Today  I'm  going  to  talk  to  you  about

a  class  of  saturated  mix- level  main effects  designs  for   even  number  of  runs.

That  sounds  like  a  mouthful,

but  we'll  get  an  understanding  of  what that  is  by  the  time  we're  finished.

If  you  see  in  my  JMP  journal  here,

here's  a  rough  idea  as  to  what the  outline  is  going  to  look  like.

Now,  at  the  beginning  of  though, I  do  have  to  call  out,

so  I  have  three  collaborators   on  this  project.

Bradley  Jones,  who  should  be  familiar to  many  of  you  at  JMP,

Dibyen  Majumdar  and  Chris  Nachtsheim.

Just  some  history  and  preliminaries .

I'll  say  usually...

If  you  see  in  there, we  talked  about  main  effects  designs.

Now,  usually,  when  we  think  of  screening  designs,

we  often  think  of  factors  being  all  at  two  levels.

Your  standard  fractional  factorial  designs  that  you  see  in  textbooks,

but  really  there's  this  big  question, what  about  nonlinear  effects?

Of  course,  it's  great  we  want  to  find  out those  most  important  main  effects,

but  what  happens   if  there  is  some  nonlinearity?

In  particular,  in  2011,

Definitive  Screening  Designs   or  DSDs  hit  the  scene.

The  big  thing  with  those  DSDs,

here  we  were  assuming   all  the  factors  were  continuous.

Each  factor  and  DSDs are  going  to  be  at  three  levels.

That  gave  some  hope  of  being  able  to  detect  quadratic  effects

when  they  were  large.

I'll  say  that  was  one of  the  big  popularity  of  DSDs.

The  designs  we're  going  to  talk  about  here.

Our  main  effect,  if  you  think  of  DSDs,

not  only  were  they  good  at  picking  up  main  effects,

we're  also  looking at  quadratics  and  interactions.

The  designs  we're  looking  at  here

are  really  main  effects  screening as  our  primary  goal.

By  main  effects  screening, I'm  saying  we  have  our  list  of  factors

and  we  want  to  find  out  which  of  those are  the  primary  drivers.

We  want  to  find  out  those significant  main  effects.

If  we're  really  lucky, we  may  get  some  quadratic  interactions.

But  again,  main  effects screening  is  the  big  thing.

What  you're  going  to  find  here   is  we're  going  to  have  a  mix

of  three- level  and  two- level  factors.

Whereas  the  DSDs   had  everything  at  three- levels,

here  we  have  more  of  this  mix   of  three  and  two- levels.

One  of  the  things  to  pay  attention  to

is  just  like  definitive screening  designs.

When  we're  talking   about  these  three-level  factors,

we're  going  to  assume that  they're  continuous.

In  particular,  that  means we're  not  looking  for  balance.

Often  you  see  in  that  title   where  we  talk  about  mixed  levels.

We're  talking  about  three and  two-level  factors.

But  I'll  say  traditionally   when  we  think  of  orthogonal  rays,

the  three-level  factors  are  categorical,

and  so  we  want  to  see  each level  an  equal  number  of  times.

What  we're  going  to  find  here   is  that  these  three-level  factors

are j ust  going  to  have  a  few  zeros.

One  of  the  other  big  things,  though,

is  that  we're  going  to  have  almost as  many  factors  as  we  do  runs.

That's  where  that  idea  of  saturated and  the  title  came  from.

Saturated  just  means  effectively  that  I have  just  as  many  factors  as  I  do  runs.

That's  where  also   that  main  effects  screening

because  now  I  have  so  many  factors,

if  you  start  considering  quadratics  interactions,

that's  an  awful  lot  of  terms.

Because  we're  looking  at  so  many  factors,

we're  just  hoping  we  can  detect those  significant  main  effects.

If  you  think  about  other  designs,

if  you  think  about  this  mix   of  three- level  and  two  levels,

now  immediately  what  might  come  to  mind

is  some  of  the  classical  Taguchi  designs, so the  L18  or  the  L36.

