Introduction to doe

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Introduction to doe

  1. 1. David.LeBlond@sbcglobal.net  
  2. 2. “Designing  an  efficient  process  with  an  effec;ve  process  control  approach  is  dependent  on  the  process  knowledge  and  understanding  obtained.  Design  of  Experiment  (DOE)  studies  can  help  develop  process  knowledge  by  revealing  rela;onships,  including  mul;-­‐factorial  interac;ons,  between  the  variable  inputs  …  and  the  resul;ng  outputs.      Risk  analysis  tools  can  be  used  to  screen  poten;al  variables  for  DOE  studies  to  minimize  the  total  number  of  experiments  conducted  while    maximizing  knowledge  gained.      The  results  of  DOE  studies  can  provide  jus;fica;on  for  establishing  ranges  of  incoming  component  quality,  equipment  parameters,  and  in  process  material  quality  aKributes.”   2
  3. 3. What  is  it?   The  ability  to  accurately  predict/control  process  responses.    How  do  we  acquire  it?   Scien;fic  experimenta;on  and  modeling.    How  do  we  communicate  it?   Tell  a  compelling  scien;fic  story.   Give  the  prior  knowledge,  theory,  assump;ons.   Show  the  model.   Quan;fy  the  risks,  and  uncertain;es.     Outline  the  boundaries  of  the  model.   Use  pictures.   Demonstrate  predictability.   3
  4. 4. Screening  Designs  •   2  level  factorial/  frac;onal  factorial  designs    •   Weed  out  the  less  important  factors  •   Skeleton  for  a  follow-­‐up  RSM  design   Response  Surface  Designs   •   3+  level  designs       •   Find  design  space   •   Explore  limits  of  experimental  region   Confirmatory   Designs   •     Confirm  Findings   •     Characterize  Variability   4
  5. 5. Key   Factors   Key   Responses  Cau;on:  EVERYTHING  depends  on  gecng  this  right  !!!   5
  6. 6. Fixed  Factors   Responses   Disint  (A  or  B)   Dissolu;on%  (>90%)   Drug%  (5-­‐15%)     Make     Disint%  (1-­‐4%)   ACE     DrugPS  (10-­‐40%)   Tablets   WeightRSD%(<2%)     Lub%  (1-­‐2%)   Day   Random  Factors   6
  7. 7. Trial   DrugPS   Lub%   Disso%     1   25   1   85   2   25   2   95   3   10   1.5   90   4   40   1.5   70   Lubricant%   2   95   90   70   1   85   10   40   DrugPS   7
  8. 8. Lubricant%   2   95   90   70   1   85   10   40   DrugPS  Disso% = 86.667 +10 × Lub% −0.667 × DrugPS +ε 8
  9. 9. ž  Previous  example  had  only  2  factors.   Ø Factor  space  is  2D.  We  can  visualize  on  paper.  ž  With  3  factors  we  need  3D  paper.   Ø Corners  even  further  away  ž  Most  new  processes  have  >3  factors  ž  OFAT  can  only  accommodate  addi;ve  models  ž  We  need  a  more  efficient  approach   9
  10. 10. True  response   • Goal:  Maximize   response   • Fix  Factor  2  at  A.  Factor  2   • Op;mize  Factor  1  to  B.   80   E   60   40   • Fix  Factor  1  at  B.   C   • Op;mize  Factor  2  to  C.   A   • Done?    True  op;mum  is   Factor  1  =  D  and     B   D   Factor  2  =  E.   Factor  1   • We  need  to   accommodate  curvature   and  interac/ons   10
  11. 11. Response   A   B   C   D   Factor  level  •  A  to  B  may  give  poor  signal  to  noise  •  A  to  C  gives  beKer  signal  to  noise  and  rela;onship  is  s;ll   nearly  linear  •  A  to  D  may  give  poor  signal  to  noise  and  completely  miss   curvature  •  Rule  of  thumb:  Be  bold  (but  not  too  bold)   11
