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 $y$.

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