Hi,
I'm
Ron
Kenett.
This
is
a
joint
talk
with
Chris
Gotwalt.
The
talk
is
on
Functional
Data
Analysis
and
Nonlinear
Regression
Models.
And
in
order
to
examine
the
options
and
what
we
get
out
of
this
type
of
analysis,
we
will
take
an
information
quality
perspective.
In
a
sense,
this
is
a
follow
up
to
a
talk
we
gave
last
year
at
the
same
Discovery
Summit.
So
I
will
start
with
simple
examples
to
introduce
FDA
and
Nonlinear
Regression.
And
then
Chris
will
cover
a complex and more
substantially
more
complex
example
of
optimization,
which
includes
mixture
experiments
designed
to
match
a
reference
profile.
So
the
story
starts
with
data
on
tablets
that
are
dissolved
and
measurements
are
done
at
different
time
points,
five
minutes,
ten
minutes,
15
minutes,
every
five
minutes,
and
then
20
minutes,
ten
minutes
later,
30,
and
then
15
minutes
later,
45
minutes.
We
have
12
tablets
that
are
our
product
and
12
tablets
that
are
the
reference.
Our
goal
is
to
have
a
product
that
matches
the
reference.
And
in
this
type
of
data,
we
have
a
profile,
and
we
consider
two
options,
FDA
and
NLR.
In
Chris' example,
we'll
talk
about
something
called
the F
2,
which
is
a
third
option
for
analyzing
this
type
of
data.
So
here's
what
it
looks
like
with
the
graph
builder.
On
the
left,
we
have
the
reference
profiles.
On
the
right,
we
have
the
tablets
and
the
test.
We
can
see
this
is
an
example
from
my
book
on
modern
industrial
statistics,
the
book
with
Shelley
Zacks,
which
is
now
in
its
third
edition.
So
on
the
left
you
can
see
there
is
a
tablet
that
seems
a
bit
different.
It's
labeled
T
5R.
And
if
we
run
a
functional
data
analysis
of
this
data,
T
5R
does
look
different.
We
see
that
the
growth
part
is
different.
It
has
a
slow
growth
but
consistent
growth.
It
does
not
have
the
shape
that
we
see
in
the
other
dissolution
curves.
This
was
done
with
a
Quadratic
B-spline
with
1
knot,
and
the
quadratic
was
in
this
case,
fitting
the
data
better
than
the
QB.
This
is
a
bit
of
an
unusual
situation.
So
because
of
the
shapes,
the
Quadratic B-spline
was
a
better
fit.
If
we
look
at
T
1R,
the
first
tablet
that
has
still
different
shape,
it
shoots
up
and
then
it
stays.
So
basically,
the
tablet
has
dissolved.
Obviously,
beyond
a
high
number
of
dissolution,
there's
not
much
left
for
the
solution.
So
T
1R
and
T
5R,
they
seem
different.
T
5R
stands
out
more
than
T
1R.
So,
yes,
T
5R
on
the
cluster
analysis
on
the
functional
principal
components
does
stand
out.
So
here
we
see
how
functional
principal
components
scatter
plot
of
the
first
two
functional
principal
components
points,
what
we
observe
visually.
And
T1R,
which
is
next
to
T
2R,
is
a
different
cluster.
We
can
proceed
with
a
nonlinear
regression
approach.
Here
we
are
fitting
a
Gompertz
three
parameter
model
with
three
parameters,
the
asymptote,
the
growth
rate,
and
the
inflection
point.
This
is
the
model
and
when
we
fit
the
profiles,
we
again
see
that
T
5R
stands
out.
So
we
have
the
same
qualitative
impression
that
we
had
with
FDA.
Now
we
have
these
three
parameters
listed
and
we
can
run
a
profiler
on
the
model
because
we
now
have
a
model.
This
is
where
T
1R
stands.
So
by
running
the
profile
on
the
different
tablets,
we
can
also
see
how
similar
or different
they
look
like.
