paired t-test with ordinal data Can I do a paired samples t-test when my data are ordinal? My data are
reading levels at time 1 and time 2, but the data levels are A, 1 2,3,4,6,8,10, 12, 14, 20, 24, 28, 30, 34, 38, 40, 50, 60, 70, 80. They
appear to be continuous but they are not. Can I still use a paired t-test? These are the reading levels from k - 8th grade. I was able to run a paired t-test on the continuous data, which was a standardized test with grade level equivalency, but I am not sure if I can also do it for the reading level data.
 A: Because this is ordinal data, the assumptions that the data follow a normal distribution will be violated. Given that the assumption of normality is violated, a typical paired t-test in this situation would at best lack sensitivity, and at worst provide spurious estimates. 
Fortunately there are non-parametric versions of the t-test which do not depend on the assumption of normality, and so are quite suitable for ordinal data. 
For this data, I would suggest the signed-rank test. It is designed for paired comparisons on non-normal data. 
Here is an example in r:  
## first construct our samples to test
# pool of possible ordinal values
# not continuous, however numerical order assumed valid
pool = c(1, 2,3,4,6,8,10, 12, 14, 20, 24, 28, 30, 34, 38, 40, 50, 60, 70, 80)   

# sample 1, randomly chosen from pool values
test1 = sample(pool, 100, replace = TRUE)

# sample 2, randomly chosen from pool values
test2 = sample(pool, 100, replace = TRUE)

# sample 3, pool values, weighted towards higher values (those at end of pool)
prob_vec = 1:length(pool)/sum(1:length(pool))
test3_weighted = sample(pool, 100, replace = TRUE, prob = prob_vec)

## run the sign rank test
# test1 vs test2 should not have significant difference, they are both chosen at random
wilcox.test(test1, test2, paired = TRUE)
# V = 1849.5, p-value = 0.1985

# test1 (or test2) vs test 3 should be significant, test 3 is weighted towards 
#  higher values
wilcox.test(test1, test3_weighted, paired = TRUE)
# V = 1221, p-value = 8.495e-05

