Hello.
My
name
is
Bradley
Jones.
I
lead
the
team
of
DoE and
reliability
in
JMP,
and
I
want
to
talk
to
you
today
about
a
family
of
orthogonal
main
effects
screening
designs
for
mixed
level
factors.
This
is
a
subject
which
I'm
really
excited
about.
We've
just
submitted
a
revision
to
the
paper
for
this
and
I'm
hoping
that
it
will
get
accepted
so
we
can
include
it
along
with
definitive
screening
designs
in
the
DoE platforms
in JMP. L
et's
get
started.
My
collaborators
for
this
work
are
Chris
Nachtsheim,
who's
a
Professor
at
the
University
of
Minnesota
Carlson
School
of
Business,
and
Ryan
Lekivetz,
who's
a
member
of
my
DoE team
at
JMP.
Here's
my
agenda.
I'm
going
to
start
with
a
little
bit
of
history
and
some
technical
preliminaries.
Then
I'm
going
to
describe
three
different
constructions
for
these
orthogonal
mixed
level
screening
designs. T
here
are
three
different
ways
that
we
can
make
them.
I'll
show
you
the
JMP
scripting
language
for
creating
these
design
sets.
That
will
only
be
necessary
until
we
can
get
them
built
into
JMP
itself.
Then
I'll
spend
a
little
bit
of
time
looking
at
the
design
properties
for
designs
constructed
under
these
three
methods.
I'll
discuss
data
analysis
for
these
designs
and
show
an
example
in
JMP,
and
then
I'll
make
a
summary
and
some
recommendations
at
the
end.
Let's
start
with
some
history
and
motivation.
The
first
screening
designs
were
fractional
factorial
designs
like
apartment
designs,
or
non
regular
fractional
factorial
designs
or
regular
fractional
factorial
designs.
For
these
designs,
every
factor
was
at
two
levels
only.
En
gineers
that
I
have
talked
to
have
felt
uncomfortable
about
these
designs
because
they
felt
that
the
world
tends
to
be
nonlinear
and
two
levels
just
isn't
sufficient
to
capture
nonlinearity
in
the
effect
of
a
factor
in
a
response.
Then
in
2011,
definitive
screening
designs
arrived
and
here
all
the
factors
were
assumed
to
be
continuous
and
each
factor
was
at
three
levels,
which
allows
you
to
fit
curves
to
the
relationship
between
factors
and
responses.
A
t
the
bottom
there
is
the
citation
for or
the
reference
for
the
paper
that
first
introduced
these
designs
in
2011.
Now,
there
are
some
pros
for
our
initial
implementation
of
DSDs
and
also
some
cons.
Let's
go
through
the
pros
first.
The
pros
are
that
at
least
in
our
original
implementation,
six
factor,
eight
factor
and
10 factor
definitive
screen
designs
had
orthogonal
main
effects,
but
we
were
unable
to
get
orthogonal
main
effects
for
more
factors.
It
turns
out
that
a
year
later,
somebody
published
a
nice
way
of
getting
orthogonal
designs
for
definitive
screening
experiments
for
every
even
number
of
factors
for
which
conference
matrices
were
available.
That
was
a
big
advance.
Another
good
thing
about
DSDs
is
that
the
main
effects are
orthogonal
two-
factor
interactions
so
that
the
estimate
of
a
main
effect
will
never
be
biased
by
any
active
two-
factor
interaction.
The
really
exciting
aspect
about
DSDs
is
that
all
the
quadratic
effects
were
estimable,
which
is
never
possible
with
screening
designs.
Even
with
centerpoints,
you
can
detect
that
there
is
a
non
linear
effect,
you
just
don't
know
where
it's
coming
from.
Then
finally,
in
DSDs,
if
there
are
six
factors
or
more,
and
only
three
of
the
factors
turn
out
to
be
important,
then
at
the
same
time
you
do
screening,
you
can
also
do
response
surface
optimization.
You
could
maybe
get,
if
you're
lucky,
a
screening
design
and
a
response
surface
design t
o
optimize
a
process
in
one
shot.