This
is
the
table
that
maps
out
the
parameters
of
this
asymptotes.
so
T
1R
growth
rate .21,
T5R, this
tablet
that
stood
out.
Growth
rate
.075,
very
slow
growth
rate,
consistent
but
slow.
The
inflection
point
is
11.5,
way
on
the
right.
So
we
can
see
through
this
parameter
values
the
difference.
We
can
also
pick
up
two
tablets
that
stand
out
for
growth
rates.
T2R
1.77,
and T8R,
almost
no
growth.
We'll
get
back
to
T
2
and
T
8
in
a
minute.
If
we
take
the
principal
components
of
these
three
parameters
space.
So
we
conceal
the
parameters
as
if
these
are
the
measurements
and
we
run
a
multivariate
control
shot.
We
can
see
T
1,
this
is
the
first
one
and
we
can
see
T
5,
this
is
the
blue
one,
the
fifth
one.
This
we
already
saw.
They
are
within
the
control
limits
of
the
T
Square
multivariate
statistical
distance
control
chart.
And
T
2
and
T
8
that
I
highlighted
before
now
stand
out
and
we
can
see
qualitatively
why.
This
is
the
model
degradation
approach
in
the
guidance
documents
that
is
used
for
modeling
dissolution
curves.
In
running
such
analysis
with
an
information
quality
perspective,
the
first
question
is
asking
what
is
the
goal
of
the
analysis?
And
then
we
can
consider
the
method
of
analysis.
Here
we're
using
nonlinear
regression
and
functional
data
analysis.
Chris
will
get
into
how
this
is
combined
with
data
derived
from
experimental
design.
We
have
a
utility
function
and
the
information
quality
is
the
utility
of
applying
a
method
F
on
data
X
conditio
n of
the
goal.
It is
evaluated
with
eight
dimensions.
And
here
Chris
again
we'll
talk
about
data
resolution
and
data
structure.
So
Chris,
the
floor
is
yours.
Thanks
Ron.
Now
I'm
going
to
give
an
example
that
is
a
little
bit
more
complicated
than
the
first
one.
In
Ron's
example,
he
was
comparing
the
dissolution
curves
of
test
tablets
to
those
from
a
set
of
reference
tablets.
In
that
situation,
the
expectation
is
that
the
curves
should
generally
be
following
the
same
path.
And
he
showed
how
to
find
anomalous
curves
that
deviate
from
the
rest
of
the
population.
In
this
second
example,
we
also
have
a
reference
dissolution
curve.
But
we
are
analyzing
data
from
a
designed
experiment
where
the
goal
is
to
find
a
formulation
in
two
polymer
additives
and
the
amount
of
force
used
in
the
tablet
production
process
that
leads
to
a
close
match
to
the
reference
splashes
the
solution
curve.
The
graph
you
see
here
shows
the
data
from
the
reference
curve
that
we
want
to
match.
To
do
this,
I'm
going
to
demonstrate
three
analyses
of
the
data
that
use
different
methods
and
models
to
find
factor
settings
that
match
the
reference
that
lead
to
a...
To
do
this,
I'm
going
to
demonstrate
three
analysis
of
this
data
that
use
different
methods
and
models
to
find
factor
settings
that
will
best
match
the
reference
curve.
In
the
first
analysis,
I'm
going
to
summarize
each
of
the
DoE
curves
down
to
a
single
metric
called
F2
that
is
typically
used
in
dissolution
curve
analysis,
a
measure
of
agreement
with
the
reference
match.
There,
I'll
use
standard
DoE
methods
to
model
that
F2
response
and
then
find
the
factor
settings
that
are
predicted
to
best
agree
with
the
reference.
In
the
second
analysis,
I'll
use
a
functional
DoE
modeling
approach
where
I
model
the
curves
using
these
blinds,
extract
functional
principle
component
stores,
and
model
them.
I'll
load
the
reference
batch
as
a
target
function
in
the
Functional
Data
Explorer
platform
and
then
use
the
FDoE
profiler
to
find
the
closest
match
recommended
by
that
model.
