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IPS279 Rense Lange
limited additional processing. Importantly, the final estimation step in no way
depends on the number of cases already processed.
PAIRS can be illustrated via the behaviour of the simplest case, i.e., binary (e.g.,
True / False) Rasch items (without using any raters), as would apply to a
standard multiple-choice type test. Here,

log(Pij / (1 - Pij)) = Tj – Di, (2)

where Pij is the probability that person j will correctly answer item i. Solving for
Pij in Equation 2 yields:
Pij = exp(Tj – Di) / (1 + exp(Tj – Di)). (3)

Let avi denote the outcome of person v taking some item i, i.e., 0 when
incorrect and 1 when correct. Consider avi = 1 given that the outcomes for two
items i and j sum to unity, i.e., avi + avj = 1:

( = 1)( = 0)

( = 1| + = 1) =

( = 1 & = 0) + ( = 0 & = 1)
= (4)

+

If we denote the number of test-takers who answer i correctly and j incorrectly
by bij, then the probability (4) can also be estimated empirically via:

= ,

+ +
Dividing by bji yields:

/ = − , and hence = − .
/ +1 − +1

Provided that both bij and bji are non-zero, taking logarithms of both sides
yields:
log ( ) = − . (5)

In other words, much of the information needed to compute the Di can
be kept updated by maintaining the appropriate bij counts in a matrix. As will
be shown below, obtaining the actual D values involves division and taking
logarithms as in Equation 5 to obtain the matrix log(R). Then, averaging the
rows of log(R) yields least-square estimates of the Di [8]. Note that no
assumptions are made concerning the distributions of the model parameters,
which supports the approach’s robustness.

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