`vignettes/blog/2020-05-27-the-information-game.Rmd`

`2020-05-27-the-information-game.Rmd`

A typical task in adaptive testing is to select, out of a precalibrated item pool, the most appropriate item to ask, given an interim estimate of ability, \(\theta_0\). A popular approach is to select the item having the largest value of the item information function (IIF) at \(\theta_0\).

When the IRT model is the Rasch model, the item response function (IRF) is \[P(X=1|\theta,b)=\frac{1}{1+\exp(b-\theta)}\] where \(b\) is the item difficulty, and the IIF can be computed as \(P(X=1|\theta,b)[1-P(X=1|\theta,b)]\). For all items, this function reaches a maximum of 0.25, encountered exactly where \(\theta=b\) and \(P(X=1|\theta,b)=0.5\). So picking the item with the maximum IIF is the same as picking the item for which a person of ability of \(\theta_0\) has a probability of 0.5 to produce the correct answer.

If we use the two-parameter logistic (2PL) model instead, the IRF is \[P(X=1|\theta,a,b)=\frac{1}{1+\exp[a(b-\theta)]}\] where the new parameter \(a\) is called the discrimination parameter, and the IIF can be computed as \(a^2P(X=1|\theta,a,b)[1-P(X=1|\theta,a,b)]\). Note that while the contribution of the product, \(P(1-P)\), remains bounded between 0 and 0.25, the influence of \(a^2\) can become quite large – for example, if \(a=5\), \(P(1-P)\) gets multiplied by 25, ten times the maximum value under the Rasch model. Selecting such an informative item gives us the happy feeling that our error of measurement will decrease a lot, but what happens to the person’s probability to give the correct response?

In our game, we have one person and two items. The person’s ability is 0, represented with a vertical gray line. One of the item is fixed to be a Rasch item with \(b=0\); its IRF is shown as a solid black curve, and its IIF as a dotted black curve. The second item, shown in red, is 2PL, and you can control its two parameters with the sliders. Initially, \(a=1\) and \(b=1\).

The purpose of the game is to find particularly awkward situations where the more informative item is the most inappropriate in the sense of:

having a difficulty as remote from \(\theta=0\) as possible;

having a probability of a correct response at \(\theta=0\) as low as possible.

Can you find the sweet spots? I have just computed the answer to the second question but I am not telling (in fact, I can tell you the value of \(P\): 0.0022, quite a bit off 0.5).