These
first
two
approaches
use
little
subject
matter
information
about
these
types
of
tablets.
In
the
third
analysis,
I'll
model
the
curves
using
a
nonlinear
model
that
was
known
to
fit
this
type
of
tablet
well
and
use
the
Curve
DoE
option
in
the
fit
Curve
platform
to
model
the
relationship
between
the
DoE
factors
and
the
shape
of
the
curve.
I
want
to
credit
Clay
Barker
for
adding
this
capability
to
JMP
Pro
16.
I
think
it
has
a
lot
of
promise
for
modeling
curves
whose
general
shape
can
be
assumed
to
be
known
in
advance
to
come
from
one
of
the
supported
nonlinear
models.
At
the
end,
verification
batches
were
made
using
the
recommended
formulation
settings
for
each
of
the
three
analyses,
and
we
compared
them
to
a
new
reference
batch.
What
we
found
was
that
the
nonlinear
regression-based
approach
led
to
the
closest
match
to
the
reference.
What
we
see
here
is
a
scatter
plot
matrix
of
the
four
factors
in
the
designed
experiment.
There
was
a
mixture
constraint
between
the
two
polymers,
as
well
as
a
constraint
on
the
total
amount
of
polymer
and
the
proportions
of
the
individual
polymers.
Here's
a
look
at
some
of
the
raw
data
from
the
experiment.
At
the
top
of
the
table,
we
have
data
from
the
reference
that
we
wish
to
match.
There
are
16
DoE
formulations
or
batches
in
the
experiment.
We
can
only
see
data
from
two
of
them
in
this
picture,
though.
There
were
six
tablets
per
formulation.
There
were
four
dissolution
measurements
per
tablet.
Here
we
see
plots
of
the
dissolution
curves
for
each
of
the
16
DoE
formulations
with
the
dissolution
curve
of
the
reference
batch
here
at
the
lower
right.
Now,
I'm
going
to
do
a
quick
preliminary
information
quality
assessment
using
the
questions
that
you'll
find
in
the
spreadsheet
that
you
can
download
from
the
JMP
user
community
page.
The
first
part
of
the
assessment
is
related
to
the
data
resolution.
In
this
case,
I
think
we're
looking
pretty
good.
The
data
scale
is
well
aligned
with
the
stated
goal
because
it's
a
design
experiment.
The
measuring
devices
seem
to
be
reliable
and
precise,
and
the
data
analysis
is
definitely
going
to
be
suitable
for
the
data
aggregation
level,
and
we'll
be
illustrating
different
kinds
of
data
aggregation
as
we
extract
features
from
these
dissolution
curves.
As
far
as
the
data
structure
goes,
we're
in
pretty
good
shape.
The
data
is
certainly
aligned
with
the
stated
goal,
we
don't
have
any
problems
with
outliers
or
missing
values,
and
the
analysis
methods
are
all
suitable
for
the
data
structure,
although
we
do
see
some
variation
in
the
quality
of
the
results
depending
on
the
type
of
analysis
we
do.
As
far
as
data
integration
goes,
this
is
a
pretty
simple
analysis.
We
have
multiple
responses,
and
we're
exploring
different
ways
of
combining
them
into
extracted
features.
So
there's
a
common
workflow
to
all
three
of
the
analysis
I'm
going
to
be
showing.
First,
we
have
to
get
the
data
into
a
form
that
is
analyzable
by
the
platform
that
we're
using.
Then
there's
around
a
feature
extraction.
Then
we
model
those
features.
That's
where
there's
a
lot
of
difference
between
the
methods,
and
then
we
use
the
profiler
in
different
ways
to
find
a
formulation
that
closely
matches
the
reference.
First,
I'm
going
to
go
over
the
F2
analysis.
F
2
is
a
standard
measure
of
agreement
of
a
dissolution
curve
relative
to
a
reference
dissolution
curve.
In
the
formula,
the
Rs
are
the
means
of
the
reference
curve
at
each
time
point,
and
the
Ts
are
the
means
of
the
non-reference
curve.
