[R] P-values Kolmogorov–Smirnov test

Fri Sep 6 07:19:41 CEST 2019

Hello,

Yesterday wasn't one of my days.
The main problem I'm seeing is that the KS statistic is meant for 
continuous data and you have counts data assumed to follow a Poisson 
distribution. This might explain the nonsense results you are getting 
from ks.test.

Have you considered a chi-squared GOF test?

Hope this helps,

Rui Barradas

Às 20:51 de 05/09/19, Boo G. escreveu:
> Hello and thanks for your patience.
> 
> As far as I understand, the paper of Marsiglia and colleagues refers to CDF samples (i.e. from a hypothetical distribution — e.g. a Poisson), while I have an ECDF sample (i.e. (pseudo-)observed data — e.g. rpois(1000, 500). In my study, I am actually comparing the statistical distribution of people per hectares in a region (~50,000 observations) with samples from that distribution.
> 
> Agree that we cannot consider at p-values for low sample size. Still, somewhere above 100, I expect to see p-values between 0 and 0.05 (rejecting the fact that the sample comes from the reference distribution). Ideally, I would try the procedure suggested by Marsiglia to compute p-values but this is far beyond my coding skills.
> 
> 
> 
> 
>> On 5 Sep 2019, at 21:21, Rui Barradas <ruipbarradas using sapo.pt> wrote:
>>
>> Hello,
>>
>> Inline.
>>
>> Às 20:09 de 05/09/19, Boo G. escreveu:
>>> Hello again.
>>> I have tied this before but I see two problems:
>>> 1) According to the documentation I could read (including the ks.test code), the ks statistic would be max(abs(x - y)) and if you plot this for very low sample sizes you can actually see that this make sense. The results of ks.test(x, y) yields very different values.
>>
>> The problem is that the distribution of Dn is very difficult to compute. From the reference [1] in the help page ?ks.test:
>>
>> 	Kolmogorov's goodness-of-fit measure, Dn , for a sample CDF has consistently been set aside for methods such as the D+n or D-n of Smirnov, primarily, it seems, because of the difficulty of computing the distribution of Dn . As far as we know, no easy way to compute that distribution has ever been provided in the 70+ years since Kolmogorov's fundamental paper. We provide one here, a C procedure that provides Pr(Dn < d) with 13-15 digit accuracy for n ranging from 2 to at least 16000.
>>
>>
>> That is why I used ks.test and its Dn and p-values. Note that n >= 2, size = 1 is not covered (p-value == 1).
>>
>> Also, the p-values distribution seem to become closer to a uniform with increasing sizes. Try
>>
>> hist(d_stat[801:1000, 3])
>>
>>
>> [1]https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.jstatsoft.org%2Farticle%2Fview%2Fv008i18&data=01%7C01%7Cgianluca.boo%40soton.ac.uk%7Cfebe94a441914aa58a9908d732363a52%7C4a5378f929f44d3ebe89669d03ada9d8%7C0&sdata=JBHkkXn6G4oLQZCV7HoqBLO4a3sMixTa16kOVFwXPlY%3D&reserved=0
>>
>>
>> Hope this helps,
>>
>> Rui Barradas
>>
>>> 2) Also in this case the p-values don’t make much sense, according to my previous interpretation.
>>> Again, I could be wrong in my interpretation.
>>>> On 5 Sep 2019, at 20:46, Rui Barradas <ruipbarradas using sapo.pt <mailto:ruipbarradas using sapo.pt>> wrote:
>>>>
>>>> Hello,
>>>>
>>>> I'm sorry, but apparently I missed the point of your problem.
>>>> Please do not take my previous answer seriously.
>>>>
>>>> But you can use ks.test, just in a different way than what I wrote previously.
>>>>
>>>> Corrected code:
>>>>
>>>>
>>>> #simulation
>>>> for (i in 1:1000) {
>>>>   #sample from the reference distribution
