# Discrimination/Slope Item Response Theory Models

In item response theory (IRT), 1-PL models, which include a slope parameter in addition to the intercept parameter, are better able to discriminate test-taker talent. Higher slopes translate to steeper sigmoid curves.

But what does a highly discriminative test mean? Why would a steeper slope yield increased learnings about test-takers?

• 1pl model has only difficulty parameter - not slope. Intercept is the one parameter. Higher slopes don't translate to steeper curves, they are steeper curves. Jun 12 at 6:18

A steeper slope means a stronger relationship between ability and the question. It means that the item (and therefore the test) is more reliable - and reliability is the inverse of measurement error. Less error = better discrimination.

You can get higher reliability / discrimination / less measurement error by having more questions or having questions with larger slope parameters.

Intuitions are useful, but it is worth understanding the math behind the model. We model the probability of answering correctly to $$i$$-th question by a person with the ability $$\theta$$, $$p_i({\theta})$$. The model can have up to three parameters $$a_i, b_i, c_i$$ and uses a logistic function.

$$p_i({\theta})=c_i + \frac{1-c_i}{1+e^{-a_i({\theta}-b_i)}}$$

The first thing to notice when trying to understand it is that if you don't use the parameter $$c_i$$ (3PL), we can ignore the logistic function. The logistic function would map real values to the probability range $$[0, 1]$$ and change the shape from linear to sigmoidal, but it won't change the ordering of the values, so high after transformation was also high before, low after was low before, etc. After the transformation, the values are easier to interpret because they can be thought of as probabilities.

What remains is

$$a_i(\theta-b_i)$$

When ability $$\theta$$ is high, the question needs to be hard $$b_i$$ to make the value low. If the question is very hard $$b_i$$, it can make the outcome very low even if the ability $$\theta$$ was high. This is moderated by the $$a_i$$ slope, if it is small, it would make smaller everything inside the brackets, if it is big, it will make it bigger. Try it with different values and observe how it behaves for better intuition.

• You didn't address this explicitly, but if $a_i$ is arbitrarily high for some question i, then a small nudge up/down in ability should correspond with a consistent change in $p(correct)$ anywhere on the ability spectrum in linear space. However, in logistic space, the relationship between $p(correct)$ and ability is inherently non-linear so a nudge is more impactful near $p=0.5$ (the inflection point of the curve) and less near the extremities. Is this correct? Jun 12 at 17:59
• @jbuddy_13 correct, this is what I meant by a logistic function making it sigmoid-shaped vs linear. But this applies mostly to the extremes (close to 0 or 1).
– Tim
Jun 12 at 18:03