The
convention
is
to
say
that
the
two
curves
are
equivalent
when
F2
is
greater
than
or
equal
to
50.
It's
important
to
point
out
that
I'm
including
this
F
2
based
analysis
not
just
as
an
example
of
a
dissolution
DoE
analysis,
but
more
broadly
as
an
example
of
how
reducing
a
response
that
is
inherently
a curve
down
to
a
single
number
leads
to
a
much
lower
quality
analysis
and
results
at
the
end
than
a
procedure
that
treats
curves
as
first-class
citizens.
So
now
I'm
going
to
share
the
F2
analysis
of
the
dissolution
DoE
data.
The
first
thing
we
have
to
do
is
calculate
the
batch
means
of
the
dissolution
curves
at
the
different
time
points.
Then
we
create
a
formula
column
that
calculates
the
F2
dissolution
curve
agreement
statistic
for
each
of
these
curves
relative
to
the
reference
batch,
and
we
modeled
the
F2
using
the
DoE
factors
as
inputs
and
use
the
profiler
to
find
the
factor
settings
that
match
the
reference.
Before
the
analysis,
we
use
the
table's
summary
feature
to
calculate
the
means
of
the
dissolution
measurements
by
batch
and
across
each
of
the
times.
We
can
save
ourselves
a
little
bit
of
work
by
using
all
of
the
DoE
factors
here
as
grouping
variables
so
they'll
be
carried
through
into
the
subsequent
table.
Now
we
have
a
17-row
data
set
and
we
hide
and
exclude
the
reference
batch.
Now
take
note
of
the
values
of
the
dissolution
means
for
the
reference
because
we're
going
to
use
those
when
we
create
a
formula
column
that
calculates
the
F2
agreement
metric
for
each
of
the
batches
relative
to
the
reference
batch
and
now
we're
going
to
be
able
to
use
this
F2
formula
column
as
a
response
to
be
modeled.
We
use
the
model
script
created
by
the
DoE
platform
to
set
up
our
model
for
us.
We
place
F2
as
our
response
variable
and
we're
going
to
analyze
this
data
today
using
the
generalized
regression
platform
in
JMP
Pro.
When
we
get
into
the
platform
we
see
that
it
has
automatically
done
a
standard
lease
squares
analysis
because
it
found
that
there
were
enough
degrees
of
freedom
in
the
data
for
it
to
do
so
and
it's
given
us
an
AICc
of
155.6.
I'm
going
to
see
if
we
can
do
better
by
trying
a
best
subsets
reduction
of
the
model
and
when
we
do
that
we
see
that
the
AICc
of
that
best
subsets
goes
down
to
136
smaller
is
better
with
the
AICc
and
the
difference
of
20
is
pretty
substantial
so
I
would
conclude
that
the
normal
best
subset
is
a
better
model
than
the
standardly
squares
one.
I'm
going
to
try
one
more
thing,
though,
and
fit
a
log
normal
distribution
with
best
subsets
to
the
data.
When
I
do
that,
the
AICc
goes
down
a
little
bit
further
to
130.6.
That's
a
modest
difference,
but
it's
good
enough
that
I'm
going
to
conclude
that
we
are
going
to
work
with
the
Lo
gNormal,
especially
because
we
know
that
we're
working
with
a
strictly
positive
response
and
the
LogNormal
distribution
fits
data
that
is
strictly
positive.
From
there,
the
analysis
is
pretty
straightforward,
so
I'm
going
to
jump
straight
ahead
to
using
the
profiler.
F2
is
an
agreement
metric
that
we
want
to
maximize.
So
we
get
into
the
profiler,
we
turn
on
Desirability
functions
and
have
them
set
to
maximize,
and
then
maximize
desirability
to
find
the
combination
of
factor
settings
that
this
model
says
gives
us
the
closest
match
to
the
reference,
and
that
would
be
at
this
combination
of
factor
settings
that
we
see
here.
