Bootstrap Interval Calibration Using New “Save Simulation Formula” Feature in JM...

Bootstrap Interval Calibration Using New “Save Simulation Formula” Feature in JMP® Pro 16 (2021-EU-30MP-744)

Hadley Myers, JMP Systems Engineer, SAS
Chris Gotwalt, JMP Director of Statistical Research and Development, SAS

The calculation of confidence intervals of parameter estimates should be an essential part of any statistical analysis. Failure to understand and consider “worst-case” situations necessarily leads to a failure to budget or plan for these situations, resulting in potentially catastrophic consequences. This is true for any industry but particularly for pharmaceutical and life sciences. Previous work has explored various methods for generating these intervals: Satterthwaite, Parametric Bootstrap and Bias-Corrected (Myers and Gotwalt, 2020 Munich), and Bias-Corrected and Accelerated (Myers and Gotwalt, 2020 Cary), which were all seen to have error rates that were too high for the small samples typical in DOE situations. Therefore, we make use of the new “Save Simulation Formula” feature in JMP Pro 16 in an add-in that improves upon these by allowing users to perform a “Bootstrap Calibration” on the Satterthwaite estimates. The add-in also includes the ability to do this for linear combinations of random components, taking advantage of another addition to JMP Pro 16. Further, we investigate a new version of the fractionally weighted bootstrap that respects the randomization restrictions of variance component models, as an alternative to the parametric bootstrap, using the new “MSA Designer” debuted at this conference.

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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.

Presenter

Hadley Myers

Files

Bootstrap Calibrated Confidence Intervals.jmpaddin