RE: Observed (yaxis) vs Predicted (xaxis) Diagnostic Plot - Scientific basis.

From: [email protected] <[email protected]> On Behalf Of Martin Bergstrand Sent: Friday, August 18, 2023 9:48 AM To: Gobburu, Joga <[email protected]> Cc: [email protected] Subject: Re: [NMusers] Observed (yaxis) vs Predicted (xaxis) Diagnostic Plot - Scientific basis. Dear Joga and all, Joga makes a valuable point that all pharmacometricians should be aware of. Standard methodology for regression assumes that the x-variable is without error (loess, linear regression etc.). Note that it is the same for NLME models i.e. we generally assume that our independent variables e.g. time, covariates etc. are without error. For DV vs. PRED plots it is common practice, even among those that do not know why, to plot PRED on the x-axis and DV on the y-axis. A greater problem with these plots is the commonly held expectation that for a "good model" a smooth or regression line should align with the line of unity. Though this seems intuitive it is a flawed assumption. This issue was clearly pointed out by Mats Karlsson and Rada Savic in their 2007 paper titled "Diagnosing Model Diagnostics''. For simple well-behaved examples you will see an alignment around the line of unity for DV vs. PRED plots. However, there are several factors that contribute to an expected deviation from this expectation: (1) Censoring (e.g. censoring of observations < LLOQ) - In this case DVs are capped at LLOQ but PRED values are not. This makes it perfectly expected that there will be a deviation from alignment around the line of unity in the lower range. (2) Strong non-linearities - The more nonlinear the modelled system is, the greater the expected deviation from the line of unity. Especially in combination with significant ETA correlations. (3) High variability - With higher between/within subject variability (e.g. IIV and RUV) that isn't normally distributed (e.g. exponential distributions) will result in an expected deviation from the line of unity. Note: this is a form of non-linearity so it may fall under the above category. (4) Adaptive designs (e.g. TDM dosing) - Listed in the original paper by Karlsson & Savic but I have not been able to recreate an issue in this case. I am rather sure that many thousands of hours have been spent on modeling trying to correct for perceived model misspecifications that are not really there. This is why I recommend relying primarily on simulation-based model diagnostics (e.g. VPCs) and as far as possible account for censoring that affects the original dataset. As pointed out by Karlsson & Savic a simulation/re-estimation based approach can also be used to investigate the expected behavior for DV vs. PRED plots for a particular model and dataset (e.g. mirror plots in Xpose). Note that to my knowledge there is yet no automated way to handle censoring in this context (clearly doable if anyone wants to develop a nifty implementation of that). If we leave the DV vs. PRED plot case, there are many other instances where we use scatter plots where it is much less clear what can be considered the independent variable and yet other cases where the assumption that the x-variable is without error is violated in a way that makes the results hard to interpret. One instance of the latter is when exposure-response is studied by plotting observed PD response versus observed trough plasma concentrations. This is already a way too long email so I will not deep dive into that problem as well. Best regards, Martin Bergstrand, Ph.D. Principal Consultant Pharmetheus AB [email protected]<mailto:[email protected]> http://www.pharmetheus.com On Thu, Aug 17, 2023 at 12:44 PM Gobburu, Joga <[email protected]<mailto:[email protected]>> wrote: Dear Friends – Observations versus population predicted is considered a standard diagnostic plot in our field. I used to place observations on the x-axis and predictions on the yaxis. Then I was pointed to a publication from ISOP https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5321813/figure/psp412161-fig-0001/) which recommended plotting predictions on the xaxis and observations on the yaxis. To the best of my knowledge, there was no justification provided. It did question my decades old practice, so I did some thinking and digging. Thought to share it here so others might benefit from it. If this is obvious to you all, then I can say I am caught up! 1. We write our models as observed = predicted + random error; which can be interpreted to be in the form: y = f(x) + random error. It is technically not though. Hence predicted goes on the xaxis, as it is free of random error. It is considered a correlation plot, which makes plotting either way acceptable. This is not so critical as the next one. 2. However, there is a statistical reason why it is important to keep predictions on the xaxis. Invariably we always add a loess trend line for these diagnostic plots. To demonstrate the impact, I took a simple iv bolus single dose dataset and compared both approaches. The results are available at this link: https://github.com/jgobburu/public_didactic/blob/main/iv_sd.html.pdf. I used Pumas software, but the scientific underpinning is agnostic to software. See the two plots on Pages 5 and 6. The interpretation of the bias between the two approaches is different. This is the statistical reason why it matters to plot predictions on the xaxis. Joga Gobburu University of Maryland This communication is confidential and is only intended for the use of the individual or entity to which it is directed. It may contain information that is privileged and exempt from disclosure under applicable law. If you are not the intended recipient please notify us immediately. Please do not copy it or disclose its contents to any other person. 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