Hi, my name is Bradley Jones.
I am the leader of the JMP DOE Group at JMP statistical software.
I'm going to talk to you today about Latin Squares,
Youden Squares, Balanced In complete Block Designs,
also known as BIBDs,
and some E xtensions for Industrial Application.
Let's get started.
What you see here is a window in Cambridge University
that shows depicts a Latin Square.
The window has seven rows of colored panes,
seven columns of colored panes, and seven different colors.
You can see the yellow, blue.
You can see that yellow appears in every row, in every column,
as well as all the other colors
also appear in every row and every column.
There's this beautiful symmetry
to the Latin Square that we see here.
As a designed experiment,
Latin Square is primarily an agricultural design.
You can think of the rows and columns as blocking factors.
For instance, if you were doing the Latin Square design in agriculture,
the rows and columns would be rows and columns of plants.
What you're doing when you make those blocking factors
is ruling out any effect of a gradient of fertility in a row
across rows or across columns.
But then the entry in each row or column is one level of a categorical factor,
that in the case of the Latin Square design,
the categorical factor has seven levels
that correspond to the seven different colors.
You can see that each of the two blocking factors
both have seven levels,
and the categorical factor also has seven levels.
There are three factors, and each of them have seven levels.
One might say, well, first,
it's rare that in an industrial experiment,
you would have three seven-level categorical factors,
or even one categorical factor at seven levels,
and two other factors that were blocking factors
at seven levels.
That makes a lot more sense in agriculture.
H ere is an example of a Latin Square design
that I created using the Balanced Incomplete Block Design tool
in the DOE menu.
You can see that there are seven blocks and seven levels, A through G,
in each row and also in each column.
The Youden Square is a Latin S quare with some rows removed.
For instance, what you're seeing here is a transpose Youden Square.
If you turned it on its side,
you would see that there are seven blocks, each of which has four levels.
Basically, this Youden Square
is created by just removing three rows from a Latin Square.
This is a little bit more like
something that you might enjoy doing in an industrial setting.
Imagine that you were doing an experiment
that you are going to run on seven days.
Each day, you could do four runs.
You would have 4 times 7 or 28 runs in all.
Each of the days,
you would be doing four of the seven levels
of some treatment factor.
But the Youden Square is not really a square.
It's more like a rectangle.
I don't know exactly how it came to come to have that name,
but it's also a special case
of a Balanced In complete Block design or BIBD.
The Youden Square is actually, I mentioned it only because
I've been asked to give the Youden a lecture
at the Fall Technical Conference this year, in October.
I wanted to show something about Youden since I'm doing that lecture.
But I really want to talk more
about Balanced Incomplete Block Designs, or B IBDs,
because they are more general type of design.
In this case, we're thinking about a seven-level categorical factor.
You can only do four runs a day,
but you worry that there might be a day- to- day effect.
The four runs a day are a blocking factor.
You have a seven- level categorical factor
that you're interested in.
Again, this is the same scenario as you would have with a Youden Square,
except that there are a lot more possibilities for creating
Balanced Incomplete Block D esigns than Youden Squares.
Here's an example of that.
Here's the B IBD with each block having four values.
There are seven blocks.
You can see that...
Here's the Incidence Matrix.
What the Incidence Matrix is,
it shows a 1 if that treatment appears in that block.
The first block, A is in the block, C is in the block, F is in the block,
and G is in the block, and you can see A, C, F, and G.
Now in this design,
each of the seven levels of the categorical factor
appears four times.
You can see that in this Pairwise Treatment Frequencies.
Also, each level of the categorical factor appears with another level
of the categorical factor two times.
Level A appears with Level B twice.
Here's one case here, in Block 2 and also in Block 5.
For every pair of factors,
they appeared in some block with any other level twice.
Now, in fact, there's one more cool thing about this design,
which isn't always guaranteed to happen, but in this case, it does.
Each treatment appears once in each possible position.
For instance, Level A appears in Block 2 in the first position,
in Block 5 in the second position,
and Block 6 in this third position, and Block 1 in the fourth position.
A ll the other levels of the various treatment effects
appear once in each position.
That means that the position,
if you wanted to make position a variable, you could have its orthogonal
to the blocking factor and also orthogonal to the treatment factor.
You can actually, in this case, have a seven-level treatment factor,
a seven-level blocking factor, and a four- level position factor.
Imagine that you are going to, again, do this experiment
in seven days with blocks of size 4 in each day,
and then in each day, you would control the position
that each treatment appears in
so that the position effect
wouldn't bias in any other main effect of the design.
