Transcript |
Hello, my name is Chris Gotwalt. My co-presenter Hadley |
Myers and I are presenting an add-in for obtaining improved |
confidence intervals on sums or linear combinations of variance |
components. This is part of a series of talks we have given as |
we work on improving and evaluating several approaches. |
Obtaining confidence intervals on sums of variance components |
is important in quality because it provides an uncertainty |
assessment on the repeatability plus the reproducibility of our |
measurement system. The problem is that when we ask for a 95% |
confidence interval, there are approximations involved and the |
actual interval coverage can be as low as 80%. |
In our previous studies, we found that two methods have |
improved coverage rates, parametric bootstrapping and |
Satterthwaite intervals, but it was still less than 95% in small |
samples. The earlier version of the add-in implemented the |
parametric bootstrap as a stopgap and Elizabeth Claassen |
implemented the Satterthwaite intervals in the Fit Mixed |
platform natively in JMP Pro 16. I want to stop here to give |
Elizabeth Claassen credit for making interval estimation of |
linear combinations of variance components so much easier and |
JMP Pro 16. She also greatly extended the Mixed Model output, |
which has made this presentation vastly easier. I'm also hoping |
that this presentation will serve as an inspiration to |
others to check these new Save commands out so they can get |
more from JMP Pro’s Mixed Model capabilities. Now we're going to |
combine the two approaches using a technique called Bootstrap |
Interval Calibration that was introduced by Loh in a 1991 |
Statistica Sinica article. Bootstrap Calibration is a very |
general procedure for improving the coverage of confidence |
intervals that can be applied to almost any parametric |
statistical model. I'm going to introduce the basic idea of |
Bootstrap Interval Calibration in the simplest terms that I |
can, and hand the mic over to Hadley, who's going to demo the |
add-in and discuss our simulation results. To make this |
simple, let's make it specific. Consider a very small nested |
Gauge R&R-type study where we want to estimate the total |
variation. We collect the data and run a nested variance |
components model with an Operator effect, a Part within |
Operator effect, and a residual effect. The software reports a |
Satterthwaite-based interval on the total. It's well known that |
this is an approximation that assumes a “large” |
amount of data is present in order for the actual coverage of |
the interval to be close to 95%. |
In small samples, the actual coverage, the probability that the |
interval procedure generates intervals that actually contain |
the true value of the estimated quantity, will tend to be less |
than 95%. Thing is the actual interval coverage is a |
complicated function of the design, true values of |
the functions, and a long list of other assumptions that are hard |
or impossible to verify. What we can do though is used the fitted |
model and their parameters to do a parametric bootstrap. When we |
do this, we know the true value of the quantity we are |
estimating because we were simulating using that value. |
We can do the simulation thousands of times. We apply the |
same model fitting process to all the simulated samples. We |
can collect the intervals from JMP and calculate how often |
they contain the generating value of the quantity that you |
were interested in. In this case we were interested in the sum of all |
the variance components, so the true value is 4.515. |
Suppose we took our original data set, took the estimates, |
use the Save Simulation Formula that is comes from Fit Mixed, |
and generated a large number of new data sets, and applied the |
same model fitting process that we applied here to each of them, |
and we collected up all of the confidence intervals that were |
reported around the total. After having done this, suppose |
that that...the estimated coverage, the estimated number |
of times that these intervals actually contained the truth, |
turned out to be 88%. |
So we wanted that 95% interval, but the Bootstrap |
procedure is telling us that the actual coverage is closer to |
88%. Now we can play a little game and we can repeat the |
Parametric Bootstrap using a 99% interval this time. So we go |
through that process, we redo all the bootstrap intervals and |
when we did the 99% interval we get an actual coverage of |
approximately 98%. Now suppose we did this game over and over |
again until we found an alpha |
with actual coverage approximately 95%. So in this |
case, suppose we did that and we ended up with finding that 97.6% |
