- [Host] This screen cast is going to introduce you
to the simplest factorial design
and that is the two squared factorial design.
We're gonna be working through an example here
to introduce you to this design.
You're an agricultural scientist
investigating productivity of wheat.
The controlled environmental conditions in your lab
allow for you to easily study the effects
of different parameters on the productivity.
So we're gonna start with
two simple predictor or regressor variables.
We're gonna start with A, which is the amount of water,
and we're gonna choose a low level and a high level.
In a two squared factorial design,
we're choosing two factors
and two levels of each of those factors.
Our second factor is going to be sunlight
and we're gonna choose two levels,
a low level and a high level.
The response is going to be the wheat produced per day.
So this is a two squared design.
Again, we have two factors, that's water and sunlight.
Each of those factors we're looking at two different levels,
a low level and a high level.
And so we can have four different combinations.
We can have low sunlight and water,
low water, high sunlight, high water, low sunlight,
and high levels of both water and sunlight.
And what you do after setting up the experiment,
you collect data, essentially your responses,
conducted at each of these combinations of our factors.
So we get our results,
and here I have on the X axis, this is our first factor.
That's A, and we have productivity on the Y axis.
And we might get something like this,
where at the low level of sunlight, this is our factor B,
you can't present two factors
and the response on a two dimensional plot like this.
But I'll show you how we can present this
as a three dimensional plot here in a moment.
At the low level of watering,
we have a certain productivity,
and at the high level of water,
we have a increased productivity.
Both of these on this blue line again, are for low sunlight.
If we take a look at the high level of our second factor B,
we can also see the same sort of increase in productivity
between the low level of watering
and the high level of watering.
And so this is sort of how we can analyze our results.
By the way, when the lines are parallel like this,
this means we have no interaction between factors A and B.
We refer to the response, so this is an actual number,
at the low level of both factors as 1 in parenthesis.
That doesn't mean it's the value of 1,
that's just what we refer to that productivity,
the actual value as.
At the upper level of the first factor A,
low level of factor B, we refer to that as little a,
at the low level of factor A,
the high level of factor B we refer to that as little b,
ab, that corresponds to a high level of both factors,
A and B.
We can also sometimes have cases
where those lines are not parallel
and this means that there's some sort of interaction.
And so when we see that,
as long as that line is significantly different in slope
than the blue line, then we have interaction,
and I'll talk a little bit more
about interaction here in a moment.
We could plot these situations on a three dimensional plot.
This is known as a response surface.
Here we've got on the X axis we have watering.
On the Y axis, we have sunlight.
On the Z axis, we have productivity.
And essentially this is a flat sheet.
The red line has a higher productivity than the blue line
but they're all sort of in the same plane.
And so this means we have no interaction.
If you do have interaction, then the red line
and blue line are not parallel to one another
and you get sort of this curved sheet.
And so basically you can think of this
as a piece of paper that kind of has a fold in it.
So let's take a more detailed look
at the two squared factorial design.
We have factor A and factor B,
A is at a low and high value,
B is at a low value and a high value,
and we collect data responses.
The actual responses are denoted by 1 in parenthesis,
little a, little b, and little ab.
So we collect those responses, two factors, two levels each.
This results in four experiments.
We can have replicates of each of these combinations.
So N, little n would be the number of replicates.
The various experimental arrangements
are referred to as treatments
and so you'll see me refer to those as treatments.
We can put this in a table.
We have our various treatments.
This essentially is our response.
And then we might have our first variable X1,
and our second variable X2.
And those correspond to our high and low values
of factors A and B.
These factors may be quantitative or qualitative.
These could be categorical in nature, just fyi.
Alternatively, we can write this
in terms of coded variables.
And this is actually more common
and this is the way that I will be doing things.
Instead of using X1 and X2,
as our actual values of the factors,
we can code them into being negative one
for their low levels
and positive one for their high levels.
So we might have our coded variables X one prime
and X two prime for A and B respectively.
So let's take a look at what we refer to as effects.
Here we've got our data plotted.
Again, we have our low sunlight in blue,
our high level of sunlight in red.
The effect of A, A is watering, remember,
is the average increase in output for both levels of B,
sunlight in going from the low level to high level of A.
If we look at this graphically,
we have an effect of A at the low level of sunlight.
Again, sunlight is factor B,
so that's X at the high level of sunlight,
factor B, we have an increase in productivity of Y,
when we go from the low to high level of watering.
And so this effect, we refer to this as the effect of A,
is just the average of X and Y here.
We can refer to X as a minus one in parentheses.
We can also refer to Y as ab minus b.
And so we can rewrite the effect like this
and sometimes you might see it written
in another way such as this.
So we can calculate the effect of factor A,
which is the average effect on productivity,
when we go from the low to high level of factor A.
The effect of B is the average increase
in output for both levels of A,
in going from the low level to high level of B.
So graphically f is the increase in productivity
when going from the low level to high level of factor B.
Remember, B is sunlight when watering is at its low level,
g here is the increase in productivity
going from the low level to high level of factor B,
when watering or factor A is at its high level
and so that's known as the effect of B.
That's just the average of f and g here.
F is b minus one, and g is ab minus a.
And so that's the effect of b.
You'll also see it sometimes written like this.
Now let's take a look at something
known as the interaction effect.
If the increase in output
when going from low to high levels of B
is not the same at the low and high levels of A
as shown here, in other words, these lines are not parallel,
then there is an interaction effect.
The interaction effect of ab is given graphically,
as the difference between g and f here.
We can substitute in for g,
g is ab minus a, f is b minus one.
And so that's known as the interaction effect.
And sometimes, again, you'll see it written differently.