In  this  talk,  we're  going  to  describe

repeated  measures  degradation   and  its  implementation  in  JMP.

I'm  going  to  present  the  background,  motivation,

some  technical  ideas  and  examples.

Then  I'll  turn  it  over  to  Peng, who  will  do  a  demonstration  showing

just  how  easy  it  is  to  apply these  methods  in   JMP 17.

Here's  an  overview  of  my  talk.

I'm  going  to  start  out  with  some motivating  examples,

and  then  explain  the  relationship  between degradation  and  failure  and  the  advantages

of  using  degradation  modeling in  certain  applications.

Then  I'll  describe  the  motivation for  our  use  of  Bayesian  methods

to  do  the  estimation.

To  use   Bayes' methods  for  estimation, you  need  to  have  prior  distributions.

I'll  talk  about  the  commonly  used, noninformative  and  weekly  informative

prior  distributions.

Also  in  some  applications,

we  will  have  informative  prior distributions  and  how  those  can  be  used.

Then  I'll  go  through  two  examples,

and  at  the  end  I'll  have some  concluding  remarks.

Our  first example  is  crack  growth.

We  have  21  notched  specimens.

The  notches  were  .9 "  deep, and  that's  like  a  starter  crack.

Then  the  specimens  are  subjected  to  cyclic  loading,

and  in  each  cycle, the  crack  grows  a  little  bit.

When  the  crack  gets  to  be  1.6 "  long, that's  the  definition  of  a  failure.

We  can  see  that  quite  a  few

of  the  cracks  have  already  exceeded that  level,  but  many  of  them  have  not.

Traditionally,  you  could  treat those  as  right  censored  observations.

But  if  you  have   the  degradation  information,

you  can  use  that  to  provide

additional  information  to  give  you a  better  analysis  of  your  data.

The  basic  idea  is  to  fit  a  model to  describe  the  degradation  paths,

and  then  to  use  that  model  to  induce a  failure  time  distribution.

Our  second  example   is  what  we  call  Device  B,

a  radio  frequency  power  amplifier.

Over  time  the  power  output  will  decrease

because  of  an  internal degradation  mechanism.

This  was  an  accelerated  test   where  units  were  subjected

to  higher  levels  of  temperature,

2 37, 195  and  150  degrees  C.

The  engineers  needed  information   about  the  reliability  of  this  device

so  that  they  could  determine  how  much redundancy  to  build  into  the  satellite.

The  failure  definition  in  this  case  was

when  the  power  output  dropped  to  -.5  decibels.

We  can  see  that  all   of  the  units  had  already  failed

at  the  higher  levels  of  temperature,

but  at  150  degrees  C, there  were  no  failures  yet.

But  there  is  lots  of  information  about  how

close  these  units  were  to  failure by  looking  at  the  degradation  paths.

Again,  we  want  to  build  a  model for  the  degradation  paths,

and  then  we  use  that  to  induce a  failure  time  distribution.

In  this  case,   the  use  condition  is  80  degrees  C.

We  want  to  know  the  time  at  which  units

operating  at  80  degrees  C, would  reach  this  failure  definition.

Once  again,  we  build  a  model for  the  degradation  paths.

We  fit  that  model,  and  then  from  that  we can  get  a  failure  time  distribution.

Many  failures  result  from  an underlying  degradation  process.

In  some  applications, degradation  is  the  natural  response.

In  those  situations, it  makes  sense  to  fit  a  model

to  the  degradation  and  then  use the  induced  failure  time  distribution.

In  such  applications,

once  we  have  a  definition  for  failure, which  we  call  a  soft  failure

because  the  unit  doesn't  actually  stop operating  when  it  reaches  that  level

of  degradation,  but  it's  close  enough to  failures  that  engineers  say  we  would

like  to  replace  that  unit at  that  point,  just  to  be  safe.

Now,  in  general,  there's  two  different

kinds  of  degradation  data, repeated  measures  degradation,

like  the  two  examples  that  I've  shown  you, and  destructive  degradation,

where  you  has  to  destroy  the  unit  in  order to  make  the  degradation  measurement.

