Spearman for sure. If your distribution is normal, results will be very similar to Pearson. You can test this by doing a few tests to convince yourself this is OK for your data.

Looking for the optimal functional form is fine, but it is advisable not to transform the data unless there is a good reason to do so. There is not a general agreement on this, it is mostly James MacKinnon's strong advice, which has proved to be right most of the time.

Hope this helps you.

Note that the normality of marginal variables is a not a reason for joint normality of all variables. You can do a copula goodness of fit test to find a dependence structure for vector of your variables. When you could find a suitable copula, you can do tests of dependence using Spearman or Kendall coefficient.

If your data are quantitative, but not normally distributed, it is best to use the sample coefficient of determination (http://www.slideshare.net/VladimirBakhrushin/ss-17868992, slides 10-13) or the correlation ratio (https://www.researchgate.net/publication/235825660_The_Methods_of_Data_Analysis_(in_Ukrainian)?ev=prf_pub, p. 107).

Book The Methods of Data Analysis (in Ukrainian)

Fitting one (maybe linear in this case) model that incorporate your parameters as independent variables is always better than multiple statistical tests. The p-values for associations with each parameter can be obtained via 1) testing of model coefficients significance, 2) ANOVA between two nested models (one fitted with the parameter and other without it) or 3) likelyhood ratio test of nested models.

Spearman for sure. If your distribution is normal, results will be very similar to Pearson. You can test this by doing a few tests to convince yourself this is OK for your data.

Looking for the optimal functional form is fine, but it is advisable not to transform the data unless there is a good reason to do so. There is not a general agreement on this, it is mostly James MacKinnon's strong advice, which has proved to be right most of the time.

Hope this helps you.

Spearman's test can be used only for monotonic dependencies. Real relationships may be more complex. Then Spearman test will lead to wrong conclusions.

Dear Magdalena, I would definitively go for Spearman, Transformation of your data is advisable only if you are able to fully interpret this new variable.

For conditions on when to use Person and when to you Spearman, you may find helpful a very nice Lesson in biostatistics from the Biochemia Medica (http://biochemia-medica.com/) Journal:

Udovicic M, Baždaric K, Bilic-Zulle L, Petrovecki M. What we need to know when calculating the coefficient of correlation? Biochemia Medica 2007;17(1):10-5. http://dx.doi.org/10.11613/BM.2007.002

This open-access Journal publishes very nice educational lessons on some very simple topics in Biostatistics.

Hope this helps!

Dear Magdalena,

As far as sample sizes are not small, you could use Pearson even if pairs are not bivariate normally distributed.

However, note that Pearson assesses linear relationships while Spearman assessess monotic relationships (see e.g. http://upload.wikimedia.org/wikipedia/commons/4/4e/Spearman_fig1.svg in Wikipedia).

Hope this helps!

Kendall coefficient is better.

Logarithm transforms can greatly distort results if much of the interesting variability is in the tails of the distribution, especially if the distribution seems to be assuming an S-shape. The transform compresses the tails and the variability is lost.

I totally agree with Adrian Otoiu.

Dear Magdalena, sure you can go for Spearman test. Data transformation is not at all a better step in common situations. If your data is including several observations of same sample, i think reepated measure ANOVA will be helpfull. However if data includes non independent data sets and the let say data X is dependent on X-1 ... so on, then only either you can go for a non parametric test or first transform your data into natural log/arcsin ets which ever is applicable prior to statistical tests.

Certainly both will give approximate value when both is used for measurement data but the best thing to do for this situation to facilitate good comparison and judgement will be to use a distribution free method , that is Spearman's rank correlation method since you have mixed distribution(normal and others).

I would like to thank all, who have taken participation in this discussion and all the suggestions and tips, which I found very helpful.

Hi, formally speaking Spearman is the best option, for it does not require data to have a normal distribution (or a high measurement level for that matter). However, I usually advise my students to get both results and at least compare. It makes them aware of the impact and differences in interpretation of both test results. Usually they are not so far apart, but Spearman tends to be more conservative and keeping the risk of type I error to the minimum.

Spearman is the Pearson correlation of the rank numbers. You could just as well compute the Pearson correlation coefficient of any monotone transformation of the rank numbers - for instance you could transform the ranks to normal quantiles. This leads to a correlation coefficient which is very very close to the standard Pearson coefficient when the data is bivariate normal, and still sensible when that is not the case. You could say: it transforms the two variables in a monotone way so that each variable is marginally normally distributed, and then takes the Pearson correlation between those.

This way you get a non-parametric or distribution free measure of association which moreover coincides exactly with the Pearson measure when the data happens to be bivariate normal. The best of both worlds.