# Comparing two curves with different x-axis points - appropriate test?

I have two curves and I want to be able to calculate the probability of these curves coming from different distributions or another appropriate statistic.

Each curve is fitted through the mean of clusters of data (x's and o's in diagram) at differing points on the x-axis, each data cluster is also non-parametrically distributed.

Is there a statistical test that would be appropriate to tell me how likely it is that the data/curves come from different distributions i.e. one is significantly different than the other?

        +   -         x
|    o        ~
|    -        x~
|    o         ~
|     -        ~~
|     --        ~~
|      -         ~
|      --         ~~~
|       -            ~~~
|       -               ~~    x
|        -o               ~~~ x
|         -                  ~~
|         o-                  x  ~
|         o-                  x   ~ ~~             x
|           --                        ~~~ ~~~~~~~~~~~
|            ----        o                         x
|                - ----  o                         x
|                      ----- ----               o
|                        o       - ---- - -  - ---
|                                               o
|
+----------------------------------------------------------+


I've looked here Comparison of two curves but I believe my problem has some distinct differences

• About all you are assuming, implicitly, seems to be single-valued curves (i.e. for each $x$, there is a single mean of $y$). So, I think the main, and possibly only, possibility is that you simulate drawing two groups randomly from your combined data, fit your curves and then do a kind of line-up comparing the "real" pattern and the simulated patterns. If the real pattern is genuine it will stand out from the others. – Nick Cox Mar 26 '14 at 11:18
• @Nick Cox Thanks for the answer although I am not sure I fully understand. Are you suggesting I create a simulated combined curve from both data sets and then use some test to see if the actual curves are different. What test would this be? – CatsLoveJazz Mar 26 '14 at 11:24
• No; I am suggesting that you simulate drawing two curves repeatedly using the same methods. See stat.wharton.upenn.edu/~buja/PAPERS/… for the flavour. Warnings: I can't see any scope for a plug-in or off-the-shelf test if the question is just "I have two curves: are they genuinely different?" which is no more a precise statistical question than "I have two friends: are they genuinely different?". However, there are newer ideas, as in the paper cited, but they usually require some custom programming. – Nick Cox Mar 26 '14 at 11:45
• @Nick Cox I understand now, thank you. I'm certainly no statistician but for the engineering work I do we typically phrase the question something like: "What is the probability that the two sets of data (curves in this case) are due to a genuine difference rather than noise" – CatsLoveJazz Mar 26 '14 at 12:02
• I am not a statistician either, but there is no free lunch in statistics. You might have that question but it is very difficult to answer. It is not far from the ultimate vague question "are my data meaningful?". – Nick Cox Mar 26 '14 at 12:11