For  many  years,  JMP  has  had

very  good  tools   for  handling  degradation  data.

I'm  focused  in  this  talk

on  the  repeated  measures  degradation  methods

that  are  being  implemented  in  JMP 17.

There  are  many  other  applications

of  repeated  measures  degradation, for  example,  LED  or  laser  output,

the  loss  of  gloss  in  an  automobile  coating and  degradation  of  a  chemical  compound,

which  can  be  measured   with  techniques  such  as  FTIR

or  any  other   measured  quality  characteristic

that's  going  to  degrade  over  time.

There  are  many  applications for  repeated  measures  degradation.

There  are  many  advantages  of  analyzing degradation  data  if  you  can  get  them.

In  particular,

there  is  much  more  information in  the  degradation  data

relative  to  turning   those  degradation  data

into  failure  time  data.

This  is  especially  true if  this  heavy  censoring.

Indeed,  it's  possible  to  make inferences  about  reliability

from  degradation  data  in  situations where  there  aren't  any  failures  at  all.

Also,  direct  observation

of  the  degradation  process   allows  us  to  build  better  models

for  the  failure  time  distribution   because  we're  closer

to  the  physics  of  failure.

Now,  several  years  ago, when  we  were  planning  the  second  edition

of  our  reliability  book, we  made  a  decision  to  use

more  Bayesian  methods  in  many  different  areas  of  application.

One  of  those  is  repeated   measures  degradation.

Why  is  that?

What  was  the  motivation  for  using  Bayes'  methods

in  these   applications?

I  used  to  think  that  the  main  motivation

for  Bayesian  methods  was  to  bring prior  information  into  the  analysis.

Sometimes  that's  true,

but  over  the  years, I've  learned  that  there  are  many

other  reasons  why  we  want to  use  Bayesian  methods.

For  example,  Bayesian  methods  do  not  rely

on  large  sample  theory to  get  confidence  intervals.

It  relies  on  probability  theory.

Also,  it  turns  out  that  when  you  use  Bayes'  methods  with  carefully  chosen,

noninformative  or  weekly informative  prior  distributions,

you  have  credible  interval  procedures that  have  very  good  coverage  properties.

That  is,  if  you  ask  for  95 %  interval,

they  tend  to  cover  what  they're  supposed  to  cover  with  95%  probability.

In  many  applications, there  are  many  non-Bayesian  approximations

that  can  be  used  to  set confidence  intervals.

When  you  do  Bayesian  inference, it's  very  straightforward.

There's  really  only  one  way  to  do  it.

Also,  Bayesian  methods  can  handle with  relative  ease,

complicated  model  data  combinations

for  which  there's  no maximum  likelihood  software  available.

For  example,  complicated  combinations

of  nonlinear  relationships, random  parameters  and  sensor  data,

the  Bayes'  methods  are  relatively

straightforward  to  apply in  these  complicated  situations.

Finally,  last  but  certainly  not  least, Bayesian  methods  do  allow  an  analyst

to  incorporate  prior  information into  the  data  analysis.

But  I  want  to  point  out that  the  revolution  we've  had

in  the  world  of  data  analysis   to  use  more  Bayesian  methods,

most  analysts  are  not  bringing

informative  prior  information into  their  analysis.

Instead,  they  use  weekly  informative  or  noninformative  priors

so  that  they  don't  have  to  defend the  prior  distribution.

But  in  many  applications,

we  really  do  have  solid  prior  information that  will  help  us  get  better  answers.

I  will  illustrate  that in  one  of  the  examples.

Bayesian  methods  require

the  specification  of  a  prior  distribution.

As  I  said,  in  many  application, analysts  do  not  want  to  bring

informative  prior  information   into  the  modeling  and  analysis.

What  that  requires  is  some  default prior

that's  noninformative or  weekly  informative.

There's  been  a  large  amount of  theoretical  research  on  this  subject

over  the  past  40  years, leading  to  such  tools  as  reference  priors,

Jeffrey's  priors, and  independent  Jeffrey's  priors

that  have  been  shown  to  have  good frequentist  coverage  properties.

