Re: Graded Response Models in IRT
- From: "John Uebersax" <jsuebersax@xxxxxxxxx>
- Date: 3 Jul 2006 01:39:18 -0700
Samejima's Graded Response Model
With a binary item, we summarize response probabilities in terms of an
item characteristics curve (ICC). Intuitively, the ICC is simply the
function that relates the probability of a positive item response given
a each possible respondent latent trait.
The ICC can, in theory, take an infinite variety of shapes. However,
we almost always expect it to be monotonically increasing (or, in
certain cases which we do not discuss here, decreasing). That is, the
greater the trait level of the respondent, the greater the probability
of a positive response. We further expect the curve to flatten out
after trait levels become very high or very low. This leads to the
typical "S"-shape of ICCs. These functions are also called ogives (a
term borrowed from carpentry where an ogive is an S-shaped detail on
the boundary of furniture or molding.)
With Gaussian response IRT models (also called normal-ogive models),
the ICC has the shape of a normal cdf. This derives from the
assumptions of (1) normally distributed measurement error, which (2)
has constant variance across all levels of the latent trait.
However, many computer programs model ICCs as logistic ogives. Two
justifications for this are made, each by a different camp. One is
that the logistic ogive is just a convenient approximation to the
normal ogive. This is questionable, since the normal ogive can be
better approximated by a simple polynomial. The second explanation is
that some theories (e.g., Rasch modeling) of how responses are made
imply that ICCs have a logistic, and not a normal-ogive shape.
When an item has multiple, graded response levels, it can no longer be
described by a single ICC. Rather, with K levels, one has K *response
functions*, each describing the probability of response level k (k = 1
.... K) given each latent trait level.
Under the same assumptions as the normal-ogive model mentioned above,
i.e., that measurement error is Gaussian with constant variance, we can
derive the k ICCs for an ordered-category item by means of k-1 normal
ogives. Let these be denoted p(2), ... p(K). These all have the same
shape, but different positions relative to the x-axis (latent trait).
Under Samejima's model:
The ICC for response level is 1 is: 1 - p(2)
The ICCs for the response levels k= 2 to K-1 are: p(k) - p(k+1)
The ICC for response level K is: p(K)
This model follows directly from the normal-ogive theory and is based
on exact probabilities of (latent trait + measurement error) for the
latent continuous variable that underlies a manifest variable falling
above each of k-1 graded thresholds.
The application of the Samejima model to logistic IRT models, however
is not exact, and seems to derive from the view of logistic ogives as
approximations to normal ogives. Note that in Rasch modeling, a
completely different system for handling ordered categories is
adopted--one based on the principles of Rasch test theory.
In any case, output from an IRT program that uses Samejima's model
should include, for each ordered-category item, K-1 thresholds. These
correspond to the "difficulty parameters" of functions p(2) ... p(K).
They can be understood as the latent trait levels at which p(k) (k = 2
.... K) is exactly .5.
You ask about how to interpret these. One thing is to consider the
relative spacing of the thresholds--comparing them both within and
between items. If, for example, two adjacent thresholds for the same
item are very close, one will find the corresponding response level
infrequently made.
Here are a couple of links with more information about Samejima's
graded response model:
io.psych.uiuc.edu/irt/modeling_poly1.asp
io.psych.uiuc.edu/siop2001/IRT%20basics.ppt
Hope this helps.
--
John Uebersax PhD
.
- References:
- Graded Response Models in IRT
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