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I am studying banks behaviour according to 6 financial rations throughout a 3-years period. I have 32 observations separated into two groups: large (6) and medium-sized banks (26). However, since in my country there are exactly 6 large banks, it does not constitute a sample, but a population, where as there are around 32 medium-sized banks (so my sample is quite large compared to the population).

To compare these two groups each year, I am using the Wilcoxon Signed Rank Test to test:

$$ H_0: \mu_{i,medium} = \mu_{i,large} $$ where $ \mu_{i,large} $ is the mean of financial ration $i$ of the $large$ group, which is a known parameter, which means I should not be comparing two samples. Is the Signed Rank Test adequate here?

However, I figure that since I have data over 3 years, it'd be better to compare these groups throughout the 3-years period as whole (or not?). What technique should I use to do so? Are there any test I should do prior to selecting a test?

Thanks

Bernardo

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I don't think you should consider $\mu_{i,large}$ to be the population mean. Even though it consists of observations from all large banks in your country. Do you care about the characteristics of the ratios outside the 3-year period? Do you care about potential measurement error in the ratios? (i.e. your observations $y_{it}$ come from $y_{it} = r_{it} + \epsilon_{it}$, where $r_{it}$ is the true value).

Also, check the assumptions for the Wilcoxon signed rank test. I think they are violated. Specifically, the data should be paired. I think the Wilcoxon rank sum test will be more appropriate, though as always you should be aware of the assumptions and the extent to which they are potentially violated.

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  • $\begingroup$ Hi, David, thanks for your thoughts! Answering your questions: 1) No, I do not care about the ratios outside the 3-year period. I am trying only to analyse whether medium banks show a signifficant different ratio when compared to large banks. 2) About the potential measurement error, I am not sure if it applies in this case. The financial ratios are based on financial statements, which we expect to be free of errors. Back to your first sentence, I am having a hard time whether or not to consider it the population mean. I'd like to "hear" why you wouldn't. $\endgroup$ – Bernardo Aug 17 '14 at 21:07
  • $\begingroup$ If you actually have the populations about which you wish to make inferences, you would not be testing. The purpose of statistical inference is to come to conclusions about unobservable population parameters, but if you have them you shouldn't treat them as samples, you just look at the values. In the case where you have sampled almost the whole population you should also be taking account of the fact that your population is small. You can't really treat the distribution as continuous (it only takes a few values) and you should account for the sampling without replacement... $\endgroup$ – Glen_b Aug 18 '14 at 0:00
  • $\begingroup$ ... I'd also be surprised if you can consider your sample 'random'. $\endgroup$ – Glen_b Aug 18 '14 at 0:01
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    $\begingroup$ We should certainly not expect financial statements to be free of errors :). But sticking to the point, @Glen_b had it: you wouldn't have two samples for a two-sample test. If you must treat the large bank ratios as population parameters then your test should be a one sample test, with the null that midsize bank parameter equals a constant (large bank parameter). But if you agree with the above note that the distribution shouldn't be treated as continuous then a t-test wouldn't work. $\endgroup$ – David Melkuev Aug 18 '14 at 2:44
  • $\begingroup$ David, I get your point about errors in financial statements (and I agree), what I meant is that it's not the usual sense of error of measurement as in measuring a distance with a instrument's error. Back to the point, that's exactly how I've approached the problem so far. I might have mistaken the test's name, but I am testing the null that midsize bank parameter equals a constant.I've heard people supporting the idea that I should consider both groups as samples or as populations, since the 2nd group comprehends almost all medium banks... $\endgroup$ – Bernardo Aug 18 '14 at 3:32

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