# linear regression when curve go up then down

I have very basic notions of statistics and I have to analyse a data sheet containing 7 points of glycemia over time in 5 groups of 6 or 7 animals.

The goal is to measure the effect of some association, so one could say groups are Control, MedA, MedB, Control+MedA and MedA+MedB (Control+MedB are not useful to test).

This is the graph I get from my points (each point is the mean of the group of animals) : I think I should use regression, but I only know linear regression. Unfortunately, in my data, values first rise and then fall, which is not linear at all.

I'm using GraphPad Prism 6, so I tried nearly every model I found, but didn't find one who can handle my "rise and fall" pattern.

So in brief, I want to

1. know if my kinetic pattern is the same in every group, and if not, which one is different and how (higher or lower).

2. compare the deltas of each points. My red curves seems to rise and fall quicker, even if the AUC may not be impacted. I would like to know if this is significative.

How could I achieve this ?

PS : I can use R and Excel if needed.

• Welcome to CV. What you are describing sounds like an Analysis of Variance (ANOVA) based on the 5 groups. ANOVA is a variant of regression. You may be interested in the mean differences in the groupings over time or in the efficacy of some treatment condition vs a control group -- we can't tell from your writeup. Regardless, one simple approach that should work would be to introduce a quadratic (squared) effect as a function of time to capture the curvilinear behavior you're observing. Judith Singer's book, Applied Longitudinal Analysis, has extensive comments on ways to approach this. Jan 25, 2016 at 10:53
• Could you describe your problem in greater detail and provide data example (e.g. data sample or plots)?
– Tim
Jan 25, 2016 at 10:54
• @Tim I added my graph, is that any clearer ? Jan 25, 2016 at 11:00
• @DJohnson there is 5 groups, one could be eventually considered as control group and others are combinaisons of treatments. I'll update my post. Jan 25, 2016 at 11:01
• Unfortunately you don't seem to have enough data to do anything but fit simple models. Are there scientific reasons to expect a turning point? What functions do people use for this kind of data? I don't see a strong signal for anything but approximate linear decline or slow exponential decline. Jan 25, 2016 at 12:43

If your curves look like fig 1. the model is a damped harmonic oscillator, see fig 2. This article may be helpful in modelling such a system:

A Mathematical Model of the Glucose-tolerance test  • Nice suggestion but I think you may have blown the OP out of the water with it. Regardless, I don't see the sinusoidal regularities in his data that your approach requires. Jan 25, 2016 at 11:37
• Yes, OP is blown :-P Nevertheless, GraphPad Prism has a built-in model called "damped sine wave", but it didn't work Jan 25, 2016 at 15:07

Such data is often fit using a nonlinear mixed effects model (aka, hierarchical model) using a compartment model for the subject-level model and a simple ANOVA model for the treatment-level model.

This is a form of random coefficients model. Instead of trying to capture the data as a series of averages within each treatment group and time point, you instead try to capture each subject's kinetics curve by condensing that data into the kinetics parameters. These parameter values then become the thing that responds to treatment and you look at differences of these values among the subjects across treatment groups.

It is an ANOVA where the focus is not on the means-over-time (which are not scientifically interesting) but on the kinetics parameters (which are scientifically interesting).

I would suggest you start by looking up nonlinear mixed-effects models, lmer4, nlme, hierarchical models, etc. Singer and Willett's book. Pinhiero and Bates' book.

The data you give is NOT the data fed to the model. You have already summarized into mean curves. The analysis will begin with each individual's curve. If I am mistaken about your data and these curves are made up of points that differ for each individual, then this model is not relevant.

I do not know your endpoint well-enough to suggest what type of compartment model you need, though. And that is usually the real trick in fitting NLME models...getting a model that actually fits the data.