Those  of  you  who  are  familiar  with  JMP,

you  may  know  that  you   can  create  Definitive  Screening  Design

but  added  two-level  factors.

The  designs  here  I  would  almost  look  at as  an  extension  to  some  of  those.

Here  we're  going  to  have a  lot  more  two-level  factors

than  you  might  want  to,  though, in  our  standard  DSDs.

I'll  say  this  is  also  an  area,  though, that's  picked  up  in  steam

a  lot  in  these  past  few  years.

You  see  some  references  here

as  to  other  authors  that  are  thinking  about  this  same  problem,

including  a  paper that  I'll  come  back  to  at  the  end.

You  see  that  last,  the  Jones, Lekivetz,  and  Nachshteim  paper.

That's  actually  related to  this  work  as  well.

Let's  see, hopefully  we're  all  in  the  same.

If  we  go  back  to  that  title.

The  saturated  just  means  we  have  a  lot of  factors  relative  to  the  run  size.

Our  mixed level  means  that  we  have some  two-level  and  three-level.

Just  to  mention  the  two- level   could  be  continuous  as  well,

where  we're  just  not  interested in  looking  at  those  quadratic  effects.

Main  effects  design  says  that's  where

our  most  important  thing  is  finding   those  significant  main  effects

and  that  even  numbers  of  runs,

it  turns  out  the  designs, if  you  remember  that  recall  the  outline,

all  these  designs  we're  looking  at  are going  to  be  an  even  number  of  runs.

Now,  when  we  think   about  building  these  designs,

so  in  some  of  these  preliminary,

I  should  really  talk about  the  building  blocks.

These  designs  are  going  to  be  built  upon

using  other  matrices   or  other  designs  in  the  literature.

Depending  on  how  familiar  you  are with  Definitive  Screening  Designs,

there's  this  idea  of  something called  a  conference  matrix.

A  conference  matrix   is  just  an  M  by  M  matrix

that  we  use  to  construct  Definitive  Screening  Designs.

One  of  the  nice  things with  conference  matrices

is  that  in  general,

they  exist  for  multiples  of  two  rows  and  columns.

I  said  in  general,

there's  a  conference  matrix   for  every  multiple  of  two

from  2 to 30,  except  for  22.

Something  to  mention  here,  though,

is  that  in  the  cases   where  a  conference  matrix  does  not  exist,

or  let's  say   if  you  have  an  odd  number  of  runs,

you  can  use  something called  a  pseudo- conference  matrix.

A  pseudo- conference  matrix  says,

I  want  to  make  it  look  as  close   as  I  can  to  a  conference  matrix,

which  has  this  special  property.

What  properties  a  conference  matrix  have?

Well,  if  I  take  C  transpose  T, or  conversely,  C C  transpose,

it's  going  to  be  M  minus  1 times  the  identity  matrix.

What  that  really  means   is  that  these  columns,

the  columns  of  the  matrix  C,  orthogonal.

If  I  take  that  cross- product of  any  two  columns,

we're  going  to  get  a  zero.

But  notice  we  have  that  M minus 1,

because  one  of  the  other  features of  the  conference  matrix

is  that  each  row  and  column  has  exactly  one  zero.

If  you  have  JMP  17.2,

so  I'll  admit  that  this  JSL first  shows  up  in  JMP  17.2,

so  we  have  this  command for  conference  matrix.

Here  I  can  just  ask  for  the  same  6x6 conference  matrix.

Again,  what's  the  special  property?

Well,  if  we  look,  so  if  I  take  C   times  C  transpose  or  conversely,

I  get  6  minus  1, which  is  5  times  the  identity  matrix.

Each  of  those  columns  is  going  to  be  orthogonal.

Again,  starting  a  JMP  17.2, you  can  create  your  own  conference  matrix

just  by  giving  it  the  order.

Here  I  wanted  a  six-run  conference  matrix, so  I  was  able  to  put  that  in.

Let's  try  it  with  eight as you'll  see.

There's  an  8  by  8  conference  matrix.

That's  one  of  the  building blocks  that  we  need.

Another  one  is  a  similar type  of  structure.