  12. 12. Trial   DrugPS   Lub%   Disso%     1   10   1   75   2   10   2   100   3   40   1   75   4   40   2   80   2   100   80   Lubricant%   1   75   75   10   40   DrugPS   12
  13. 13. Lubricant%   2   100   80   1   75   75   10   40   DrugPS  Disso% = 43.33 +0.667 × DrugPS +31.667 × Lub% −0.667 × DrugPS × Lub% +ε 13
  14. 14. ž  Model  non-­‐addiKve  behavior   ›  interacKons,  curvature  ž  Efficiently  explore  the  factor  space  ž  Take  advantage  of  hidden  replicaKon   14
  15. 15. Planar:  no  interac;on   Non-­‐planar:  interac;on   Y = a + b ⋅ X1 + c ⋅ X 2 Y = a + b ⋅ X1 + c ⋅ X 2 + d ⋅ X 1 ⋅X2 15
  16. 16. 16
  17. 17. 17
  18. 18. 18
  19. 19. 2   A   B   Trial   DrugPS   Lub%   Disso%  Lub%   1   10   1   C   2   10   2   A   1   C   D   3   40   1   D   10   40   4   40   2   B   DrugPS   B +D A +C A   B   MainEffectDrugPS = − 2 2 C   D   A +B C +D A   B   MainEffectLub% = − 2 2 C   D   C +B A +D A   B   InteractionEffectDrugPS×Lub% = − 2 2 C   D   19
  20. 20. Uncoded  Units   Coded  Units   Trial   DrugPS   Lub%   Trial   DrugPS   Lub%   1   10   1   1   -­‐1   -­‐1   2   10   2   2   -­‐1   +1   3   40   1   3   +1   -­‐1   4   40   2   4   +1   +1  •  Coding  helps  us  evaluate  design  proper;es  •  Some  sta;s;cal  tests  use  coded  factor  units  for  analysis   (automa;cally  handled  by  sotware)  •  Easy  to  convert  between  coded  (C)  and  uncoded  (U)  factor  levels   U − Umid C= ⇔ U = C(Umax − Umid ) + Umid Umax − Umid 20
  21. 21. +1  A   B   Trial   DrugPS   Lub%   DrugPS Disso%     *Lub%  Lub%   1   -­‐1   -­‐1   +1   C   2   -­‐1   +1   -­‐1   A   -­‐1   C   D   -­‐1   +1   3   +1   -­‐1   -­‐1   D   DrugPS   4   +1   +1   +1   B   Disso = a a = (+ A + B + C + D) / 4 +b × Lub% b = MEDrugPS / 2 = (−A + B − C + D) / 4 +c × DrugPS c = MELub% / 2 = (+ A + B − C − D) / 4 +d × Lub% × DrugPS d = IEDrugPS×Lub% / 2 = (−A + B + C − D) / 4 +ε 21
  22. 22. Disso = a + b × Lub + c × DrugPS + d × Lub × DrugPS + εž  It  is  obtained  through  the  “magic”  of  regression.  ž  b  measures  the  “main  effect”  of  Lub  ž  c  measures  the  “main  effect”  of  DrugPS  ž  d  measures  the  “interac;on  effect”  between  Lub  and   DrugPS   Ø  if  d  =  0,  effects  of  Lub  and  DrugPS  are  addi;ve   Ø  if  d  ≠  0,  effects  of  Lub  and  DrugPS  are  non-­‐addi;ve  ž  ε  represents  trial  to  trial  random  noise   22
  23. 23. +1   +1   +1   Lub%   Lub%  Lub%   -­‐1   -­‐1   -­‐1   -­‐1   +1   -­‐1   +1   -­‐1   +1   DrugPS   DrugPS   DrugPS  Trial   DrugPS   Lub%   Trial   DrugPS   Lub%   Trial   DrugPS   Lub%   1   -­‐1   -­‐1   1   -­‐1   -­‐1   1   -­‐1   -­‐1   2   -­‐1   +1   2   -­‐1   0   2   -­‐1   -­‐1   3   +1   -­‐1   3   +1   0   3   +1   +1   4   +1   +1   4   +1   +1   4   +1   +1   Inner  product:            +1-­‐1-­‐1+1=0                                                +1+0+0+1=2                                        +1+1+1+1=4   23
  24. 24. 24
  25. 25. Dissolu;on  (%LC)   1%  Lubricant   2%  Lubricant   90   10   40   DrugPS   25
