tell you that at least two

groups were different from each other. But it

won’t tell you what groups were different.

A two-way ANOVA is, like a one-way ANOVA, a

hypothesis-based test.

It examines the influence of two different categorical

independent variables on one continuous dependent

variable.

The two-way ANOVA not only aims at assessing the

main effect of each independent variable but also if

there is any interaction between them.

Thinking again of our walruses, researchers might use a

two-way ANOVA if their question is: “Are walruses

heavier in early or late mating season and does that

depend on the gender of the walrus?”

In this example, both “month in mating season” and

“gender of walrus” are factors –i.e, there are two

factors. Further, each factor’s number of groups must

be considered – for “gender” there will only two

groups “male” and “female”.

The two-way ANOVA therefore examines the effect of

two factors (month and gender) on a dependent variable

– in this case weight, and also examines whether the

two factors affect each other to influence the

continuous variable.

Your dependent variable – here, “weight”, should

be continuous – that is, measured on a scale which

can be subdivided using increments (i.e. grams,

milligrams)

Your two independent variables – here, “month”

and “gender”, should be in categorical,

independent groups.

Sample independence – that each sample has

been drawn independently of the other

samples

Variance Equality – That the variance of data

in the different groups should be the same

Normality – That each sample is taken from a

normally distributed population

There are three pairs of null or alternative hypotheses for

the two-way ANOVA. Here, for walrus experiment, where

month of mating season and gender are the two

independent variables.

H0: The means of all month groups are equal

H1: The mean of at least one month group is different

H0: The means of the gender groups are equal

H1: The means of the gender groups are different

H0: There is no interaction between the month and gender

H1: There is interaction between the month and gender

The key differences between one-way and two-way

ANOVA are summarized clearly below.

1. A one-way ANOVA is primarily designed to enable the

equality testing between three or more means. A two-way

ANOVA is designed to assess the interrelationship of two

independent variables on a dependent variable.

2. A one-way ANOVA only involves one factor or

independent variable, whereas there are two independent

variables in a two-way ANOVA.

3. In a one-way ANOVA, the one factor or

independent variable analyzed has three or more

categorical groups. A two-way ANOVA instead

compares multiple groups of two factors.

4. One-way ANOVA need to satisfy only two

principles of design of experiments, i.e. replication and

randomization. As opposed to Two-way ANOVA,

which meets all three principles of design of

experiments which are replication, randomization, and

local control.

If you are comparing means between more than two groups,

why not just do several two sample t-test to compare their

mean from one group with the mean from each of the other

group?

Before ANOVA, this was the only option available to

compare means between more than two groups.

The problem with the multi t-test approach is that as the

number of groups increases, the number of two sample t-

test also increases.

As the number of test increases the probablity of making a

type1 error also increases.

The F-test can assess the equality of variances. F-tests are

named after its test statistic, F, which was named in honor

of Sir Ronald Fisher.

The F-statistic is simply a ratio of two variances. Variance

is the square of the standard deviation.

To use the F-test to determine whether group means are

equal, it’s just a matter of including the correct variances in

the ratio. In one-way ANOVA, the F-statistic is this ratio:

F = variation between sample means / variation within

the samples

The technique of analysing variance in case of single

variable and in case of two variable is similar.

In both cases a comparision is made between the variance

of sample means with the residual variance.

However, in case of single variable, the total variance is

divided into two two parts only,viz..,

Variance between the samples and the variance within the

samples

The later variance is the residual variance. In case of

two variables the total variance is divided in three

parts viz..,

1. Variance due to variable no.1

2. Variance due to variable no.2

3. Residual variance

This is particularly applicable to experiment otherwise

difficult to implement such as is the case in clinical trails.

In the bioequivalence studies the similarities between the

samples will be analyzed with ANOVA only.

Pharmacovigillance data can also be evaluated using

ANOVA.

Pharmacodynamic data can also be evaluated with ANOVA

only.

That means we can analyze our drug is showing

pharmacological action or not’

Pharmacy

Biology

Microbiology

Agriculture

Statistics

Marketing

Business research

Finance

Mechanical calculation