You
have
to
be
lucky,
of
course,
you
have
to
have
three
or
fewer
active
effects
or
factors.
The
cons
of
the
initial
implementation
of
definitive
screening
designs
is
that
first,
they
couldn't
accommodate
categorical
factors.
Secondly,
they
couldn't
accommodate
any
blocking.
Thirdly,
some
researchers
have
pointed
out
that
for
detecting
small
quadratic
effects,
that
is,
quadratic
effects
where
the
size
of
the
effect
is
about
the
same
order
as
the
error
standard
deviation,
there
is
low
power
for
these
estimates,
detecting
them
successfully.
Of
course,
if
the
quadratic
effect
is
big,
then
of
course
you
can
detect
it,
especially
if
it's
three
times
as
big
as
the
error standard
deviation.
Now,
after
the
original
publication
of
DSDs,
we
were
well
aware
that
it
was
a
problem
that
we
couldn't
accommodate
two
level
categorical
factors.
So
we
wrote
a
new
paper
in
Journal
of
Quality
Technology
in
2013
that
showed
how
to
incorporate
2-
level
categorical
factors.
Then
in
2016,
we
wrote
another
paper
in
technometrics
that
showed
how
to
block
definitive
screen
designs
using
orthogonal
blocks,
blocks
that
are
orthogonal
to
the
factors.
W
e
were
trying
step-
by-
step
to
address
the
cons
that
were
associated
with
original
implementation
of
this
methodology.
Another
thing
that
we
noticed
was
that
people
were
having
a
little
bit
of
trouble
knowing
how
to
analyze
definitive
screening
designs.
We
invented
an
analysis
technique
that
was
based
on
our
understanding
of
the
structure
of
a
DSD.
It
took
particular
advantage
of
the
special
structure
of
a
DSD
to
make
the
analysis
sensitive
to
that
structure.
Rather
than
trying
to
use
some
generic
model
selection
tool
like
stepwise
or
lasso
or
one
of
those.
This
made
it
possible
for
a
non-expert
in
model
selection
to
use
this
out
of
the
box
automated
technique
for
analyzing
a
definitive
screening
design.
That
was
in
2017,
but
there's
still
problems.
First,
we
did
write
that
paper
that
added
categorical
factors
to
a
DSD,
but
if
you
had
more
than
three,
the
quality
of
the
design
went
down,
and
that
was
undesirable.
In
fact,
if
you
had
too
many
categorical
factors,
things
didn't
look
good
at
all.
That
was
an
issue.
A
gain,
quadratic
effects
have
been
pointed
out
to
have
low
power
if
they're
small.
The
purpose
of
this
talk
is
to
introduce
a
new
family
of
designs
that
addresses
these
issues. H
ere
we
go.
First,
I
have
to
do
some
technical
preliminaries
to
explain
what
we
need
to
have
the
ability
to
do
in
order
to
build
these
designs.
I'm
going
to
start
out
by
talking
about
conference
matrices.
The
conference
matrix
is
the
tool
that
the
second
paper
that
discussed
DSDs
in
2012
and
introduced
all
the
orthogonal
DSDs
for
twelve
factors,
14
factors,
16
factors,
and
so
on.
They
use
conference
matrices
to
make
that
happen.
I
need
to
show
you
what
a
conference
matrix
is.
You
can
see
here,
conference
matrix,
that
is
four
factors
with
four
runs,
and
there
are
zeros
on
the
diagonal
elements
and
ones
and
minus
ones
off
the
diagonal.
T
he
cool
thing
about
a
conference
matrix
is
that
if
you
multiply
the
transpose
of
conference
matrix
with
the
conference
matrix,
you
get
an
identity
matrix
times
the
one
minus
the
number
of
rows
in
the
design.
I
t's
an
orthogonal
design.
Now,
conference
matrices
exist
when
the
number
of
rows
and
columns
is
equal.
They're
square
matrices
and
they
only
exist
when
the
number
of
rows
and
columns
is
an
even
number.