  26. 26. y = a + bA + cB + dC + eAB + fAC + gBC + hABC + ε •  Average  Number  of   Number  of   •  Main  Effects  Factors  (k)   Trials  (df  =   •  2-­‐way  interac;ons   2k)   •  Higher  order   0   1   interac;ons  (or   1   2   es;mates  of  noise)   2   4   3   8   4   16   5   32   6   64   26
  27. 27. Main Effects Trial   I   A   B   C   D=AB   E=AC   F=BC   ABC   1   +   -­‐   -­‐   -­‐   +   +   +   -­‐   2   +   +   -­‐   -­‐   -­‐   -­‐   +   +   3   +   -­‐   +   -­‐   -­‐   +   -­‐   +   4   +   +   +   -­‐   +   -­‐   -­‐   -­‐   5   +   -­‐   -­‐   +   +   -­‐   -­‐   +   6   +   +   -­‐   +   -­‐   +   -­‐   -­‐   7   +   -­‐   +   +   -­‐   -­‐   +   -­‐   8   +   +   +   +   +   +   +   +   y = a + bA + cB + dC + eD + fE + gF + ε•  Can  include  addi;onal  variables  in  our  experiment  by  aliasing  with   interac;on  columns.  •  Leave  some  columns  to  es;mate  residual  error  for  sta;s;cal  tests   27
  28. 28. Trial   I   A   B   C   AB   AC   BC   ABC   1   +   -­‐   -­‐   -­‐   +   +   +   -­‐   2   +   +   -­‐   -­‐   -­‐   -­‐   +   +   +1 3   +   -­‐   +   -­‐   -­‐   +   -­‐   +   4   +   +   +   -­‐   +   -­‐   -­‐   -­‐   C 5   +   -­‐   -­‐   +   +   -­‐   -­‐   +   +1 B 6   +   +   -­‐   +   -­‐   +   -­‐   -­‐   -1 -1 7   +   -­‐   +   +   -­‐   -­‐   +   -­‐   -1 A +1 8   +   +   +   +   +   +   +   +   y = a + bA + cB + dC•  Create  a  half  frac;on  by  running  only  the  ABC  =  +1  trials  •  Note  confounding  between  main  effects  and  interac;ons  •  Compromise:  must  assume  interac;ons  are  negligible  •  In  this  case  (not  always)  design  is  “saturated”  (no  df  for  sta;s;cal   tests).   28
  29. 29. •  “I=ABC”  for  this  23-­‐1  half  frac;on  is  called  the  “Defining  Rela;on”   •  Note  that  “I=ABC”  implies  that  “A=BC”,  “B=AC”,  and  “C=AB”.  •  3-­‐way  interac;ons  are  confounded  with  the  intercept  •  Main  effects  are  confounded  with  2-­‐way  interac;ons  •  The  number  of  factors  in  a  defining  rela;on  is  called  the  “Resolu;on”  •  This  23-­‐1  half  frac;on  has  resolu;on  III  •  We  denote  this  frac;onal  factorial  design  as  2III3-­‐1   29
  30. 30. •  I=ABCD  for  this  24-­‐1  half  frac;on  is  called  the  Defining  Rela;on  •  Note  that  I=ABCD  implies   •   A=BCD,  B=ACD,  C=ABD,  and  D=ABC.   •   AB=CD,  AC=BD,  AD=BC   •   Main  effects  are  confounded  with  3-­‐way  interac;ons   •   Some  2-­‐way  interac;ons  are  confounded  with  others.  We  like  our  screening  designs  to  be  at  least  resolu;on  IV  (I=ABCD)   30
  31. 31. Number  of  Factors   2   3   4   5   6   7   8   9   10   11   12   13   14   15   4   Full   III                           6     IV                           8     Full   IV   III   III   III                  Number  of  Design  Points   12       V   IV   IV   III   III   III   III   III           16       Full   V   IV   IV   IV   III   III   III   III   III   III   III   20                     III   III   III   III   III   24                 IV   IV   IV   IV   III   III   III   32         Full   VI   IV   IV   IV   IV   IV   IV   IV   IV   IV   48             V   V                 64           Full   VII   V   IV   IV   IV   IV   IV   IV   IV   96                 V   V   V           128             Full   VIII   VI   V   V   IV   IV   IV   IV   31
