Transformation of Rasch model logits for enhanced interpretability.

backgroundThe Rasch model allows for linear measurement based on ordinal item responses from rating scales commonly used to assess health outcomes. Such linear measures may be inconvenient since they are expressed as log-odds units (logits) that differ from scores that users may be familiar with. It can therefore be desirable to transform logits into more user-friendly ranges that preserve their linear properties. In addition to user-defined ranges, three general transformations have been described in the literature: the least measurable difference (LMD), the standard error of measurement (SEM) and the least significant difference (LSD). The LMD represents the smallest possible meaningful unit, SEM relates the transformed scale values to measurement uncertainty (one unit on the transformed scale represents roughly one standard error), and LSD represents a lower bound for how coarse the transformed scale can be without loss of valid information. However, while logit transformations are relatively common in the health sciences, use of LMD, SEM and LSD transformations appear to be uncommon despite their potential role.

methodsLogit transformations were empirically illustrated based on 1053 responses to the Epworth Sleepiness Scale. Logit measures were transformed according to the LMD, SEM and LSD, and into 0-10, 0-100, and the original raw score (0-24) ranges. These transformations were conducted using a freely available Excel tool, developed by the authors, that transforms logits into user-defined ranges along with the LMD, SEM and LSD transformations.

resultsResulting LMD, SEM and LSD transformations ranged 0-34, 0-17 and 0-12, respectively. When considering these relative to the three user-defined ranges, it is seen that the 0-10 range is narrower than the LSD range (i.e., loss of valid information), and a 0-100 range gives the impression of better precision than there is, since it is considerably wider than the LMD range. However, the 0-24 transformation appears reasonable since it is wider than the LSD, but narrower than the LMD ranges.

conclusionsIt is suggested that LMD, SEM and LSD transformations are valuable for benchmarking in deciding appropriate ranges when transforming logit measures. This process can be aided by the Excel tool presented and illustrated in this paper.