Measurement Scales and Psychological Statistics: Empirical Science or Metaphysics?
Townsend & Ashby (Psychol. Bull. 96: 394-401) note that measurement involves assigning
numbers to objects in a way that “interesting qualitative empirical relations among the objects are
reflected in the numbers themselves.” For example (mine) I may put each object on the left side of
a scale and count the number (X) of one grain weights I must put on the right to balance the scale.
When each object has been so “measured” I can predict things such as which way the scale will
tip (or if it will balance) when I put one or more objects on the left and another on the right.
T & A go on to note that two scientists may “by chance” devise two different assignmentsystems (scales), each correctly reflecting the “empirical relational structure” of interest. For
example (mine), you may use a balance with 1.5 grain weights. Both of our scales are interval
(and ratio), that is, a linear function of the “true” magnitudes, and thus we may relate them to one
another with a linear function, Your weight = 0 + (2/3)my weight.
We might also appropriately use “parametric” statistics (given normal data, etc.) with either
scale. If we both wanted to compare one group of heavy objects with another group of light
objects (to make inferences about the populations from which they were drawn) we might both
transform (linear) our data into standard error units by doing a t-test. My interval data are: 175,
260, 340, 425  475, 560, 640, 725. I find M1 = 300, M2 = 600, t 
600  300
 3.96 .
11483  11483
4
Your interval data are 116 2/3, 173 1/3, 226 2/3, 283 1/3  316 2/3, 373 1/3, 426 2/3, 483 1/3.
You find M1 = 200, M2 = 400 (my means  2/3) and within groups variances of (2/3)2 times mine,
so t 
400  200
 3.96 . We get exactly the same result, and would with any other interval
51035
.  51035
.
4
measure of mass/weight (assume gravity constant).
God has told me that the true scores are 17.5, 26, 34, 42.5  47.5, 56, 64, 72.5. Thus, my
scale was a linear transformation of the Truth, X = 0 + 10T, as was yours, Y = 0 + 6.667T. To
see this more clearly, you might try plotting either of the above sets of interval data against the
true scores. That is, we both had interval data, and we get the same t that we would get using the
God-given true scores (check this out if you doubt it).
Now suppose we used only ordinal scales, scales where the relationship between the true
magnitudes and our measurements is monotonic, but not necessarily linear. Lets say my scale
can be specified by the function X = 100(log T). My ordinal data then are: 124, 141, 153, 163 
168, 175, 181, 186. My t 
177.5  145.25
 3.49 , not the value computed with the true scores or
2816
.  60.3
4
with linear transformations of the true scores. Note that the transformation has also introduced
considerable heterogeneity of variance.
Suppose your ordinal data were 1, 2, 3, 4; 10, 11, 12, 13. Note that this is a monotonic
transformation of the true scores, the true order is preserved. Your t 

Copyright 2000, Karl L. Wuensch - All rights reserved.
115
.  2.5
 9.86 .
1667
.
 1667
.
4
scales2.doc
Clearly, the t obtained, and the conclusion reached, depend on the nature of the transformation. If
the transformation is linear, the results will be the same as those that would be obtained if we had
the true scores. Thus, if we have interval data, we can claim the results of our t-tests faithfully
reflect differences in true scores.
There is one big hitch in all this. How do we know whether our scale is indeed a linear
transformation of the truth? If we ask God to tell us, why not also ask her what the whole truth is
and dispense with all this silly measurement and inferential statistics? Lord (Amer. Psychol. 8:
750-751) told us that “The numbers do not know where they came from.”
I tell you that we don’t know that either, and without such knowledge we never know whether
the data are interval or not. We may feel very confident that our numbers represent a linear
transformation of some true magnitudes when working with something as seemingly concrete as
mass, but “psychophysical” scientists tell us the relationship between the physical and the
psychological is not linear - recall Fechner’s logarithmic law and Stevens’ power law. What is the
“reality” we wish to measure- “physical” or “psychological”? Are our measurements a linear
transformation of “true scores” in one of these realities? How could we ever know? And what
about more “abstract” constructs such as intelligence? If IQ test A scores are not a linear
transformation of IQ test B scores (their Pearson r is not 1.0), is that because of “random error” or
because one is a linear transformation of the truth (interval scale) and the other (or both) a
nonlinear, monotonic transformation (ordinal)? If the latter, which scale is interval, which ordinal?
Why not just say that the two IQ tests measure somewhat different things? The things they
measure are mere abstractions anyway, so just say that the true scores they reflect are those that
are a linear transformation of our measurements. That is, not only is intelligence what an IQ test
measures, but it is a linear function of what an IQ test measures, and different IQ tests likely
measure different intelligences. Let us be logical, and let us be positivist, and do our statistics
without worrying about scale of measurement. All this unresolvable dispute about the nature of
these not empirically knowable realities belongs not in statistics, but is best left to ontology,
transcendental psychometrics, and other metaphysical domains.
While I do not consider the distinction between ordinal and interval data often to be important
when choosing inferential statistics, I do consider the distinction between categorical (discrete,
nominal) and measurement or quantitative (continuous, ordinal and above) to be important. For
example, I think it would be foolish to use r to correlate political conservatism (measured on a
continuous scale) with party affiliation (Democrat, Independent, Republican, coded 1, 2, 3),
although we shall later learn how to use dummy variable coding to incorporate categorical
variables into our general linear model.
Copyright 2000, Karl L. Wuensch - All rights reserved.