Now
the
F2
analysis
is
complete,
and
we're
going
to
go
into
the
second
analysis,
the
functional
DoE
analysis.
For
this
analysis,
we're
going
to
work
with
the
data
in
a
stacked
format
where
all
of
the
dissolution
measurements
have
been
combined
into
a
single
column,
and
we
have
a
time
column
as
well.
The
first
thing
we
do
is
go
into
the
Functional
Data
Explorer
platform.
In
the
platform
launch,
we
put
dissolution
as
our
response,
time
is
our
X,
the
batch
column
as
our
ID,
and
we
supply
the
four
DoE
factors
as
supplementary
variables.
Once
we're
in
the
platform,
we
take
a
look
at
the
data
using
the
initial
data
plot.
This
particular
data
set
doesn't
need
any
clean
up
or
alignment
options,
but
we
are
going
to
go
ahead
and
load
the
reference
dissolution
curve
as
a
target
function.
For
relatively
simple
functions
like
these,
I
typically
use
B
-Splines
for
my
functional
model.
When
we
do
that,
we
see
our
B-Spline
model
fit,
and
the
initial
fit
that
has
come
up
is
a
cubic
model
that
is
behaving
poorly.
It's
interpolating
the
data
points
well,
but
kind
of
doing
crazy
things
in
between
them.
So
I'm
going
to
change
from
the
default
recommended
model
over
to
a
quadratic
Spline
model
instead
of
the
cubic
one.
We
do
that
by
simply
clicking
on
Quadratic
over
here
in
the
right
of
the
B-Spline
model
fit.
We'll
see
that
this
quadratic
model
fits
the
data
well.
A
functional
principal
components
analysis
is
automatically
calculated,
and
we
see
that
the
Functional
Data
Explorer
platform
has
found
three
functional
principal
components.
The
leading
one
is
very
dominant,
explaining
97.
9 percent
of
the
functional
variation.
And
it
looks
like
this
is
a
level
set
up
or
down
kind
of
shape
component.
The
second
one
looks
like
a
rate
component,
and
the
third
one
almost
looks
like
a
quadratic.
Looking
a
little
closer
at
this
quadratic
B-Spline
model
fit,
we
see
that
this
model
is
fitting
the
individual
dissolution
curves
pretty
well.
So
now
we're
ready
to
do
our
functional
DoE
analysis.
Each
of
our
individual
dissolution
curves
has
been
approximated
now
by
an
underlying
mean
function
common
to
all
the
batches
plus
a
batch
dependent
FPC
score
times
the
first
eigenfunction,
plus
another
batch dependent
FPC
score
times
the
second
eigenfunction,
and
so
on
with
the
third
one.
What
we're
going
to
do
is
set
up
individual
DoE
models
for
each
of
these
functional
principal
component
scores
as
responses
using
our
DoE
factors
as
inputs.
The
Functional
Data
Explorer
platform,
of
course,
makes
all
this
simple
and
kind
of
ties
it
up
into
a
bow
for
us.
And
when
I
say
that,
it
ties
it
up
in
a
bow
for
us,
what
I
really
mean
is
the
FDoE
profiler.
So
this
pane
here
shows
our
predicted
trajectory
of
dissolution
as
a
function
of
time,
and
then
we
can
see
how
that
trajectory
would
change
by
altering
the
DoE
factors.
That
relationship
with
the
DoE
factors
comes
from
these
three
generalized
regression
models
for
each
of
our
functional
principal
component
scores.
If
we
want,
we
can
open
those
up
and
we
can
look
at
the
relationship
between
the
DoE
factors
and
that
functional
principal
component
score,
and
we
could
even
alter
the
model
by
moving
around
to
other
ones
in
the
solution
path.
I
just
want
to
point
out
that
it's
possible
to
change
the
DoE
model
for
an
FPC
score.
In
the
interest
of
time,
I'm
just
going
to
have
to
move
on
and
not
demonstrate
that,
though.