There's
a
conference
matrix
for
every
even
number
of
rows
and
columns
from
two
rows
or
and
two
columns
to
30
rows
and
30
columns,
except
for
the
case
where
the
number
of
rows
and
columns
is
22.
It's
actually
been
proven
that
the
case
where
there
are
2
2 rows and colum
ns,
the
conference
matrix
has
been
proven
not
to
exist.
So
there's
no
way
to
construct
one,
sadly,
although
I
can't
prove
that
result
myself.
Okay,
the
next
thing
I
need
to
talk
about
is
something
called
a
Kronecker product,
which
uses
that
circular
symbol
with
an
X
in
the
middle
of
it,
so
that
when
you
see
that
in
an
equation,
it
means
you
want
to
make
a
Kronecker
product.
The
Kronecker product
is
also
called
a
direct
product,
and
in
fact,
JMP
scripting
languages,
language
JSL makes
Kronecker products
of
matrices
using
the
direct
product
command,
not
the
Kronecker
product
command.
The
Kronecker product
of
a
vector
one
stacked
on
top
of
negative
one
with
a
conference
matrix
stacks
C
on
top
of
negative
C
as
below.
What
the
Kronecker
product
does
is
for
every
element
in
the
first
matrix,
it
substitutes
that
element
times
the
second
matrix.
So
one
times
the
conference
matrix
is
just
the
conference
matrix,
and
negative
one
times
the
conference
matrix
is
negative
of
the
conference
matrix.
Basically,
a
Kronecker
product
of
one
minus
one
with
a
conference
matrix
just
stacks
the
conference
matrix
on
top
of
itself.
So
here's
a
case
where
I
did
just
that.
You
have
a
four
by
four
conference
matrix
on
top
of
its
fold
,
which
is
also
four
by
four,
and
if
you
were
to
add
a
row
zero,
you'd
have
a
four
factor
definitive
screening
design.
Conference
matrices
are
useful,
Kronecker products
are
also
very
useful
for
constructing
designs,
as
it
turns
out.
I
have
a
few
more
preliminaries
to
go
over.
Let
me
talk
a
little
bit
about
Hadamard matrices.
Hadamard matrices
are
also
square
matrices,
but
they're
constructed
of
ones
and
minus
ones.
We
have
Hadamard matrices
built
into
JMP
for
every
multiple
of
four
runs
or
four
rows
and
four
columns
up
to
668
rows
and
668
columns.
So
every
multiple
of
four,
we
can
support
a
Hadamard
matrix.
That
well
known.
Hadamard designs
are
the
PlackettâBurman
designs
and
the
two
level
fractional
factorial
designs.
These
are
both
Hadamard matrices.
Hadamard was
a
French
mathematician
who
lived
in
the
late
19th
century,
and
he
invented
this
idea.
I
f
I had my matrix as
m
rows,
it's
transposed
times
itself.
Is
m
times
the
identity
matrix.
T
hat
means
the
Hadamard
matrix
is
orthogonal,
number
one,
and
number
two,
it
has
the
greatest
possible
information
about
the
rows
and
the
columns
in
the
matrix
that's
possible,
given
that
you're
using
numbers
between
negative
one
and
one.
They're
very
valuable
tools
for
constructing
designs.
Now
we
have
everything
that
we
need
to
show
how
to
construct
these
new
designs.
We
call
them
orthogonal
mix
level
designs
or
OMLs.
They're
mixed
level
because
half
of
the
columns
or
half
of
the
factors
are
three
levels.
Therefore,
the
three
level
continuous
factors
and
half
of
the
columns
or
factors
are
two
levels,
and
they're
for
categorical
factors
or
for
continuous
factors
for
which
we're
not
worried
about
nonlinear
effects.
Here's
the
first
method
for
constructing
one
of
our
OMLs.
If
C
sub
K
is
a
k
by
k
conference
matrix
and
H
sub
2k
is
a
2k
Hadamard
matrix,
so
H
sub
2K
is
a
Hadamard matrix
which
has
twice
as
many
rows
and
columns
as
the
conference
matrix
has.
