# RE: [NMusers] VPCs confidence intervals?

From: Ken Kowalski <kgkowalski58_at_gmail.com>
Date: Thu, 14 Mar 2019 17:01:39 -0400

Hi All,

I know there is a lot of confusion about the distinction between a confiden=
ce interval and a prediction interval. Here is a layperson’s way o=
f making the distinction.

A confidence interval makes inference on a population parameter which is fi=
xed (never changes) regardless of any sample data that is collected to esti=
mate the parameter (if you repeatedly sampled an infinite number of observa=
tions to obtain the population value by definition you would get the same p=
opulation value for each sample with an infinite sample size) . Thus, the =
confidence interval only reflects the uncertainty in the estimate of that p=
arameter.

In contrast, a prediction interval makes inference on a statistic for a fut=
ure sample set of data. That statistic will vary from sample to sample and=
hence must also take into account the sampling variation as well as the pa=
rameter uncertainty. A prediction interval can be thought of as a confiden=
ce interval of the prediction of some statistic from a future sample. That=
is, both a confidence interval and a prediction interval have a confidence=
level associated with them. In the case of the confidence interval, the c=
onfidence level is the coverage probability that the interval will contain=
the true value of the population parameter if one were to repeat the exper=
iment an infinite number of times. In the case of the prediction interval,=
the confidence level is the coverage probability that the interval will co=
ntain the future sample mean (of a finite sample size) if one were to repea=
t the experiment an infinite number of times.

There is another type of statistical interval in addition to confidence and=
prediction intervals and that is a tolerance interval. A tolerance interv=
al can be thought of as a confidence interval that a specified proportion o=
f the individual responses will be contained within the interval. For exam=
ple, we can calculate a 95% tolerance interval to contain 90% of the observ=
ed data (i.e., we are 95% confident that the interval will contain 90% of t=
he individual observations). Tolerance intervals are more common in a manu=
facturing setting where it is important to produce an item to some specific=
ation within some tolerance limits. Nevertheless, there is a certain VPC p=
lot that we often generate that is somewhat akin to a tolerance interval. =
When we summarize our simulated data for VPCs and summarize the 5th and 95t=
h percentiles of the individual responses this is more akin to a tolerance =
interval to contain 90% of the observed individual data. In contrast, when=
we summarize the sample mean or median from say 1000 simulated trials and =
calculate the 5th and 95th percentiles across the 1000 trials that is more =
akin to a prediction interval for that statistic (e.g., sample mean or samp=
le median). Note however, the intervals obtained as percentiles of a sampl=
e statistic across trials (i.e., prediction interval) or sample observation=
s across individual subjects (i.e., tolerance interval) don’t have =
valid coverage probabilities for repeated experiments unless they take into=
account parameter uncertainty.

Kind regards,

Ken

From: Bill Denney [mailto:wdenney_at_humanpredictions.com]
Sent: Thursday, March 14, 2019 2:17 PM
To: Ken Kowalski <kgkowalski58_at_gmail.com>
Subject: RE: [NMusers] VPCs confidence intervals?

Hi Ken,

Thanks for that good clarification!

Bill

From: Ken Kowalski <kgkowalski58_at_gmail.com <mailto:kgkowalski58_at_gmail.com> =
>
Sent: Thursday, March 14, 2019 2:01 PM
To: 'Bill Denney' <wdenney_at_humanpredictions.com <mailto:wdenney_at_humanpredic=
tions.com> >; 'Soto, Elena' <elena.soto_at_pfizer.com <mailto:elena.soto_at_pfize=
r.com> >; nmusers_at_globomaxnm.com <mailto:nmusers_at_globomaxnm.com>
Subject: RE: [NMusers] VPCs confidence intervals?

Hi All,

I know what Bill is trying to say but it is not quite accurate the way he s=
tates it.

A prediction interval makes inference on a statistic based on a future samp=
le such as a sample mean of a future set of data. In contrast, a confidenc=
e interval makes inference on a parameter such as the population mean which=
is a fixed number. A prediction interval takes into account both the unce=
rtainty in the existing data used to estimate the population parameter as w=
ell as the sampling variation to make inference on a sample statistic (e.g.=
, sample mean for a future trial). A confidence interval only takes into =
account the uncertainty in the existing data used to estimate the parameter=
. Based on the Law of Large Numbers, the population mean can be thought =
of as taking the sample mean of an infinite sample size (i.e., sampling the=
entire population). For this reason, a prediction interval with an infini=
te sample size will collapse to a confidence interval.

