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Checking if the residues are normal ad nauseum? 
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#1
Feb2313, 10:27 PM

P: 557

If I am checking whether my data fits a curve C1, I have to check to see whether the residues R1_{n} are normally distributed, which is checking R1_{n} against a normal curve C2, giving me residues R2_{m}; which must be normally distributed, that is, must be checked against a normal curve C3, giving me residues R3_{p}, and so on ad nauseum. Where does this end?



#2
Feb2413, 05:51 AM

Mentor
P: 11,589

The normal distribution of your residues is not necessary  if their mean is 0 and their standard deviation is 1, you are done. Any deviation from a normal distribution there would indicate some weird (nongaussian) uncertainties for the individual data points.



#3
Feb2413, 10:17 AM

Sci Advisor
P: 3,252

It is possible that you could invent a statistical hypothesis test based on residuesofresidues. To compare the utility of that test to the customary tests, people would look a the "power" of your test. The "power" of a test is complicated to define. It isn't a single number. It is a curve or surface that depends on how you parameterize the shape of the nonnormal distributions that you consider. 


#4
Feb2613, 05:15 AM

P: 557

Checking if the residues are normal ad nauseum?
Thanks for the answers, mfb and Stephen Tashi. (Sorry for the delayed response.) Apparently statisticians rely quite a bit on "hm, looks OK". (I'm not at all a statistician, which you can certainly tell from my beginner's questions; I'm more used to those strange places in mathematics where correlation is a yes/no affair unless you are doing perturbation theory. On the other hand, prior assumptions are the heart and soul of mathematics: "Er, well, let's call (N, <) consistent, and have done with it.")
More seriously: the statistical test I had in mind for the beginning set of points was the Pearson's correlation coefficient or something similar, where the residues should (I think) be more or less normally distributed, because otherwise (it appears at first glance at the formula) one could construct some wild mismatch between data and a line yet come up with a high r^{2}. It might even not be too difficult to construct such with a 0 mean and sd=1. But as was pointed out, such a counterexample would probably look weird. (Something like Anscombe's quartet.) Or, to a blind computer, there would be other tests (which I haven't got to yet in my selfstudy of statistics) to check if it was weird. But then I was not sure about a test for the following steps to check data (residues) against normality; your answers indicate that there is none. Interesting. 


#5
Feb2613, 09:41 AM

Sci Advisor
P: 3,252

In the standard sort of statistics ("frequentist" statistics) people do sometimes employ several hypothesis tests to analyze data. (The Wikipedia has an article about this under the topic of "Multiple Comparisons", which I haven't read carefully.) Applying statistics to real life data is a subjective matter. The nature of hypothesis testing is that it is a procedure for producing a decision, not a proof that the decision is correct. In most cases, all that can be quantified is the probability of making the wrong decision given that the "null hypothesis" is assumed to be correct. (From the point of view of a proof, if one assumes the null hypothesis is true then there is nothing to decide about whether it is true or not.) 


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