Patient_Reported_Outcome_Measures_in_Rheumatic_Diseases

47

binary item (i.e., right/wrong, true/false, agree/disagree) is determined by the indi-
vidual’s trait level and the difficulty of the item. One way of expressing the Rasch
model is in terms of the probability that an individual with a particular trait level will
correctly answer an item that has a particular difficulty. This is often presented as [ 56 ]:

PX

e

e

is si

si

()= = + si

()-

1 |,qb 1 ()-

qb

qb

where:
Xis refers to response (X) made by subject s to item i.
θ(theta)s refers to the trait level of subject s.
β(beta)i refers to the difficulty of item i.
Xis = 1 refers to a “correct” response or an endorsement of the item.
e is the base of the natural logarithm (i.e., e = 2.7182818 ...), found on many
calculators.
So, P(Xis = 1|θ[theta]s, β[beta]i) refers to the probability (P) that subject s will
respond to item i correctly or in a particular way. The vertical bar in this statement
indicates that this is a “conditional” probability. The probability that the subject will
correctly respond to the item depends on (i.e., is conditional upon) the subject’s trait
level (θ[theta]s) and the item’s difficulty (β[beta]i). In an IRT analysis, trait levels
and item difficulties are usually scaled on a standardized metric, so that their means
are 0 and the standard deviations are 1.
A slightly more complex IRT model is called the two-parameter logistic model
(2PL) because it includes 2 item parameters. The difference between the 2PL and

Table 2.1 Commonly used item response theory (IRT) models

IRT model

Item response

format Model characteristics

Rash/one parameter

logistic model

Dichotomous Discrimination power equal across all

items. Threshold varies across items

Two parameters

logistic model

Dichotomous Discrimination and threshold

parameters vary across items

Graded response model Polytomous Ordered responses. Discrimination

varies across items

Nominal model Polytomous No pre-specified item response order.

Discrimination varies across items

Partial credit model Polytomous Discrimination power constrained to

be equal across items

Rating scale model Polytomous Discrimination equal across items.

Distance between item threshold

steps equal across items

Generalized partial

credit model

Polytomous Generalization of the partial credit

model that allows discrimination to

vary across items

2 A Guide to PROMs Methodology and Selection Criteria