We
have
diagnostic
plots,
the
most
important
one
probably
being
the
Actual
by Predicted
Plot.
This
has
our
plotted
dissolution
measurements
on
the
Y-axis
and
the
predicted
dissolution
values
using
the
functional
DoE
model.
And
as
always,
we
want
to
see
that
plot
have
data
points
tight
along
the
45
degree
line.
And
in
this
case,
I
think
this
model
looks
pretty
good.
We
don't
want
to
see
any
patterns
in
our
residuals,
and
I'm
not
seeing
any
bad
ones
here.
So
this
model
looks
pretty
good,
and
we're
going
to
work
with
it.
So
I've
already
explained
how
this
pane
right
here
represents
the
predicted
dissolution
curve
as
a
function
of
time
and
the
individual
DoE
factors.
Now,
these
other
two
rows
here
are
because
we've
loaded
the
reference
as
a
target
function.
So
this
row
is
the
difference
of
the
predicted
dissolution
curve
from
the
target
reference
curve.
And
then
the
bottom
pane
here
is
the
integrated
distance
of
the
predicted
curve
from
the
target.
When
we
maximize
desirability
in
this
profiler,
it
gives
us
the
combination
of
factor
settings
that
minimize
this
integrated
distance
from
the
target.
So
I'm
going
to
do
that
by
bringing
up
maximize
desirability.
And
now
we
see
the
results
of
the
functional
DoE
analysis,
where
we
have
identified
.725
of
polymer
A
275 of
polymer
B,
a
total
polymer
of
about
.17,
and
a
compression
force
of
about
1700
minimize
the
distance
between
our
predicted
curve
and
the
reference.
Now,
we've
done
two
analyses.
Both
of
those
analyses
have
recommended
that
we
go
to
the
lowest
setting
of
polymer
A
and
the
highest
setting
of
polymer
B.
They
differ
in
their
recommendations
for
what
total
polymer
amount
to
use
and
how
much
compression
force
to
use.
The
third
analysis
I'm
going
to
do
is
the
Curve
DoE
analysis.
This
is
going
to
be
structured
pretty
similar
in
some
ways
to
the
functional
DoE
analysis,
in
that
we're
going
to
use
the
same
version
of
the
data
where
dissolution
measurements
are
all
in
one
column
and
we
have
a
time
column.
But
we
don't
have
a
built-in
target
function
option
in
the
Fit
Curve
platform
yet.
So
the
first
thing
we
have
to
do
is
fit
just
the
reference
batch
and
save
its
prediction
formula
back
to
the
table.
Then
we
do
a
Curve
DoE
analysis,
which
is
largely
similar
to
a
functional
DoE
analysis
in
that
we're
extracting
features
from
the
curves
modeling
the
curves.
Then
we
go
to
the
graph
profile
to
find
settings
that
best
match
the
reference.
The
nonlinear
model
that
we're
going
to
be
using
is
a
three
parameter
Weibull
Growth
Curve,
which
has
a
long
history
in
the
analysis
of
dissolution
curves.
Weibull Growth Curve
have
an
asymptote
parameter
A
that
represents
the
value
as
time
goes
to
infinity.
There's
what's
called
the
inflection
point
parameter
that
I
see
is
a
scaling
factor
that
kind
of
stretches
out
or
squeezes
in
the
entirety
of
the
curve.
And
then
there's
also
a
growth
rate
parameter
that
dictates
the
shape
of
the
curve.
What
I
think
is
really
valuable
about
using
this
model
relative
to
the
functional
DoE
model
or
the
F2
type
analysis
is
that
we're
going
to
be
modeling
features
extracted
from
the
data
that
have
real
scientific
meaning,
especially
the
asymptote
and
inflection
point
parameters.
Now
Curve
DoE
analysis
doesn't
have
a
target
matching
capability
like
the
Functional
Data
Explorer.
So
we
begin
the
analysis
by
excluding
all
of
the
DoE
rows
in
the
data
table.
These
are
represented
with
the
set
column
equal
to
A.
