From: Mats Karlsson <mats.karlsson@biof.uu.se>

Subject: PREDvsDV

Date: Mon, 01 Feb 1999 19:42:28 +0100

Dear all,

It is customary to use as goodness-of-fit plots PRED (or IPRED) versus DV. Can anyone give a reason to prefer this over DV versus PRED (or IPRED)? It seems to me that the latter is to be preferred (if one wants to make a distinction). The reason is the possibility to add a regression line/smooth in the DV versus PRED plot. For DV versus PRED such a line should be reasonably close to the line of identity, whereas for the PRED versus DV plot such a line has no meaning. Of course there are situations, e.g. with extensive censoring, where also for the DV versus PRED plot a regression line cannot be expected to be close to the line of identity.

Best regards,

Mats

From: "Piotrovskij, Vladimir [JanBe]" <VPIOTROV@janbe.jnj.com>

Subject: RE: PREDvsDV

Date: Tue, 2 Feb 1999 08:31:46 +0100

As we plot residuals versus predictions, not measurements, DV should be plotted against predictions, nor vice versa.

Best,

Vladimir

From: "Bonate, Peter, Quintiles" <pbonate@qkcm.quintiles.com>

Subject: more on PREDvsDV

Date: Tue, 2 Feb 1999 07:37:10 -0600

In my mind, the choice of whether to plot Pred vs. DV or DV vs Pred is really quite arbitrary. I believe that the use of Pred vs. DV is common practice because then one can then plot a regression line using DV as the independent variable and Pred as the dependent variable. But it must be kept in mind that any regression line (whether it is least squares, loess, etc.) between these two variables is an artifact because however you plot them, they are both subject to error. Thus, the regression coefficient will be attenuated compared to the true value of the slope. A more proper method would be to use an error-in-variables regression, such as simulation extrapolation or maximum likelihood.

PETER L. BONATE, PhD.

Clinical Pharmacokinetics

Quintiles

POB 9627 (F4-M3112)

Kansas City, MO 64134

phone: 816-966-3723

fax: 816-966-6999

From: Chuanpu Hu 301-827-3210 FAX 301-480-2825 <HUC@cder.fda.gov>

Subject: Re: more on PREDvsDV

Date: Tue, 02 Feb 1999 09:42:14 -0500 (EDT)

I share with Mats' belief that a regression line/smooth in the DV versus PRED plot, not PRED vs. DV, is a more sensible choice. The reason is that the assumption in a regression line is that the "x variable" is fixed, and the "y variable" has errors, not vice versa.

It seems to me that using PRED vs. DV may have historical reasons. The S-plus nonlinear mixed effect routine NLME plots DV versus PRED, among other diagnostic plots. I often find myself turn the PRED vs. DV plot up side down in my head.

Chuanpu

----------------------------------------------

Chuanpu Hu, Ph.D.

Quantitative Methods and Research

Food and Drug Administration

Room 15B-45

5600 Fishers Lane (HFD-705)

Rockville, MD 20857

Phone: (301)827-3210

Fax: (301)480-2825

-----------------------------------------------

From: "Stephen Duffull" <sduffull@fs1.pa.man.ac.uk>

Subject: DV vs PRED

Date: Fri, 9 May 1997 16:04:48 +0100

It seems more logical to me to plot DV vs PRED assuming you want to do some sort of regression analysis (otherwise it wouldn't seem to matter). [Whether this regression tells you much is a different story.] It is presumed in the modelling that the DV is distributed around the expectation with some unknown but estimable variance. Therefore, despite that there are inherent errors in estimation of the expectation it would seem more appropriate to consider these

errors to be less and hence the expectation should treated as the independent variable.

Kind regards

Steve

=====================

Stephen Duffull

School of Pharmacy

University of Manchester

Manchester, M13 9PL, UK

Ph +44 161 275 2355

Fax +44 161 275 2396

From: "Bonate, Peter, Quintiles" <pbonate@qkcm.quintiles.com>

Subject: RE: more on PREDvsDV

Date: Tue, 2 Feb 1999 09:28:26 -0600

Chuanpu Hu is correct when he states that regression models assume that the independent variable is fixed and the dependent variable has errors, not vice-versa. That is correct for classical functional models, but this is a structural model where both X and Y are random and have error associated with them. That is why plotting one vs. the other is really quite arbitrary. Ordinary regression models will result in artifactual parameter estimates. Any type of fitted line should be simply a visual aid to examine goodness-of-fit and no attempt should be made to do any type of hypothesis testing on the values of the parameter estimates, unless the parameter estimates have been "corrected" for errors in X.

If a regression line must be fitted using a linear model, it is best to keep the variable with the smallest error as the independent variable. In this case that is DV.

To test whether the errors in X are sufficient to significantly affect the values of the ordinary regression parameter estimates, see the papers by Davies and Hutton (1975) Biometrics 62: 383-391 and Hodges and Moore (1972) Applied Statistics 21: 185-195.

Also, the paper by Linnet (1990) Stat Med 9: 1463-1470 is a useful exposition for how to perform error in variables regression for the simple linear model.

PETER L. BONATE, PhD.