One  of  my  recent   and  current  research  areas

is  to  try  and  make   these  ideas  operational

in  practical  problems, particularly  in  the  area  of  reliability.

A  simple  example  of  this  is  if  you  want to  estimate  a  location  parameter

in  the  log  of  a  scale  parameter, a  flat  prior  distribution  leads

to  credible  intervals  that  have exact  coverage  properties.

That's  very  powerful.

Also,  flat  prior  distributions  can  be  well approximated  by  a  normal  distribution

with  a  very  large  variance,  and  that  leads to  weekly  informative  priors.

Again,  it's  somewhat  informative,

but  because  the  variance  is  very  large, we  call it  weekly  informative.

The  approach  that  I've  been  taking to  specify  prior  distributions  is  to  find

an  unconstrained  parameterization, like  the  location  parameter

and  the  log  of  the  scale  parameter  that  I  mentioned  above,

and  then  use  a  noninformative

or  weekly  informative  flat  or  normal  distribution

with  very  large  variances  as  the  default  prior.

Then  it's  always  good  idea  to  use  some

sensitivity  analysis  to  make  sure  that  the prior  are  approximately  noninformative.

That  is,  as  you  perturb  the  prior  parameters,

it  doesn't  affect  the  bottom  line  results.

JMP  uses  very  sensible, well-performing  methods  to  specify

default  prior  distributions   that  are  roughly  in  line

with  what  I've  described  here.

Having  those  default  prior  distributions

makes  the  software  user-friendly,

because  then  the  user  only has  to  specify  prior  distributions

where  they  have  informative  information that  they  want  to  bring  in.

Here's  just  an  illustration  to  show that  as  the  standard  deviation

of  a  normal  distribution  gets  larger, you  approach  a  flat  prior  distribution.

Now,  as  I  said,  in  some  applications, we  really  have  prior  information

that  we  want  to  bring  in  that  is informative  prior  information.

When  we  have  such  information,

we  will  typically  describe  it with  a  symmetric  distribution,

like  a  normal  distribution, although  some  people  prefer  to  use

what  we  call   a  location- scale  t  distribution

because  they  have  longer  tails.

In  most  applications  where  we  have  this

informative  prior  information, it's  only  on  one  of  the  parameters.

Then  we're  going  to  use  noninformative,

or  weekly  informative  prior  distributions

for  all  of  the  other  parameters.

Let's  go  back  to  alloy  a.

What  we're  going  to  do  is  we're  going to  fit  a  model  to  the  degradation  paths,

and  then  use  that  model  to  induce a  failure  time  distribution.

Now  if  you  look  in  an  engineering  textbook

on  fatigue  or  materials  behavior,

you'll  learn  about  the   Paris  crack-g rowth  model.

It's  always  nice  to  have  a  model that  agrees  with  engineering  knowledge.

JMP  has  implemented this   Paris crack-growth model.

Here's  the  way  it  would appear  in  a  textbook.

Then  on  the  right  here  we  have the  JMP  implementation  of  that.

It's  one  of  the  many  models  that  you  can choose  to  fit  to  your  degradation  data.

Now,  c  and  m  here, which  in  JMP  is  c  and  b2,

are  materials  parameters, and  they  are  random  from  unit  to unit.

The  K  function  here  is  known  as a  stress  intensity  function.

For  the  crack  we're  studying  here,

the  stress  intensity  function has  this  representation.

Now  this  is  a  differential  equation,

because  we've  got  a  of  t  here,  and  also  a  here,

you  can  solve  that  differential  equation   and  get  this  nice  closed  form.

This  is  the  model that's  being  fit  within  JMP.

Again  the  parameters   b1  and  b2 will  be  random  from  unit  to  unit.

Now  here's  the  specification of  the  prior  distribution.

