- [Instructor] Now that you've been introduced
to blocking effects,
in this screencast, I'm going to explain how you can account
for blocking and something known
as confounding in the two to the K design.
Ideally, you would run all the experiments
in a two to the K factorial design
under homogenous conditions.
The ideal is often not possible.
Experiments will often be blocked according
to some extraneous factor.
I've talked about these
in that previous screencast on blocking.
But we can have multiple equipment setups,
different personnel, different raw materials,
and the temporal conditions can change,
and there are other factors.
Block size may be smaller
than the number of runs in a complete replicate.
And in analyzing the results,
the block effects will be confounded
with certain factor effects.
And I'll explain what this means here in a few minutes.
Let's go back and revisit an example that I've talked
about a couple of times.
You're an agricultural scientist investigating productivity
of wheat.
We have two different factors, watering and sunlight,
and the response is wheat produced per day.
So you go out there, you collect some data
with these four different experimental combinations
and here's what we get.
Now, what if there are two combines that harvest the wheat?
So a combine is an actual tractor that harvests the wheat.
So maybe we have an older one up here on the left,
and we have a brand new super efficient one down here
on the right.
And in order for you to harvest the wheat on time
you have to have half of it be harvested
by this old one, and half of it harvested by the newer one.
There's a couple of different ways that we can split
our experimental runs into the old combine
and the new combine.
In other words, how we can block the experiments.
So here's one, we could harvest the ones on the left here
with the old combine and the ones on the right
with the new combine.
This means that the experiments with low water
would be harvested by the old combine, and those
with high water would be harvested by the new combine.
And you might be thinking that there's already some issues
with this setup.
Or we could do something similar,
where we harvest the top two experiments with the old one,
and the bottom two with the new.
And finally, we could put the two different blocks
on the corners of this two squared design.
And it turns out that this is the best arrangement
because each block, the old combine and the new combine,
has a low and a high value of A
and a low and a high value of B.
Any effects will cancel each other out, at least
for determining main effects which are most important.
So let's look at the math behind this.
If we went with this arrangement where we put
the left two of this square in the old combine block
and the right two in the new combine block,
we could compute the contrasts.
And here we're defining this term delta.
Delta is our blocking effect.
It's the increase in productivity
between the old combine and the new combine.
So even if we harvested wheat
from the same exact field that had been grown
under the same exact conditions,
the new combine would have a delta increase
in productivity over the old combine.
That's what this blocking factor or delta is.
We could compute the contrast here
for A, B, and the interaction effect.
And you notice here that at least in two of these,
in the second two, we have a positive delta here,
which cancels with the negative delta here.
And in AB we have a positive delta
that cancels with a negative delta.
So for B and AB, the blocking effect cancels out,
but we're left with the blocking effect delta
effecting the main contrast of A.
And this is bad because what we've done is we've confounded,
we've confounded the blocking effect delta
with a main effect of A.
Let's look at the third alternative
that we considered where we put the two different blocks
on opposite corners of this two squared design.
We can compute the contrasts of A, B and AB.
It turns out for the first two of these, the deltas cancel.
But when we look at the interaction effect AB,
this is confounded with the blocking effect.
For a two squared design
it's a lot better to put the confounding
of the blocks on the higher order interaction.
In other words AB, than it is to
confound the blocks with our main effects, A and B.
In general, we want to confound the blocking effect
with the highest order interaction that we can.
And in this case, for a two squared design, that's AB.
In order to do this, we can set up our table
and we look at the highest order interaction.
So that's AB here.
For the plus ones in the AB column
we put that into one of the blocks.
And for the negative ones in the AB column
we put that in the second block.
And so this is how we can easily split the design
into two blocks.
Now, this becomes a little bit more complicated
when we split into more than two blocks,
but Minitab can do this nicely for us.
Let's take a look at a two cubed design.
How should we choose which experimental treatments go
in each block?
Well, it turns out that this design,
with the blocks on opposite corners of this cube,
is the most efficient design.
Let's take a look at some example contrasts.
The main effect of A would be given by this formula.
AB and ABC are given by these formulas.
If we take a look at this
we see that for the main effect of A
we have two positive deltas and two negative deltas.
So the blocking effect is not confounded
with the main effect for A.
If we look at AB,
we see that we have two positives
and two negatives for the blocking effect.
And so AB is not confounded with blocks.
But if we look at ABC, again
what we wanna do is confound the blocking effect
with the highest order interaction.
For a two cubed design that's ABC.
We see that overall we have a negative four delta.
So we've confounded the blocking effect with ABC,
which is good.
And the main effects and binary effects
for this two cubed design are not confounded by blocks.
So that's why we chose to do this design.
Because we've confounded ABC with the blocking effect,
we don't know if the ABC effect that we observe,
if we observe an ABC effect, is due to an actual effect
or is it from the blocking effect.
The way that we choose these blocks,
we set up our two cubed table
and then I've already sorted the ABC column.
I've got the ones on the top
and the negative ones on the bottom.
And so what we do is we assign
to a principle block, the top four, and the bottom four,
where ABC are all negative.
For those ones, we assign those to the secondary block.
And again, by doing this
we're confounding the blocking effect
with the ternary effect, the highest order interaction,
and all other effects are independent of the blocks.
Let's take a look at an example in your textbook.
We're not gonna solve this completely
but we're gonna set it up.
Researchers reported on an experiment to minimize variations
in blood glucose levels.
They had three factors.
Juice intake before exercise, a low and a high level,
amount of exercise on a Nordic track, cross-country skier,
10 or 20 minutes, and delay between time of juice intake
0 or 20 and the beginning of the exercise period.
So we have three factors.
Each are present at a low and high level.
Unfortunately, due to feasibility issues,
all tests cannot be conducted by the same scientists.
Four runs must be performed in the morning
by the morning scientists,
and four runs must be performed in the afternoon
by the afternoon scientists.
It is expected that the two scientists
have slightly different lab techniques.
In this case, we should think blocks
because of the different scientists.
So we're gonna assign four of the runs to the morning,
and four to the afternoon.
And there's a prudent way to do this.
The question is, if we wish to block for the time of day,
how should we choose which experiments to conduct
in the morning versus the afternoon?
We can go back to our two cubed table here where
we've sorted by column ABC.
We're gonna choose little a, little b, little c,
and ABC to be the morning experiments,
that's in our principle block.
And then the secondary block will be the bottom ones here.
So it turns out that this second one is A,
the third one here is B, and the fifth one here is C.
The last one is ABC.
And so we would put then those four
in the morning and the other four in the afternoon.
So that's how we would assign these blocks
for this experiment.
Let's look at a more complicated design.
You wish to create and analyze a two to the fifth
unreplicated design but can only perform
eight experiments each day.
How many blocks, and the blocks are gonna be days
because each day might have slightly different temperature,
humidity, things like that, would be required?
Two to the fifth is 32.
Eight experiments each day.
That means we have to have four days.
Your book does explain how to assign blocks when
you have more than two blocks,
but we're just gonna have minitab do this
for us when we have more complicated designs,
and you will explore this in an upcoming workshop.