1. David.LeBlond@sbcglobal.net

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. 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. 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. Key
Factors
Key
Responses
Cau;on:
EVERYTHING
depends
on
gecng
this
right
!!!
5

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. 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. Lubricant%
2
95
90
70
1
85
10
40
DrugPS
Disso% = 86.667
+10 × Lub%
−0.667 × DrugPS
+ε
8

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. 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. 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. 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. 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. ž Model
non-­‐addiKve
behavior
› interacKons,
curvature
ž Efficiently
explore
the
factor
space
ž Take
advantage
of
hidden
replicaKon
14

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

17. 17

18. 18

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. 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. +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. Disso = a + b × Lub + c × DrugPS + d × Lub × DrugPS + ε
ž 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. +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

25. Dissolu;on
(%LC)
1%
Lubricant
2%
Lubricant
90
10
40
DrugPS
25

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. 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. 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. • “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. • 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. 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. 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. ReplicaKng
1
measurement
batch
3
batches
per
batch
producKon
Repeated
3
measurements
1
batch
per
batch
measurement
33

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. 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. 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
2
0.01
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. 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. 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. 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. 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

42. Confounding Table
I = ABCDE
Blk = AB = CDE
A = BCDE
B = ACDE
C = ABDE
D = ABCE
E = ABCD
AC = BDE
AD = BCE
AE = BCD
BC = ADE
BD = ACE
BE = ACD
CD = ABE
CE = ABD
DE = ABC
42

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. 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

46. 46

47. 47

48. 48

49. 49

50. Source DF Adj MS F P
Blocks 1 2.21 0.11 0.745
Disint 1 0.30 0.01 0.905
Drug% 1 2.94 0.15 0.707
Disint% 1 0.30 0.01 0.905
DrugPS 1 1174.45 58.93 0.000
Lub% 1 258.61 12.98 0.004
Curvature 1 32.68 1.64 0.225
Res Error 12 19.93
2.179
is
the
1-­‐α/2
th
quan;le
of
the
t-­‐
distribu;on
having
12
df.
50

51. Source DF Adj MS F P
Blocks 1 0.01090 0.51 0.487
Disint 1 0.03751 1.77 0.208
Drug% 1 0.00847 0.40 0.539
Disint% 1 0.08282 3.91 0.071
DrugPS 1 0.00189 0.09 0.770
Lub% 1 2.10586 99.46 0.000
Curvature 1 0.21198 10.01 0.008
Res Error 12 0.02117
51

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. 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 + ε
Disso = a + b × Lub% + c × DrugPS + d × Lub% × DrugPS + ε
53

54. Response
Factor
54

55. Response surface
design
Factorial or
fractional factorial
screening design
55

56. 56

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

59. 59

60. 60

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. 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

64. Y = a + b ⋅ DrugPS + c ⋅ Lub% + d ⋅ DrugPS2 + e ⋅ Lub%2 + f ⋅ Drug ⋅ PSLub% + ε
64

65. 65

66. 66

67. 67

68. Source DF Adj SS Adj MS F P
Blocks 2 2.27 1.13 0.48 0.625
Regression
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 0.000
Residual 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. Source DF Adj SS Adj MS F P
Blocks 2 0.02341 0.01171 0.41 0.671
Regression
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 0.778
Residual 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. 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. • 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. Estimated Regression
Coefficients for Disso% using
data in uncoded units
Term Coef
Constant 105.321
DrugPS -0.478970
Lub% 6.62343
DrugPS*DrugPS 0.0130426
Lub%*Lub% -0.959956
DrugPS*Lub% -0.497745
S = 1.49153 PRESS = 83.4051
R-Sq = 97.76% R-Sq(pred) =
95.79% R-Sq(adj) = 97.20%
72

73. Estimated Regression
Coefficients for WtRSD using
data in uncoded units
Term Coef
Constant 4.66698
DrugPS -0.0293187
Lub% -2.96608
DrugPS*DrugPS 0.000623945
Lub%*Lub% 0.763118
DrugPS*Lub% -0.00161165
S = 0.164211 PRESS = 0.850996
R-Sq = 83.93% R-Sq(pred) =
74.65% R-Sq(adj) = 79.92%
73

74. Acceptable
performance
more 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

76. 76

77. Global Solution
DrugPS = 11.2121
Lub% = 1.93939
Predicted Responses
Disso% = 100.002 ,
desirability = 1.000
WtRSD = 1.500 ,
desirability =
0.117927
Composite Desirability
= 0.343404
77

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 WtRSD
Point Fit SE Fit 95% CI 95% PI
1 1.49952 0.0683772 (1.35689, 1.64216) (1.12848,
1.87057)
78

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. 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. 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. Contour
Profiling
and
overlay
for
design
space
idenKficaKon
Monte-­‐Carlo
SimulaKon
to
determine
effect
of
poor
factor
control
on
future
batch
failure
67
rate
82

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. 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 CA
5. 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=53118
6. 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.pdf
7. ICH Expert Working Group (2008) GUIDELINE on PHARMACEUTICAL DEVELOPMENT Q8(R1)
Step 4 version dated 13 November 2008
8. ICH Expert Working Group (2005) Guideline on QUALITY RISK MANAGEMENT Q9 Step 4
version dated 9 November 2005
9. FDA CDER/CBER/CVM (November 2008) Draft Guidance for Industry Process Validation:
General Principles and Practices (CGMP)
Thank You!!
84