# Associating non-linear three-time-point change with a continuous variable

I would be incredibly grateful for help or advice regarding my following project:

• I have 3 time points (0, 30, 120 min) and complete data for about $n=500$ subjects for a continuous variable $M$. Individual time courses in $M$ over the 3 time points are highly variable and mostly non-linear. Thus, representing individuals by mean-M for each time point is not appropriate.

• What I want to analyse is: How does the longitudinal change in $M$ over the 3 time points predict a third variable (measured a 0min), called $IR$?

• In a third step, I would run the above analyses for about 200 variables $M$ in order to find out which among them (in terms of their longitudinal change) are associated with $IR$.

I am stuck to find an accurate summary value per individual (like a beta coefficient from linear regression if the Time-x-M association was linear) that I could use to predict $IR = a + (\text{summary_for_time_course_in_M})\times{b}$.

Equally, I am finding it hard to find an adequate model that captures the longitudinal association time-x-M in the first place: The associations are mostly non-linear and there is high inter-individual variation in curves.

Could anyone possibly help me with this or make suggestions? That would be really helpful!

P.S. As an analogy: If the change from 0 to 30 to 120min in $M$ was a straight line, I could calculated the slope Beta per individual. I could then in the second model use beta to predicted $IR$. Having done that for the 200 M-variables, I would rank them (by, e.g., F statistics, % Var explained, regression coefficient F-statistic), apply multi-testing correction, and end up with a select list of top M-variables significantly associated in time-course with $IR$.

This approach does not work, however, as the time-x-M association are not linear and variable between individuals.

• I doubt this question is answerable in its current (vague) form. What are M & IR, eg? How could you possibly know something is "highly nonlinear" w/ just 3 points? Can you add more information about your situation, your data & your goals? Commented Apr 12, 2015 at 4:12
• Hi, M is the intensity of a metabolite (measured by Mass spectrometry originally). IR is insulin resistance (measured as a continuous variable). It is an epidemiolgical study, where 500 subject had blood taken at 0/30/120 min that where each analysed by MS to evaluate the intensity of M. At time 0min, all subject also had the insulin resistance measured (once). There are about 200 metabolites and the question is: Which among them are significantly associated in their time course with insulin resistance? Commented Apr 12, 2015 at 4:26
• Non-Linear means: E.g., for person 1, the intensity of M over three time points is 11-20-23. For person 2, it may be 22-19-43 etc. By non-linear, I mean "not a straight line over all three time points". But linear model between, 0/30, 30/120 are possible. Thanks! Commented Apr 12, 2015 at 4:29
• With only 3 data, you can't differentiate between linear w/ high residual variance from curvilinear w/ 1 'bend' or 2 'bends'. You just don't have enough information. Is there some prior theory that describes these, or did you try a mixed effects model to see what it might show? Commented Apr 12, 2015 at 4:32
• Hi gung, Unfortunately there is no prior hypothesis as to how the metabolite intentities change over time. I am generally not too confident with OLS regression modeling, as the individual time course vary quite a bit (i.e. comparing Spaghetti vs. mean plots). Does mixed effect modeling account for interindividual variations in time course? By mixed effects models, I have always understood combining random and fixed effects (I worked a lot with meta-analysis). Can you illustrate, what you mean in this context - thanks a lot! Commented Apr 12, 2015 at 6:28