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.