So
I
select
a
cell
there,
select
matching
cells,
and
then
hide
and
exclude
those
rows
so
that
I
only
have
the
reference
batch
not
excluded.
Then
I
go
to
the
Fit
Curve
platform,
load
it
up,
get
in
there,
fit
the
Weibull
growth
model,
and
then
I
save
that
prediction
formula
back
to
the
table.
Once
we
complete
the
Curve
DoE
analysis,
we're
going
to
compare
the
Curve
DoE
prediction
formula
to
this
reference
predictor
to
find
combinations
of
the
factor
settings
that
get
us
as
close
to
this
curve
as
possible.
So
now
we
unhide
and
exclude
the
DoE
batches,
go
back
into
the
Fit
Curve
platform,
just
like
the
Functional
Data
Explorer
platform.
We're
going
to
load
up
the
DoE
factors
as
supplementary
variables.
Now
that
we're
in
the
platform,
we
can
fit
our
Weibull
growth
model.
The
initial
fit
here
looks
pretty
good.
Looks
like
we're
capturing
the
shape
of
the
dissolution
curves.
One
thing
I
like
to
do
next
is
to
make
a
parameter
table.
This
creates
a
data
table
with
our
fitted
down
the
near
regression
parameters.
I
like
to
look
at
these
in
the
distribution
platform
to
see
if
there
are
outliers
in
there
or
anything
unusual.
I
also
like
to
look
at
the
patterns
I
see
in
the
multivariate
platform
just
gives
you
a
better
sense
of
what's
going
on
with
the
nonlinear
model
fit.
Once
we
know
that
everything
is
looking
pretty
good,
we
can
do
our
curve
DoE
analysis
and
this
looks
very
much
like
the
functional
DoE
analysis
from
before.
We
have
a
profiler
that
shows
the
relationship
between
dissolution
and
time
and
how
that
relationship
changes
as
a
function
of
our
DoE
factors.
And
then
we
also
have
a
generalized
regression
model
for
each
of
those
three
parameters
that
we
can
take
a
look
at
individually.
The
first
thing
I
would
do
before
trying
to
use
the
model
in
any
way
is
look
at
the
Actual
by Predicted
Plot,
so
that's
what
we
see
here.
This
is
the
predicted
values
incorporating
both
the
time
model
from
the
mean
function
and
the
eigen
functions,
as
well
as
the
DoE
models
on
the
nonlinear
regression
parameters.
This
looks
pretty
good
because
there
is
a
fairly
easy
interpretation
for
the
viable
growth
model
parameters.,
It
can
be
useful
and
interesting
to
open
up
the
individual
model
fits
for
these
parameters.
For
example,
here
are
the
coefficients
for
the
inflection
point
model.
Because
inflection
point
is
a
strictly
positive
quantity,
a
LogNormal
best
subsets
model
has
been
fixed
with
the
data
by
the
generalized
regression
platform.
We
see
that
the
mixture
manifests
have
been
forced
in
and
that
the
compression
forced
by
polymer
A
interaction
is
the
only
other
term
in
the
model.
What
this
means
is
that
if
we
hold
the
polymer
proportions
constant
and
increase
the
compression
force,
we
would
expect
a
larger
value
of
the
inflection
point.
One
would
observe
this
as
a
tablet
that
takes
longer
to
dissolve,
which
is
exactly
what
we
would
expect
to
have
happen.
We
can
save
the
curve
DoE
prediction
formula
back
to
the
table
and
we
can
see
in
all
its
gory
detail
how
the
model
for
asymptote
inflection
point
and
growth
rate
are
combined
with
time
to
come
up
with
our
overall
prediction
for
the
dissolution
curve
based
on.
Fortunately,
with
junk
we
don't
have
to
look
at
the
formula
too
closely
though,
because
we
have
profilers
that
let
us
see
the
relationships
in
a
visual
way
rather
than
an
algebraic
one.