It's  called  a  Hadamard  Matrix.

The  difference  is  a  conference  matrix had  values  of  negative  one,  zero,  and  one.

A  Hadamard  matrix  just  has values  of  plus  and  minus  one.

A  Hadamard  matrix  exists for  most  multiples  of  four.

In  particular,  this  is  where   when  you  hear  Hadamard  matrix,

another  thing  you'll  hear about  often  times  is  an  orthogonal  array.

In  particular  for  a  Hadamard  matrix, it  has  that  same  property.

If  I  take  it  by  its  transpose,

in  this  case,   I  get  N  times  the  identity  matrix.

If  you  recall  the  conference  matrix,

it  was  that  M  minus  1,

the  order  because  we  had  one  zero  in  each  row  and  column.

Here  it's  going  to  be  N  times  the  identity  matrix.

The  idea  here  is  that  if  I  take  any  pair  of  columns,

we  have  that  concept of  orthogonality  for  any  pair  of  columns.

Similar  to  the  conference  matrix,

we  actually  have  a  special  command in  JSL  for  constructing  a  Hadamard.

Let's  take  a  look  here.

If  you  notice,  let's  just  say  Hadamard  8,

this  is  going  to  give  me an  8-run  Hadamard  matrix.

If  we  see  about  taking  a  Hadamard,

the  strand  rows,  there  we  get eight  times  the  identity  matrix.

Again,  so  what  that  means, if  I  take  any  pair  of  columns,

they're  going  to  be  orthogonal, which  you  can  actually  already  see.

Just  pretend  that  this  first one  is  like  an  intercept.

You  see,  all  of  these  are  balanced.

I'm  going  to  get  that  idea of  orthogonality.

We're  actually  almost  there with  our  building  blocks.

The  last  piece  that  we  need   is  something  called  the  Kronecker  product.

What  you'll  see  is  that  throughout   we  may  not  even  really  need

to  think  of  it  in  terms  of  a   Kronecker product.

Just  often  when  we  create  these  designs, that's  the  way  we  like  to  think  of  these.

You'll  see  a   Kronecker product is  denoted  by  this  symbol.

It  looks  like  a  multiplication with  a  circle  around  it.

In  JMP  or  another  definition  for   Kronecker product

is  called  the  direct  product.

It  just  happened  it's  a  convenient way  to  construct  designs.

All  of  this,   Kronecker product   or  direct  product  is,

if  I  take  a  matrix  A  and  the   Kronecker product  of  B,

I  take  each  element  of  A

and  apply  it  to  the  entire  matrix  B  over  and  over  again.

Where  this  comes  in  handy,

let's  see  where  you  may  have seen  something  like  this  before.

Again,  we  have  a  direct  product, a JSL  command.

Let's  say  if  I  just  wanted  to  start with  that  6x6  conference  matrix.

Now  let's  say  if  my  matrix  A, so  think  here  in  this   Kronecker product  A,

I  just  take  these  2  by  1,

so  two  rows  plus  one  and  a  minus  one,

and  direct  product  with  C.

What  do  I  get  here?

Well,  this  is  effectively  if  I  added  a  center  run,

this  would  be  a  13-run  DSD   and  six  factors.

By  taking  this  one  and  minus  one,

so  with  the   Kronecker product    up  here  with  this  plus  one,

I  get  that  conference  matrix  C, and  down  below  I  get  negative  C.

This   Kronecker product   is  just  a  convenient  way

to  think  about  things like  what  we  might  call  a f oldover.

With  those  preliminary  is  done,

now  we  can  actually  start  talking about  these  different  constructions.

The  first  method  is  the  most, I'll  say  the  nicest  of  all  of  them.

What  we're  going  to  say,   this  is  where  our  run  size

is  going  to  be  a  multiple  of  eight.

What  are  we  doing  here?

Well,  here  we're  going  to  start.

We're  going  to  have  a  conference  matrix of  order  M  equals  4K.

Again,  remember, the  conference  matrices

tend  to  exist  as  long   as  we  have  an  even  number.