  32. 32. Trial   DrugPS   Lub% Disso%     2 98,102 88,82 1   10   1   76   Lub% 2   10   2   98   3   40   1   73   4   40   2   82   1 76,84 73,77 5   10   1   84   10 40 6   10   2   102   7   40   1   77   DrugPS 8   40   2   88   FiKed  model  is  based  on  averages   SDindividual SDaverage = number of replicates 32
  33. 33. ReplicaKng   1  measurement  batch   3  batches   per    batch  producKon  Repeated   3  measurements   1  batch   per    batch  measurement   33
  34. 34. Trial   DrugPS   Lub% Disso%   ReplicaKon     1.  Every  operaKon  that   1   10   1   76   contributes  to  variaKon  is   2   10   2   98   3   40   1   73   redone  with  each  trial.   4   40   2   82   2.  Measurements  are   5   10   1   84   independent.   6   10   2   102   3.  Individual  responses  are   7   40   1   77   analyzed.   8   40   2   88   RepeKKon  Trial   DrugPS   Lub% Disso%   1.  Some  operaKons  that     contribute  variaKon  are  not   1   10   1   76, 84   redone.   2   10   2   98, 102   3   40   1   73, 77   2.  Measurements  are  correlated.   4   40   2   82, 88   3.  The  averages  of  the  repeats   should  be  analyzed  (usually).   34
  35. 35. ž Frac;onal  factorial  designs  are  generally  used  for   “screening”  ž Sta;s;cal  tests  (e.g.,  t-­‐test)  are  used  to  “detect”  an   effect.  ž The  power  of  a  sta;s;cal  test  to  detect  an  effect   depends  on  the  total  number  of  replicates  =  (trials/ design)  x  (replicates/trial)  ž If  our  experiment  is  under  powered,  we  will  miss   important  effects.  ž If  our  experiment  is  over-­‐powered,  we  will  waste   resources.  ž Prior  to  experimen;ng,  we  need  to  assess  the  need   for  replica;on.   35
  36. 36. 2 2 ⎛ σ ⎞N = (#points  in  design)(replicates/point) ≅ 4 z1−α + z1−β ( 2 ) ⎜ ⎟ ⎝ δ ⎠ σ  =  replicate  SD   δ    =  size  of  effect  (high  –  low)  to  be  detected.   α  =  probability  of  false  detec;on   β  =  probability  of  failure  to  detect  an  effect  of  size  δ α z1-­‐α/2   β z1-­‐β 20.01   2.58   0.1   1.28   ⎛ σ ⎞0.05   1.96   N ≅ 16 ⎜ ⎟ 0.2   0.85   ⎝ δ ⎠0.10   1.65   0.5   0.00   •  While  not  exact,  this  ROT  is  easy  to  apply  and  useful.   •  Commercial  sotware  will  have  more  accurate  formulas.   36
  37. 37. 2 2 ⎛ σ ⎞ ( N = (#points  in  design)(replicates/point) ≅ 4 z1−α + z1−β 2 ) ⎜ ⎟ ⎝ δ ⎠ Disso%   WtRSD   Replicate  SD   σ 1.3   0.1   Difference  to  detect   δ 2.0   0.2   False  detecKon  probability   α 0.05   0.05   z1-­‐α/2   1.96   1.96  DetecKon  failure  probability   β 0.2   0.2   z1-­‐β 0.85   0.85   Required  number  of  trials   N   13.3   8   37
  38. 38. Run A B C D E Confounding Table 1 - - - - + I = ABCDE 2 + - - - - A = BCDE 3 - + - - - B = ACDE 4 + + - - + C = ABDE 5 - - + - - D = ABCE 6 + - + - + E = ABCD 7 - + + - + AB = CDE 8 + + + - - AC = BDE 9 - - - + - AD = BCE 10 + - - + + AE = BCD 11 - + - + + BC = ADE 12 + + - + - BD = ACE 13 - - + + + BE = ACD 14 + - + + - CD = ABE 15 - + + + - CE = ABD 16 + + + + + DE = ABC 38
  39. 39. ž  Sta;s;cal    test  for  presence  of  curvature  (lack  of  fit)  ž  Addi;onal  degrees  of  freedom  for  sta;s;cal  tests  ž  May  be  process  “target”  secngs  ž  Used  as  “controls”  in  sequen;al  experiments.  ž  Spaced  out  in  run  order  as  a  check  for  drit.   39
