# [Rd] stats::line() does not produce correct Tukey line when n mod 6 is 2 or 3

Serguei Sokol sokol at insa-toulouse.fr
Mon May 29 12:21:25 CEST 2017

```The problem or actual R implementation relies on an assumption
that median(x[i] | x[i] <= quantile(x, 1/3)) == quantile(x, 1/6)
which reveals not to be true despite very trustful appearance.

If we continue with the example of x=y=1:9
then quantile(x, 1/6)=2.5 (here quantile() is taken in C-code sens, not R's one)
while median(y[i] | x[i] <= quantile(x, 1/3))=2
On the other sample's side we've got 7.5 and 8 for x and y respectively.
Hence the slope (8-2)/(7.5-2.5)=1.2

To get a correct version of this, one should calculate x robust points in the same way as the y's,
i.e. xb=median(x[i] | x[i] <= quantile(x, 1/3)) and xt=median(x[i] | x[i] >= quantile(x, 2/3))

Best,
Serguei.

Le 29/05/2017 à 10:02, peter dalgaard a écrit :
> A usually trustworthy R correspondent posted a pure R implementation on SO at some point in his lost youth:
>
> https://stackoverflow.com/questions/3224731/john-tukey-median-median-or-resistant-line-statistical-test-for-r-and-line
>
> This one does indeed generate the line of identity for the (1:9, 1:9) case, so I do suspect that we have a genuine scr*wup in line().
>
> Notice, incidentally, that
>
>> line(1:9+rnorm(9,,1e-1),1:9+rnorm(9,,1e-1))
> Call:
> line(1:9 + rnorm(9, , 0.1), 1:9 + rnorm(9, , 0.1))
>
> Coefficients:
> [1]  -0.9407   1.1948
>
> I.e., it is not likely an issue with exact integers or perfect fit.
>
> -pd
>
>
>
>> On 29 May 2017, at 07:21 , GlenB <glnbrntt at gmail.com> wrote:
>>
>>> Tukey divides the points into three groups, not the x and y values
>> separately.
>>
>>> I'll try to get hold of the book for a direct quote, might take a couple
>> of days.
>>
>> Ah well, I can't get it for a week. But the fact that it's often called
>> Tukey's three group line (try a search on *tukey three group line* and
>> you'll get plenty of hits) is pretty much a giveaway.
>>
>>
>> On Mon, May 29, 2017 at 2:19 PM, GlenB <glnbrntt at gmail.com> wrote:
>>
>>> Tukey divides the points into three groups, not the x and y values
>>> separately.
>>>
>>> I'll try to get hold of the book for a direct quote, might take a couple
>>> of days.
>>>
>>>
>>>
>>> On Mon, May 29, 2017 at 8:40 AM, Duncan Murdoch <murdoch.duncan at gmail.com>
>>> wrote:
>>>
>>>> On 27/05/2017 9:28 PM, GlenB wrote:
>>>>
>>>>> Bug: stats::line() does not produce correct Tukey line when n mod 6 is 2
>>>>> or
>>>>> 3
>>>>>
>>>>> Example: line(1:9,1:9) should have intercept 0 and slope 1 but it gives
>>>>> intercept -1 and slope 1.2
>>>>>
>>>>> Trying line(1:i,1:i) across a range of i makes it clear there's a cycle
>>>>> of
>>>>> length 6, with four of every six correct.
>>>>>
>>>>> Bug has been present across many versions.
>>>>>
>>>>> The machine I just tried it on just now has R3.2.3:
>>>>>
>>>> If you look at the source (in src/library/stats/src/line.c), the
>>>> explanation is clear:  the x value is chosen as the 1/6 quantile (according
>>>> to a particular definition of quantile), and the y value is chosen as the
>>>> median of the y values where x is less than or equal to the 1/3 quantile.
>>>> Those are different definitions (though I think they would be
>>>> asymptotically equivalent under pretty weak assumptions), so it's not
>>>> surprising the x value doesn't correspond perfectly to the y value, and the
>>>> line ends up "wrong".
>>>>
>>>> So is it a bug?  Well, that depends on Tukey's definition.  I don't have
>>>> a copy of his book handy so I can't really say.  Maybe the R function is
>>>> doing exactly what Tukey said it should, and that's not a bug.  Or maybe R
>>>> is wrong.
>>>>
>>>> Duncan Murdoch
>>>>
>>>>
>> 	[[alternative HTML version deleted]]
>>
>> ______________________________________________
>> R-devel at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-devel

--
Serguei Sokol
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