Non-parametric alternative to simple t-test I have five numeric variables of two populations (each of them with 60 individuals) and for each of those five variables I want to know if there is difference in the means.
I was trying to use a simple t-test for this (the t.test R function), but let me explain my concerns to see if it's possible.
One variable of one population do not pass the Shapiro normality test.
Any of the five variables passed Levene's test for 0.05, only one for 0.01 (but is the one containing the not normal distribution in one variable).
Even with all that, would it be a good choice to use the t-test to evaluate the means? What could be a non-parametric alternative that suits my problem?.
 A: One thing to keep in mind- outside of some contexts in physics, no process in nature will generate purely normally distributed data (or data with any particular nicely behaved distribution).  What does this mean in practice?  It means that if you possessed an omnipotent test for normality, the test would reject 100% of the time, because your data will essentially always only be, at best, approximately normal. This is why learning to ascertain the extent of approximate normality and its possible effects on inference is so important for researchers, rather than relying on tests.  
A: *

*The t-test does not assume normality of the dependent variable; it assumes normality conditional on the predictor. (See this thread: Where does the misconception that Y must be normally distributed come from?). A simple way to condition on your grouping variable is to look at a histogram of the dependent variable, splitting the data on your grouping variable.

*Normality tests, like the Shapiro-Wilk test, may not be that informative. Small deviations from normality may come up as significant (see this thread: Is normality testing 'essentially useless'?). However, given your small sample size, this probably is not an issue. Nonetheless, it does not really matter (practically speaking) if normality is violated, but the extent to which it is violated.

*Depending on how non-normal your data might be, you probably do not have much to worry about. The general linear model (of which the t-test is a part) is more robust to violations of the normality assumption than to other assumptions (such as independence of observations). Just how robust it is has been discussed on this website before, such as in this thread: How robust is the independent samples t-test when the distributions of the samples are non-normal?. There are many papers looking at how robust this method is to violations of normality, as well (as evidenced by this quick Scholar search: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C33&q=t-test+robust+to+nonnormality&btnG=).

*You might as well run a nonparametric test and see if your conclusions differ; it costs essentially nothing to do so. The most popular alternative is the Mann-Whitney test, which is done with the stats::wilcox.test function in R (http://stat.ethz.ch/R-manual/R-devel/library/stats/html/wilcox.test.html). There are a lot of good introductions to this test on the internet—I would Google around until a description of it clicks with you. I find this video, calculating the test statistic by hand, to be very intuitive: https://www.youtube.com/watch?v=BT1FKd1Qzjw. Of course, you use a computer to calculate this, but I like knowing what's going on underneath.