Likewise,  for  the  Hadamard  matrix,

that's  where  we're  looking for  multiples  of  four.

Here  I'm  going  to  take  a  conference  matrix

of  order  M  and  a  Hadamard  matrix   of  the  same  order

where  we're  assuming  here  that  both  exist.

What  I'm  doing  here is  I'm  going  to  fold over.

You  can  actually  express  this as  a   Kronecker product,

but  here  I  find  it  more  convenient just  to  write  it  this  way.

You  see,  this  first  part  looks  like a  Definitive  Screening  Design.

I'm  taking  a  conference matrix  and  folding  it  over.

Then  on  the  other  side, I'm  replicating  this  Hadamard  matrix.

I'm  taking  a  copy  of  that.

What  do  we  get  with  this  design?

Well,  these  first  M  columns  that  are formed  from  the  conference  matrix,

that's  going  to  give  us M  three-level  factors.

The  remaining  M  minus  1  are  all  going  to  be  two-level  factors.

This  C  part  is  going  to  be   for  three-level,

this  H  part  for  two-level.

What  do  we  end  up  with?

Well,  what  we  did, we  basically  doubled  these  here.

We're  going  to  have  two  M-runs and  two  M  minus  1  factors.

Let's  take  a  look  at  what this  might  look  like.

Let's  see  an  example  here.

Let's  take  my  C.

I'm  going  to  just  create an  8  by  8  conference  matrix.

First,  let's  construct these  three-level  columns.

I'm  going  to  take  that  direct  product.

Again,  in  this  case,  I  want  that  foldover, I  want  C  and  minus  C.

Let's  take  a  look  at  what  C  looks  like.

Again,  this  is  just that  foldover  structure  on  that.

One  thing  here,  you  noticed  I  said  the  remaining  M-1  column

for  the  two-level  factors.

The  reason  for  that...

Let's  take  a  look  at  the  Hadamard  matrix  of  Order  8.

If  I'm  going  to  replicate  this, if  I'm  going  to  copy  this,

well,  this  first  column  here   is  going  to  be  for  the  intercept.

I  don't  want  to  put  that as  one  of  my  design  factors

if  it  never  changes,

if  it's  constant   throughout  the  entire  thing.

If  you  notice  here,  I'm  going to  just  drop  the  intercept.

Now  I  have  an  8  by  7  design.

If  you  notice,  I'm  going  to  use the  direct  product  again  here.

But  instead  of  with  the  conference  matrix

where  I  was  using  one  and  minus  one, I  just  want  to  make  a  copy

of  that  Hadamard  matrix  H  without  the  intercept.

Let's  give  that  a  look.

We  can  take  a  look  here  on  this  matrix

where  you  see  that  the  one,  one,  one.

You  can  actually  see  where it  gets  just  copy  it  again.

It's  just  the  same  matrix stacked  on  top  of  itself.

If  I  concatenate  all  of  those  together, you  can  see  I  have  a  16  by  15.

I  can  actually  just  create  that  data  table.

I  have  this  design  now, a  16-run  design  with  15  factors.

The  first  eight  of  those  are  three  level and  the  remaining  seven  are  at  two  levels.

Let's  just  take  a  look.

Let's  go  to  design  diagnostics

and  let's  see  what  this  looks like  and  evaluate  design.

You  can  see  I  just  created  a  main  effects.

This  might  be  hard  to  see, so  this  looks  pretty  messy,

but  you  can  already  see  there's a  special  structure  with  these  designs.

One  thing  I  want  to  point  out, let's  get  rid  of  the  alias  terms.

Let's  just  look  at  the  correlations here  with  these  main  effects.

You  can  see  in  this  case,

actually,  all  my  main  effects are  orthogonal  to  each  other.

One  thing  to  point  out  here,

because  I  was  using  that  Hadamard  matrix as  a  building  block,

those  two-level  designs,  we  have all  that  nice  orthogonality  there.

You'll  notice  this  fractional  increase  in  confidence  interval  length

is  a  little  bit  higher.

Why  is  that?

