Item Response Theory models individual responses to items by estimating individual ability (theta) and item difficulty (diff) parameters.
This is an early and crude attempt to capture this modeling procedure. A better procedure is to use `irt.fa`

.

```
irt.person.rasch(diff, items)
irt.0p(items)
irt.1p(delta,items)
irt.2p(delta,beta,items)
```

a data.frame with estimated ability (theta) and quality of fit. (for irt.person.rasch)

a data.frame with the raw means, theta0, and the number of items completed

- diff
A vector of item difficulties --probably taken from irt.item.diff.rasch

- items
A matrix of 0,1 items nrows = number of subjects, ncols = number of items

- delta
delta is the same as diff and is the item difficulty parameter

- beta
beta is the item discrimination parameter found in

`irt.discrim`

William Revelle

A very preliminary IRT estimation procedure.
Given scores xij for ith individual on jth item

Classical Test Theory ignores item difficulty and defines ability as expected score : abilityi = theta(i) = x(i.)
A zero parameter model rescales these mean scores from 0 to 1 to a quasi logistic scale ranging from - 4 to 4
This is merely a non-linear transform of the raw data to reflect a logistic mapping.

Basic 1 parameter (Rasch) model considers item difficulties (delta j): p(correct on item j for the ith subject |theta i, deltaj) = 1/(1+exp(deltaj - thetai)) If we have estimates of item difficulty (delta), then we can find theta i by optimization

Two parameter model adds item sensitivity (beta j): p(correct on item j for subject i |thetai, deltaj, betaj) = 1/(1+exp(betaj *(deltaj- theta i))) Estimate delta, beta, and theta to maximize fit of model to data.

The procedure used here is to first find the item difficulties assuming theta = 0 Then find theta given those deltas Then find beta given delta and theta.

This is not an "official" way to do IRT, but is useful for basic item development. See `irt.fa`

and `score.irt`

for far better options.

`sim.irt`

, `sim.rasch`

, `logistic`

, `irt.fa`

, `tetrachoric`

, `irt.item.diff.rasch`