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I have a paired data consisting of $N$ = 421 samples. I would like to find out if there is statistical significance between the paired samples. Since, I do not know the underlying data distribution I select the Wilcoxon Signed Rank test for my task. I use the SciPy stats module in Python to do the job for me [0]. This function returns the $z$ statistic (under the large-sample approximation that the signed-rank statistic is normally distributed) and $p$ value for the two-sided test. I received the following values as output

$z = 4788.5$

$p = 1.00530788183* 10^{-8}$

The Null Hypothesis $H_o$ Wilcoxon Signed Rank test [1,2,3] states that there is no difference between the paired samples. Since, the $p \leq 0.99$ , I reject $H_0$ as my null hypothesis at 1% confidence interval and conclude that the difference between the two paired samples is statistically significant.

I have two questions -

$Q1$ Is my choice of Wilcoxon Signed Rank test a correct choice. If not, what should I use?

$Q2$ Are my conclusions from the significance test correct? If not, can you please explain why they are not correct?

[0] http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wilcoxon.html#scipy.stats.wilcoxon [1] http://blog.excelmasterseries.com/2010/10/wilcoxon-signed-rank-test-in-excel-for_13.html

[2] Please help me interpret these Wilcoxon signed-rank test results

[3] http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test

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  • $\begingroup$ Since $p\le 0.99$, you reject $H_0$ at the 1% level of confidence, not the 99% level! (But since $p$ is actually much smaller than $0.99$, and indeed very small indeed, it's safe to reject $H_0$ anyway.) $\endgroup$ – whuber Oct 9 '12 at 22:00
  • $\begingroup$ @whuber Thanks for the reply. Indeed I should reject $H_0$ at the 1% level confidence. I am confused though - at what level is rejecting better 99% or 1%? $\endgroup$ – Dexter Oct 9 '12 at 22:16
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Common practise is to compare p-value with three levels - 0.05, 0.01 and 0.001. Since your p-value is less than each of them, you have to choose the smallest one, so you should conclude that differences are significant and p<0.001. Roughly speaking: The smaller the p-value, the more significant differences are.

Since we do not know distribution of your data, we do not also know which test should you use. But you have quite large sample, so there is high chance that parametric test can be appropriate (t-test for paired data).

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  • $\begingroup$ Thanks! I do not know the distribution of my data either and therefore I chose to use the Wilcoxon Signed Rank Test as it makes no assumption on the underlying data sample distribution. $\endgroup$ – Dexter Oct 10 '12 at 8:53

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