I've  illustrated  here  two  different

prior  distributions, the  default  prior  in  JMP,

and  the  prior  distribution   that  we  used  in  the  second  edition

of  our  reliability  book,   which  we  call  SMRD2,

statistical  methods  for  reliability  addition  two.

Now  the  way  we  specify  prior  distributions

in  SMRD2  and  JMP  is  doing  this  as  well, is  with  what  I  call  a  99 %  range.

For  example,  we  say  that  we're  going

to  describe  the  mean  of  the   b1  parameter

by  a  normal  distribution that  has  99 % of  the  probability

between  -15  and  22.

That's  a  huge  range.

That  is  weekly  informative.

Then  we  have  similar,  very  wide  ranges for  the  other  mean  parameter  here.

Then  for  the  Sigma  parameter  JMP,

following  usual  procedures  for  these  uses  a  half   Cauchy distribution,

which  has  a  long  upper  tail,

and  therefore,  again,   is  weekly  informative.

Now  in  our  reliability  book, we  used  much  tighter  ranges.

But  interestingly,  the  two  different   prior  distributions  here

give  just  about  the  same  answer,   because  both  are  weekly  informative.

That  is,  the  ranges  are  large  relative  to,

let  us  say,  the  confidence  interval

that  you  would  get  using  non-Bayes'  methods.

Now,  in  addition  to  specifying  the  prior  distributions,  which  again,

JMP  makes  very  easy  because  it  has  these  nice  default  priors,

you  also  have  the  ability  to  control the   Markov chain Monte Carlo  algorithm.

The  only  default  that  I  would  change  here

is  typically  I  would  run more  than  one  chain.

I  changed  the  one  here  to  four.

The  reason  for  doing  that  is  twofold.

First of  all,  in  most  setups, including  JMP,

you  can  run  those  simultaneously,  so  it doesn't  take  any  more  computer  time.

The  other  thing  is  we  want  to  compare those  four  different  Markov  chains

to  make  sure  that  they're giving  about  the  same  answers.

We  call  that  mixing  well.

If  you  see  a  situation  where  one of  those  chains  is  different

from  the  others,   that's  an  indication  of  a  problem.

If  you  have  such  a  problem,

then  the  usual  remedy  is  to  increase  the  number  of  warmup  laps,

which  is  set  to  be  10  by  default,  but  you  can  increase  that.

What  that  does  it  allows  JMP  to  tune

the  MCMC  algorithm  to  the  particular problem  so  that  it  will  sample  correctly

to  get  draws  from  the  joint  posterior  distribution.

In  all  of  my  experiences  using  JMP, and  Peng  has  suggested  that  he's  had

similar  experiences   that  by  increasing  that  high  enough,

with  any  examples  that  we've  tried, JMP  will  work  well.

But  with  10  for  most  applications, that  is  a  sufficiently  large  number.

Here's  the  results.

Here's  a  table  of  the  parameter  estimates.

Well,  typically  in  reliability  applications,

we're  not  so  much  interested   in  the  estimates  themselves.

We're  going  to  be  interested  in  things

like  failure  distributions, which  we  look  at  in  a  moment.

Then  in  this  plot,

we  have  estimates  of  the  sample  paths

for  each  of  the  cracks.

Again,  you  can  see  the  failure  definition  here.

As Peng  will  show  you,

JMP  makes  it  easy  to  look at  MCMC  diagnostics.

It's  always  a  good  idea  to  look

at  diagnostics  to  make  sure everything  turned  out  okay.

What  you  do  is  you  export  posterior  draws  from  JMP,

and  then  JMP  has  set  up  there  some  scripts  to  create

these  various  different  diagnostics.

For  example,  there's  a  script  to  make a  trace  plot  or  a  time  series  plot.

I  always  like  to  compare  those for  the  different  chains.

Then  there's  another  one  to  make   what  we  call  a  pairs  plot,

or  scatterplot  matrix  of  the  draws.

That's  what  we  see  here.

Then  as  I  said,  we  can  use  those  draws

to  generate  estimates   of  the  failure  time  distribution.

JMP  implements  that  by  using the  distribution  profiler  here.

