# [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 15:28:12 CEST 2017

```Sorry, I have seen it too late that we had different tab width in the original file and my editor.
Here is the patch with all white spaces instead of mixing tabs and white spaces.

Serguei.

Le 29/05/2017 à 15:13, Serguei Sokol a écrit :
> Here is an attached patch.
>
> Best,
> Serguei.
>
> Le 29/05/2017 à 12:21, Serguei Sokol a écrit :
>> 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
>>
>>
>
>
>
> ______________________________________________
> R-devel at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-devel

--
Serguei Sokol
Ingenieur de recherche INRA
Metabolisme Integre et Dynamique des Systemes Metaboliques (MetaSys)

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