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

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.