When the purpose is to select a limited group of examinees, such as only the top 10%, the widely used Classical Test Theory (CTT) and Item Response Theory (IRT) item discrimination indices may be biased because they estimate how well the entire score range is discriminated.

In this case, a special case of Brennan’s index is proposed by Arikan & Aybek (2022) to identify items for selecting a limited number of high-achieving examinees. Inspired by Brennan’s formula, this index calculates the difference between the performance of the upper 10% and lower 90% of the group on the item. It's important to note that the proportion defining the upper group is adjustable based on user requirements and is not fixed at 10%.

The formula is expressed as:

B10−90 = pU10 − pL90

where pU10 represents the proportion of students in the upper 10 percentile who answered the item correctly and pL90 denotes the proportion of students in the lower 90 percentile who answered the item correctly.

This index is designed for criterion-referenced tests or absolute assessments where a specific cut-off score is used to make decisions.

The index is calculated using the following formula:

BI = p(A) - p(B)

where p(A) is the proportion of examinees above the cut-off score who answered the item correctly, and p(B) is the proportion of examinees below the cut-off score who answered the item correctly (Brennan, 1972).

The discrimination index is a measure of how well a question discriminates between high and low scorers. It's calculated in the following steps:

1. Rank all students based on their total test score.
2. Identify the upper 27% and the lower 27% of students.
3. For each item, calculate the proportion of students in the upper group who answered correctly (U) and the proportion in the lower group who answered correctly (L).
4. The discrimination index is D = U - L.

The item-total correlation is the correlation between a scored item and the total test score. Higher positive values indicate that the item is more consistent with the overall test.

The item-rest correlation is the correlation between a specific item and the total score of all other items in the scale, excluding the specific item. It helps to determine how well an item fits with the other items in the test.