An interval based on VPCs is more akin to a prediction interval since it ta=
kes into account the sampling variation based on a finite sample size, howe=
ver, one cannot assign a valid coverage probability (confidence level) to t=
his interval unless it also takes into account the parameter uncertainty. =
With VPCs applied to existing data (i.e, an internal VPC) it is customary t=
o not take into account this parameter uncertainty so many refer to such pr=
ediction intervals as degenerate as they place 100% certainty on the model =
parameter estimates used to obtain the VPC predictions. One could potent=
ially call these intervals ‘degenerate prediction intervals�=
� but I tend to just call them ‘VPC intervals’ (e.g., a 9=
0% VPC interval) so as to avoid misperception that these prediction interva=
ls have a statistically valid coverage probability. However, when VPCs are=
applied to an independent dataset not used in the development of the model=
, it is often advised to take into account the parameter uncertainty when p=
erforming the VPCs to essentially reflect the trial-to-trial uncertainty of=
the independent data not used in the estimation of model (i.e., refitting =
the same model to a new set of trial data will not give the same set of est=
imates and hence reflects trial-to-trial variation). In this setting, wher=
e the VPCs take into account both the parameter uncertainty and sampling va=
riation to predict on an independent (e.g., future) dataset, then one is on=
more solid ground to refer to these VPC intervals as prediction intervals =
with valid coverage probabilities.

Kind regards,

Ken

Kenneth G. Kowalski

Kowalski PMetrics Consulting, LLC

Email: <mailto:kgkowalski58_at_gmail.com> kgkowalski58_at_gmail.com

Cell: 248-207-5082

From: owner-nmusers_at_globomaxnm.com <mailto:owner-nmusers_at_globomaxnm.com> [=
mailto:owner-nmusers_at_globomaxnm.com <mailto:owner-nmusers_at_globomaxnm.com> ]=
On Behalf Of Bill Denney
Sent: Thursday, March 14, 2019 1:10 PM
To: Soto, Elena <elena.soto_at_pfizer.com <mailto:elena.soto_at_pfizer.com> >; nm=
users_at_globomaxnm.com <mailto:nmusers_at_globomaxnm.com>
Subject: RE: [NMusers] VPCs confidence intervals?

Hi Elena,

VPCs are accurately called prediction intervals not confidence intervals. =
The difference is that a prediction interval shows what you would expect fo=
r the next individual in a study while a confidence interval shows what you=
would expect for the result of a statistic (often confidence intervals of =
a mean are shown). With many VPCs, the confidence interval of the median a=
nd the confidence interval of the 5th and 95th percentiles are shown.

Also, when the lines indicate the median, 5th, and 95th percentiles of the =
simulations, that is the 90% prediction interval since it is the middle 90%=
of the data (not the 95% confidence interval).

Thanks,

Bill

From: owner-nmusers_at_globomaxnm.com <mailto:owner-nmusers_at_globomaxnm.com> <=
owner-nmusers_at_globomaxnm.com <mailto:owner-nmusers_at_globomaxnm.com> > On Beh=
alf Of Soto, Elena
Sent: Thursday, March 14, 2019 12:49 PM
To: nmusers_at_globomaxnm.com <mailto:nmusers_at_globomaxnm.com>
Subject: [NMusers] VPCs confidence intervals?

Dear all,

I have a question regarding visual predictive checks (VPCs).

Most of VPCs used now, include a line representing the median and 5th and 9=
5th percentiles of the data values and an area around the same percentiles =
that is commonly define as the 95% confidence interval (of the simulations)=
.

But is it correct, from the statistical point of view, to call confidence i=
nterval to this area? And if this is not the case how should we define them=
?

Thanks,

Elena Soto

Elena Soto, PhD

Pharmacometrician

Pharmacometrics, Global Clinical Pharmacology

Global Product Development

Pfizer R&D UK Limited, IPC 096

CT13 9NJ, Sandwich, UK

Phone : +44 1304 644883

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Received on Thu Mar 14 2019 - 17:01:39 EDT

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