  40. 40. Complete  RandomizaKon:    •  Is  the  cornerstone  of  sta;s;cal  analysis  •  Insures  observa;ons  are  independent    •  Protects  against  “lurking  variables”  •  Requires  a  process    (e.g.,  draw  from  a  hat)  •  May  be  costly/  imprac;cal  Restricted  RandomizaKon:  •  “Difficult  to  change  factors  (e.g.,  bath  temperature)  are  “batched”  •  Analysis  requires  special  approaches  (split  plot  analysis)  Blocking:  •  Include  uncontrollable  random  variable  (e.g.,  day)  in  design.  •  Assume  no  interac;on  between  block  variable  and  other  factors  •  Excellent  way  to  reduce  varia;on.  •  Rule  of  thumb:  “Block  when  you  can.  Randomize  when  you  can’t  block”.   40
  41. 41. 41
  42. 42. Confounding TableI = ABCDEBlk = AB = CDEA = BCDEB = ACDEC = ABDED = ABCEE = ABCDAC = BDEAD = BCEAE = BCDBC = ADEBD = ACEBE = ACDCD = ABECE = ABDDE = ABC 42
  43. 43. StdOrder  RunOrder  CenterPt  Blocks  Disint  Drug%  Disint%  DrugPS  Lub%  11  1  1  2  A  5  1.0  10  2.0  13  2  1  2  A  5  4.0  10  1.0  19  3  0  2  A  10  2.5  25  1.5  15  4  1  2  A  5  1.0  40  1.0  18  5  1  2  B  15  4.0  40  2.0  14  6  1  2  B  15  4.0  10  1.0  20  7  0  2  B  10  2.5  25  1.5  16  8  1  2  B  15  1.0  40  1.0  17  9  1  2  A  5  4.0  40  2.0  12  10  1  2  B  15  1.0  10  2.0  9  11  0  1  A  10  2.5  25  1.5  7  12  1  1  B  5  4.0  40  1.0  1  13  1  1  B  5  1.0  10  1.0  2  14  1  1  A  15  1.0  10  1.0  4  15  1  1  A  15  4.0  10  2.0  3  16  1  1  B  5  4.0  10  2.0  10  17  0  1  B  10  2.5  25  1.5  5  18  1  1  B  5  1.0  40  2.0  8  19  1  1  A  15  4.0  40  1.0  6  20  1  1  A  15  1.0  40  2.0   43
  44. 44. RunOrder  CenterPt  Blocks  Disint  Drug%  Disint%  DrugPS  Lub%  Disso%  WtRSD  1  1  2  A  5  1.0  10  2.0  100.4  1.6  2  1  2  A  5  4.0  10  1.0  103.0  2.1  3  0  2  A  10  2.5  25  1.5  88.8  1.6  4  1  2  A  5  1.0  40  1.0  94.3  2.3  5  1  2  B  15  4.0  40  2.0  78.9  1.6  6  1  2  B  15  4.0  10  1.0  102.9  2.0  7  0  2  B  10  2.5  25  1.5  90.9  1.4  8  1  2  B  15  1.0  40  1.0  91.8  2.2  9  1  2  A  5  4.0  40  2.0  76.3  1.4  10  1  2  B  15  1.0  10  2.0  103.4  1.6  11  0  1  A  10  2.5  25  1.5  89.9  1.8  12  1  1  B  5  4.0  40  1.0  91.8  2.2  13  1  1  B  5  1.0  10  1.0  101.2  2.2  14  1  1  A  15  1.0  10  1.0  101.8  2.6  15  1  1  A  15  4.0  10  2.0  102.5  1.4  16  1  1  B  5  4.0  10  2.0  100.3  1.5  17  0  1  B  10  2.5  25  1.5  91.2  1.6  18  1  1  B  5  1.0  40  2.0  76.3  1.3  19  1  1  A  15  4.0  40  1.0  92.4  2.1  20  1  1  A  15  1.0  40  2.0  76.8  1.6   44
  45. 45. 45
  46. 46. 46
  47. 47. 47
  48. 48. 48
  49. 49. 49
  50. 50. Source DF Adj MS F PBlocks 1 2.21 0.11 0.745Disint 1 0.30 0.01 0.905Drug% 1 2.94 0.15 0.707Disint% 1 0.30 0.01 0.905DrugPS 1 1174.45 58.93 0.000Lub% 1 258.61 12.98 0.004Curvature 1 32.68 1.64 0.225Res Error 12 19.93 2.179  is  the  1-­‐α/2   th  quan;le  of  the  t-­‐ distribu;on  having   12  df.   50
  51. 51. Source DF Adj MS F PBlocks 1 0.01090 0.51 0.487Disint 1 0.03751 1.77 0.208Drug% 1 0.00847 0.40 0.539Disint% 1 0.08282 3.91 0.071DrugPS 1 0.00189 0.09 0.770Lub% 1 2.10586 99.46 0.000Curvature 1 0.21198 10.01 0.008Res Error 12 0.02117 51