Then
if
we
stack
a
conference
matrix
on
top
of
its
foldover
and
then
replicate
that,
we
get
this
matrix
DD,
which
is
just
C
negative
C,
C
negative
C,
all
stacked
on
top
of
each
other.
DD
is
two
DSDs
stacked
above
each
other,
minus
the
two
center
runs
that
DSDs
normally
have.
Now,
HH
is
because
DD ends
up
having
4k
runs
because
there
are
four
conference
matrices
with
K
rows
and
columns
stacked
on
top
of
each
other.
So
there
are
4k
rows
in
this
design
and
K
columns.
HH
is
just
Hatamard
matrix
stacked
on
top
of
its
fold
over
design.
Since
H
has
two
k
rows
and
columns
already,
stacking
it
on
top
of
itself
makes
it
have
4k
rows
just
like
the
DD
matrix
has.
It
turns
out
that
you
can
just
concatenate
these
two
matrices
horizontally
to
make
an
orthogonal
multilevel
design.
The
DD
part
of
it
all
has
three
levels
per
factor.
And
the
HH
part
of
it
has
two
levels
per
factor.
And
you
can
see
that
there
are
k,
three
level
factors
and
two
k
two
level
factors.
Therefore, a
4k
row
design
you
can
have
as
many
as
3k
columns.
For
example,
if
your
design
K
was
six,
you'd
have
24
rows
and
18
columns.
Six
of
the
factors
would
be
three
levels
and
twelve
of
them
would
be
two
levels.
Now
you
have
way
more
two-
level
factors
and
you
haven't
lost
any
of
the
features
of
the
definitive
screening
design.
The
main
effects
of
this
design
are
orthogonal
to
each
other.
Here's
an
example
where
I
constructed
an
OML
from
a
6 by 6
conference
matrix
and
a
12 by 12
Hadamard
matrix.
You
can
see
there
are
24
rows
in
this
matrix
and
18
columns.
The
first
six
of
them
are
the
six,
three-
level
columns,
and
the
next
twelve
are
the
twelve
two-
level
columns.
Now,
of
course,
you
don't
need
to
use
every
column
of
this
design.
You
could
still
use
this
design
even
if
you
had
say,
four
or
five
three
level
factors
and
seven
two
level
factors.
You
just
remove
five
of
the
two
level
factors
and
a
couple
of
the
three
level
factors,
and
it's
sort
of
arbitrary
which
ones
you
might
remove.
Here's
the
second
construction
approach
here.
C
sub
K
is
a
K
by
K
conference
matrix,
and
H
sub
K
is
a
K
by
K
Hadamard
matrix.
Now
we're
going
to
create
DD
the
same
way
we
did
before.
DD
is
just
a
definitive
screening
design
stacked
on
top
of
itself,
minus
the
two
synergies.
HH
is
a
replicated
Hadamard
matrix
on
top
of
the
same
Hadamard
matrix
folded
over
twice.
Now
if
you
look
at
the
two
columns
of
ones
and
minus
ones,
you
might
notice
that
those
two
vectors
are
orthogonal
to
each
other.
That's
what
makes
this
particular
construction
really,
really
powerful.
In
this
case,
the
design
has
4K
rows
and
only
two
K
columns.
K
of
the
factors
are
at
three
levels
and
K
factors
are
at
two
levels.
The
number
of
runs
in
this
design
is
twice
the
number
of
columns.
But
that's
still
a
very
efficient
number
of
runs
given
the
number
of
factors.
It's
the
same
effect
of
definitive
screening
designs
in
fact.
Definitive
screening
designs
have
twice
as
many,
twice
as
many
plus
one
runs
than
factors.
Here's
an
example
created
using
a
4 by 4
conference
matrix
and
a
4 by 4 H
adamard
matrix.
When
you
stack
them
on
top
of
each
other
four
times,
you
get
eight
columns
and
16
rows.
Columns
A
through
D,
you
can
see
are
three
levels
because
you
can
see
those
zeros
and
there
are
four
zeros
in
every
column.
I
should
point
out
that
if
you
had
a
definitive
screening
design,
there
are
only
three
zeros
in
each
column.