Well,  that's  because we  have  these  three-level  factors.

Those  first  factors  are  at  three  levels.

What  that  means  is  that if  there  is  a  quadratic  effect,

now  this  is  giving  me  some hope  of  detecting  that.

Again,  don't  forget,  we're  already at  16  runs  and  15  main  effects.

If  we  start  thinking   about  quadratic  effects  and  interactions,

we  really  have  to  hope  for  those  large  effects

when  it  comes  to  doing  any  model  selection.

If  you  think  of  a  traditional  design where  we  only  had  two  levels,

in  that  case,  we  would  have  no  hope

of  being  able  to  detect any  quadratic  effect.

That's  that  first  construction.

In  some  sense,  that's  where everything  works  out  nicely.

I  have  a  conference  matrix  and  I had  a  run  matrix  available  to  me.

That  gives  me  a  run  size that's  a  multiple  of  eight.

Now  what  happens   if  I  don't  have  one  of  those?

Let's  say  my  design  is  going to  be  a  multiple  of  four.

The  run  size  is  going to  be  a  multiple  of  four.

Let's  assume  now  that  I  have  a  conference  matrix  available  to  me,

but  maybe  I  don't  have  a  Hadamard  matrix.

If  you  recall  before,

where  for  my  Hadamard,   I  need  it  to  be  a  multiple  of  four,

the  original  run  size,  so  that  when  I doubled  it,  it  was  a  multiple  of  eight.

Instead,   maybe  if  I  don't  have  a  Hadamard  matrix,

I  could  use  something  like a  D-optimal  main  effects  plan.

This  would  just  be if  I  went  into  custom  design.

Let's  say  if  I  wanted  for  six  runs, I  would  go  into  custom  design

and  say  I  have  five  main  effects,   five  factors,

and  I  want  six  runs  for  a  main  effects.

This  construction  actually  looks a  lot  like  it  did  in  method  1.

The  only  real  difference  is  instead of  that  Hadamard  matrix,

now  I'm  going  to  be  using  this D-optimal  main  effects  plan.

But  it  turns  out  to  be the  same  thing  here.

I'm  going  to  have  those  first  M  columns for  the  three-level  factors,

the  remaining  M  minus 1  for  the  two-level.

I'm  really  at  that  idea  of  saturation

because  by  the  time  I  factor in  the  intercept,

I'm  at  two  M-runs  and  2 M  minus  1  factors.

Let's  take  a  look   at  how  this  one  might  look.

In  this  case,

I'm  going  to  go   with  a  10-run  conference  matrix,

which  again,   because  that's  a  multiple  of  two,

I  can  create  a  conference  matrix.

The  one  thing  to  pay  attention  to  here,  though,

because  we're  at  10- runs,

there's  not  going  to  be  a  Hadamard  matrix  of  order  10  available  to  us.

Let's  construct  though  first  though, let's  just  fold  over  that  CC.

Again,  that's  just that  DSD-like  structure.

Now  in  this  case, because  I  don't  have  that  command,

and  so  this  is  where  you could  decide  how  to  do  this.

You  may  actually  even  have  your  own design  that  you  want  to  use  in  this  case.

I'll  say  in  this  case, I'm  showing  a  D-optimal  main  effects.

You  may  want  to  use  something  l ike  a  Bayesian  D-optimal

where  two-factor  interactions  are  if  possible.

But  so  in  this  case,  all  that  I've  done

is  I've  taken  the  D-optimal, 10-run  design  for  nine  factors,

and  I've  just  happened to  include  the  intercept  here.

If  you  go  into  Custom  Design,  this  is what  you  get  from  the  model  matrix.

The  model  matrix  actually   includes  the  intercept  by  default.

That's  in  fact  where  I  had  this  from  here.

Again,  I'm  going  to  drop.

Let's  drop  that  intercept  column.

Again,  here  I'm  just  replicating the  exact  same  thing  again.

I've  just  folded  over,  not  folded  over.

I've  just  replicated   the  exact  same  thing  twice.

It'd  be  hard  to  see  as  to  where it  was  replicated  here.