We  can  estimate  fraction  failing  is  a  function  of  time

for  any  given  number  of  cycles.

Now  let's  go  to   the  Device  B RF  Power A mplifier,

again,  an  accelerated  repeated measures  degradation  application.

We're  going  to  need  a  model that  describes  the  shape  of  the  paths

and  the  relationship  the  temperature has  on  the  rates  of  degradation.

Again,  the  use  condition  in  this  application  is  80  degrees  C,

and  we're  going  to  want  to  estimate

the  failure  time  distribution at  80  degrees  C.

In   SMRD2,  this  is  the  way  we  would  describe  the  path  model

that  fits  device  b.

We  call  this  an  asymptotic  model,

because  as  time  gets  large, we  eventually  reach  an  asymptote.

In  this  equation, X  is  the  transformed  temperature.

We  call  it an  Uranus  transformation  of  temperature.

X 0  is  an  arbitrary  centering  value.

Beta  one  is  a  rate  constant for  the  underlying  degradation  process.

Beta  three  is  the  random  asymptote.

Those  two  parameters, the  rate  constant  and  the  asymptote,

are  random  from  unit  to  unit, and  we're  going  to  describe

that  randomness  with  a  joint or  bivariate  lognormal  distribution.

Beta  two,  on  the  other  hand,  is  a  fixed, unknown  parameter  that  is  the  effect

of  activation  energy  that  controls  how temperature  affects  the  degradation  rate.

This  is  where  the  X0  comes  in.

Typically  we  choose  X 0  to  be  somewhere

in  the  range  of  the  data

or  at  a  particular  temperature  of  interest,

because  beta  one  would  be the  rate  constant  at  X 0.

Again,  there's  a  large  number of  different  models  that  are  available.

Here  is  how  you  would  choose this  particular  asymptotic  model.

This  corresponds  to  the  same equation  we  have  in SMRD 2.

The  only  difference  is  that  JMP  uses a  slightly  different  numbering  convention

for  the  parameters.

That  was  done  to  be  consistent

with  other  things  that  are already  in  JMP  elsewhere.

Again,  we  have   to  specify  prior  distributions,

but  JMP  makes  that  easy

because  they  provide  these  defaults, these  weekly,  informative  defaults.

Here  I  have  the  default  that  JMP would  have  that  we're  going  to  use

if  we  did  not  have  any  prior information  to  bring  in.

I'm  going  to  do  that  analysis,

but  I'm  also  going  to  bring  in  the  information  that  engineers  have.

In  particular,  we  only have  information  for  b3

and  so  that's  being  specified  here.

But  all  the  other  entries  in  the  table   are  exactly  the  same  as  the  JMP  default,

again,  making  it  really  easy to  implement  these  kinds  of  analyses.

Here's  the  results.

Once  again,  here  we  have  a  table  giving  the  parameter  estimates,

credible  intervals  and  so  forth.

In  this  plot,  again,

we  have  estimates  of  the  sample  paths for  all  of  the  individual  units.

Again,  we  have   the  failure  definition  here,

but  what  we  really  want  are  estimates  of  the  failure  time  distribution

at  80  degrees  C.

Again,  we're  going  to  do that  by  using  a  profiler.

On  the  left  here  we  have  the  estimate

of  fraction  failing  is  a  function  of  time at  80  degrees  C  for  the  default  priors.

On  the  right  we  have  the  same  thing,

except  that  we've  used  the  informative prior  distribution  for  B 3.

Immediately  you  can  see   that  prior  information  has  allowed  us

to  get  much  better  estimation  precision.

The  confidence  interval is  much  more  narrow.

Interestingly, the  point  estimate  of  fraction  failing

actually  increased  when  we   brought  in  that  prior  information.

In  this  case,  the  prior  information would  allow  the  engineers  to  get

a  much  better  estimate

of  fraction  failing  as  a  function  of  time.

Then  to  make  that  important  decision

about  how  much  redundancy to  build  into  the  satellite.

Let  me  end  with  some  concluding  remarks.