Having
an
extra
zero
makes
the
power
for
detecting
a
quadratic
effect
a
little
higher
than
for
the
definitive
screened
design.
That's
the
second
construction
method.
The
third
construction
method
is
very
similar
to
the
second,
except
that
you
have
two
different
ways
of
adding
Hadamard
matrices
to
the
example.
Here
we
have
the
DD
part
is
the
same
as
the
first
two
construction
methods.
The
HH
part,
there
are
two
HH
things.
One
with
the
vector
1, 1
-1
, -1
,
and
the
other
one
with
the
vector
1,
-1
,
-1, 1
.
The
three
vectors
that
are
ones
and
negative
ones
are
all
orthogonal
to
each
other.
That
yields
an
orthogonal
main
effects
design.
In
this
case,
the
third
construction
again
has
4K
rows
and
3K
columns,
that
is
K
factors
at
three
levels
and
two
K
factors
at
two
levels.
Those
are
the
three
methods.
Here's
an
example
of
that
construction
with
using
a
4 by 4
conference
matrix
and
a
4 by 4
Hadamard
matrix.
The
result
is
a
twelve-
column
design
with
16
rows.
Twelve
factors
in
16
rows.
Very
efficient
design
for
looking
at
twelve
factors.
It's
also
orthogonal
for
the
main
effects.
Main
effects
are
orthogonal
to
two-
factor
interactions.
Now
I
want
to
show
you
three
scripts
for
creating
these
designs
using
JSL.
In
the
meantime,
before
we
drop
this
methodology
into
JMP,
you
can
create
these
designs
with
a
very
simple
JSL
script.
The
first
command
is
creating
a
conference
matrix,
in
this
case
conference
matrix
with
six
rows
and
six
columns.
Then
D
is
the
direct
product
of
the
vector 1, -1
,
1,
-1
and
C.
That
gives
you
the
matrix
DD
that
we
saw
in
our
constructions.
Then
eight
is
a
Hadamard
matrix
with
twelve
rows
and
twelve
columns.
Notice
that
twelve
is
two
times
six.
We
were
requiring
that
for
the
first
construction,
the
Hadamard
matrix
has
to
have
twice
as
many
rows
and
columns
as
the
conference
matrix.
We
make
HH
by
multiplying
1, -1
by
using
the
direct
product,
the
Carnegie
Product
Construction.
That
gives
you
a
24-
run
HH
design.
The
H
thing
has
24
runs
and
twelve
columns.
Then
the
last
step
is
to
horizontally
concatenate
D
and
HH
to
produce
ML
which
I
just
shortened,
shortened
OML
to
ML.
Then
the
as
table
command
makes
a
table
out
of
that
matrix.
The
OML
that
we
just
created
has
24
rows
and
18
columns.
Six
of
the
columns
are
factors
at
three
levels,
and
twelve
of
the
factors
are
at
two
levels.
Now,
the
six
in
the
first
line
can
be
replaced
by
eight, 10,
12, 14
up
to
30,
except
for
22.
The
twelve
in
the
third
line
must
be
twice
whatever
number
you
put
in
the
first
line.
You
can
use
this
construction
to
create
all
kinds
of
OML
designs
just
by
changing
the
numbers
in
the
first
and
third
columns.
Here's
the
second
construction
method
script.
I
start
again
with
the
conference
matrix.
This
time
I'm
doing
a
conference
matrix
of
four
rows
and
four
columns.
D
is
a
direct
product
of
1, -1
, 1, -1
and
C.
That's
the
same
as
before.
This
time
I
make
H
be
a Hadamard
of
four
instead
of
twice
the
number
in
the
first
line.
I
have
to
have
a
vector
with
four
elements
to
direct
product
with
H.
I use
1, 1, -1, -1,
use
the
chronicle
product
of
that
or
the
direct
product
in JMP and
JSL
speak.
I
get
a
design
that
has
16
rows
and
eight
columns
by
horizontally
concatenating
that
double
vertical
line
thing
horizontally
concatenates
two
matrices
and
then
the
S
table
makes
the
table
out
of
it.
