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This is a general question. In my case study I have measures of performance for 4 groups. For the same groups I have another large set of measure, which in our opinion can influence performance.

My question is: which statistics should we use to try describing an association between predictors and the multiple performance metrics?

In other cases, before trying anything more complicated statistics, I would have gone for correlation (either Pearson's or Spearman's) and multiple linear regression. However, here it does not make al lot of sense with 4 observations.

I know I cannot prove causality with this data. However, what would you use, at least to discuss a possible association? And/or which kind of descriptives should I present?

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I would rank groups according to performance and according to each predictor and see if any predictor gives the same ranking as performance. If yes, I'd call it (them) the best predistor(s). If not, I'd seek for rankings that need the smallest number of permutations to match performance's ranking.

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You may want to consider giving an estimate of the population size, to put the results of your case study in context and enable your readers to relate your four cases to the size of the larger group they are part of (especially if that group is small as doing so can provide a justification for why metrics of your four cases should be considered - I am assuming they should be).

Also, you could consider reporting your findings alongside findings from similar, more representative samples, in order for your readers to get an explicit sense of data from a similar process.

As the limited sample size results in extremely low statistical power, I'd discourage you from making any comparative claims in terms other than hypotheticals.

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