Non-parametric tests exist to cater for what you just described above. Like the first poster highlighted, just use the alternative tests suggested, since they are 'distribution-free tests'. If the two means you're comparing are dependent, instead of using paired t test, you should use Wilcoxon signed-ranks test.

At the end of the day, you can compare your results (both from parametric and non-parametric tests) but you have to highlight the difference, often using superscripts and defined under the result tables.

Maybe you could describe why your data is non-normal... What does the actual distributions look like? It would help for giving advice. Given that you talk about biomarkers, I assume that "treatment" would conform to an approximate normal distribution, while the controls would be skewed due to data being concentrated at low values - this would be perfectly normal and expected in many cases, since some biomarkers should not be expressed, or be expressed at very low levels, in control environments. When data cannot be negative the only possible spread of the data would be upwards from 0, and there would be skew (i.e. some control data SHOULD be non-normal - otherwise something is wrong...). Log-transformation should help in some cases (many biological processes follow log-normality), but maybe a gamma-distribution might fit better sometimes. In any case, I would assume that you are actually interested in the actual difference in their central tendency, not only different in rank-order. Actual difference probably tells you something about the severity of your treatment. Hence, I would suggest that you don't do a non-parametric test, which often just test rank-order.

Also, I would actually suggest that you plot a graph with your raw data and the different transformations you apply (put raw data on x-axis, and transformed values on the y-axis - make different series for different transformations). This way, you would see and learn what each transformation does with your data, and you would not have to blindly apply every transformation in future projects. Usually, there is a theoretically suitable transformation that can be figured out before data analysis. For instance, square root transformation will typically compress your data much less than a log-transformation (unless you go for some weird log-base like 1.1), which means that if your data is positively skewed after log-transformation, applying sqrt-tranformation is very unlikely to produce a better result...

Thanks Joacim...and Adam, you may wish to present your results in 'waterfall' graph. It will tell a beautiful story of your findings.

I hope that the following links can provide you with the answers you are looking for.

Conference Paper Clustering Heterogeneous Data Sets

Article Using the linear mixed model to analyze nonnormal data distr...

As a rule of thumb, when the two sets of data reveal different distribution characteristics , we should check whether the number of participants in the parametric , homogeneous group is equal to or more than that of the non-parametric group. Under such circumstances, we should use a parametric test. Apparently, if the number of participants in parametric group is less than that of the non- parametric sample, then we should use a non parametric statistical test.

Best of luck,

R. Biria

Reza Biria I found that quite useful, could you provide some references that I could check and cite about it? Thanks!