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I have a sample of 9 patients. If I evaluate the correlation with the Spearman's test between some variables I measured in time 0 and then I repeat the same test with the same variables in time 1, is it normal that the results of "significant correlation" change?

In particular I find several significant correlation in the pre treatment, and just one significant correlation in the post, with p=0.038, which however was not significant in the other phase. What does imply the fact that all these variable are correlated in the pre and not in the post?

Should I instead perform a correlation test with all the values in a single vector, without considering pre and post phases?

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  • $\begingroup$ Note that p-values and whether tests are significant or not can generally be quite unstable, particularly with such a small dataset. For example, if the null hypothesis is true, p-values are (ideally, in a continuous situation, but even here probably approximately) uniformly distributed, so that a certain p-value comes out as 0.7 once and then 0.1 should not surprise anyone. Also if you compute enough p-values, some will automatically be significant without meaning anything. With a sample of 9 patients and several variables in the first place I recommend not to rely on p-values and tests. $\endgroup$ Jul 20 '21 at 20:54
  • $\begingroup$ Putting pre- and post-phase values together would only be correct if there were no difference between pre and post. Chances are you don't want to assume that because in that case it would be pointless to run pre- and post-phases in the first place. I think you better face the truth that 9 is too small a sample size to make safe statements about multivariable relationships. The best you can do is a good visualisation and some descriptive statistics, I'd think. $\endgroup$ Jul 20 '21 at 20:58
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On the theoretical side, variables may change between two measurement timepoints, either because of measurement variation, longitudinal variation, or just because of regression to the mean. Particularly if you have only a few observations, as you mention, it is not surprising to see some variation.

The meaning of your finding is dependent on the change in correlation (e.g. from rho=0.99 to 0.1 or rho=0.2 to 0.1). Note that the P-value of a Spearman correlation is the probability of finding a coefficient rho as high as you have found (or higher), if in reality no correlation exists (check Understanding the p-value in Spearman's rank correlation). It is a function of the sample size (n) and the correlation coefficient (r).

Coming back to your practical problem. Why did you correlate the variables in the first place - did you have an hypothesis about it? Would it be likely that any of the variables is influenced by the treatment that you mention? Do you have a biological rationale for the change? I would emphasize that you should plot the changes that you see to interpret them adequately. If you are unsure and it is feasible, collect more data.

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  • $\begingroup$ Thanks for your answer. I'm going to see the link you gave me. In the mean time I add that, yes, the hypotesis is that some variables I measure, e.g. the movement time to perform the task, gets better (in this example decreases) with the rehabilitation. The other variables I want to check if are correlated with the previous, are some clinical scale which also change with the treatment (the better the patinents are, the bigger is the score). Can I ask you also if it would be correct to study the correlation without distinguish between pre and post, as if I test it with 18 samples? $\endgroup$
    – jnj_it
    Jan 3 '17 at 18:06
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    $\begingroup$ I would suggest the following as being clearer. 1) make a boxplot of the variable that you expect to change before and after treatment and test the variable before vs after with a paired t-test or Wilcoxon rank test (depending on if your variable is normally distributed or not) 2) if you are interested if the change (before-after) in variable A and B are correlated: calculate the difference of the variables before and after (before minus after, or before divided by after) and plot against the other variable's difference calculated in the same way. you can do a correlation test now! $\endgroup$
    – David
    Jan 4 '17 at 13:28

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