# How to compare two plots - an alternative for AUC?

Below I attached an example which I would like to analyze.

Let's for the example purposes assume, that these plots show the activity of two different drugs which were tested. On the Y axis we have "intensity" of this drug and on the X axis we have a period of time how long is a lifetime of this drug. At this moment I use integrate.xy function to compare those plots, but I am looking for some alternative.

For this experiment the height (intensity) is more important than the time it was in the system. Is there any way to compare those curves with an assumption of "intensity" importance? Or is the AUC the best solution?

What kind of function should be used for it in R ?

• Since intensity of the drug and half-life of the drug are two very different concept (at least in a medical thinking) I think the best way to discuss these results would be a different presentation of both the two variables. Regarding the techical implementation of this since there are no further details I'm not sure of which would be the best method. – GGA Feb 8 '17 at 10:45
• Usage of this example was more about giving you an idea what these curves show. My real data is not connected to medical field at all. I am looking for a function which can be used to compare these plots and is something different than AUC. If it focuses more on the height of the plot than width that's just a plus. – Rechlay Feb 8 '17 at 10:54
• @Rechlay: I have no idea what you might've taken offence at in any of the comments - please read Be nice & assume people are trying to help you. – Scortchi - Reinstate Monica Feb 8 '17 at 12:56
• I am not sure if I constructed my question incorrectly or why I was referred to take a "tour". I would like to know if there is something wrong with my question. Making such comment like Tavrock did suggests something. – Rechlay Feb 8 '17 at 13:07

Short answer, yes, there are methods of so doing. Longer answer; it is unclear what and how you obtained the parameters so that at present, without that clarification; Half-life of what when?, Effect measured how?, it is not possible to formulate an unambiguous answer to the question asked.

Information needed:

1) How is effect measured? Usually pharmacokinetic data is concentration (e.g., in blood plasma) versus time. Among other possibilities, it is sometimes assumed that drug effect is proportional to concentration, sometimes not, sometimes drug effect is measured directly in time (not especially in half-life of time).

2) How is the half-life measured? Half-life of a drug is not constant in time, e.g., see this. That is, a half-life depends on when it is measured, and the longer one waits to measure it, the longer that half-life is. Perhaps you want to measure the mean residence time of the drug, which is at least theoretically more invariant with elapsed time?

3) Some other important facts: route of administration of drug; oral, intravenous, subcutaneous. Kinetics: zeroth order, first order, Michaelis-Menten, hydrophylic, lipophylic, molecular size, route of elimination, metabolites and their drug effect activity.

Make you a deal, you answer my questions, I will attempt to answer yours.

EDIT following comment by OP

You asked a question of moderate difficulty--(What is)...an alternative to AUC? Note further, you asked about drugs, and then related that your concern was something else entirely. I can answer the question in the context of drugs, which should give you a very good idea of what happens in general, but, is not a general answer, nor is it specific to your unstated application. Please do not go and change the context again, it is exhausting enough to give an answer specific to the question as is currently stated.

In some situations, AUC of drug plasma concentration versus straight time (and not half-life) is the conventional first approximation to drug effect following bolus intravenous dosing for first order drug kinetics. Not all elimination is first order (i.e., proportional to concentration) as some eliminations are enzyme-saturable, see Michaelis-Menton, and some drug effects are not proportional either (see below). Moreover, beware! To quote George Box, "Essentially, all models are wrong, but some are useful." For example, a fat-tailed Pareto distribution, if the exponent is small enough, has no mean residence time. The best, but still likely incorrect, current modelling hinting at why this might be true has to do with a dispute concerning right tail heaviness of $C(t)$, which as herein is evaluated properly via survival functions, and appears in Plos One (acknowledgement: With apologies, my own work.).

There have been attempts to modify AUC calculations for practical application to patient care, read Clavert et al$^1$ and Newell et al$^2$ for starters. However, current modelling is not always straightforward, see anesthesia models$^3$.

There have been fairly predictive but only approximate drug-effect models presented under the banner of PKPD.

You also ask if the intensity of drug effect can be more important than AUC. Indeed, it can be, and sometimes is enormously different. Rather than use the formal language of binary outcomes versus continuous proportional effect plus blah blah blah, let's consider a simple example. A mathematically trivial but devastating example is intravenous injection of potassium chloride. That is, if KCl concentration is high enough, the heart stops beating, death ensues, and utility of AUC is discounted by a lack of stationarity, which is another assumption needed for the application of AUC.

1 Calvert AH, Egorin MJ. Carboplatin dosing formulae: gender bias and the use of creatinine-based methodologies. Eur J Cancer. 2002;38:11-6.

2 Newell DR, Pearson A, Balmanno K, Price L, Wyllie R, Keir M, Calvert A, Lewis I, et al. Carboplatin pharmacokinetics in children: the development of a pediatric dosing formula. The United Kingdom Children's Cancer Study Group. J Clin Oncol. 1993;11:2314-23.

3 Hughes MA, Glass P, Jacobs JR. Context-sensitive half-time in multicompartment pharmacokinetic models for intravenous anesthetic drugs. Anesthesiology. 1992;76:334-41.

• Can you share what kind of the methods you are talking about ? As I stated in my question For this experiment the height (intensity) is more important than the time it was in the system. Is there any way to compare those curves with an assumption of "intensity" importance? Or is the AUC the best solution? So I am mostly focused on a height of a curve but width is important as well so I can't look only on height. Can you give me some examples about such methods and I will read about them and try to find what's best for my data. Just ignore the example with "drug" I used... – Rechlay Feb 20 '17 at 12:06
• Well, yes, I can, as above, but hold on to $\hat{your}$. – Carl Feb 20 '17 at 17:51

The function which has to be used is depended on the application. If we are talking about drugs efficiency I would go for AUC. Have you thought about taking total intensity ? Maybe that would be ok for you.