  It is shown that the psychometric test reliability, based on any true-score
model with randomly sampled items and conditionally independent errors,
converges to 1 as the test length goes to infinity, assuming some fairly
general regularity conditions. The asymptotic rate of convergence is given by
the Spearman-Brown formula, and for this it is not needed that the items are
parallel, or latent unidimensional, or even finite dimensional. Simulations
with the 2-parameter logistic item response theory model reveal that there can
be a positive bias in the reliability of short multidimensional tests, meaning
that applying the Spearman-Brown formula in these cases would lead to
overprediction of the reliability that will result from lengthening the tests.
For short unidimensional tests under the 2-parameter logistic model the
reliabilities are almost unbiased, meaning that application of the
Spearman-Brown formula in these cases leads to predictions that are
approximately unbiased.
