7.7 Full information statistics
Full-information statistics compare model-predicted probabilities of all item response patterns with the observed proportions.
If a test has 5 items, there are \(2^5\) possible item response patterns.
Let \(p_y\) and \(\hat{\pi}_y\) be the observed and predicted proportions of item response pattern \(y\).
After fitting a model to the data, we have \[\hat{\pi}_y=\sum_{c=1}^{2^K}L(y|\alpha_c)p(\alpha_c),\]
where, \(\hat{\pi_y}\) is the model based proportion for response vector \(y\).
| Item.1 | Item.2 | Item.3 | Item.4 | Item.5 | observed | predicted | |
|---|---|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 | 0 | ||
| 2 | 1 | 0 | 0 | 0 | 0 | ||
| 3 | 0 | 1 | 0 | 0 | 0 | ||
| 4 | 0 | 0 | 1 | 0 | 0 | ||
| 5 | 0 | 0 | 0 | 1 | 0 | ||
| 6 | 0 | 0 | 0 | 0 | 1 | ||
| 7 | 1 | 1 | 0 | 0 | 0 | ||
| 8 | 1 | 0 | 1 | 0 | 0 | ||
| 9 | 1 | 0 | 0 | 1 | 0 | ||
| 10 | 1 | 0 | 0 | 0 | 1 | ||
| 11 | 0 | 1 | 1 | 0 | 0 | ||
| 12 | 0 | 1 | 0 | 1 | 0 | ||
| 13 | 0 | 1 | 0 | 0 | 1 | ||
| 14 | 0 | 0 | 1 | 1 | 0 | ||
| 15 | 0 | 0 | 1 | 0 | 1 | ||
| 16 | 0 | 0 | 0 | 1 | 1 | ||
| 17 | 1 | 1 | 1 | 0 | 0 | ||
| 18 | 1 | 1 | 0 | 1 | 0 | ||
| 19 | 1 | 1 | 0 | 0 | 1 | ||
| 20 | 1 | 0 | 1 | 1 | 0 | ||
| 21 | 1 | 0 | 1 | 0 | 1 | ||
| 22 | 1 | 0 | 0 | 1 | 1 | ||
| 23 | 0 | 1 | 1 | 1 | 0 | ||
| 24 | 0 | 1 | 1 | 0 | 1 | ||
| 25 | 0 | 1 | 0 | 1 | 1 | ||
| 26 | 0 | 0 | 1 | 1 | 1 | ||
| 27 | 1 | 1 | 1 | 1 | 0 | ||
| 28 | 1 | 1 | 1 | 0 | 1 | ||
| 29 | 1 | 1 | 0 | 1 | 1 | ||
| 30 | 1 | 0 | 1 | 1 | 1 | ||
| 31 | 0 | 1 | 1 | 1 | 1 | ||
| 32 | 1 | 1 | 1 | 1 | 1 |