Careful measurement is aimed at ascertaining the extent or quantity of one particular attribute of something. In practice, this ideal of unidimensional measurement is never achieved. It proves impossible to isolate entirely one attribute from another, e.g., length of steel rods and temperature, or time and very high velocities. In the social sciences, measurement of unintentional combinations of attributes is frequently encountered. This can mislead the unwary into supposing that two different things are the same.
Figure 1 illustrates a situation in which Measure 1 is intended to quantify one attribute and Measure 2 another. Henry& Alred (1996) encountered similar situations during their investigation of Depression and Anxiety. Measure 1 was intended to measure anxiety, but included some depression-sensitive items. Measure 2 was intended to measure depression, but included some anxiety-influenced items. Thus Measure 1 covaried with Measure 2 because both quantify combinations of the two attributes. Usually combinations of two different attributes are noticed because item misfit statistics can classify the items according to their predominant attribute. Also, calibrating the items with samples with different proportions of the two attributes causes obvious shifts in the calibrations.
Suppose, however, that the two attributes have been clearly defined and each measure quantifies only its intended attribute. What if unidimensionality has been achieved, and yet the now clearly unidimensional measures still covary? Can this happen? Unidimensionality does not mean "one dimension totally unlike any other". Rather, unidimensionality means one dimension comprising more or less of the same one quality, however it is defined. This quality may be closely related to other qualities. Height and weight are usually considered distinct qualities, and measured as such, but tall adults are generally heavier than short adults, so that measures of height and weight will covary. Thus measures may covary because the underlying qualities being measured overlap.
Look again at depression and anxiety. These are commonly observed to occur together and may well be manifestations of some more basic psychological state. Their relationship may be that depicted in Figure 2. Measure A focuses sharply on Anxiety, and Measure D focuses sharply on Depression. Both measures are unidimensional, but because the circumstances causing manifestation of the two attributes overlap, the measures covary.
When measures are discovered to covary across attributes, the test developer faces the challenge of constructing items that probe the desired attribute at a more demanding level than its companion attribute. For arithmetic word problems, the arithmetic task must be markedly more demanding than the reading comprehension task. But this requires care. Give arithmetic word problems written in English to a French speaker, and the test no longer measures arithmetic, but rather ESL proficiency.
Based on Henry D. & Alred K. (1996) Measuring Depression in Children with Achenbach's Checklist. MOMS, December 1996.
Diagnosing Measure Covariance. Henry D., Alred K. Rasch Measurement Transactions, 1997, 11:1 p. 556
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