7.7 Full information statistics

Full-information statistics compared 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 $${\hat{\pi }}_y=\sum _{c=1}^{2^K}L\left(y|{\alpha }_c\right)p\left({\alpha }_c\right)$$

TABLE 7.1: An illustration

	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