  52. 52. Disso%  •  Only  DrugPS  and  Lub%  show  significant  main  effects  •  Plot  of  Disso%  residuals  vs  predicted  Disso%  shows  systema;c   paKern.  •  The  residual  SD  (4.5)  is  considerably  larger  than  expected  (1.3)  WtRSD  •  Only  Lub%  shows  a  sta;s;cally  significant  main  effect  •  Curvature  is  significant  for  WtRSD  Therefore  •  Only  DrugPS  and  Lub%  need  to  be  considered  further  •  The  other  3  factors  can  fixed  at  nominal  levels.  •  The  predic;on  model  is  inadequate.  Addi;onal  experimenta;on   is  needed.   52
  53. 53. Trial   DrugPS   Lub%   Disso%   1   10   1   C   2   10   2   A   2   A   F   B   3   40   1   D   Lub%   4   40   2   B   G   I   H   5   25   1   E   1   C   E   D   6   25   2   F   10   40   7   10   1.5   G   DrugPS   8   40   1.5   H   9   25   1.5   I  Disso = a + b × Lub% + c × DrugPS + d × Lub% × DrugPS + e × Lub%2 + f × DrugPS2 + εDisso = a + b × Lub% + c × DrugPS + d × Lub% × DrugPS + ε 53
  54. 54. Response   Factor   54
  55. 55. Response surfacedesignFactorial orfractional factorialscreening design 55
  56. 56. 56
  57. 57. •     “Cube  Oriented”   •       3  or  5  levels  for  each  factor  In  3  factors     Factorial  or                                Center  Points       +   FracKonal  Factorial                      +            Axial  Points   =   Central  Composite  Design   57
  58. 58. 58
  59. 59. 59
  60. 60. 60
  61. 61. Std  Run  Center  Block  Disint  Drug%  Disint%  DrugPS  Lub%  Disso%  WtRSD  Order  Order  Point  11  1  1  2  A  5  1.0  10  2.0  100.4  1.6  13  2  1  2  A  5  4.0  10  1.0  103.0  2.1  19  3  0  2  A  10  2.5  25  1.5  88.8  1.6  15  4  1  2  A  5  1.0  40  1.0  94.3  2.3  18  5  1  2  B  15  4.0  40  2.0  78.9  1.6  …  10  17  0  1  B  10  2.5  25  1.5  91.2  1.6  5  18  1  1  B  5  1.0  40  2.0  76.3  1.3  8  19  1  1  A  15  4.0  40  1.0  92.4  2.1  6  20  1  1  A  15  1.0  40  2.0  76.8  1.6  21  21  -­‐1  3  A  10  2.5  10  1.5      22  22  -­‐1  3  A  10  2.5  40  1.5      23  23  -­‐1  3  A  10  2.5  25  1.0      24  24  -­‐1  3  A  10  2.5  25  2.0      25  25  0  3  A  10  2.5  25  1.5      26  26  0  3  A  10  2.5  25  1.5       61
  62. 62. Std  Run  Center  Block  Disint  Drug%  Disint%  DrugPS  Lub%  Disso%  WtRSD  Order  Order  Point  11  1  1  2  A  5  1.0  10  2.0  100.4  1.6  13  2  1  2  A  5  4.0  10  1.0  103.0  2.1  19  3  0  2  A  10  2.5  25  1.5  88.8  1.6  15  4  1  2  A  5  1.0  40  1.0  94.3  2.3  18  5  1  2  B  15  4.0  40  2.0  78.9  1.6  …  10  17  0  1  B  10  2.5  25  1.5  91.2  1.6  5  18  1  1  B  5  1.0  40  2.0  76.3  1.3  8  19  1  1  A  15  4.0  40  1.0  92.4  2.1  6  20  1  1  A  15  1.0  40  2.0  76.8  1.6  21  21  -­‐1  3  A  10  2.5  10  1.5  101.8  1.7  22  22  -­‐1  3  A  10  2.5  40  1.5  84.0  1.7  23  23  -­‐1  3  A  10  2.5  25  1.0  96.7  2.1  24  24  -­‐1  3  A  10  2.5  25  2.0  82.8  1.4  25  25  0  3  A  10  2.5  25  1.5  92.3  1.5  26  26  0  3  A  10  2.5  25  1.5  91.9  1.2   62
  63. 63. 63
  64. 64. Y = a + b ⋅ DrugPS + c ⋅ Lub% + d ⋅ DrugPS2 + e ⋅ Lub%2 + f ⋅ Drug ⋅ PSLub% + ε 64
  65. 65. 65
  66. 66. 66
  67. 67. 67
