Hi,
my
name
is
Colleen
McKendry,
and
I
am
a
senior
statistical
writer
at
JMP,
but
I
also
like
to
play
around
with
functional
data.
This
project
is
on
measurement
systems
analysis
for
curve
data.
First,
I'm
just
going
to
give
a
very
brief
background
on
MSA
studies
in
general.
MSA
studies
determine
how
well
a
process
can
be
measured
prior
to
studying
the
process
itself.
It
answers
the
question,
how
much
measurement
variation
is
contributing
to
the
overall
process
variation.
Specifically,
the
Gage
R&R
method,
which
is
what
I'll
be
using
in
my
analysis,
determines
how
much
variation
is
due
to
operation
variation
versus
measurement
variation.
You
can
use
a
Gage
R&R
crossed
MSA
model
when
you
have
both
a
part
and
an
operator.
The
model
you
can
see
here
for
your
measurement
Y
sub
I
J
K,
that's
going
to
be
the
Kth
measurement
made
by
the
Jth
operator
on
the
Ith
part.
In
this
model,
you
have
a
mean
term,
a
random
effect
that
corresponds
to
the
part,
a
random
effect
that
corresponds
to
the
operator,
and
a
random
effect
that
corresponds
to
the
interaction
or
cross
term.
You
also
have
an
error
term.
This
is
simply
a
random
effects
model,
and
all
of
these
random
effects
are
normally
distributed
random
variables
with
mean
zero
and
some
corresponding
variance
component.
When
you
fit
this
model,
you
can
use
that
to
estimate
the
variance
components
and
then
use
those
variance
component
estimates
to
calculate
the
percentage
gage
R&R
using
the
formula
shown
there.
In
a
standard
MSA
study,
all
of
your
measurements
are
going
to
be
single
points.
But
what
happens
if
that's
not
the
case?
What
if
instead
you're
measuring
something
like
a
curve?
That
question
was
the
motivation
behind
this
project.
There
was
a
client
of
JMP
that
was
a
supplier
of
automotive
parts,
and
they
had
a
customer
that
specified
that
a
part
needed
to
have
a
specific
force
by
distance
curve.
Obviously,
the
client
wanted
to
design
their
product
to
match
the
customer
specified
curve.
In
order
to
do
that,
they
wanted
to
run
a
functional
response
DOE
analysis
and
JMP
to
design
their
product
in
order
to
do
so.
However,
before
spending
money
on
that
experiment,
they
wanted
to
perform
an
MSA
on
their
ability
to
measure
the
parts
force.
There
are
a
lot
more
details
about
the
actual
data
and
this
problem
specifically
in
an
earlier
2020
white
paper
titled
Measurement
Systems
Analysis
for
Curved
Data.
If
you
want
any
more
details,
look
that
up.
It
should
be
on
the
community.
This
in
this
graph,
that's
what
the
data
looks
like.
On
the
Y-axis,
we
have
force,
and
on
the
X-axis,
we
have
distance.
It
looks
like
there
are
only
10
curves
in
this
graph,
but
there
are
actually
250
total
curves.
There's
just
some
clustering
going
on.
There
are
10
different
parts,
five
different
operators,
and
five
replications
per
part
operator
combination.
A
little
bit
about
this
data,
obviously,
these
measurements
are
curves
and
not
points.
The
data
was
collected
evenly
spaced
in
time,
but
not
evenly
spaced
in
distance.
There
were
some
earlier
projects
that
tried
a
few
different
ways
to
perform
some
type
of
MSA
study
on
this
data.
They
used
some
functional
components,
but
stayed
pretty
true
to
a
standard
MSA.
When
I
looked
at
this
data,
I
wanted
to
take
a
true
functional
approach
because
I
have
a
background
in
functional
data.
Functional
data
analysis
is
useful
for
data
that
are
in
the
form
of
functions
or
curves.
There
are
many
techniques
to
handle
unequally
spaced
data,
a
lot
of
which
are
available
in
the
Functional
Data
Explorer
platform
in
JMP.
My
goal
was
to
combine
functional
data
methods
with
traditional
MSA
methods
to
perform
some
type
of
functional
measurement
systems
analysis.
