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I have 2 datasets (nonpaired), P1 & P2, and I want to know 'scale parameter' of one to the other, aka P1=a*P2.

As both datasets are highly nonnormal, I am performing a Mann-Whitney Test on the data. I take the log of the data, do the MW test, get the Hodges-Lehmann shift estimate (HL) and then take the antilog of the HL shift estimate to get the scale estimate. I do the same thing with confidence intervals.

This all seems to work fine, until I deal w/ datasets which are positive & negative, since the log of a negative number is imaginary, and then everything needless to say explodes...

What method can I use to get the estimate for positive & negative data?

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Your method seems a complicated way of answering a simple question.

Even though your data are not paired, you can still compare them quantile-to-quantile in a quantile-quantile plot. See http://en.wikipedia.org/wiki/Q%E2%80%93Q_plot Note that is possible even if your sample sizes differ. Your assumption that one differs multiplicatively from another would result in a linear configuration through the origin with appropriate slope on the corresponding plot. (If you don't get that, your assumption is in doubt.)

Non-normality here is of secondary concern, unless what it means is that outliers make comparison of scale tricky. But a quantile-quantile plot should let you see which quantiles might be used to estimate scale robustly, e.g. 97.5% point $-$ 2.5% point, 75% point $-$ 25% point. Other measures of scale are also possible. Then your estimate of the proportionality factor $a$ is the ratio of scale measures.

There is no assumption here about sign of values.

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  • $\begingroup$ Hi Nick, thank you for your suggestion. My only concern with QQ plots (besides not being too confortable with them) is that they can better visualize differences within the data, but they cannot establish rigourous statistical significance to those differences? Would it be valid to use the MW test to establish significance and then QQ plots to get the scale parameter? $\endgroup$
    – DankMasterDan
    Commented Dec 3, 2013 at 18:49
  • $\begingroup$ I don't see the connection between your questions. Mann-Whitney can't tell you whether in terms of distributions variables $x, y$ differ following $y = ax$, as it only looks at ranks. What question do you think Mann-Whitney answers? $\endgroup$
    – Nick Cox
    Commented Dec 3, 2013 at 18:59
  • $\begingroup$ I thought MW will tell you if two distributions differ by a shift, ie y=x+a. That is why I first take the log of the data so y'=log(y) & x'=log(x). Thus, MW will say y'=x'+a which means, e^y'=e^log(y)=y = e^(x'+a)=e^(logx)e^a=e^ax. Thus, y=e^ax, so the scale estimate is e^a $\endgroup$
    – DankMasterDan
    Commented Dec 3, 2013 at 19:10
  • $\begingroup$ The MW gives an estimate for the increase of the median (a property of the rank). By using logs, we get a the increase of the median of the scale parameter. The estimate for tghe increase of the median is the Hodges-Lehman estimator (en.wikipedia.org/wiki/Hodges%E2%80%93Lehmann_estimator), which is essentially the median of all pairwise differences $\endgroup$
    – DankMasterDan
    Commented Dec 3, 2013 at 19:38
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You could do an Ansari-Bradley test or a Siegel-Tukey test ... or indeed almost any suitable scale test.

To estimate the scale, you can scale one sample until the test statistic reaches its expected value under the null hypothesis, and you can get an interval for the scale by the same method as generating an interval for the shift parameter in the Mann-Whitney.

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