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What is the cleanest, easiest way to explain someone the concept of Kolmogorov Smirnov Test? What does it intuitively mean?

It's a concept that I have difficulty in articulating - especially when explaining to someone.

Can someone please explain it in terms of a graph and/or using simple examples?

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  • $\begingroup$ What it does or how it does? $\endgroup$ – gunes Jun 12 '20 at 8:25
  • $\begingroup$ @gunes What it does, If possible both what and how it does!! $\endgroup$ – Pluviophile Jun 12 '20 at 8:30
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    $\begingroup$ Is cumulative distribution function simple English? If not, this is a difficult challenge. The simplest explanation would be terms of a graph. $\endgroup$ – Nick Cox Jun 12 '20 at 8:40
  • $\begingroup$ Are you familiar with the cumulative distribution function of a distribution and the empirical (ie.e sample) cdf? $\endgroup$ – Glen_b Jun 12 '20 at 10:29
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    $\begingroup$ Or the diagrams already on site, such as the two-sample KS test diagrams here $\endgroup$ – Glen_b Jun 12 '20 at 11:50
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The Kolmogorov-Smirnov test assesses the hypothesis that a random sample (of numerical data) came from a continuous distribution that was completely specified without referring to the data.

Here is the graph of the cumulative distribution function (CDF) of such a distribution.

Figure 1: Graph of the standard normal CDF from -3 to 3

A sample can be fully described by its empirical (cumulative) distribution function, or ECDF. It plots the fraction of data less than or equal to the horizontal values. Thus, with a random sample of $n$ values, when we scan from left to right it jumps upwards by $1/n$ each time we cross a data value.

The next figure displays the ECDF for a sample of $n=10$ values taken from this distribution. The dot symbols locate the data. The lines are drawn to provide a visual connection among the points similar to the graph of the continuous CDF.

Figure 2: Graph of an ECDF

The K-S test compares the CDF to the ECDF using the greatest vertical difference between their graphs. The amount (a positive number) is the Kolmogorov-Smirnov test statistic.

We may visualize the KS test statistic by locating the data point situated furthest above or below the CDF. Here it is highlighted in red. The test statistic is the vertical distance between the extreme point and the value of the reference CDF. Two limiting curves, located this distance above and below the CDF, are drawn for reference. Thus, the ECDF lies between these curves and just touches at least one of them.

Figure 3: CDF, ECDF, and limiting curves

To assess the significance of the KS test statistic, we compare it--as usual--to the KS test statistics that would tend to occur in perfectly random samples from the hypothesized distribution. One way to visualize them is to graph the ECDFs for many such (independent) samples in a way that indicates what their KS statistics are. This forms the "null distribution" of the KS statistic.

Figure 4: Many ECDFs, displaying a null distribution

The ECDF of each of $200$ samples is shown along with a single red marker located where it departs the most from the hypothesized CDF. In this case it is evident that the original sample (in blue) departs less from the CDF than would most random samples. (73% of the random samples depart further from the CDF than does the blue sample. Visually, this means 73% of the red dots fall outside the region delimited by the two red curves.) Thus, we have (on this basis) no evidence to conclude our (blue) sample was not generated by this CDF. That is, the difference is "not statistically significant."

More abstractly, we may plot the distribution of the KS statistics in this large set of random samples. This is called the null distribution of the test statistic. Here it is:

Figure 5: Histogram of 200 KS test statistics

The vertical blue line locates the KS test statistic for the original sample. 27% of the random KS test statistics were smaller and 73% of the random statistics were greater. Scanning across, it looks like the KS statistic for a dataset (of this size, for this hypothesized CDF) would have to exceed 0.4 or so before we would conclude it is extremely large (and therefore constitutes significant evidence that the hypothesized CDF is incorrect).


Although much more can be said--in particular, about why KS test works the same way, and produces the same null distribution, for any continuous CDF--this is enough to understand the test and to use it together with probability plots to assess data distributions.


In response to requests, here is the essential R code I used for the calculations and plots. It uses the standard Normal distribution (pnorm) for the reference. The commented-out line established that my calculations agree with those of the built-in ks.test function. I had to modify its code in order to extract the specific data point contributing to the KS statistic.

ecdf.ks <- function(x, f=pnorm, col2="#00000010", accent="#d02020", cex=0.6,
                    limits=FALSE, ...) {
  obj <- ecdf(x)
  x <- sort(x)
  n <- length(x)
  y <- f(x) - (0:(n - 1))/n
  p <- pmax(y, 1/n - y)
  dp <- max(p)
  i <- which(p >= dp)[1]
  q <- ifelse(f(x[i]) > (i-1)/n, (i-1)/n, i/n)

