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yes, there are ... too many ! See Mann Whitney Wilcoxon as a starter

but non normality is not always sooo bad when using t-test because the important thing is the normality of the mean

• this normality of the mean is guaranteed if the sample is drawn from a normal law ; that's the easy part
• now, CLT also tells you that the distribution of the mean will be asymptotically normal (if you can assume finite first two moments ... with refinements on the convergence from Barry-Esseen bound if required ...) so using the t-test in the case of non-normal samples boils down to deciding whether you are or not in this asymptotic regime ; there are three main ingredients here, the size of the sample (your sample must be large), the "fattiness" of the distribution of your sample (the fatter the tail, the slower the convergence to the asymptotic regime) and the confidence level (the higher the confidence level, the higher the size of your sample since you are probing further into the tail of the distribution of the mean)

see here

http://thestatsgeek.com/2013/09/28/the-t-test-and-robustness-to-non-normality/

http://rctdesign.org/techreports/arphnonnormality.pdf

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for t-test with large samples, non normality is usually a much milder issue than non independence !

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What kind of data do you have? How exactly are the data obtained? Are these counts, concentrations, proportions, precentages, rates, survival times, ... something else?

Btw: there is no "proper way to detect outliers" for the purpose of removing them and using the rest. Outlier tests are made to select these outliers for further analysis (and not to select the rest!). The problem is that the outlier-test may oversee a relevant value - and you have no idea how likely this is! - and that it can also signify values as outliers that would not impact your analysis - and again you have no idea how likely this is!

Using a statistic to "identify" and remove outliers is rather stupid, because:

either outliers are rare - in this case their removal does not have any great impact, or the are not rare - in which case the crucial question is: why are there so many "outlieres"?, and would I trust the rest of the data at all? The latter case is usually an indication of a fawlty method or a considerably wrong model.

There are only two reasons to signify outliers: the value is "out of range", like impossibly (or unphysiologically) high or low, or I know external reasons why I should not trust this value.

The following link is a relatively comprehensive list of common non-parametric edit: "analogues" for parametric tests.

The tests for ordinal or interval data are the non-parametric tests.

But note that some of these test do have certain assumptions about the distributions of the data, particularly in the interpretation of the results.  (E.g. Can a significant result be interpreted as a difference in medians?)

http://www.ats.ucla.edu/stat/stata/whatstat/

edit: at the time of reviewing (2020) the link is not working.

I believe you can use non-parametric tests instead of transforming your data. Mann Whitney test has been mentioned earlier by Fabrice Clerot. I think that would answer your question.

The important things already have been said: do not throw out data, use non-parametric tests. They are not inferior in detecting significant differences (at least in the most times).

@Johannes:

I know that this is a commonly applied strategy (if your data does not fit to your assumptions that are required for the test -> change the test). But in my understanding this is a rather "strange" strategy. The t-test is about the expected values. I would hope that researchers apply a t-test when they are just interested in this (I know the truth is that reserchers don't care about that - they want to satisfy the reviewers demaning such tests). The expected values have a particular meaning, a particular interpretation. And when this is relevant, then this should be analyzed. Doing a different test instead means to throw away the idea behind the test and simply substitute it by some other. If this is not any problem then I wonder what the value of the test is anyway (I mean: scientifically; apart from simply getting it published, what I do not consider a scientific argument).

It is like if you want to go to a particular party to talk to particular persons you will meet there. The hosts assume that their guests come evening wear. If you have a look at yourself and find that your clothes do not meet this assumption well (and you don't have a change to change your dress) ... so would you just go so some other party where the hosts don't care about the dress code? You can report that you have been to a party, but you missed to have the opportunity to talk to those guests you inteded to meet.

I would think it is better to still go to the party I intended to go to, at the risk of getting funny comments. Knowing this I can more carefully interpret the comments I get from the people I met.

Hmmm, not sure if my point is clear from this funny allegory, but it was worth to try...

Is it correct to interpret allegory in the following way?: 

When we get data that should have been normally distributed but are not, then despite of this use a t-test. Indeed I would very much like to apply for example a two-way- ANOVA even when some of the subgroups show no exact normal distribution, but the data usually is normally distributed. When working with biological materials, this case is not so seldom.

However, I still don´t understand why it is bad to use a Mann-Whitney test instead of a t-test. In most cases, the result of such tests are almost the same.

Very interesting thoughts, but I fear that the reviewers of the jounals dot´care too much about this...

"When we get data that should have been normally distributed but are not, then despite of this use a t-test."

This is my opinion: yes.

You do the test with a particular purpose in mind. In case of the t-test you are interested in the expected values (or the expected difference). The test is not exact when the distribution (of the residuals) is not normal, but if this is the case (no normal residuals) the t-test is still the only test we have that adresses the expected difference.

"... show no exact normal distribution"

There is nothing like an "exact normal distribution" of real data. This is always "just" an approximation - the normal distribution is an idealized model. Exact or not is not the question. The question is: is the approximation good enough for the purpose?

For biological data it usually is not really sensible to ask for absolute differences. Biology usually works on a "proportional scale", that is: the relevance or size of an effect depends on the value. It is only the relative effect that really counts. Example: our eye does not measure if there are 100 photons per second more hitting the retina. The effect depends on the intensity of the light that is already there. In absolute darkness, even a small couple of photons will make a huge difference, but in bright sunshine even a billion of photons won't make a difference at all. The same applies to physilogical parameters, to gene activity, protein concentrations, growth, almost anything I can imagine. So what actually counts is the relative effect (difference), not the absolute effect.

An easy way to adress such relative effects is by log-transforming the quantity. On the log scale such relative effects turn into additive effects that can then be analyzed with the standard methods (like t-test, or linear models in general).

"However, I still don´t understand why it is bad to use a Mann-Whitney test instead of a t-test."

Is the result the same? What is the result? The p-value. But the p-values refer to different statistical hypotheses. They may be numerically similar, but they have a different (background) meaning. Another allegory: I may ask someone a question, say: "Is the spin of cobalt an integer number?" - The person's aswer is "no". Unfortunately it turns out that the person you asked is not a chemist, so you may be sceptical about this answer. To get a reliable answer you instead ask a different question, like "is cobalt a halogen?". The answer is again "no". Different questions, similar answer. Is this a good reason to interchange the questions?

But even disregarding all this - you can only analyze interactions when you look at quantitative effects (not at ranks). And such interactions are often the main thing. And ranks don't allow you to efficiently analyze experiments with a more complicated replication structure (like, typically, cell-culture experiments where you have several levels of replication, etc.).

"I fear that the reviewers of the jounals dot´care too much about this..."

I am SURE that the reviewers don't give a damn to all this. They want to see some "p

do not trim

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