interpretation from loess graph

I have measured events X and Y in two strains (A and B) of same species. After calculating the values of these X and Y, I compared them with wild type values (X and Y) to calculate fold change. I have plotted these fold changes of X and Y in A and B, and curve was fitted with LOESS.

Now for event X (top plot) LOESS approximates to straight line, but for event Y (bottom plot), the smaller values (Chromosome size < 600 kb) seem to be elevated.

My question is by just showing this plot can I claim that smaller chromosomes get more event-Y than larger chromosomes (chromosome size > 600 kb) or do I need to do statistical tests to say this ?? If so what tests do you suggest ?

Sorry I am poor when it comes to advanced statistics. Hope I have clearly stated my problem.

Welcome to the site.

You can say that smaller chromosomes get more Y-event than larger ones based on this graph; if you want to say that they get significantly more then you need some sort of statistical test.

If you are using R then the rms package offers tests of splines and loess fits.

If you are using SAS there is PROC LOESS.

Several interesting things seem to be going on in your two graphs; you didn't ask about this, but did you note that for X, the A curve is higher and for Y the B curve is? This might be interesting.

• Thank you very much for the reply. Our hypothesis is that fold change would be same across all chromosomes, but smaller chromosomes seems to be more active for event-Y. So it will be nicer to show difference in terms of significance level. Thanks again for the suggestion, I will check out rms (I am an R user). Nov 13, 2013 at 11:53
• rms was written by @FrankHarrell who often comments here. He has good documentation, but I also recommend his book "Regression Modelling Strategies". If you are going to be doing this sort of work, that book is worth owning. Nov 13, 2013 at 11:58
• Thank you for suggestion. I will be doing this kind of work. I will definitely look into it. Nov 13, 2013 at 12:03

(1) You can't use lowess as it is a data exploration tool. The degree of flexibility of the function is set by the user. So if you play around with the function you'll see that you get dramatically different results depending on the bandwidth you select. (in R this is "f" the smoother span)

(2) There are a number of smoothing functions that derive the degree of flexibility from the data rather than being set by the user. As a result they can be used in this situation. These include restricted splines, GAM, and MARS (in the Earth package). But there are many more. I prefer GAM and MARS and would use MARS in this situation, or GAM if I was planning on publishing. Restricted splines are likely the most accepted/commonly seen method.

(3) You can sometimes just show that the smoothing functions look different: "Relationship Between Hospital Readmission and Mortality Rates for Patients Hospitalized With Acute Myocardial Infarction, Heart Failure, or Pneumonia" JAMA In the case of MARS this is dramatic and might be sufficient.

(4) You do have a sparse data issue. In the flexibility of the functions above are derived from the data having more data might be better (give you less biased estimates).

(5) You could avoid all this, but I don't think it will be that popular, doing something like: creating a 5 level categorical variable from chromosome size "size_cat" regress fold ~ size size_cat in the first graph both size and size_cat will be non-significant. In second graph both will be significant. And you could say size=captures linear relationship size_cat non-linear relationship

Maybe not? Might have to think about that some more. Needs a bit of polish.