Repeated  measures  degradation  analysis

is  important  in  many reliability  applications.

It  is  also  important  in  many  other  areas  of  application,

like  determining  expiration  dates

for  products  like pharmaceuticals  and  foodstuffs.

When  will  the  quality  be

at  such  a  low  level  that  the  customers  no  longer  happy?

Also,  in  certain  circumstances,

we  can  bring  prior  information into  the  analysis,  potentially  allowing  us

to  lower  cost  of  our degradation  experiments.

JMP  17  has  powerful, easy- to- use  methods  for  making

lifetime  inferences  from  repeated measures  degradation  data.

Now  I'm  going  to  turn  it  over  to  Peng,

and  he's  going  to  illustrate  these  methods to  you  using  a  demonstration  in  JMP.

Thank  you,  Professor.

Now,  the  demo  time.

The  purpose  of  a  demo  is  to  help  you to  begin  exploring  the  stable  art  approach

to  analyze  repeated measures  degradation  data.

I  will  show  you  how  to  locate   the  sample  data  tables  using  JMP,

how  the  information  is  organizing  in  the  report

and  highlight  some  important information  that  you  need  to  know.

First,  there  are  three  repeated  measures degradation  data  samples

among  JMP  sample  data  tables.

Alloy  a, d evice  b, are  two  examples  with  embedded  scripts.

The  s laser  does not  have an embedded script.

Alloy  a  is  an  example without  an  x  variable.

Device  b  is  example  with  an  x  variable.

To  find  them  go  to  the  help and  click  sample  index.

Try  to  find  the  outline  node  called reliability/ survival.

Unfold it  and  should  see a lloy  a  is  here, and  device  b  is  here.

To  find  GAS  laser,

you  need  to  go  to  the  sample  data  folder

on  your  computer  by  clicking  this  button, open  the  sample  data  folder.

Now  I'm  going  to  open  alloy a.

Then  I'm  going  to  analyze  menu,  reliability  and  survival,

choose  the  repeatedly  measure  degradation.

Now  we  see  the  launch  dialog.

Our  assign  length,  crack  lengths  goes  to  y,

specimen  goes  to  label  system  ID and  million  cycles  go  to  times.

Then  I'm  going  to  click  okay.

This  is  the  initial  report.

It's  a  fifth linear  models for  individual  batches.

I'm  going  to  select  the  third model here  by  clicking  this  video  button.

This  fifth  initial  model  of  a  Paris  model  of  this  alloy a  data.

I'm  going  to  click  this,  go  to  Bayesian  estimation  button

and  generate  a  configuration  interface.

Here  we  see  the  model  formula.

Here  are  the   default  settings  for  the  priors.

Then  we  are  not  going  to  change anything  right  now,

and  we  are  just  going  to  use the  default  prior  to  fit  our  model.

Now  I'm  going  to  click

this  fit  model  button  here   and  let  it  run.

Then  I'm  going  to  explain what  are  in  the  report.

In  the  end,  how  to  get the  failure  distribution  profiler.

Now  I'll  click  the  button, the  algorithm  start  run

and  the  progress  dialogue  said  that  the  first step  is  tuning.

The  underlying  algorithm  will  go  through  some

round  of  warm  up laps, is  procedure.

The  algorithm  is  trying  to  learn  the  shape of  the  posterior  distribution,

for  example,  where  the  peak, how  wide  is  the  span,  et cetera.

In  the  end,  they  will  try  to  figure  out

what  is  a  good  thinning  value  to  draw samples  from  the  posterior  distribution,

such  that  the  samples  are   as  little  auto correlated  as  possible.

Then  the  algorithms  enter  the  second  step.

The  dialogue  says,  this  step  is collecting  posterior  samples.

In  this  step,  an  automatic thinning  of  80  is  applied.

The  dialogue  shows  how  much  time in  total  the  algorithm  had  been  running.

In  the  second  stage,  the  dialogue also  shows  expected  completion  time.

By  such,  I  hope  it  can  help  users

to  adjust  to  their   expectation  accordingly.