Clinical Pharmacokinetics

Quintiles

POB 9627 (F4-M3112)

Kansas City, MO 64134

phone: 816-966-3723

fax: 816-966-6999

From: michael_looby@sandwich.pfizer.com

Subject: DV vs PRED: a question

Date: Tue, 2 Feb 1999 16:22:25 -0000

This is an intriguing discussion! I have naively used PRED vs DV plots with a regression line superposed as a simple visual aid to examine goodness-of-fit. Following Mats' email, I have looked back over some plots and notice that DV Vs PRED plots generally correspond well to the line of identity but the PRED vs DV plots have a smaller slope, i.e. above the identity line as smaller values and below it at high values. My question is whether this will always be the case?

Regards

Mick Looby

From: "Bonate, Peter, Quintiles" <pbonate@qkcm.quintiles.com>

Subject: RE: DV vs PRED: a question

Date: Tue, 2 Feb 1999 14:02:57 -0600

Mick Looby writes that slope(Pred vs. DV) > slope(DV vs. Pred) in general. He asks is this always the case. The answer (as always) is it depends - it depends on how good your model is. For models with pretty good fits, Mick's observation will be the case and here goes my attempt to explain why.

For notation, let (Pred vs. DV) mean Pred is the Y-variable and DV is the x-variable. This is different than the way NONMEM plots scatter plots, but is consistent with SAS notation. Please note in my earlier notes that I refer to Y vs. x. I think there was some confusion regarding my notation.

For the simple linear regression where X has no error then Y = B0 + B1*x + e where e~(0, sigma^2). Suppose now that x cannot be directly observed, but W can be, where W=X+U and U has mean 0 and variance phi^2. Then Y = B0' + B1'*(X+U) + e. The ordinary least squares estimate of B1 is not B1, but instead B1' = L*B1, where L = sigma^2/(sigma^2 + phi^2). Notice that L ranges from 0 to 1. Thus B1' < B1, always. This is why it is said that error in variables attenuates regression parameters.

Predicted values have more variation than dependent variables because predicted values not only have variation due to DV, they also have variation due to model uncertainty. Thus, sigma^2 > phi^2.

Now suppose model uncertainty is small. For the case where Pred vs. DV (Y vs. X) is plotted, L is near 1 and OLS estimates are good estimators of B1. For the case where DV vs. Pred is plotted, L = phi^2/(sigma^2 + phi^2) and L is near 0. Thus B1(Pred vs. DV) > B1(DV vs. Pred).

Now suppose model uncertainty is large, then for the case where Pred vs. DV is plotted L will be small and B1 will be a poor estimate of the true B1. When DV vs. Pred is plotted, L will be near 1 and B1 will be a good estimate for the true value.

Since as pharmacokineticists our models always have small model uncertainty, then the former situation occurs predominantly and Mick Looby's observation is proven.

I have to go do some real work now. Hopefully this will settle matter things for a while.

PETER L. BONATE, PhD.

Clinical Pharmacokinetics

Quintiles

POB 9627 (F4-M3112)

Kansas City, MO 64134

phone: 816-966-3723

fax: 816-966-6999

From: LSheiner <lewis@c255.ucsf.edu>

Subject: PRED v DV

Date: Tue, 02 Feb 1999 12:36:54 -0800

Regarding the PRED vs DV debate:

I am a little puzzled by Peter Bonate's emphasis on regression relationships. No one is going to regress PRED on DV or vice-versa: DV has presumably already been used as optimally as possible to generate PRED. The issue of errors in variables, which is admittedly important in estimating structural models, does not apply here since there is no structural model to be created.

Rather, the sole issue is which plot serves the model diagnostic purpose better? As I remarked to Mats separately, there is no question that PRED is appropriate on the abscissa when, for example, residuals are to be plotted. In the case of PRED & DV, the choice is arbitrary, although as Mats point out, if one wants to put a smooth through the points, then since E(Y|PRED) should equal PRED, the mismatch of the smooth from the line of identity is diagnostic only when PRED is on the abscissa. Perhaps we have a tradition of DV on the abscissa because we are trying to see if there is something amiss with the model (PRED) conditional on the data (DV), and in regression, we tend to put the variable on which we are conditioning on the abscissa.

LBS

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Lewis B Sheiner, MD

Professor: Lab. Med., Biopharm. Sci., Med.

Box 0626

UCSF, SF, CA

94143-0626

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email: lewis@c255.ucsf.edu

From: "Bonate, Peter, Quintiles" <pbonate@qkcm.quintiles.com>

Subject: yet another reply to pred vs. DV

Date: Tue, 2 Feb 1999 15:38:34 -0600

I think Mats has done a great job of summarizing what must have been a slow morning for most of us. A couple of comments, Lew Sheiner was puzzled over my emphasis on regression relationships. I never meant to imply that someone would want to regress Pred vs. DV or DV vs. Pred and examine the regression parameter estimates. What I was trying to do was show why Mick Looby got the results that he observed and to make the case that at best a regression line should be for visual inspection only. Second, I partially disagree with Mats. Pred will be fixed when the model is certain and the covariates are known with certainty. Since this may not be the case, then pred will be random. Granted it may not be random with mean zero as we traditionally think of random variates but it will have some center of mass which should be pred.

PETER L. BONATE, PhD.

Clinical Pharmacokinetics

Quintiles

POB 9627 (F4-M3112)

Kansas City, MO 64134

phone: 816-966-3723

fax: 816-966-6999