When measuring attributes of entities, we strive to keep our
measurement objective. Subjective measures (direct evaluation using
for instance the scale: 1 (good) 2 3 4 5 (bad), are used when no formal procedures of measurement can be applied. Subjective measures are of course useful, as long as we understand their imprecision.
Sometimes an attribute cannot be directly measured but consists of
a number of sub-attributes that can be measured. Assume for instance that transportation is described by the pairs (cost, speed) and that we prefer low cost and high speed. To choose the best way of
transportation it must be possible to rank order the transportation
alternatives but this is not always possible ((10, 4), (12, 6)).
This is an example of a multiobjective or multiple criteria decision making (MCDM) problem.
What about scales and meaningfulness for indirect measurement
involving a model?
Example: Density d is an indirect
measure of mass m and volume:
d = m/V.
Mass and volume have obviously ratio scales and transformations
m' = am and V' = bV, a,b >0 giving
m'/V' = (am)/(bV) = (a/b)m/V = cd = d',
where c = a/b > 0, so that density is also ratio.
Generally an indirect measure will be no richer than any of its components.
Example: An indirect measure of testing efficiency T
is D/E, where D is the number of defects discovered
and E is the effort in person moths. Here D is an
absolute scale measure and E is on the ratio scale. Since
absolute is richer than ratio, it follows that T is a ratio
scale measure. The acceptable rescalings of T arise from
rescalings of E into other measures of effort (person days,
person years, etc.).