>>>>   m_2 <-m_1[(sample(nrow(m_1), size=i, prob=p_1, replace=F)),]
>>>>   m_2 <-m_2[order(m_2$d_1),]
>>>>   d_2 <- m_2$d_1
>>>>   p_2 <- m_2$p_1
>>>>
>>>>   #weighted ecdf for the reference distribution and the sample
>>>>   f_d_1 <- ewcdf(d_1, normalise=F)
>>>>   f_d_2 <- ewcdf(d_2, 1/p_2, normalise=F, adjust=1/length(d_2))
>>>>
>>>>   #kolmogorov-smirnov distance
>>>>   x <- f_d_1(d_2)
>>>>   y <- f_d_2(d_2)
>>>>   ht <- ks.test(x, y)
>>>>   d_stat[i, 2] <- ht$statistic
>>>>   d_stat[i, 3] <- ht$p.value
>>>> }
>>>>
>>>>
>>>> Hope this helps,
>>>>
>>>> Rui Barradas
>>>>
>>>> Às 19:29 de 05/09/19, Rui Barradas escreveu:
>>>>> Hello,
>>>>> I don't have the algorithms at hand but the KS statistic calculation is more complicated than your max/abs difference.
>>>>> Anyway, why not use ks.test? it's not that difficult:
>>>>> set.seed(1234)
>>>>> #reference distribution
>>>>> d_1 <- sort(rpois(1000, 500))
>>>>> p_1 <- d_1/sum(d_1)
>>>>> m_1 <- data.frame(d_1, p_1)
>>>>> #data frame to store the values of the simulation
>>>>> d_stat <- data.frame(1:1000, NA, NA)
>>>>> names(d_stat) <- c("sample_size", "ks_distance", "p_value")
>>>>> #simulation
>>>>> for (i in 1:1000) {
>>>>> #sample from the reference distribution
>>>>> m_2 <-m_1[(sample(nrow(m_1), size=i, prob=p_1, replace=F)),]
>>>>> d_2 <- m_2$d_1
>>>>> ht <- ks.test(d_1, d_2)
>>>>> #kolmogorov-smirnov distance
>>>>> d_stat[i, 2] <- ht$statistic
>>>>> d_stat[i, 3] <- ht$p.value
>>>>> }
>>>>> hist(d_stat[, 2])
>>>>> hist(d_stat[, 3])
>>>>> Note that d_2 is not sorted, but the results are equal in the sense of function identical(), meaning they are *exactly* the same. Why shouldn't they?
>>>>> Hope this helps,
>>>>> Rui Barradas
>>>>> Às 17:06 de 05/09/19, Boo G. escreveu:
>>>>>> Hello,
>>>>>>
>>>>>> I am trying to perform a Kolmogorov–Smirnov test to assess the difference between a distribution and samples drawn proportionally to size of different sizes. I managed to compute the Kolmogorov–Smirnov distance but I am lost with the p-value. I have looked into the ks.test function unsuccessfully. Can anyone help me with computing p-values for a two-tailed test?
>>>>>>
>>>>>> Below a simplified version of my code.
>>>>>>
>>>>>> Thanks in advance.
>>>>>> Gianluca
>>>>>>
>>>>>>
>>>>>> library(spatstat)
>>>>>>
>>>>>> #reference distribution
>>>>>> d_1 <- sort(rpois(1000, 500))
>>>>>> p_1 <- d_1/sum(d_1)
>>>>>> m_1 <- data.frame(d_1, p_1)
>>>>>>
>>>>>> #data frame to store the values of the siumation
>>>>>> d_stat <- data.frame(1:1000, NA, NA)
>>>>>> names(d_stat) <- c("sample_size", "ks_distance", "p_value")
>>>>>>
>>>>>> #simulation
>>>>>> for (i in 1:1000) {
>>>>>> #sample from the reference distribution
>>>>>> m_2 <-m_1[(sample(nrow(m_1), size=i, prob=p_1, replace=F)),]
>>>>>> m_2 <-m_2[order(m_2$d_1),]
>>>>>> d_2 <- m_2$d_1
>>>>>> p_2 <- m_2$p_1
>>>>>>
>>>>>> #weighted ecdf for the reference distribution and the sample
>>>>>> f_d_1 <- ewcdf(d_1, normalise=F)
>>>>>> f_d_2 <- ewcdf(d_2, 1/p_2, normalise=F, adjust=1/length(d_2))
>>>>>>
>>>>>> #kolmogorov-smirnov distance
>>>>>> d_stat[i,2] <- max(abs(f_d_1(d_2) - f_d_2(d_2)))
>>>>>> }
>>>>>>
>>>>>>
>>>>>>      [[alternative HTML version deleted]]
>>>>>>
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>>>>>> and provide commented, minimal, self-contained, reproducible code.
>>>>>>
>>>>> ______________________________________________
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>>>>> and provide commented, minimal, self-contained, reproducible code.
>