# Can bootstrap be used to replace non-parametric tests?

I am fairly new to statistics. The concept of bootstrapping has been confusing to me.

I know that normality of the sampling distribution is required to use certain tests such as the t-test. In cases when the data are not normally distributed, by requesting "bootstrapping" in t-tests in SPSS would this circumvent the problem of non-normality? If so, is the t-statistic that is reported in the output based upon the bootstrapped sampling distribution?

Also, would this be a better test compared to using non-parametric tests like Mann-Whitney or Kruskal-Wallis in cases where I have non-normal data? In situations when the data are not normal and I am using bootstrap I would not report the t-statistic: right?

## 1 Answer

The bootstrap works without needing assumptions like normality, but it can be highly variable when the sample size is small and the population is not normal. So it can be better in the sense of the assumptions holding, but it is not better in all ways.

The bootstrap samples with replacement, permutation tests sample without replacement. The Mann-Whitney and other nonparametric tests are actually special cases of the permutation test. I actually prefer the permutation test here because you can specify a meaningful test statistic.

The decision on which test to use should be based on the question being answered and knowledge about the science leading to the data. The Central Limit Theorem tells us that we can still get very good approximations from t-tests even when the population is not normal. How good the approximations are depends on the shape of the population distribution (not the sample) and the sample size. There are many cases where a t-test is still reasonable for smaller samples (and some cases where it is not good enough in very large samples).

• Thanks that is helpful. So if I use bootstrapping then I would only report the p-value and CI without any test statistic, is this correct? – JC22 Jun 14 '13 at 22:35
• (+1) Would you by any chance have a reference or a link regarding Mann-Whitney and permutation tests? That's very interesting but not obvious to me! – Gala Jun 14 '13 at 22:39
• @JC22 You should report a test statistic (whatever statistic you bootstrap); a bootstrap test based on a mean will be different from one based on a trimmed mean, for example. – Glen_b Jun 15 '13 at 1:03
• @GaëlLaurans For an example of generating the exact (permutation) distribution of the Wilcoxon rank-sum test statistic (equivalent to Mann-Whitney) and of the Kruskal-Wallis test statistic, see this answer. – caracal Jun 15 '13 at 9:51
• @GaëlLaurans, on reference is: Different Outcomes of the Wilcoxon—Mann—Whitney Test from Different Statistics Packages Reinhard Bergmann, John Ludbrook & Will P. J. M. Spooren Journal: The American StatisticianVolume 54, Issue 1, February 2000, pages 72-77 – Greg Snow Jun 16 '13 at 4:42