This
second
construction
has
16
rows
and
eight
columns.
There
are
four
factors
at
three
levels
and
four
factors
at
two
levels.
The
four
in
the
first
and
third
lines
can
be
replaced
with
any
even
number
for
which
a
conference
matrix
exists.
I
need
to
correct
myself.
The
conference
matrix
has
to
be
a
multiple
of
four
in
order
for
this
to
work,
because
the
Hadamard
matrix
is
a
multiple
of
four.
Here's
the
last
construction
method.
Again,
we
have
a
conference
matrix
of
four,
but
it
could
be
four,
it
could
be
eight,
or
twelve
or
16.
We
make
a
direct
product
of
this
vector
of
ones
and
negative
ones
and
C
to
get
the
replicated
definitive
screening
design.
Here
we
create
a
Hadamard
matrix
of
four,
but
we
have
two
different
direct
products.
The
first
one
where
we're
making
a
chronicle
product
of
1,
1,
-1, -1
with
H,
and
the
second
one
we're
making
a
chronicle
product
or
direct
product
of
1, -1
,
-1
,
1
and
H.
Those
are
two
different
matrices
and
they
happen
to
be
orthogonal
to
each
other.
Then
we
horizontally
concatenate
all
three
of
these
matrices
and
make
a
table
from
that.
This
design
now
has
16
rows
because
it's
four
runs
in
the
conference
matrix
times
four.
You
have
16
rows
and
twelve
columns.
There
are
four
factors
at
three
levels
and
eight
factors
at
two
levels.
Here
are
three
very
easy
JSL
scripts.
To
make
these
designs,
I'll
put
JSL
into
the...
When
it
goes
into
the
JMP
community,
I'll
add
the
JSL.
I'll
also
add
several
examples
of
these
OML
designs
that
you
can
use.
Now
I
want
to
talk
about
a
little
bit
about
the
properties
of
these
designs.
Here
we
see
the
design
properties
for
method
one
and
the
colour
map
and
the
correlations
shows
that
there
are
no
correlations
between
any
of
the
12
factors
in
this
or
actually
18
factors
in
this
design.
The
three-
level
factors
are
about
10%
less
efficiently
estimating
than
the
main
effects
than
the
two-
level
factors.
That's
because
of
the
zeros
in
each
of
those
columns.
That
doesn't
help
you,
the
zeros
don't
help
you
estimate
main
effects.
Now
I
want
to
show
you
the
alias
matrices
for
this
design
construction
method.
You
can
see
that
there
are
a
lot
of
main
effects
that
are
uncorrelated
with
two-
factor
interactions,
but
there
are
also
a
lot
of
main
effects
that
are
correlated
with
two-
factor
interactions.
The
three-
level
factors
main
effects
are
not
alias
with
any
of
their
two
factor
interaction.
The
same
is
also
true
of
the
two-
level
factors.
Their
main
effects
are
not
alias
with
their
two-
factor
interactions
because
both
sides
of
this
design
are
constructed
from
fold-
over
designs.
W
e
see
that
there's
quite
a
bit
of
potential
aliasing
of
main
effects
from
active
two-factor
interactions.
In
some
sense,
method
one
is
a
little
riskier
to
use
than
the
other
methods.
Here
are
the
design
properties
for
method
two.
You
can
see
that
here
I'm
making
a
design
that
has
16
factors
and
eight
columns, I mean
16
rows f
or
eight
factors.
The
three-
level
factors
have
15%
longer
confidence
intervals
than
the
two-
level
factors.
Again,
that
is
because
those
four
factors
all
have
four
zeros.
Four
of
the
16
runs
are
zero
instead
of
1
or
-1.
The
cool
thing
about
the
second
design
construction
is
that
none
of
the
main
effects
is
correlated
with
any
two-
factor
interactions.
That
has
many
of
the
desirable
effect
or
characteristics
of
a
definitive
screening
design.
There's
a
lot
of
orthogonality
between
pairs
of
two-
factor
interactions,
but
there
are
also
some
correlations.
You
can
see
here
are
some
correlations,
here
are
some,
and
so
on.