But  again,  all  that  we've done  is  just  made  a  copy.

I  made  a  copy  of  that  doptN10m9  twice, concatenate  those  together.

Let's  create  the  data  table  again.

Again,  20- runs  and  19  factors.

Again,  keep  in  mind  that  my  first  10  factors

are  all  at  three  levels,  the  remaining  at  two.

Let's  take  a  look at  evaluate  design  again  here.

I'm  going  to  put  all  of  these  factors  in.

In  this  case,  before,

let's  go  directly  to  remove  those  alias  terms.

Let's  take  a  look  at  the  color  map.

This  still  looks  like a  pretty  good  color  map  to  me.

What  do  we  notice  here versus  the  last  one?

One  of  the  biggest  differences  is,

well,  here  our  three  levels  are  still going  to  be  orthogonal,

and  that's  because  we  were using  a  conference  matrix.

The  three- levels   and  the  two- levels  are  orthogonal.

In  particular,  my  three-level  factors are  orthogonal  to  the  two-level  factors.

But  because  I  was  using  that  10-run  design,

it  turns  out  we  can't   get  perfect  orthogonality

for  the  10-run  design  with  nine  factors.

If  we  look  among  those  two-level  factors, we  have  some  small  correlations  there.

One  thing  to  point  out,

there  does  exist  a  Hadamard  matrix  of  order  20.

In  some  sense,  we  are  taking  a  little bit  of  a  hit  on  those  two-level  factors.

But  if  you  take  a  look, the  cost  of  using  this  type  of  design

versus,  let's  say, everything  at  two  levels,

we  have  about  a  5%  increase in  the  confidence  interval  length.

But  the  nice  thing  here  now  is  that  now we  have  those  factors  at  three  levels.

If  we  really  are  worried   about  quadratic  effects

appearing  in  these  first  ones, now  we  have  a  chance  of  detecting  those

at  a  small  price  to  that  estimation efficiency  for  those  main  effects.

That  was  great  in  the  case

that  we  still  had  a  conference  matrix   that  existed,  but  no  Hadamard.

This  last  method,

this  is  where  we  don't really  fit  into  either  of  those  cases.

In  this  case  now,

we're  talking  about  a  run  size  is  going  to  be  a  multiple  of  two.

In  this  case,  what  we  have  is  we  don't have  a  conference  matrix  available  to  us.

This  is  where  we're  going  to  use that  pseudo- conference  matrix.

I'll  say  if  you're   really  interested  in  that,

I'll  go  back  to  these  preliminary.

In  the...

Let's  see.

Actually,  it  was  the  original  DSD  design.

Sorry,  this  2011 Definitive  Screening  Design.

If  you  also  look,  I  have  it in  the  list  of  references  at  the  end.

Let's  see, those  who  are  particularly  interested.

This  original  class  of  three-level  design for  Definitive  Screening

in  the  presence  of  second- order  effects.

This  paper  was  written  before  they   were  aware  of  the  existence

of  this  idea  of  conference  matrices.

In  that  paper,  they  talk  about  a  general   purpose  algorithm

for  creating  something that  looks  like  a  conference  matrix.

You  set  zeros  along  the  diagonal,

and  then  the  rest  of  the  values  are  going to  be  plus  and  minus  one,

where  you're  trying  to  make a  main  effects  D-optimal  design.

These  pseudo- conference  matrices,   as  they  were,

they  look  like  a  conference  matrix.

You  can  use  that  algorithm  when the  conference  matrix  doesn't  exist.

Really,  what  it's  trying  to  do  is to  drive  it  to  look  as  close  as  it  can.

If  you  remember  a  conference  matrix,

if  you  take  C  transpose, you  get  zeros  and  the  off  diagonal

for  perfect  orthogonality,

it's  going  to  be  trying  to  drive  it  to  look  like  that

where  I  can't  make  it  perfectly.

Similar  to  the  case  we  had  for  method  2,

our  T  is  going  to  be a  D-optimal  main  effects  plan.

What  do  we  end  up  with  here?

