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.

翻译：研究表明，基于随机抽样项目和条件独立误差的任何真分数模型，在满足若干一般性正则条件下，心理测量测验信度会随着测验长度趋于无穷大而收敛至1。其渐近收敛速率由斯皮尔曼-布朗公式给出，且该结论无需项目平行、潜变量单维性甚至有限维假设。基于双参数逻辑回归项目反应理论模型的仿真实验发现：短多维测验的信度可能存在正向偏差，这意味着在这些情况下应用斯皮尔曼-布朗公式会高估延长测验后的信度；对于双参数逻辑模型下的短单维测验，其信度近乎无偏，即此时应用该公式可得到近似无偏的预测结果。