Solved – How to read result from Kolmogorov-Smirnov one-sample test in SPSS

kolmogorov-smirnov testp-valuepoisson distributionspssstatistical significance

I'm newbie and bit lost… Can someone help me read this result from SPSS 1-sample Kolmogorov-Smirnov test (poisson distribution).

                                        SUBKEY1   SUBKEY2  SUBKEY3  SUBKEY4    SUBKEY5

N                                128       128       128      128    128

Poisson Parametera,,b     Mean          .4609     .4609    .4922     .5156  .4922

Most Extreme Differences  Absolute  .092      .092     .103      .113   .103

                      Positive  .079      .079     .088      .095   .088

                      Negative     -.092     -.092    -.103     -.113      -.103

Kolmogorov-Smirnov Z                1.037    1.037    1.171     1.276   1.171

Asymp. Sig. (2-tailed)              .233      .233     .129      .077   .129

From the result above, I define (in simple words):

Ho = satisfied
Ha = not satisfied

but I don't know how to specify the level significance for this test and I don't know whether have to read Asymp. Sig. (2-tailed) (p-value) or Kolmogorov-Smirnov Z (D value) to be compare with critical value…

Best Answer

If $\alpha = .05$ is conventional in your field, I would simply state, "A one-sample Kolmogorov-Smirnov test failed to reject the null hypothesis that the data followed the Poisson distribution (D = .092, .092, .103, .113, and .103 respectively for variables 1-5, N = 128 each, and p > .05 each)." You can also report that you set $\alpha = .05$ in your methods section. Note that $Z \approx \sqrt{n} D$ in the SPSS output and that the "Asymp. Sig." (p-values) in the output should be based on critical values of the Kolmogorov distribution to which $Z$ converges in distribution (see here for a description). However, you don't need to find or report these critical values -- I would just report the test statistics and p-value in the manner above.

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Solved – Two sample Kolmogorov-Smirnov test and p-value interpretation

If you are using the traditional 0.05 alpha level cutoff then all but group 3 are significantly different from your full group. It is a little easier to see this if the p-values are not in scientific notation ( you can use options(scipen=5) in R to make this less likely). Also group 1 becomes non-significant for some adjustments for multiple tests. You should consider whether that adjustment applies in your case or not. Also note that the groups that are not significant could be different, just low power.

But that just means that any differences, however small, are not easily explained by chance. It could be that your groups are close enough for practical purposes. It is usualy more meaningful to plot the data to see how different the distributions are. You could use the qqplot function as one approach. The vis.test function in the TeachingDemos package for R gives another approach.

One possible hitch is if your groups are part of the "Full" data set as well, then you don't have the independence assumed (but given the sample sizes, I am not sure how much this would affect things). You could address this by taking random samples from the full data set and computing the KS-distance for each (ignore the p-value), then compare where your actual data falls relative to the random samples.

Most of this comes down to what question you really want answered, many of the exact distributional tests answer a different question than the researcher is really interested in.

Kolmogorov-Smirnov Test – How to Perform a Kolmogorov-Smirnov Two-Sample Test

I am assuming you are asking because the Suanshu help page reports in reference to the K-S distribution, "This is not done yet." Luckily, it is very easy to do in R. If x and y are your two samples, ks.test(x,y) returns the test statistic and pvalue. For example,

> x <- rnorm(50)
> y <- runif(30)
> ks.test(x, y)    
        Two-sample Kolmogorov-Smirnov test    
data:  x and y 
D = 0.5, p-value = 9.065e-05
alternative hypothesis: two-sided

By default, it will compute exact or asymptotic p-values based on the product of the sample sizes (exact p-values for n.x*n.y < 10000 in the two-sample case), or you can specify this option with a third argument, exact=F or exact=T. Exact p-values are calculated using the methods of Marsaglia, et al. (2003), which the Suanshu documentation also cites. Some large sample approximations are given here, although I don't have a proper citation. Lastly, if you don't want to install R, there are web calculators for the two-sample K-S test, although I don't know if they use the same algorithm as R because the one I found only reported three decimal points for the p-value.

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