Using the Guttman Scale to Define and Estimate Measurement Error in Items over Time: The Case of Cognitive Decline and the Meaning of “Points Lost”

We used a Guttman model to represent responses to test items over time as an approximation of what is often referred to as "points lost" in studies of cognitive decline or interventions. To capture this meaning of "point loss", over four successive assessments, we assumed that once an item is incorrect, it cannot be correct at a later visit. If the loss of a point represents actual decline, then failure of an item to fit the Guttman model over time can be considered measurement error. This representation and definition of measurement error also permits testing the hypotheses that measurement error is constant for items in a test, and that error is independent of "true score", which are two key consequences of the definition of "measurement error"--and thereby, reliability--under Classical Test Theory. We tested the hypotheses by fitting our model to, and comparing our results from, four consecutive annual evaluations in three groups of elderly persons: a) cognitively normal (NC, N = 149); b) diagnosed with possible or probable AD (N = 78); and c) cognitively normal initially and a later diagnosis of AD (converters, N = 133). Of 16 items that converged, error-free measurement of "cognitive loss" was observed for 10 items in NC, eight in converters, and two in AD. We found that measurement error, as we defined it, was inconsistent over time and across cognitive functioning levels, violating the theory underlying reliability and other psychometric characteristics, and key regression assumptions.