Story 16 — Analysis of Variance Factorial Designs

Now it will be easy for us to

understand two-way ANOVA.

An example of a two-way

ANOVA

A psychiatrist wanted to see

whether a combination of wine

and vitamin C may have an effect

on depression.

He randomly selected 10 male

patients, and also 10 female

patients, and randomly assigned

them in two groups: wine group, or

vitamin C group.

.

The layout of this experiment is

presented in the next table:

Look at subject 1. This subject is

influenced by two variables. Male

gender, and also wine. The score

of depression that he will give, will

be the result of these two factors.

For this reason, we call this type of

experiment a two-factor

experiment. The same, of course,

holds for all subjects. They are, in

a way, under crossfire. Two

factors hit them.

The layout above can also be

given in a more abstract form.

Variable A is gender, variable B is

nutrition. Each variable has two

levels, a1 a2 and b1 b2

We say: We have two variables,

A and B. A is gender, B is nutrition.

Each of these two variables has

two levels. a1, a2, and b1, b2.

Because in this experiment we

use 2 variables with 2 levels each,

we call this experiment 2 x 2

factorial. We read this as follows:

two by two factorial.

The ANOVA summary table for

two-factor experiments is the

following:

ANOVA SUMMARY TABLE

Two-Way, 2x2 Factorial

* Also called interaction

Things are getting complicated, I

hear you say.

I say: You already know everything

in this new ANOVA.

Our approach of understanding

the concepts and not memorizing

formulas has paid out.

Why do we have two Between

terms, A and B? you say.

Because here we have two

variables: gender, and nutrition,

i.e., A and B. We want to know if

gender (being male or female) has

an effect, and also if nutrition (wine

or vitamin C) has an effect.

Remember, the Between term is

the term that senses the effects of

our treatments.

The Within term we also know. It is

the variance of each group

separately. The sum of these

variances.

The Total term we also know. It is

simply the variance of all scores

without regard to what group they

came from.

The interaction term is a Between

term for cells taken diagonally:

mean for a1b1+a2b2 and mean

a1b2+a2b1. Look at the layout to

visualize this.

What is new here is the concept of

the interaction term. We need to

develop this concept, so we get a

gut feeling for it.

When you give two treatments to

subjects, one of the things you

want to see is whether the two

variables interact with each other. 

To begin developing the concept

of interaction, let us consider a

simple experiment:

We give 5 mg of an anti-anxiety

drug, such as diazepam, and find

that this results in an increase in

the time patients sleep. This

increase is 2 hours.

Using different subjects, we find

that 200 ml of wine increase sleep

time by 1 hour.

Now if we give both 5 mg of

valium and 200 ml of wine, is it

sure that we will get 3 hours

increase in sleep time? Perhaps

yes, perhaps no. We know that

drugs may interact and produce

dramatic results, if given together.

You may have heard of cases in

which diazepam taken together

with alcohol caused coma, and

even death, because of

potentiation.

Students find the concept of

interaction difficult. For this reason

I will give an example later.

For the purposes of calculation of

this term in the ANOVA, there is no

problem. The df, as you would

expect, is the df of A x the df for B.

The SS you can calculate by

subtraction. SS total-(SS Between
 

A+SS Between B+SS within).

Alternatively, you can compute the

SS for AxB the same way you

calculated the between term, but

here calculate two means

diagonally, i.e. 

mean for

a1b1+a2b2 

and mean for

a2b1+a1b2. 

Then we proceed with

the calculation of the variance of

these means.

The type of ANOVA design we are

discussing here is called factorial,

because in designing the

experiment we produce all

possible combinations.

In the above example we have:

Male - Wine, Male Vitamin C

Female - Wine, Female Vitamin C

Read this several times, it sounds

like a nursery rhyme. There is a

symmetry in it.

Visualizing the layout of

factorial designs

You will often come across

experiments that use these

designs, and if you go to graduate

school there is good chance you

will use them in your research.

We need to be able to visualize

the designs in order to understand

and evaluate them. A key

part of the task of a scientist is to

be able to critically evaluate the

research of others. Regrettably,

even reputable journals publish

research that is not sound.

We have considered so far a 2x2

design. How do we visualize this?

We see two characters (forget that

it is the number 2 here) separated

by the symbol x which stands for

times.

We have two things, two

variables, we therefore write down

A also B.

A                B

Now we look again at 2x2 and this

time pay attention to what number

we have. Here we have 2.

We therefore write

A

a1 a2

Then we look at the number after

the x. It is also 2 (mind you it does

not have to always be 2, it can be,

4, 10 any number).

We therefore write

B

b1 b2

To sum up:

a1b1 a1b2

a2b1 a2b2

Read this several times, it sounds

like a nursery rhyme. There is a

symmetry in it.

This is how we visualize a 2x2

factorial:

a1b1      a1b2

a2b1      a2b2

Now let us consider this: 2x3 a1b2

a2a2b2

How do we visualize this? We

see two characters (forget that it is

the numbers 2 and 3 here)

separated by the symbol x which

stands for times. We have two

things, two variables, we therefore

write down A and also B.

A           B

Now we look again at 2x3 and this

time pay attention to what

numbers we have. Before x we

have 2. We therefore write:

A

a1 a2

Then we look at the number after

the x. It is 3 (mind you, it does not

have to always be 3, it can be 4,

10, any number).

We therefore write:

B

b1 b2 b3

This is how we visualize a 2x3

factorial:

a1b1      a1b2   a1b3

a2b1      a2b2   a2b3

Now let us consider this: 2x3x5

How do we visualize this? We see

three characters (forget that it is

the numbers 2 and 3 and 5 here)

separated by the symbol x which

stands for times. We have three

things, three variables, we

therefore write down A, B, 

also C.

A     B     C 

Now we look again at 2x3x5 and

this time pay attention to what

numbers we have. Before x we

have 2.

We therefore write

A

a1 a2

Then we look at the number after

the x. It is 3 (mind you, it does not

have to always be 3, it can be, 4,

10, any number).

We therefore write

B

b1 b2 b3

Then we look at the third

number after the x. It is 5 (mind

you it does not have to always be

5, it can be, 6, 28, any number).

We therefore write

C

c1 c2 c3 c4 c5