## Friday, March 11, 2016

### From reliability (CTT) to test information (IRT) and back

Test information in item response theory (IRT) is defined as:
$I(t_j)=\frac{1}{\textrm{Var}(\hat{t}_j|T=t_j)}$

The denominator is basically the squared value of the standard error of measurement (SEM). The SEM is defined in classical test theory (CTT) as:

In IRT, we assume T to have a variance of 1 in the population, so let's say Sxx = 1. (Although the variance of theta is not necessarily the same as the variance of the observed test scores. I just do not know what the observed version of theta would be in IRT.). Then:

It follows that:

and