In this video we continue the prior video by considering the effect of confounding.
That prior video was all about the technical details behind confounding.
We know that confounding can confuse us, but I'll show how some knowledge
about our systems can be used to recover useful information.
But first, right at the beginning of the course remember how I always asked you to think
about how each factor might impact the outcome?
Now you are going to see why I made you do that.
Consider the case where we are treating water, and we have three factors.
The first is the chemical added to treat the water.
The second is temperature.
And the third is stirring speed.
A full set of experiments would require eight runs, but if I can only afford four
or five experiments, I should run a half fraction.
Prior experience with the system might lead me
to believe there will be no significant interaction
between temperature and stirring speed.
I would like to get a good estimate of the chemical factor added.
Remember the premise that chemical Q was twice the cost of chemical P?
In that case, I don't want the effects of the chemical to be confounded
with other effects in the system.
So if I would like this good estimate of the chemical added factor,
by that I mean I don't want it to be confounded with other large effects, a natural choice is
to alias the interaction between any other two factor interaction, temperature
and stirring speed in this case, with the chemical effect.
So now when I plan my experiments, I could rather assign A as the temperature,
B as the stirring speed, and factor C, the chemical factor, can be set as A times B. Notice
that I'm free to assign the letters to my factors in any way,
and I can choose the assignment that causes the least problems given
that I know confounding will occur.
Now factor C, that chemical effect, will be confounded with the AB interaction,
but I've used my knowledge of the system,
that I know that that AB interaction is going to be small.
So I'm pretty sure that the effect of C, the chemical effect, will be a good estimate
of that chemical effect and not be confounded with the AB interaction.
Right there is where I've demonstrated good use of educated guessing and smart assumptions.
Notice that you may not get this assignment
of letters correct the first time, but that's quite okay.
I often tell my students, it feels like I spend more time planning my experiments
than actually doing them.
That's because usually we only have one chance to do them,
but I've many chances to plan them on paper.
If I don't like the confounding pattern I get the first time, I can simply reassign my letters
and I can do that as many times as I like until I get the desired confounding pattern.
What's really important to notice here is that at no point
in this video have we used any of the y-values.
What this means is that this analysis can be done before you run any experiments.
You must use your brain and some educated guessing before you start the work.
Remember in the water treatment example,
each experiment costs $10,000, so we can't simply repeat them.
Now back to that trade-off table.
You might be curious about the other entries and how they were found.
Those entries here are found so that you can minimize the confounding
and recover the most amount of information for a given row and column combination in the table.
So that's a bit of the detail behind half-fractions.
We're going to have plenty of practice with these tables coming soon.
Here's one more important pointer about half-fractions.
They are often more suitable than a full factorial design
when you are trying to learn more about a system.
In other words, for that very first set of runs that you're trying.
What if you had four factors to investigate and did the full set of 16 runs?
You do them, you take the samples, and you send them a laboratory for analysis.
A few days later you get the results,
only to find out there was a problem with your experimental system.
What a waste.
It would have been cheaper to perform just eight runs, send the results for analysis
to discover the problem, then you can still spend the remaining budget
on those other eight runs to recover most of your results.
What about the other case?
What if you had done those first eight runs and there was no problem?
Well, you haven't lost anything.
You can quickly go do the analysis on those first eight runs,
and if you are satisfied you can stop.
If you want, you can go do the other eight runs.
We call those the complementary half fraction, and it's totally your choice to do that,
depending on how you want to spend the rest of your money.
Trying to visualize graphically what a half fraction as well
as its complementary half fraction looks like is only feasible for a system of 3 factors: A, B,
and C. First we would do the runs with the open circles, and complete all our analysis to find
out which of the factors is significant, and my how much they affect the outcome.
Recall, we would use a Pareto plot for this.
Then, let's say things looked promising, and we got approval to do the other half-fraction,
then you would come back and do the runs with closed circles.
This concepts extends naturally to a system with 8 + 8 = 16 experiments,
or 16 + 16 = 32 experiments, and so on.
In other words, the experiments on the diagonal of the trade-off table.
Obviously the savings are more impressive for the larger systems.
To end off: half fractions are a great example of what we call an experimental building block.
A piece of work that we start with and decide to add on top of if we choose.
You should be asking questions about how to use this table.
Here's a suggestion, in your the experimental system you thought of,
back in an earlier module, consider how you would have planned a reduced set of runs.
How would you have used this table in your system?
Make sure you can answer that.