How to interpret Location parameters in Generalised Partial Credit Model?

I am struggling to interpret the location parameters in generalised partial credit models. Say you have location parameters $$a_1$$,$$a_2$$ and $$a_3$$. My professor said that for the item to be accepted then they must be monotonic i.e. $$a_1$$<$$a_2$$<$$a_3$$ which I think has to do with the monotonicity assumption of such models however I am not sure.

Your understanding is on the right track. In the context of the Generalized Partial Credit Model (GPCM), a type of Item Response Theory (IRT) model, the location parameters $$\alpha_1, \alpha_2, \alpha_3$$ are important for interpreting item characteristics.

Here's a brief explanation:

Location Parameters: In the GPCM, these parameters represent the difficulty of achieving a certain score level on an item. For instance, $$\alpha_1$$ might represent the difficulty of moving from a score of 0 to 1, $$\alpha_2$$from 1 to 2, and so on.

Monotonicity: The requirement that $$\alpha_1 < \alpha_2 < \alpha_3$$ is indeed related to the monotonicity assumption. This assumption states that as a person's ability increases, their probability of achieving a higher score on the item also increases. In practical terms, it wouldn't make sense for an item to be easier to get a score of 2 than a score of 1, which is why these parameters need to be ordered (monotonic).

In summary, your professor's emphasis on the monotonic nature of these parameters is well-founded as it relates to the logical progression of difficulty within an item and the fundamental assumptions of the GPCM. For more information on the assumptions of IRT models, I highly recommend Thissen & Steinberg (1986).

References

Thissen, D., & Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51(4), 567-577.