What I just talked to you about in the BIBD
is that in this BIBD, there are the seven-level
categorical factor, that's a treatment factor.
There are seven different possible treatments that you might have.
You could imagine that you could have
seven different lots of material, for instance.
You would think of the lot of material as being different lot of material,
might be a different treatment.
The blocking factor is day.
You're going to run the experiment over seven days,
and you're going to run for four runs in each day.
Then within each day, there's a position.
The time order of the position of the run
isn't going to affect any other estimate of either day or treatment.
Now, in industrial experiments,
having only a categorical factor and two blocking factors is a rare thing.
I'm thinking,
what if I wanted to add some factors to this experiment,
say, four continuous factors?
I can make design with four continuous factors,
a seven-level categorical factor, a seven-level blocking factor,
and four- level position factor using the custom designer.
But I wouldn't necessarily get that beautiful symmetric structure
of the BIBD on the categorical factors and the blocking factors.
S uppose I want to keep that beautiful structure
and just add the four continuous factors.
That is an extension of the BBD
that might be more appropriate to an industrial experiment.
Here's an example of that.
I have four continuous factors.
I have 6 degrees of freedom for blocks,
6 degrees of freedom for treatment, and 3 degrees of freedom for order.
Because in a categorical factor,
you have one fewer degrees of freedom, then you have levels.
You can see that the main effects of the continuous factor
are all orthogonal to each other.
They're orthogonal blocks, they're orthogonal treatments,
and they're only slightly correlated with the order variable.
Let me point out to...
I'm going to leave the slideshow,
and move to JMP here.
Here is the JMP BIBD capability.
You can find it under special Purpose, Balanced Incomplete Block Design,
so DOE then Special Purpose and Balanced Incomplete Block Design.
I chose that.
I defined a treatment variable that has seven treatments, A through G.
I made the block size here.
Let's suppose I want blocks of size 4 and seven blocks.
That's my design here.
Now I have the picture
that I showed you before in the slideshow.
Here's the blocking factor as seven blocks.
Each has four elements.
This is the Incidence Matrix,
which shows which treatment is applied in which block.
If it's applied, it's 1, and if it's not applied at 0.
You can see that each treatment appears four times in the design,
in each treatment, or each level pair appears twice in some block of the design.
Finally, we have the Positional F requencies
that shows each treatment
appears in each position in the design.
Here's the table of the Balanced Incomplete Block Design.
Now what I want to do is I want to create a design experiment
that forces this set of factors into the design.
I can do that in the custom designer,
but I have a script that does it automatically.
I'm going to run the script,
and it's going to do 10,000 random starts of the custom designer behind the scenes.
Then you'll see the resulting design pop up
as soon as it's finished getting through all these 10,000 random starts.
Here is the design,
and you can see that the factors are my four continuous factors:
the block, the treatment, and the order effect.
These are the covariate factors that came from this table here.
There are 28 rows in this table.
I'm calling these factors covariate factors
because I'm forcing them into the design as they are.
I've already created the design, and it's matched up
the four continuous factors and all of their rows
with the Balanced Incomplete Block Design
that have the treatment, the block, and order variable.
Now I can show you the table of the design.
What I've done is I've sorted this table by the order call,
because I want,
for instance, the first block, I want the order to go 1, 2, 3, 4,
and the second block again, 1, 2, 3, 4, and so forth.
I'm controlling the order of the runs in a non-random way,
but I've now made order be
orthogonal to treatment an orthogonal block.
When I evaluate this design,
one thing I want to show you is how well I can estimate
the continuous factors
compared to an absolutely completely orthogonal design,
a completely orthogonal design,
the fractional increase in the confidence interval
would be 0 here.
What we see here are numbers that are 0 .01 or 0.011,
which is to say that a confidence interval
for the main effect of factor 1 is 1 % longer
than the confidence interval would be
if you could make a completely orthogonal design for this case.
I'm going to select all of these
and remove these terms so that I can show you
the correlation cell plot without a bunch of noise.
This is the correlation cell plot for this design,
showing the orthogonality of the main effects
of the four continuous factors.
The block variable is orthogonal to them,
the treatment variable is orthogonal to them,
and the only thing that's not orthogonal
to the four continuous factors is the order effect.
But the order effect is orthogonal to the blocks in the treatments.
There's very minimal correlation.
That correlation is leading to almost no loss of information or increase
in variance of the continuous factors in the design.
The result of doing it this way is a much simpler design structure
so that analysis of this design will be easier
for even a novice in design to do.
That is all I have for you today.
Thanks for your attention.
