This is an interactive demonstration to help you better understand the 3 parameter Item Response Function:
Pij() = P(ui=1 | , ai, bi, ci) = ci + (1 - ci) / [1 + exp(-1.7 ai ( - bi)]
which states that the probability of a correct response to item i for examinee j, Pij() or P(ui=1) is a function of three item parameters, ai, bi, ci, and one examinee ability parameter . You can create imaginary test items by specifying the three item
parameters ai, bi, ci. The computer will then generate two curves. The first curve will be the Item Response Function for your item as defined by equation (1). The
computer will calculate the probability of a correct response to your item for values of ranging from -3.0 to +3.0 and then plot against the probability of a correct response given .
The second curve will be the Item Information Function,
I()= P()' d / [P() (1 - P()],
which shows the information provided by your item at each value of and is based on the Item Response Function. The computer will calculate values of I() for ranging from -3.0 to +3.0 and will then plot I() against .
What to observe
Suggested activities: (You may want to print this page rather than scrolling up and down.)
- The horizontal axis is the ability scale, ranging from very low (-3.0) to very high (+3.0). The vertical axis the is the probability of responding correctly to this item (defined by the three item parameters) given = .
- The lower asymptote is at ci. This is the probability of a correct response for examinees with very little ability (e.g. = -2.0 or -2.6). The curve has an upper asymptote at 1.0; high ability examinees are very likely to respond correctly.
- The bi parameter defines the location of the curve's inflection point along the theta scale. Lower values of bi will shift the curve to the left; higher to the right. The bi does not effect the shape of the curve.
- The ai parameter defines the slope of the curve at its inflection point. The curve would be flatter with a lower value of ai; steeper with a higher value. Note that when the curve is steep, there is a large difference between the probabilities of a correct response for a) examinees whose ability is slightly below (left) of the inflection point and b) examinees whose ability is slightly above the inflection point. Thus ai denotes how well the item is able to discriminate between examinees of slightly different ability (within a narrow effective range).
- Try different values for the ai parameter
- Changing values of ai changes the shape of the item response function and does not change it's location. Thus, the ai parameter and bi parameter are independent of each other.
- The higher the value of ai, the more information the item provides within its effective range.
- When the item discrimination is zero, the item response function is flat. Everyone has the same probability of a correct response and the item yields little information. Items with a low ai are usually not very good items and are eliminated from the item pool.
- When the item discrimination is low (ai<.75) the item response function is nearly linear and only increases slightly.
- When the item discrimination is high (ai>1.0) the item response function is S-shaped, steep in the middle, and the item yields a great deal of information within a range of ability. The item also will yield very little information outside of that useful range.
- The value of the item discrimination parameter must be quite large (ai>1.75) before the item response function is very steep.
- The value of the ai parameter can be negative. However, this results in a monotonically decreasing item response function. People with high ability have a lower probability of responding correctly than people of low ability. Such bad items are usually quickly weeded out of an item pool.
- Try different values for the bi parameter
- Changing values of bi changes the location of the item response function and does not change it's shape.
- Changing values of bi does not change the shape of the information function. It does change the location of the effective range for the information function.
- When the item difficulty is less than 0 (the mid point), the probability of a correct response is greater than .5 for most examinees. These are relatively easy items.
- When the item difficulty is greater than 0, the probability of a correct response is less than .5 for most examinees. These are relatively hard items.
- The probability of a correct response when ability (theta)= bi is 0.5.
- When bi is very low (-2.5) only the upper part of the curve appears on the chart. If the chart were extended to include a wider range of ability, then the entire curve would appear.
- Try different values for the ci parameter
- Changing values of ci changes the lower asymptote of the item response function, does not change it's location, and does not change its shape for the upper half of the item response function.
- When the guessing parameter is 0 (as in a competition item), examinees with very little ability a low probability of a correct response. They will have almost zero probability if the item has moderate discrimination and is located alway from the bottom tail of the ability distribution.
- When the guessing parameter is high, the probability of a correct response is less than .5 for most examinees.
- Increasing ci reduces the amount of Information at all levels of theta. The point of maximum information moves to the right as ci increases.
From: Rudner, Lawrence M. (1998). An On-line, Interactive, Computer Adaptive Testing Mini-Tutorial, http://edresearch.org/scripts/cat