To
solve
our
problem
of
finding
the
combination
of
factors
that
give
us
the
dissolution
curve
that
would
be
closest
to
the
reference,
I
created
a
formula
column
that
calculated
the
percentage
difference
of
the
predicted
curve,
taking
into
consideration
the
DoE
factors
from
the
reference.
The
last
step
of
the
analysis
is
to
bring
up
this
percent
difference
response
in
the
profiler
that
is
under
the
graph
menu,
being
sure
to
check
the
Expand
Intermediate
formulas
option.
This
led
to
a
profiler
where
we're
able
to
see
the
percent
difference
from
the
reference
as
a
function
of
time
and
the
DoE
factors,
I've
shared
the
region
where
the
difference
is
less
than
one percent
in
green.
By
manually
adjusting
the
factors,
I
was
able
to
find
settings
where
the
predicted
curve
is
less
than
one percent
from
the
reference
across
all
time
values.
This
looks
really
good,
but
in
practice
I
bet
that
this
is
overly
optimistic.
Here
we
see
the
optimal
values
of
the
factor
settings
for
all
three
analyses.
The
curve
DoE
analysis
is
in
the
interior
of
the
range
for
the
polymers.
The
optimal
value
for
total
polymer
is .
16
which
is
close
to
the
functional
DoE
analysis
result,
and
compression
force
is
in
between
the
optimal
values
recommended
by
the
F2
analysis
and
the
functional
DoE
analysis.
After
this,
we
made
new
formulations
based
on
the
recommended
factor
settings
from
each
of
these
models
and
measured
their
dissolution
curves
as
well
as
took
a
new
set
of
measurements
from
the
reference.
Here
we
see
a
summary
of
the
final
results
from
the
verification
runs.
The
new
reference
distribution
curve
is
in
black,
and
the
curve
DoE
in
green
is
the
closest
curve
to
it,
followed
by
the
F
DoE
curve
in
blue.
The
result
of
modeling
F2
is
in
red,
and
it
did
the
poorest
overall.
This
should
perhaps
not
be
too
surprising.
The
F2
metric
was
the
simplest,
reducing
the
data
down
to
a
single
metric
and
did
the
poorest.
The
functional
DoE
model
had
to
empirically
derive
the
shapes
of
the
curves
and
then
model
three
features
of
those
shapes,
essentially
using
more
of
the
information
in
the
data.
The
curve
DoE
led
to
the
best
formulation
because
it
used
the
data
efficiently
via
some
prior
knowledge
about
the
parametric
form
of
the
dissolution
curves.
We
see
that
the
results
of
the
F2
based
analysis
are
not
equivalent
with
the
new
reference
patch,
while
the
approaches
that
treated
curves
as
first
class
objects
are
equivalent.
What
this
means
is
that
the
approach
would
have
required
at
least
another
round
of
DoE
runs,
and
so
an
inefficient
analysis
leads
to
an
inefficient
use
of
time
and
resources.
I'm
going
to
close
the
presentation
with
a
retrospective
Info
Q
Assessment
of
the
Results.
Overall,
we
found
that
the
curved DoE
prediction
generalized
the
best
to
new
data,
but
was
the
most
difficult
analysis
to
perform.
I
want
to
note
that
if
we
didn't
have
a
known
nonlinear
model
to
work
with
that
fit
the
data
well,
we
could
not
have
done
that
analysis.
The
functional
DoE
analysis
and
the
F
2 based
approach
can
be
used
more
broadly
in
other
situations.
The
profiler
leads
to
excellent
communication
scores
for
all
three
analyses.
The
ability
to
see
how
the
shape
of
the
dissolution
curve
changes
with
the
DoE
factors
in
the
functional
and
curve-based
approaches
leads
me
to
give
them
a
better
communication
score.
I
see
the
curve
DoE
approach
having
the
highest
communication
score
by
a
little
bit
because
we're
directly
modeling
more
meaningful
parameters
than
the
functional
DoE approach.
That's
all
we
have
for
you
today.
I
want
to
thank
you
for
your
time,
interest,
and
attention.