Sometimes  excessive,  long  expected completion  time  is  a  sign  of  a  problem.

Then  we  wait  a  little  bit, and  the  algorithm  should  finish  soon.

Okay,  now  the  algorithm  has  finished.

Let's  see  what's  in  the  report.

First is  the  completion  time.

If  we  left  your  computer  run  over  time,

you  may  want  to  know in  the  morning  on  next  day.

The  second  is  a  copy  of  your  settings, including  priors,  number  of  iterations,

random  seed  and  other  thing shows.

Third part  is  posterior  estimates.

Be it a  summary   or  the  posterior  samples,

there  are  two  links  on  the  site  to  allow you  to  export  posterior  samples.

I'm  going  to  emphasize  the  first link.

One  purpose  of  using  these  first link

to  export  posterior  examples  is  to  inspect  potential  problems.

Two  main  concerns  are:  convergence  and  effective  sample  size.

Let's  look  at  it.

The  table  have  parameters in  individual  columns.

Each row  a  posterior  sample.

There  are  several  embedded  scripts.

The  most  important  one  is  the  first one.

I'm  going  to  click this  green  triangle  to   run the  script.

The script simply  run  time  series  on  individual  columns

and  show  their  time  series  plot.

In  the  context  of  MCMC, this  plot  is  known  as  the  trace  plot.

What  do  we  see  here?

What  I  call  good  results.

The  series  are  stationary   and  no  significant  auto correlation.

Loosely  speaking,  when  I  say  stationary  in this  context,  I  specifically  mean  plots

looks  like  these.

They  are  straight  equal with  band  of  random  dots.

Okay,  let  me  close the  report  and  the  table.

We  are  seeing  good  results  here.

Also  the  data  and  a fitted model also  shows  the  results  is  good.

Now  we  are  ready  to  ask  for  a fitted time  distribution  profiler.

To  do  that,  go  to  the  report  outline  node  menu

and  select  show  like distribution  profilers.

Most  entries  in  this  dialogue has  sensible  default  values

and  then  we  only  need  to  supply  one  of  the failure  definition.

I'm  going  to  enter   1.6   to  this  upper  failure  definition.

Before  I'm  going  to  click okay, I'm  going  to  reduce  this  number

of  SMRD2   realizations  to  5,000 to  save  me  some  time.

Then  I'm  click  on  okay.

This  is  also   a  computational  intensive  procedure,

but  not  as  expensive  as  MCMC  in  general.

It  should  finish  quickly.

You  can  use  the  profilers  to  get

the  failure  probability and  the  quantile  estimates.

I'm  not  going  to  elaborate  further,

because  profiler  is  a  very  common and  important  feature  in  JMP.

Okay,  this  is  the  end   of  the  demonstration,

and  you  are  ready  to  explore  by  yourself.

But  here  are  a  couple  of  tips that  might  be  useful.

First,  before  you  save   the  script  to  the  table,

go  to  check  this   save  posterior  to  script  option

before  you  save  the  script  to  the  table.

By  this,  next  time   you  run  the  save  script,

software  will  bring  back  the  fitted  model  instead  of  going

through  the  lengthy   MCMC  procedure  once  again.

The  second  thing  that  I  want to  bring  to  your  attention  is

we  have  seen  good  examples,  good  results, but  there  are  bad  ones.

Here  are  some  bad  examples.

This bad  example  means either  fail  to  converge,

or  there  are  high  auto correlations.

To  address  them  my  first suggestion  is  to  increase

the  number  of  warm up  laps.

Second  suggestion   where  we  turn  off  auto-thinning

and  apply  a  large  thinning  value, a  thinning  number  manually.

If  those  suggestions  don't  work, it's  likely  that  the  model

or  its  configurations  are  not appropriate  for  the  data.

You  may  need  help.

Okay,  this are  all  we would  like   to  illustrate  this  time,

and  I  hope  you  can  start  to  use  this information  to  explore

the  state  of  art  approach  to  analyze  repeated  measures  degradation  data.

Thank  you.