The posterior distribution, showing complete distributions of the difference of means (right middle), the difference of standard deviations, the effect size (right bottom), and posterior predictive check (right upper).

Bayesian estimation supersedes the t test.

John K. Kruschke, Journal of Experimental Psychology: General, 2013, v.142(2), pp.573-603. (doi: 10.1037/a0029146)

Abstract:

Bayesian estimation for two groups provides complete distributions of credible values for the effect size, group means and their difference, standard deviations and their difference, and the normality of the data. The method handles outliers. The decision rule can accept the null value (unlike traditional t tests) when certainty in the estimate is high (unlike Bayesian model comparison using Bayes factors). The method also yields precise estimates of statistical power for various research goals. The software and programs are free, and run on Macintosh, Linux, and Windows platforms.

The article:

Get the article here^* or at doi: 10.1037/a0029146.
^*Your click on this link constitutes your request to me for a personal copy of the linked article, and my delivery of a personal copy. Any other use is prohibited.

Videos:

Watch the video, and this additional video that includes discussion of sequential testing. (Both presented at the 2012 Psychonomic Society meeting.)

And here is a video about how to install the software and run it.

Software (named "BEST" for "Bayesian estimation"):

•  Demo version that runs in a web browswer:
      A web app created by Rasmus B��th provides the primary results of BEST without any need to install new software. Just paste in your data and click a button. See the blog post. (03 January 2013)

•  Full programs that accompany the article:
      Get the BEST programs from this zip file (updated 02 December 2015). Be sure to unzip (extract) the zip file after it is saved on you computer. For information about running the programs, see the comments at the top of the file BESTexample.R.
      The programs now include a version for estimating the parameters of a single group. The 1-group version is named BEST1G.R. See the example in BEST1Gexample.R and this blog post.
      For information about installing the software (R, JAGS, rjags, and RStudio), see the instructions at this blog post.
      A video about software installation and running an example is here.

•  Re-packaged version for R:
      There is a re-packaged version of the BEST programs with some enhanced functionality and better adherence to R-world conventions, created by Mike Meredith. Please see the simple installation instructions and examples at this blog post. (05 June 2013)

•  Programmed in Python:
      A version for Python has been created by Andrew Straw and is available at this link.

•  For Matlab:
      A version for Matlab has been created by Nils Winter and is available at this link.

Learn more:

For a complete tutorial about Bayesian methods, see the book.

You can make comments at the blog. Search the blog with the phrase "Bayesian estimation".