# Item information in IRT

According to item information curves, item information for a 2PL IRT model is

$I(\theta)=a^2_i p_i(\theta) q_i(\theta)$

1. To determine $p_i(\theta)$ and $q_i(\theta)$, do you just use the observed response pattern for the item, e.g., 1110 implies $p_i(\theta)=0.75$? Do you use the responses of all individuals or just one?

2. Why doesn't item difficulty affect item information?

3. (Perhaps this is the same as 2.) Using this definition of item information, how can you plot item information as a function of individual proficiency $\theta_p$?

-

## 1 Answer

1. Having fitted the IRT, you use the predicted probabilities from it. The pattern for a given item is not informative unless you know the abilities of the students who responded to it. This may have been a difficult question, but the first three students were bright enough to answer it, while the fourth one who failed it may have been a median student, but this could have been above his or her head.

2. Item difficulty appears in the expressions for $p_i(\theta)$. In fact, the product $p_i(\theta)q_i(\theta)$ is maximized when $\theta=$ difficulty, as then both $p_i(\theta)=q_i(\theta)=1/2$.

3. Uhm... you just plot $I(\theta)$ as a function of $\theta$?

-

I must be missing something obvious. Could you give the formula for $p_i(\theta)$? – Jack Tanner Nov 12 '12 at 18:24
Ah, figured it out. $p_i(\theta) = \frac{1}{1 + exp(a_i(b_i - \theta_p))}$. – Jack Tanner Nov 12 '12 at 20:05
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 13, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9480159282684326, "perplexity_flag": "middle"}