  68. 68. Source DF Adj SS Adj MS F PBlocks 2 2.27 1.13 0.48 0.625Regression Linear DrugPS 1 1331.87 1331.87 567.73 0.000 Lub% 1 340.61 340.61 145.19 0.000 Square DrugPS*DrugPS 1 27.39 27.39 11.68 0.003 Lub%*Lub% 1 0.14 0.14 0.06 0.811 Interaction DrugPS*Lub% 1 222.98 222.98 95.05 0.000Residual Error 18 42.23 2.35 Lack-of-Fit 7 25.15 3.59 2.32 0.103 Pure Error 11 17.07 1.55 68
  69. 69. Source DF Adj SS Adj MS F PBlocks 2 0.02341 0.01171 0.41 0.671Regression Linear DrugPS 1 0.00118 0.00118 0.04 0.842 Lub% 1 2.31351 2.31351 80.72 0.000 Square DrugPS*DrugPS 1 0.04980 0.04980 1.74 0.204 Lub%*Lub% 1 0.09743 0.09743 3.40 0.082 Interaction DrugPS*Lub% 1 0.00234 0.00234 0.08 0.778Residual Error 18 0.51589 0.02866 Lack-of-Fit 7 0.28587 0.04084 1.95 0.154 Pure Error 11 0.23003 0.02091 69
  70. 70. StaKsKcal  Significance?   Model  Term   Disso%   WtRSD   DrugPS   P   P   Lub%   P   P   DrugPS2   P   P   Lub%2   ?   DrugPS  ×  Lub%   P   P   Lack  of  Fit   ?  Y = a + b ⋅ DrugPS + c ⋅ Lub% + d ⋅ DrugPS2 + e ⋅ Lub%2 + f ⋅ Drug ⋅ PSLub% + ε 70
  71. 71. •  The  simplest  model  that  explains  the  data  is  best   (Occam’s  razor,  rule  of  parsimony)   •  Eliminate  “least  significant”  terms  one  at  a  ;me   followed  by  re-­‐analysis   •  Always  eliminate  highest  order  terms  first   •  Don’t  eliminate  lower  order  terms  which  are   contained  in  significant  higher  order  terms   •  Any  exis;ng  theory  or  prior  knowledge  trumps  these   rules.   ?  Y = a + b ⋅ DrugPS + c ⋅ Lub% + d ⋅ DrugPS2 + e ⋅ Lub%2 + f ⋅ Drug ⋅ PSLub% + ε 71
  72. 72. Estimated RegressionCoefficients for Disso% usingdata in uncoded unitsTerm CoefConstant 105.321DrugPS -0.478970Lub% 6.62343DrugPS*DrugPS 0.0130426Lub%*Lub% -0.959956DrugPS*Lub% -0.497745S = 1.49153 PRESS = 83.4051R-Sq = 97.76% R-Sq(pred) =95.79% R-Sq(adj) = 97.20% 72
  73. 73. Estimated RegressionCoefficients for WtRSD usingdata in uncoded unitsTerm CoefConstant 4.66698DrugPS -0.0293187Lub% -2.96608DrugPS*DrugPS 0.000623945Lub%*Lub% 0.763118DrugPS*Lub% -0.00161165S = 0.164211 PRESS = 0.850996R-Sq = 83.93% R-Sq(pred) =74.65% R-Sq(adj) = 79.92% 73
  74. 74. Acceptableperformancemore likely •  Difficult  to  do  with  >  2  factors   •  Does  not  take  into  account     •  es;ma;on  uncertainty   •  correla;on  among  responses   •  variability  in  control  of  factor   levels   •  variability  in  the  underlying  true   model  over  ;me   74
  75. 75. 75
  76. 76. 76
  77. 77. Global SolutionDrugPS = 11.2121Lub% = 1.93939Predicted ResponsesDisso% = 100.002 ,desirability = 1.000WtRSD = 1.500 ,desirability =0.117927Composite Desirability= 0.343404 77
  78. 78. Predicted Response for New Design Points Using Model for Disso%Point Fit SE Fit 95% CI 95% PI 1 100.002 0.621070 (98.7063, 101.297) (96.6316, 103.372)Predicted Response for New Design Points Using Model for WtRSDPoint Fit SE Fit 95% CI 95% PI 1 1.49952 0.0683772 (1.35689, 1.64216) (1.12848,1.87057) 78
  79. 79. 1.  Number  of  trials  ≥  Number  of  model  coefficients  2.  Each  coded  column  adds  to  0  (balance)  3.  Inner  product  of  any  2  coded  columns  =  0  (orthogonality)  4.  Use  resolu;on  V  (or  at  least  IV)  for  screening  designs  5.  Factor  ranges  are  bold  (but  not  too  bold)  6.  Incorporate  process  knowledge  &  sequen;al  strategies  7.  Assure  adequate  sample  size  (power)  8.  Randomize  processing  order  9.  Block  when  you  cannot  randomize  10.   Incorporate  tests  for  model  adequacy  (e.g.,  center  points)  11.   Avoid  PARC  (Planning  Ater  Research  is  Complete)   79