when we asked for a 97.6% interval, we actually got |
something like a 95% coverage. |
Then what we can do is set 1 minus alpha to 0.976 using the Fit |
Model launch dialogue, set alpha option and will get an |
interval that has been Bootstrap Calibrated to have |
approximate coverage 95%. This is still an approximation. There |
is still a simulation component to it, as well as a deeper |
underlying approximation that is extraordinarily hard to analyze, |
but it can be made easy to use, and this is where Hadley comes |
in. Now I'm going to hand it over to him and he will demo the |
add-in and go over the simulations that he did that |
show that we are able to get better coverage rates than |
before by applying Bootstrap |
Calibration to Satterthwaite intervals on linear combinations |
of variance components. Take it away Hadley. |
Thank you very much, Chris, and hello to everyone watching online |
wherever you are. So I'm going to start out by showing you how the add-in |
works and how you can use it to calculate Bootstrap Calibrated |
confidence limits for random components in Mixed Models in |
JMP Pro 16. |
And from there we'll take a step back. We'll see how the add-in |
makes these calculations and I'll highlight some of the additions |
to Mixed Models in JMP Pro 16 that allow it to do that. |
From there, I'll show you the results of some simulation |
studies to give you an idea about how accurate this |
interval estimation method is, the Bootstrap Calibration |
method, and how it compares to some of the other methods for |
calculating confidence limits, as well as the situations where it's |
more or less accurate and some of the limitations and things you |
should be aware of if you're going to be applying it. |
We’ll discuss possibilities for improvements in future work |
just briefly, and from there, I'll conclude by showing you the |
new MSA Designer, Measurement Systems Analysis Designer, |
available from the DOE menu in JMP Pro 16 so that you can quickly |
and easily design and analyze your own MSA Gauge R&R |
studies. So let's start out by looking at this data set. |
This is one that I pulled from the sample data files. |
I'm going to run this Fit Mix script here that I've saved. So |
what we've got here are our random estimates, estimates |
for a random components. |
Now. |
it could be that you want to, for some reason, calculate an |
intermediate total, for example Operator and Part nested with |
Operator, or the three of these, |
you know, Operator and residual. |
So to calculate those is very simple, we simply add these |
estimates, but what's not so simple is to determine those |
confidence limits. |
There's a new feature in Mixed Models that's been added in 16. |
The linear combination of |
The linear combination of variance components feature |
right here, and so what you can do is you can click that. |
You can choose the combination of variance components that you're interested in, |
and you can press done. So now we have an estimate for those. |
Components as well as their |
confidence limits. So, |
what I'm going to do now is I'm going to take this one step |
further and I'm going to calculate the Bootstrap |
Calibrated Satterthwaite estimates and I'm going to do that by |
going to my add-ins and clicking the Bootstrap Calibrated |
confidence intervals there. So from here we can estimate the |
number of simulations. |
2500 is a |
recommended number to the default number. It's also the |
default number in some of the other simulation platforms and |
in JMP Pro. I'm going to choose |
this one. But one thing to note is that it takes some |
time to be able to do this, and so in the interest of time |
what I'm going to do is I'm going to stop it early. |
And here we have our |
calibrated intervals, calibrated upper and lower confidence |
limits added to the report. |
So let's take a step back and see what happened there. |
I'm going to go ahead and add this again. |
Now, one thing that the add-in |
does, as soon as you run it, |
is it adds |
this simulation formula to the data table, so you can see the |
simulation formula here. |
When the add-in is closed, |
the simulation formula disappears. |
The simulation formula there takes advantage of another |
feature that's been added to... |
to the Mixed Models platform in JMP Pro, and that is the Save |
Simulation Formula feature |
here. So what this would allow you to do is to save the |
simulation formula and then to |
use that, for example, to simulate |
these values here. So, we can swap out our “Y” with our |
new simulation formula, |
and go ahead and run that. So when you run the add-in, this is |
all done in the background. But this is how the add-in goes |
about calculating these intervals. So I'm going to stop |
