What I want is some tool that will test an ordinal, discrete variable against a specific probability distribution, say the variable might take 4 values, 1, 2, 3, 4, and my expected probability distribution is that 10% will be 1, 50% will be 2, 30% will be 3 and 10% will be 4.
What program, or application, need I use to check if an empirical dataset comes from the expected distribution, or if it deviates significantly from it? How do I do it?
You can start with Pearson's chi-squared test. It is implemented in R, as a function chisq.test. Here is the example with fictitious data:
set.seed(1)
#Generate some discrete variable
y<-rpois(30,1)
#Tabulate the values
table(y)
y
0 1 2 3 4
10 12 5 2 1
##Calculate the theoretical probabilities of the values
p<-dpois(0:3,1)
p<-c(p,1-sum(p))
> p
[1] 0.36787944 0.36787944 0.18393972 0.06131324 0.01898816
##Do actual test. You need to supply the table and the corresponding probabilities
chisq.test(table(y),p=p)
Chi-squared test for given probabilities
data: table(y)
X-squared = 0.5693, df = 4, p-value = 0.9664
Message d'avis :
In chisq.test(table(y), p = p) :
l'approximation du Chi-2 est peut-être incorrecte
See the p-value. If it is bigger than 0.05, your data conforms to the expected probability distribution.
This is just an example, but it will get you started.
Not a formal test, but to help you get a feel for how well your data matches a distribution you may want to look at a hanging rootogram, there are implementations of this in the vcd package and latticeextra pakages for R.
Also note that any formal test (the chi-squared is appropriate here as @mpiktas posted) can only rule out distributions, it can never prove that your data came from a given distribution, just show that there is not enough data to rule it out.
