Page 53 - Special Topic Session (STS) - Volume 4

STS560 James Houran et al.
• The left-hand side represents the ratio of Pijk over Pik(k-1), i.e., the log
odds of observing a rating in category k relative to probability of
observing one in category k-1.
• Tj represents the trait level (or ability) of person j, and most applications
focus on the Tj only.
• Di is the “difficulty” of the item, or the magnitude of the trait level
needed to elicit the response k. Higher values of D indicate that higher
trait levels are needed to obtain a higher rating.
• In general, higher D yields lower ratings and higher T will yield higher
ratings.
• As the trait level Tj increases we expect to see the categories 0, 1, 2, ….,
m to be used in this order – at least probabilistically. The Fik denote the

locations at which the ratings k and k-1 are equally likely, and such Fik
are typically referred to as “step-values.”
• Although the following assumes that items share the same step-values,
the subscript i indicates that the step-values are actually allowed to
vary across items. We use the convention that ∑k Fik = 0 and it is
mathematically convenient to define Fi0 = 0.
• Note that all parameters vary along the same latent trait variable and
they are expressed in the same unit of measurement, i.e., the logit as
defined by the left-hand side of Equation 1.

3. Example
The various definitions are illustrated in Figure 2 which shows the Pijk of
each rating k for a rating scale with four categories (“Disagree Completely,”
“Disagree Somewhat,” “Agree Somewhat,” and “Agree Completely”) across the
underlying latent Rasch dimension. In the example, Di = 0 and Fk = {0, -1.37,
0.14, 1.23}, and these step-values are shown relative to Di. It can be seen that
for very low trait levels the rating “Disagree Completely” is virtually certain, but
that “Disagree Somewhat” becomes equally likely at -1.37 logits and the latter
is more likely above this point. Similarly, “Disagree Somewhat” and “Agree
Somewhat” are equally likely at 0.14 logits, and “Agree Somewhat” and “Agree
Completely” are equally likely at 1.23 logits. Note that the Pijk sum to unity
across the various categories. For instance, at -0.9 logits (solid vertical line) the
probabilities of the 4 categories (in order) are about 0.31, 0.45, 0.21, and 0.03,
respectively.

4. Background
Obtaining images such as Figure 2 requires solving for Pijk in Eq. 1:

exp ∑ =0 [ − − ]
= ∑ exp ∑ =0 [ − − ] = 0, 1, … , (2)
=0

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