this early, once again in the interest of time. |
And now we see here the samples |
estimated for each. |
simulation. And so how the add-in works is |
it takes all of these. |
And it calculates new estimates for the upper and |
lower Satterthwaite intervals from this |
estimate and this standard error, swapping out different |
values for alpha. So what we're aiming for 0.05, right? |
So that we get 95% upper and lower limits, and what it |
does is it finds |
an alpha value that results in 95% coverage, that is 95% hits |
and 5% misses, |
swaps that in, that's how you get your calibrated intervals. |
So I hope you enjoyed seeing that. I hope you find it useful. |
We've done some simulation studies and what we found out |
is that |
the intervals, which you can see here for four operators |
and 12 days as our random components, |
we've achieved misses of about |
7%, so a 92.8 hit ratio. Now this is better than all of the |
others, including this, so the linear combination, which is |
simply the standard Satterthwaite interval |
calculated on the combination of linear components, as well as the |
Bootstrap quantiles, the bias-corrected intervals in the |
bias-corrected and accelerated intervals, but as you'll see |
these intervals improve, all of them, |
as you increase your number of Operators from |
4 to 8 and the number of Days from 12 to 24. So increasing the |
levels of these random components |
result in much better, much more |
accurate estimates for the confidence limits, and so much so |
that we now have |
a method here |
that is equivalent, |
just, to an |
alpha value of .05. |
So. this improvement in performance of course, comes at |
a cost, and one of those costs is the length of the intervals. |
And so you can see here, |
that with our |
Bootstrap calibrated, well with all of our intervals in fact, |
that when we have increasing number of Operators, |
that the length of the interval is much more |
bundled closer to 0 than it is when you've got smaller |
number of Operators. You can see that this tails out much |
further, so that's this blue area here. That's true for |
all of them, but it's especially true for the |
Bootstrap Calibrated interval. You can see this |
long tail here. |
On average, you're going to get longer lengths using this |
method, but you have a more accurate method. |
Exploring that a little bit deeper, you can see here |
that this increase in length is true for four Operators, as |
well as eight Operators, and it is significant. |
Statistically significant. |
The other thing that I looked at, |
is the effect of adding repetitions, so the difference |
between two repetitions and five repetitions, and what you'll see |
here is that there really is no |
difference. So looking across |
the different sets of combinations from |
four Operators and two reps to four Operators and five reps, about 6 |
measurements total versus 3 measurements, we really don't |
gain anything. All of these are equivalent to each other. |
So that's something to be aware of, that you see improvements in |
accuracy when increasing the number of Operators, and you |
don't see improvements when increasing the number of |
repetitions. |
One thing that I'd like to mention as a |
possibility to improve upon these results is the |
Fractional Random Weight Bootstrap, which we would have |
liked to have been able to implement for this in time for |
this conference. We weren't able to do that, to take this |
and to apply it to random variance components, and so we |
hope to be able to do that in future work and perhaps even |
see an improvement upon the Bootstrap Calibrated interval. |
And then the other thing that I'd like to highlight before I |
go is the new |
MSA designer that's been added to JMP 16, and so |
from here what we can do is we can very quickly |
create our own design in order to be able to |
perform our own MSA or |
Gauge R&R analysis. And so let's see, I'll do this with three |
Operators and Five parts. |
I'll label these |
A, B and C. |
And we'll do one repetition of each. So that's two |
measurements total. |
So here we've got a table with our |
design. What I can do is I can press this button to very |
quickly send that to the different operators, have |
them fill out their parts, send that back to me. |
And then I can add |
those results together. |
So I'll just sort this because I've got another table over here |
where I've done this ahead of time. So I'll just add these |
values over there. And now from the scripts within the table we |
can quickly and easily do our own Measurement Systems Analysis |
and Gauge R&R. So I hope you found this useful. I hope you |
continue to enjoy the talks at this conference. Thank you very |
much for listening. |