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.