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- R is the lingua franca of statistical programming. Despite a steep learning curve, I find R to be a more natural language than statistical software like Stata or SAS. Plus R is free, open-source and has a broad community of users. Pretty much any statistical method you need has a dedicated R package; and if it doesn't, you can always create one!
- Julia is a relative newcomer, introduced in 2012. It's a high-level and dynamic language that an R or Matlab user will find approachable. In some benchmark tests, Julia outperforms R in terms of speed, especially when extensive looping is required. Julia does not yet have the broad network of libraries familiar to R or Matlab users, but it's gaining functionality quickly. For programs requiring R functionalities, users can always call R using Rif.jl.
- Stan is a programming language for Bayesian inference. Written in C++, Stan runs much faster than most user-written samplers in R, Julia, Python, et cetera. The language looks quite similar to BUGS or JAGS, though Stan uses Hamiltonian Monte Carlo sampling methods using the No U-Turn Sampler (NUTS). I'm a big fan of Stan for most problems, though there are still some wrinkles that the developers are ironing out (like pretty slow sampling for multinomial logistic regression models). You can call Stan from R, Julia, Python, and Matlab, and shinyStan adds nice visualization functionality.
- Stata is a software program for data analysis. For beginners at statistics, Stata makes analysis quite easy. It's a spreadsheet-based program with point-and-click capability and pretty simple programming language. For students wishing to enter industry (with the exception of technology companies, perhaps), knowledge of Stata often makes one more marketable. I use Stata frequently for quick data cleaning, especially for large survey data. For actual programming of advanced problems, however, I recommend R or Julia. Also: Stata is for profit, i.e., not free.