My
solution
was
to
create
a
functional
random
effects
model
by
expanding
the
functional
model
using
eigen
function
expansion,
rewriting
that
as
a
random
effects
or
a
mixed
model
if
you
had
any
fixed
effects
also,
and
then
estimating
the
variance
components
associated
with
the
part
and
operator
terms.
To
go
a
little
bit
into
the
model
notation.
For
your
functional
model,
you
have
Y
sub
I
J
K,
but
this
time
at
a
particular
distance,
D,
to
account
for
the
functional
nature
of
the
data.
You're
going
to
have
a
functional
mean
term,
a
functional
random
effect
that
corresponds
to
the
part,
a
functional
random
effect
that
corresponds
to
the
operator,
and
a
functional
random
effect
that
corresponds
to
the
cross
term
and
also
your
error
term.
Here,
when
you
do
the
model
expansion,
it's
a
little
mathy,
but
essentially,
instead
of
having
one
variance
component
associated
with
the
part
and
one
variance
component
associated
with
the
operator,
you
now
have
multiple
variance
components
associated
with
each
of
those
things.
That's
going
to
account
for
the
functional
nature.
When
you're
fitting
the
model
and
estimating
the
variance
components,
like
I
said,
now
you're
going
to
have
this
set
of
variance
components
that
you
can
sum
together
to
estimate
the
functional
variance
component
for
part
and
the
same
thing
for
operator
and
the
cross
term.
Once
you
have
all
those
individual
variance
components,
you
can
use
those
to
estimate
the
%
gage
R&R
just
like
in
a
standard
MSA.
How
do
I
do
this
in
JMP?
It's
a
multi
step
process
that's
outlined
here,
and
there
are
some
more
details
in
other
slides.
But
essentially,
I
estimate
the
mean
curve
in
FDE
and
obtain
the
residual
curves.
I
then
model
the
residual
curves
in
FDE
to
obtain
the
eigen
functions
needed
for
the
eigen
function
expansion
of
the
functional
model
and
save
those
eigen
functions
to
the
original
data
table.
I'm
going
to
use
those
saved
eigen
functions
in
FitMix
to
create
a
random
effects
model
or
a
mixed
model
if
you
also
have
fixed
effects
in
your
data.
I'm
going
to
use
nesting
of
the
eigen
function
formula
columns
and
also
the
par
and
operator
variables
to
define
the
appropriate
model
specifications.
This
is
what
your
fit
model
window
would
look
like.
Once
I
did
all
that
for
this
data,
I
was
able
to
estimate
the
variance
components
and
calculate
the
%
gage
R&R,
which
in
this
case
was
3.3030.
This
indicated
an
acceptable
measurement
system
according
to
some
ranges
that
were
defined
in
this
paper
by
Baren
team.
That
was
it
for
the
data
analysis
for
my
part.
This
result
was
actually
very
similar
to
a
worst-case
scenario
that
was
obtained
in
a
presentation
in
2019.
It
would
be
interesting
to
know
if
that
was
a
coincidence
or
if
the
results
would
be
similar
for
different
data
as
well.
Some
thoughts
that
this
project
provoked.
Should
we
add
a
functional
random
effect
for
ID
to
capture
the
within
function
correlation
across
distance?
This
type
of
functional
random
effect
is
actually
really
important
in
functional
data
and
is
a
big
benefit
of
accounting
for
the
functional
nature
of
the
data.
Unfortunately,
in
this
data
in
particular.
Anytime
I
created
a
model
with
this
term,
the
corresponding
variance
components
were
zero,
so
it
didn't
really
capture
anything
extra,
but
it
would
be
interesting
to
see
if
it
could
be
useful
in
different
types
of
data.
I
also
think
it
would
be
interesting
if
we
could
calculate
a
confidence
interval
for
the
%
gage
R&R.
There
were
also
some
minor,
not
issues,
but
brought
up
questions
of
the
residuals
in
the
random
effects
model.
I
observed
a
cyclical
nature
in
those.
That's
not
always
great.
I
don't
think
it
was
a
huge
deal,
but
I
would
like
to
have
a
good
reason
for
why
that
was
the
case.
That's
it.
Thanks
for
listening.
If
you
want
more
details
on
this
project,
it's
very
similar
to
a
full
30-minute
talk
that
I
presented
at
Discovery
Europe,
and
so
that
video
is
on
the
community
as
well.
Thank
you.