  # if (dp != ks.test(x, f)$statistic) stop("Incorrect.")

  plot(obj, col=col2, cex=cex, ...)
  points(x[i], q, col=accent, pch=19, cex=cex)
  if (limits) {
    curve(pmin(1, f(x)+dp), add=TRUE, col=accent)
    curve(pmax(0, f(x)-dp), add=TRUE, col=accent)
  }
  c(i, dp)
}
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  • $\begingroup$ @Maximilian see ks.test. $\endgroup$ – whuber Jun 12 '20 at 15:02
  • $\begingroup$ I’m aware of ks.test i was keen to see the implementation. $\endgroup$ – Maximilian Jun 12 '20 at 15:11
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    $\begingroup$ @Maximilian: Read the code. It's what I used. $\endgroup$ – whuber Jun 12 '20 at 15:46
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    $\begingroup$ At the R console, type ks.test It will respond by displaying the code. $\endgroup$ – whuber Jun 12 '20 at 17:16
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    $\begingroup$ @Max and Pluvio, I have posted the code I think you might be most interested in. $\endgroup$ – whuber Jun 12 '20 at 17:21
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The one-sample Kolmogorov-Smirnov test finds the largest vertical distance between a completely specified continuous hypothesized cdf and the empirical cdf.

The two-sample Kolmogorov-Smirnov test finds the largest vertical distance between the empirical cdfs for two samples.

Unusually large distances indicate that the sample is not consistent with the hypothesized distribution (or that the two samples are not consistent with having come from the same distribution).

These tests are nonparametric in the sense that the distribution of the test statistic under the null doesn't depend on which specific distribution was specified under the null (or which common distribution the two samples are drawn from).

There are "one-sided" (in a particular sense) versions of these tests, but these are relatively rarely used.

You can do a Kolmogorov-Smirnov test with discrete distributions but the usual version of the test (i.e. using the usual null distribution) is conservative, and sometimes very conservative. You can (however) obtain new critical values for a completely specified discrete distribution.

There is a related test when parameters are estimated in a location-scale family* (or a subset of location and scale), properly called a Lilliefors test (Lilliefors did three tests for the normal case and a test for the exponential case). This is not distribution-free.

* up to a monotonic transformation

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You are looking for the maximum deviation of empirical CDF (built from observations) from the theoretical values. By definition it can't be larger than 1.

Here's a plot for a uniform distribution CDF (black) and two stylized candidate CDFs (red):

enter image description here

You see that your candidate CDF can't be over the theoretical by more than $D^+$ or below it by more than $D^-$, both of which are bounded in magnitude by 1.

The empirical CDF $S_n$ for the purpose of this test is $S_i=i/N$. Here we sorted the sample $x_i$ where $i=1,\dots,N$ so that $x_i<x_{i+1}$. You compare it with a theoretical CDF $F_i=F(x_i)$, then you have set of deviations $D^+_i=\max(0,S_i-F_i)$.

However, that's not what's amazing about KS statistic. It is that the distribution of $\sup_{x\in(-\infty,\infty)} D^+$ is the same for any distribution of data set! To me that's what you need to get intuitively if you can.

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  • $\begingroup$ Your final remark is important, albeit tangential to the question. But this result is unsurprising, because any continuous monotonic increasing transformation of the distribution (applied at the same time to the data) will not change the KS statistic, since geometrically it changes only the $x$ coordinates, leaving all vertical distances unaffected. Because any continuous CDF can transformed (via its probability integral transform) to a uniform distribution, your picture of the uniform CDF is completely general. $\endgroup$ – whuber Jun 12 '20 at 21:11
  • $\begingroup$ BTW, your labels "$D^+$" and "$D^-$" may be confusing to readers, because these terms are usually used for the largest deviations between data and the hypothesized CDF rather than between the CDF and its theoretical limits of $1$ and $0.$ $\endgroup$ – whuber Jun 12 '20 at 21:13
  • $\begingroup$ @whuber agreed that notation is confusing, will improve it. $\endgroup$ – Aksakal Jun 12 '20 at 21:17
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I find it helpful to think of the two CDFs, whether population of empirical, as dancing around each other but staying close. Dance partners can spin around each other but will stay two armslengths of each other, right? When two people are further apart than that, they probably aren't dancing with each other.

ONE-SAMPLE

In the one-sample (goodness-of-fit) test, we assume that the data come from some distribution that has a particular CDF. The data also have an empirical CDF. If we are right, then the CDF of the data should dance around the CDF of the assumed distribution but stay close. If the dance partners get too far apart (in vertical distance), then we see that as evidence against our assumption.

TWO-SAMPLE

In the two-sample test, we assume that two data sets come from the same distribution. If that is the case, then the two empirical CDFs should dance around each other but stay fairly close. If the dance partners get too far apart (again, in vertical distance), then we see that as evidence against our assumption.

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