Again,  our  first  M  columns  are   still  going  to  be  three-level  factors,

the  remaining  M minus 1   for  the  two- level.

Again,  we're  still  at  that  case of  saturation

by  the  time  we  factor  in  the  intercept.

The  cost  here  is  that  we're  not  going to  get  the  nice  orthogonality

that  we  may  have  had  in  method  1  and  2.

Let's  take  a  look  at  how this  one  works  here.

My  C  in  this  case,

and  I'll  do  that  CC.

What  I  started  with  here

was  a  pseudo- conference  matrix  of  order  nine.

Nine  is  not  a  multiple  of  two.

I  have  to  do  something  that  looked like  a  pseudo- conference  matrix.

Let's  just  take  a  look  here.

Let's  actually  take  a  look  and  see  what  that  C  transport  C  looks  like.

You  notice  I  can't  get   that  perfect  orthogonality,

but  instead  I  have  eight  along  the  diagonal

and  then  these  plus and  minus  ones  on  the  off  diagonals.

We're  close.

I  have  that  CC  that  was  just  doing the  same  thing,  folding  that  matrix  over.

Likewise,  so  T, that's  going  to  be  of  the  same  order.

I  took  the  nine-run, eight-factor,  D-optimal  design.

In  this  case,  I've  actually already  removed  the  intercept.

We're  just  going  to  replicate that  design  again.

I  have  this  18  by  8.

I'm  going  to  combine   those  two  pieces  together.

Let's  take  a  look  at  the  table.

This  is  a  particularly  difficult design  to  generate  in  general.

In  this  case,  I  don't  have  a  nice  number.

My  run  size  is  18.

Now  I  have  these  first  nine  factors at  three  levels

and  the  remaining  eight  at  two  levels.

Not  surprisingly,  there  is  a  cost  of  this.

Again,  let's  just  remove  those  alias  terms  again

and  take  a  look  at  our  color  map.

You  see  now,  my  two  and  three  levels are  actually  orthogonal  to  each  other,

but  the  three  levels  have a  small  correlation  among  them.

That  was  where,  remember,  if  you  recall,

we  saw  that  plus  and  minus one  and  that  off  diagonal.

Likewise  for  the  two  levels.

We  actually  have  one   quite  high  correlation  here,

but  in  general,  so  about  0.1.

In  this  case,  we  don't  even  have an  orthogonal  design  to  compare  it  to.

We  actually  have  a  couple  of  factors that  we  may  be  worried  about a  bit,

have  a  little  bit of  a  larger  fractional  increase.

Those  three  levels  will  say,  well,  we can't  get  those  to  be  perfect  as  it  is.

This  fractional  increase  is  compared   to  the  hypothetical  orthogonal  design,

the  orthogonal  array, which  doesn't  even  exist.

We're  paying  a  small  price,  but  we  still have  some  generally  nice  properties.

The  three  levels  are  orthogonal  to  the  two  levels,

and  we've  minimized that  correlation  in  general.

In  some  sense,  those  are those  three  methods.

The  nice  thing  is  depending   on  the  run  size  that  you  have

and  depending  on  the  number  of  factors, this  methodology  or  this  class  of  designs

really  gives  you  some  flexibility in  the  number  of  runs.

Of  course,  if  you  can  afford   a  multiple  of  eight  or  a  multiple  of  four,

the  properties  are  going   to  look  a  lot  nicer.

But  I  mean,  when  runs  are  really  expensive,

we  still  have  this  method  3,

which  is  going  to  give you  a  reasonable  design.

With  that,  I  just  want  to  give some  final  thoughts  here.

I  have  a  link.

If  you  take  a  look   at  this  journal  afterwards,

I  have  a  link  to  a  presentation   from  JMP  Discovery  Europe

from  Bradley  Jones  that  was on  orthogonal  mix  level  designs.

The  designs  in  this  presentation, the  orthogonal  mix  level  designs,

will  almost  look  very  similar to  what  was  presented  here.

The  designs  presented  in  this  presentation are  for  when  you're  closer  to  saturation.

