Example:

We want to test how does the regularity of
feeding with a liquid fertilizer affects the growth of tomato plants?” Thus, 20
plants were selected be fed at 5 different rates. The total 100 plants should
be assigned at random to the 5 feeding rates.

If we
introduce another tomato plant, which has, been genetically modified from our
original strain. We repeat the same set-up that we used for the first plant
with the second.

We are
varying two factors: the level of feeding and type of tomato.

Now it is a **two-factor
design (two-way design)**.

If we have
set up the experiment identically for the two strains, we have a **fully-crossed
factored (fully crossed) design**.

We have all
possible combinations of the two factors. We have individuals of both strains
at all feeding rates.

If we could
only apply three of the feeding rates to the new strain, but still used all
five for our original strain, then our design would be **incomplete design**.

The statistical
analysis of fully crossed design is easy to do, whereas analyzing incomplete
designs is significantly more difficult.

Avoid
incomplete designs whenever possible

An n-factor
or n-way design varies n different independent factors and measures response to
these manipulations.

Example

If we want to
examine the effects of both diet and exercise regime on dog’s health, then it
would be a two-factor design. If we consider three different diets and four
different exercise regimes then there are three levels to factor “diet” and
four levels to factor “exercise”.

It would be a
3X4 two-factor (way) design

For the
tomato example:

It is a fully
cross-factored, two factor design with five levels for factor One and two for
factor two.

5X2 fully
crossed two way design

Thus we have
10 treatment groups, and as long as we have the same number of plants in each
treatment and more than one plant in each treatment then we have a fully
replicated and balanced design.

Give example
of three factor or four factor design.

If we want to
investigate whether factor **A** is influenced by independent factors **B**
and **C**. if the value of **A** is affected by the value of **B**
when the value of **C** is kept constant, then we can say that there is a
main effect due to factor **B**. Similarly if **A** is affected by **C**
when **B** is kept constant, then there is a main effect due to **C**. If
the effect of **B** on **A** is affected by the value of **C**, or
equivalently if the effect of **C** on **A** is affected by the value of **B**,
then there is an interaction between the factors **B** and **C**.

Why not do
two separate one-way design for the two different
plants?

The answer

1)
Two-way design allows us to
investigate the main effects in a single experiment (whether growth rate
increases with feeding rate in both species and whether one species grows
faster then the other when both are given the same feeding rate).

2)
It allows us to examine the
interaction effect (whether the two species differ in the effect that feeding
rate has on their growth).