  80. 80. 1.  Use  graphics  (picture  =  1,000  words)  2.  Always  verify  model  assump;ons  (normality,  independence,   variance  homogeneity)  3.  In  model  reduc;on,  follow  rules  of  hierarchy  tempered  by  prior   process  knowledge    4.  Use  coded  factor  levels  in  judging  sta;s;cal  significance  of   model  coefficients.  5.  Consider  predic;on  uncertainty  when  iden;fying  op;mal  factor   secngs  6.  Take  advantage  of  curvature  &  interac;ons  when  choosing   op;mal  factor  secngs  7.  Always  perform  independent  trials  to  confirm  predic;ons.   80
  81. 81. Minitab   Surface Plot of Hard%RSD Overlaid Contour Plot of Hardness...Hard%RSD• General  purpose  stat  package   Lower Bound Upper Bound White area: feasible region 3.0 Hardness 19.5 20.5 Hard%RSD 0 7• User  friendly   20 Water(L) 2.5 15 Hard%RSD 10• Good  learning  tool   3.0 2.0 5 5 2.5 Water(L) 7 9 2.0 11 MixTime(min) 13 15 17 6 11 16 MixTime(min)  JMP  • General  purpose  stat  package  • Excellent  for  DOE  &  SPC  • Very  advanced  features   • Monte-­‐Carlo  simula;on  of  DOE  models   • Good  D-­‐op;mal  design  features  • May  need  sta;s;cal  support  for  some  features    Design  Expert  • Exclusive  focus  on  DOE  (may  want  addnl  tools)  • I  have  not  used  but  my  impression  is  very  good   81
  82. 82. Contour  Profiling   and  overlay  for     design  space  idenKficaKon  Monte-­‐Carlo  SimulaKon  to  determine  effect  of  poor  factor  control  on  future  batch  failure   67rate   82
  83. 83. • Robust  design  &  Taguchi  designs  • Mixture  (e.g.,gasoline  blend)  and  constrained  designs  • D-­‐op;mal  designs  and  custom  augmenta;on    • Bayesian  approaches   • Probability  of  mee;ng  specifica;ons   • mul;ple  correlated  responses   • incorpora;on  of  prior  knowledge  • Variance  component  analysis  &  Gage  R&R  • Split-­‐plot  experiments   83
  84. 84. 1.  Box, G. E. P.; Hunter, W. G., and Hunter, J. S. (1978). Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. John Wiley and Sons.2.  Montgomery D (2005) Design and analysis of experiments, 6th edition, Wiley.3.  Myers R, Montgomery D, and Anderson-Cook C (2009) Response surface methodology, Wiley.4.  Diamond W (1981) Practical Experiment Designs, Wadsworth, Belmont CA5.  Altan S, et al (2010) Statistical Considerations in Design Space Development (Parts I-III) PharmTech Nov 2, 2010. Available on line at http://www.pharmtech.com/pharmtech/author/ authorInfo.jsp?id=531186.  Conformia CMC-IM Working Group (2008) Pharmaceutical Development case study: “ACE Tablets”. Available from the following web site: http://www.pharmaqbd.com/files/articles/ QBD_ACE_Case_History.pdf7.  ICH Expert Working Group (2008) GUIDELINE on PHARMACEUTICAL DEVELOPMENT Q8(R1) Step 4 version dated 13 November 20088.  ICH Expert Working Group (2005) Guideline on QUALITY RISK MANAGEMENT Q9 Step 4 version dated 9 November 20059.  FDA CDER/CBER/CVM (November 2008) Draft Guidance for Industry Process Validation: General Principles and Practices (CGMP) Thank You!! 84

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