A note on stochastic ordering of the latent trait using the sum of polytomous item scores

In contrast to dichotomous item response theory (IRT) models, most well-known polytomous IRT models do not imply stochastic ordering of the latent trait by the total test score (SOL). This has been thought to make the ordering of respondents on the latent trait using the total test score questionable and throws doubt on the justifiability of using nonparametric polytomous IRT models for ordinal measurement. We show that a broad class of polytomous IRT models has a weaker form of SOL, denoted weak SOL, and argue that weak SOL justifies ordering respondents on the latent trait using the total test score and, therefore, the use of nonparametric polytomous IRT models for ordinal measurement.