What statistical test should be used when sample size is small (n = 13)

I have collected data from employees at 13 branches of a bank and also administered customer satisfaction scale to customers of each of the branches. As a result, I am unable to have pairs of observations for each case of data at the individual level. To deal with this, I have contemplated aggregating by computing the mean for each branch to obtain branch level data to enable me pair observations (data) for each branch. I want to know if this is appropriate and whether because of the small sample size a Spearman's $\rho$ is better than Kendall $\tau$. However, further reading has revealed additional techniques (such as Gower index of agreement and Zegers' identity coefficient); are these contenders as well?

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It's going to be hard for us to answer this question in it's current state. Can you say more about the nature of your data? What do you want to learn from this investigation? It may be that a multilevel model would be appropriate, but I can't tell. – gung Aug 5 '12 at 3:19
Generally when testing hypotheses sample size does not dictate the best test. What is known about the data determines whether not parametric or nonparametric tests are preferred. In some situations there exist uniformly most powerful tests. Those tests are best for every sample size. What chnages in small samples is that even the most powerful test has low power. – Michael Chernick Aug 5 '12 at 4:21
What is the aim of the research? Is it to identify a relationship between the employee data (which is what, by the way) and customer satisfaction? If so, a correlation coefficient of any kind is unlikely to be that helpful. – Peter Ellis Aug 5 '12 at 23:45

Since you want to use Spearman's $\rho$ or Kendall $\tau$ I think you scale is at least ordinal.

Note that in Spearman's $\rho$ we have ranking of each individuals under two characters.Next take their difference ,suqarring ,summing and you know take some steps and ultimately get Spearman's $\rho$.

But in Kendall $\tau$ we take each pair of individuals and care about the order of their ranks .

Clearly to compute Kendall $\tau$ we need more time than Spearman's $\rho$ but Kendall $\tau$ give us better idea about correlation than Spearman's $\rho$. If you have time then you must need to use Kendall $\tau$ otherwise you can use Spearman's $\rho$. It is totally depends on you.

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If you have data from multiple employees and/or customers at each branch, then you have multilevel data and should probably use a multilevel model rather than taking the average at each branch.

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Spearman ρ and Kendall τ should give similar results with your N of 13. But with high correlations the Spearman tends to give somewhat higher correlation values and smaller p values (i.e., more/higher statistical significance) - - based on two quick Monte Carlo runs I just did with r= 0 and r=.9 and N=13.

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