Item response theory for psychologist
Embreston & Reise
Notes：
Se: The standard error of measurement describes expected score fluctuations due to error
Not only is the standard error of measurement basic to describing the psychometric quality of a test, it is also critical to individual score interpretations. The confidence intervals defined by the SE can guide score interpretations in several ways.
CTT: SE=s(1-rtt)1/2
IRT: rtt=1-(s2q/s2)
s2q: averaged squared standard errors; s2: trait score variance.
Item response theory is model-based measurement for trait level is estimated in the context of an IRT model.
Assumptions
1 item characteristic curves have a specific form
2 local independence has been obtained: controlling for person and item parameters, the probability of solving any item is independent of the outcome of any other item.
Unlike CTT, the trait level that corresponds to a response pattern is determined from a search process rather than from a simple computation.
A person’s response pattern is “scored” by finding the trait level that gives the maximum likelihood.
Maximum likelihood method：
1 represent the likelihoods of a response pattern under various trait levels
2 a search process that yields the trait level that gives the highest likelihood
Sufficient statistic: no additional information is required from the data for estimation
In Rasch（1PL）model，total score is a sufficient statistic and the response inconsistency will lower the likelihoods of all trait levels for the response pattern.
An important feature of IRT models is that trait level estimates with invariant meaning may be obtained from any set of items. That is, given the item parameters, trait level may be estimated for a person’s response pattern from any set of items.