I  would  say  these   orthogonal  mix-level  designs

from  the  previous  discovery, and  we  actually  have  a  paper  on  that.

That's  this  Jones  Lekivetz,  and  Nachtsheim

that  was  in  JQT,  Journal  of  Quality  Technology.

Those  designs  work  very  well

when  you  have  about  half  the  number   of  factors  relative  to  the  run  size,

which  sounds  a  lot  like  DSDs,

but  it's  when  you  have more  two-level  factors.

In  the  design  presented in  that  presentation,

you  can  go  up  to   about  three-quarters  of  the  run  size.

The  designs  presented  here

are  really  when  you're  closer to  that  saturation,

where  you  say,  well,  no, I  really  have  a  lot  of  factors

that  I'm  interested  in, and  the  runs  are  really  expensive.

The  designs  here  to  fill  that  last  gap when  you  want  to  get  close  to  saturation.

I  showed  you  these  designs  look  nice,

but  can  they  actually  do  anything when  it  comes  to  model  selection?

I'll  say  here.

Again,  we  have  run  a  lot of  simulations  on  these.

This  isn't  doing  it  properly, this  is  just  for  a  single  realization.

Let's  say  here

my  factors,  so  call  one,  call  five, call  13  and  14,

I've  chosen  it  as  being  significant.

In  base  JMP,  you  might  want to  use  something  like  stepwise.

Let's  take  a  look.

Here  I've  done  a  reasonable  job.

I  may  have  ended  up  with  one  extra  factor,

but  in  some  sense,  I'd  still  be

quite  happy  that  I've  been  able to  pick  up  those  effects  there.

Let's  try  this  same  thing with  generalized  regression.

If  we  take  a  look,  the  same  thing.

Let's  see,  1,  5,  13,  and  14.

If  we  go  back,  1,  5,  13,  and  14.

In  this  case,

both  of  you  have  JMP  Pro  with  generalized regression  or  Fit  stepwise,

both  were  able  to  pick  up those  main  effects  that  we  had  in  there.

Just  to  show  you  now, if  you  have  a  large  quadratic  effect,

I'll  say  because  compared  to  DSDs   where  we  had  that  center  point  run,

this  we  only  have  two  zeros.

We  don't  even  have  a  third   for  a  center  point

and  we  have  a  lot  more  factors.

Detecting  quadratic  effects is  still  going  to  be  difficult.

But  let's  just  pretend  we  have  a  large  one.

Here  I  have  again, the  1,  5,  13,  and  14.

But  now  I've  also  added  a  quite  large quadratic  effect  for  the  column  one.

Let's  just  take  a  look  here.

Let's  try  to  fit  stepwise  again.

Now  I've  actually  just  added those  quadratics  in  the  model.

Again,  I  have  one  extra  term,

but  if  you  see,  even  in  this  case,  it  did actually  pick  up  that  quadratic  effect.

Not  only  did  it  pick  up  the  correct  main  effects,

it  did  detect  the  quadratic  effect  that  I  had.

Let's  try  the  same  thing.

Let's  see  the  model  launch.

It  looks  like  I  already   have  it  shown  here.

The  same  thing.

You  see  the  1, 5,  13,  14,

and  generalized  regression  was  also able  to  pick  up  that  quadratic  effect.

I'll  say  I  would  not  expect  to  anticipate

that  you  can  detect   that  many  quadratic  effects.

But  even  if  you  look  at  your  residual plots  or  your  main  effects  plots,

even  those  two  zeros   give  you  some  indication

that  maybe  I  do  want  to  follow  up

and  take  a  look  at  those  factors  a little  bit  deeper  for  quadratic  effects.

Where  again, in  your  traditional  screening,

if  you're  only  doing  things  at  two- level, you  wouldn't  have  the  chance  to  do  that.

With  that,  again,  I  will  post  this journal  where  this  video  is  located,

but  I'll  also  flash  up  these  references  at  this  time.

With  that,  thank  you  for  taking  the  time to  watch  this  video,

and  please  share  any  messages  you  have  in  the  community  below.