Finally,
the
design
properties
for
method
three
show
that
three-
level
factors
are
a
little
less
efficiently
estimated
than
two
-level
factors
15%,
15.5%.
We
can
see
that
there's
some
aliasing
between
main
effects
and
two-
factor
interactions,
but
not
as
much
as
for
the
design
construction
number
one.
In
terms
of
risk,
this
accommodates
more
factors
with
less
risk
than
the
first
construction
method.
I'd
like
now
to
compare
DSD
to
an
orthogonal
main
effect
or
mixed-
level
design.
You
can
make
a
DSD
with
eight
factors,
eight,
three-
level
factors,
and
that
would
have
17
runs.
I
f
you
get
rid
of
the
centre
run,
you
would
have
a
run
that's
directly
comparable
with
a
multi-
level
16-
run
design
that
you've
seen
in
design
construction
too.
Now,
if
we
compare
the
efficiency
for
estimating
main
effects,
the
definitive
screening
design
is
only
91%
D
efficient
with
respect
to
the
mixed-
level
design,
the
G
efficiency
is
92%,
the
A
efficiency
is
roughly
90%
and
the
I
efficiency
is
roughly
82%.
Y
ou
can
see
that
the
fraction
of
the
design
space
plot
shows
that
the
curve
for
the
mixed-
level
design
is
below
the
curve
for
the
definitive
screen
design
pretty
much
everywhere.
This
design
is
clearly
preferable
to
the
defendant
screen
design
for
estimating
main
effects
at
least.
Now
I
want
to
talk
about
data
analysis
and
use
an
example.
I've
created
a
design
using
the
second
construction
and
I
created
a
Y
vector
by
adding
random
normal
errors
to
a
specific
function
with
both
main
effects
and
two-
factor
interactions
and
rounding
it
to
two
decimal
places.
The
true
equation,
the
true
function
is
this
one
here.
It
has
A, B, E
and
F
main
effects
are
all
active
and
the
AB, BE
and
EF
two-
factor
interactions
are
all
active.
That's
a
function
without
error.
I
added
normal
random
errors
with
a
standard
deviation
of
one.
What
I
used
was
since
this
design
can
be
fit
using
the
Fit
Definitive
Screening
Design
platform
within
JMP,
that's
what
I
used.
Here
you
see
that
the
Fit
Definitive
Screening
Design
finds
all
seven
real
effects
and
doesn't
find
any
spurious
effects.
It
gets
the
exact
correct
set
of
effects.
The
deviation
between
the
true
parameter
values
and
the
estimated
parameter
values
are
pretty
small.
For
example,
the
true
parameter
value
for
factor
A
is
2.03
and
its
estimate
is
2.45
plus
or
-35.
That's
one
standard
deviation
so
it's
just
a
little
bit
more
than
one
standard
deviation
from
its
true
value.
Here
that
the
true
value
of
the
coefficient
of
B
is
3.88
and
I
get
3.94,
which
is
very
close
to
its
exact
correct
value.
You
can
see
for
yourself
the
estimated
value
of
the
root
mean
squared
error
is
1.2
and
the
true
amount
of
random
error
I
added
was
exactly
one.
You
can
see
again,
this
analysis
procedure
has
really
chosen
the
exact
correct
analysis.
Now
I
want
to
do
a
little
JMP
demo
that
shows
basically
the
actual
by
predictive
plot
that
you
see
there
below.
Then
this
plot
shows
that
the
residuals
don't
have
any
indication
of
any
problem
as
well.
I'm
going
to
leave
PowerPoint
for
a
second
and
just
go
to
JMP.
Here's
my
data,
here's
the
function
with
no
error.
I
can
show
you
that,
that's
just
this
formula
that
I
showed
you
in
the
slide.
Then
here's
the
data
where
I've
added
random
error
to
each
of
these
values
with
it.
Then
what
delta
is,
is
the
difference
between
the
prediction
formula
of
Y
and
the
Y
with
no
error.
These
values
are
how
far
we
missed
the
true
value
of
Y
for
every
point.
