# [R] significance test interquartile ranges

peter dalgaard pdalgd at gmail.com
Sun Jul 15 12:17:15 CEST 2012

```On Jul 14, 2012, at 19:58 , Schaber, Jörg wrote:

> Dear Peter,
>
> thanks for your clarifications. Sample size is around 200 in each group. Would that justify your approach?

It's certainly better than 10...

I did a small check on the IgM data from the ISwR package (298 obs.) and found something somewhat amusing: Discretization effects can kick in rather profoundly with data sets of that magnitude.

The IgM data are discretized to 1 decimal digit, which is fairly common for "continuous" data in practice

> table(IgM)
IgM
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9   1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8   2 2.1
3   7  19  27  32  35  38  38  22  16  16   6   7   9   6   2   3   3   3   2
2.2 2.5 2.7 4.5
1   1   1   1
> summary(IgM)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.100   0.500   0.700   0.803   1.000   4.500
> IQR(IgM)
[1] 0.5

However, if we want to look at the sample distribution of a quantile, we get some curious effects as the variation of the estimate is close to the discretization error. Try a simple bootstrap sample from the empirical CDF:

> medians <- replicate(10000,median(sample(IgM,replace=T)))
> table(medians)
medians
0.6 0.65  0.7 0.75  0.8
13    6 9035  179  767

However, if we smoothen the empirical CDF by adding a little noise, we do get something that does look passably (although not perfectly) gaussian:

> x <- IgM + runif(IgM, -.05,.05)
> medians2 <- replicate(10000,median(sample(x,replace=T)))
> hist(medians2)
> qqnorm(medians2)

Interestingly, adding noise has the counterintuitive effect of reducing the standard error of the medians:

> sd(medians)
[1] 0.02748966
> sd(medians2)
[1] 0.02347363

(It's not _that_ counterintuitive given that the definition of the median isn't quite the same for discrete data.)

Back to the IQR. You can do much the same thing:

> iqrs <- replicate(10000,IQR(sample(IgM,replace=T)))
> table(iqrs)
iqrs
0.3 0.375   0.4  0.45 0.475   0.5  0.55 0.575   0.6
60    42  3885     7   640  5100     3    87   176

or, use the smoothed one replacing IgM by x (defined above).

Now, what if we wanted to compare two IQRs? I'll cheat and reuse the same ECDF for both groups.

> i1 <- replicate(10000,IQR(sample(IgM,replace=T)))
> i2 <- replicate(10000,IQR(sample(IgM,replace=T)))
> qqnorm((i1-i2)/sd(i1-i2))
> mean(abs(i1-i2)/sd(i1-i2) < 2)
[1] 0.9698

So, not really all that bad, but it is a bit fortuitous given the discreteness of the distribution.

Same thing with the x comes out quite a bit nicer

> ix1 <- replicate(10000,IQR(sample(x,replace=T)))
> ix2 <- replicate(10000,IQR(sample(x,replace=T)))
> qqnorm((ix1-ix2)/sd(ix1-ix2))
> mean(abs(ix1-ix2)/sd(ix1-ix2) < 2)
[1] 0.9546

So, my conclusion would be that yes, you can use bootstrap techniques with data of that size, but you need to watch out for discretization effects by checking the bootstrap sample distributions and you might want to add a little smoothing-noise for stability.

As always with bootstrapping, beware that the simulation is never done under the null hypothesis, one merely hopes that the distribution of the resampled estimates around the observed estimate is sufficiently similar to that of the estimator around the true estimate that it can be used for tests and confidence intervals, implicitly using a location-shift argument. This gets particularly dubious when there are discretization effects because the jumps occur at values that do not depend on the parameters.

(Pragmatically speaking, you might not be interested at all in differences in IQR which are comparable to discretization error, though.)

>
> I found a couple of more tests for scale on continous variables, ie.
> Mood Test
> Ansari-Bradley Test (that one is also implemented in R)
> Klotz Test
> Conover Test
>
> Would one of those be suitable to test for different dispersion (e.g. IQR or the like) in non-normal distributions?
>

That is what they were designed to do... I'm not all that well acquainted with them, but given what I have seen from that general area and period, they should likely be studied with a critical eye to hidden assumptions. Quite a lot of work has been published with the general structure of "let's do some sensible transformations of data and apply a nonparametric test, then call the whole procedure assumption-free" (in those days, 1950s and 1960s, essentially, computer simulations were not readily available to show people the error of their ways...).

--
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com

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