Compare covariance ratio I am using a nonparametric item response theory method called Mokken scale analysis. One of the coefficients that is calculated is called the item scalability coefficient, and it is calculated as the normed covariance between the score a person gets on an item and their total score (minus that from the item of interest), like this:
Hi = cov(Xi, R(i))/covmax((Xi, R(i))
Where:
Hi is the scalability of item i, 
Xi is the score on item i, and 
R(i) is the "restscore" - calculated as the total score across the set of items minus the score from item i
There is an R package that calculates the standard errors for scalability coefficients (called "mokken").
I want to know if I can somehow use the standard errors to do a significance test to compare two scalability coefficients to see if they are significantly different (something like an F-test or a t-test)
 A: I don't know of any formal test for this, but you could form a bootstrap test to see how often the two groups provide the observed behaviour under re-sampling. That would give a non-parametric distribution of the difference between the scalability coefficients as well as a $p$ value for the null hypothesis that they are equal.
For instance, consider the following R code which constructs a bootstrap test given some arbitrary group variable. At the end, an empirical $p$-value is formed from the bootstrapped standard error for the test statistic, though you could form the same conclusion from the bootstrapped confidence interval:
library(mokken)
data(acl)
Communality <- acl[,1:10]
set.seed(1)
group <- sample(c('Group1', 'Group2'), nrow(Communality), TRUE)

scale_diff <- function(dat, index, group){
    dat <- dat[index, ]
    group2 <- group[index]
    H1 <- coefH(dat[group2 == 'Group1', ], se = FALSE)$H
    H2 <- coefH(dat[group2 == 'Group2', ], se = FALSE)$H
    H1 - H2    
}

library(boot)
bt <- boot(Communality, scale_diff, R=1000, group=group)
print(bt)

## ORDINARY NONPARAMETRIC BOOTSTRAP
## 
## 
## Call:
## boot(data = Communality, statistic = scale_diff, R = 1000, group = group)
## 
## 
## Bootstrap Statistics :
##        original       bias    std. error
## t1* -0.03212578 -0.001394881   0.0391486

z <- abs(bt$t0) / sd(bt$t)
p <- pnorm(abs(z)) 
p

## [1] 0.7940661