If
I
do
the
Fit
Definitive
Screening
Design
of
Y,
I
get
what
I
showed
you
before
and
I
get
the
correct
main
effects
and
also
the
correct
two
factor
interactions.
Then
when
I
combine
them,
I
get
the
correct
model
with
the
correct
art,
very
close
to
the
true
estimate
of
sigma.
If
I
run
this
model
using
Fit
model,
this
is
the
actual
by
predicted
plot
I
get.
This
is
the
residual
plot
I
get.
Here's
the
prediction
profiler
that
shows
the
predictive
value
of
Y.
You
can
see
that
if
you
look
at
the
slope
of
the
line
of
B
and
see
as
I
change
the
value
of
A,
the
slope
of
B
changes.
If
I
change
this
B,
the
slope
of
A
changes,
if
I
change
E,
the
slope
of
F
changes.
This
indicates
interactions
happening.
If
I
wanted
to
maximize
this
function,
I
would
choose
the
high
value
for
each
of
the
factors.
Then
one
of
the
things
I
did
was
I
created
a
profiler
that
where
I
look
at
this
is
the
true
function,
this
is
my
predictive
formula
and
this
is
the
difference
between
those
two
functions.
This
is
the
setting
that
leads
to
the
largest
difference
between
the
predictive
value
and
the
true
value
of
the
function.
That's
what
I
wanted
to
show
you in
JMP
and
I'll
move
back
to
my
slides
now.
Let
me
summarize,
we've
talked
about
definitive
screen
designs
with
their
pros
and
cons.
I
then
introduced
the
idea
of
a
Chronicler
product
and
showed
how
to
construct
these
orthogonal
multilevel
designs
in
three
different
ways.
I
showed
you
the
JSL
scripts,
I
shared
the
JSL
scripts
for
constructing
these
designs.
You
can
use
this
script
by
changing
the
numbers
in
the
first
and
third
lines
to
make
designs
that
with
increasingly
large
numbers
of
runs.
I
talked
about
the
statistical
properties
of
these
designs
and
particular
showed
that
their
orthogonality,
but
also
in
the
case
of
design
construction
two,
not
only
are
they
orthogonal
for
the
main
effects,
but
the
main
effects
are
orthogonal
to
the
two
factor
interactions.
Design
construction
two
only
exists
for
designs
that
have
a
multiple
of
16
rows
and
columns,
which
is
a
slight
disadvantage
compared,
there's
more
flexibility
with
the
other
approaches.
Then
finally
I
showed
how
to
analyze
these
designs.
Let
me
make
a
couple
of
recommendations.
Design
construction
method
two
is
the
safest
approach
because
of
all
of
the
orthogonality
involved
and
the
fact
that
the
two
factor
interactions
are
uncorrelated
with
main
effects.
I
pointed
out
already
that
you
don't
have
to
use
all
the
columns.
You
can
create
one
of
these
designed
and
then
throw
away
certain
columns
in
order
to
accommodate
the
true
number
of
factors
that
you
have.
The
advantage
of
doing
that
is
that
you
will
also
get
better
estimates
of
the
error
variance.
Then
it's
important
to
remember
that
the
three
level
factors
are
for
continuous
factors
only.
It
wouldn't
make
sense
to
have
three
level
categorical
factors
for
these
columns
because
there
are
far
fewer
zero
elements
than
ones
and
plus
ones.
A
couple
of
more
things.
Quadratic
effects
it
turns
out,
are
slightly
better
estimated
by
an
orthogonal
main
mixed
level
design
than
a
DSD.
But
if
you
wanted
to
improve
the
quadratic
effect
estimation,
you
could
add
two
rows
of
zeros
to
the
continuous
factors.
Those
would
be
like
center points.
Then
for
the
categorical
factor,
you
can
choose
any
vector
of
plus
ones
that
has
plus
ones
and
minus
ones
in
it.
Then
the
other,
the
second
row
would
have
just
the
fold
over
of
the
first
row
in
the
ones
and
minus
ones.
That
is
all
I
have
for
you
today.
Thank
you
very
much
for
your
attention.
