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# Absolute zero measurements

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I need your help finding a research issue, problem, or opportunity that uses data that has absolute zero measurements, such as interval and ratio level data.

The class is on Research and Evaluation

Can you also give me the definitions for absolute zero measurements and interval and ratio level data??

I also need a description of the research issue.

Can your response be in paragraphs and can you give me the sources.

#### Solution Preview

Version 3, Sep 14, 1997
Warren S. Sarle
SAS Institute Inc.
SAS Campus Drive
Cary, NC 27513, USA

URL: ftp://ftp.sas.com/pub/neural/measurement.html

Version 1 originally published in the Disseminations of the International Statistical Applications Institute, volume 1, edition 4, 1995, Wichita: ACG Press, pp. 61-66.
Copyright 1995, 1996, 1997 by Warren S. Sarle, Cary, NC, USA.

Permission is granted to reproduce this article for non-profit educational purposes only, retaining the author's name and copyright notice.

Contents
What is measurement theory?
What is measurement?
Why should I care about measurement theory?
What are permissible transformations?
What are levels of measurement?
Is measurement level a fixed, immutable property of the data?
Isn't an ordinal scale just an interval scale with error?
What does measurement level have to do with discrete vs. continuous?
Don't the theorems in a statistics textbook prove the validity of statistical methods without reference to measurement theory?
Does measurement level detemine what statistics are valid?
But measurement level has been shown empirically to be irrelevant to statistical results, hasn't it?
What are some more examples of how measurement level relates to statistical methodology?
Are there other theories of measurement?
What's the bottom line?
References
What is measurement theory?
Measurement theory is a branch of applied mathematics that is useful in measurement and data analysis. The fundamental idea of measurement theory is that measurements are not the same as the attribute being measured. Hence, if you want to draw conclusions about the attribute, you must take into account the nature of the correspondence between the attribute and the measurements.
The mathematical theory of measurement is elaborated in:

Krantz, D. H., Luce, R. D., Suppes, P., and Tversky, A. (1971), Foundations of measurement, Vol. I: Additive and polynomial representations, New York: Academic Press.

Suppes, P., Krantz, D. H., Luce, R. D., and Tversky, A. (1989), Foundations of measurement, Vol. II: Geometrical, threshold, and probabilistic respresentations, New York: Academic Press.

Luce, R. D., Krantz, D. H., Suppes, P., and Tversky, A. (1990), Foundations of measurement, Vol. III: Representation, axiomatization, and invariance, New York: Academic Press.

Measurement theory was popularized in psychology by S. S. Stevens, who originated the idea of levels of measurement. His relevant articles include Stevens (1946, 1951, 1959, 1968).

For a recent discussion of measurement theory and statistics, see Hand (1996).

What is measurement?
Measurement of some attribute of a set of things is the process of assigning numbers or other symbols to the things in such a way that relationships of the numbers or symbols reflect relationships of the attributes of the things being measured. A particular way of assigning numbers or symbols to measure something is called a scale of measurement.
Suppose we have a collection of straight sticks of various sizes and we assign a number to each stick by measuring its length using a ruler. If the number assigned to one stick is greater than the number assigned to another stick, we can conclude that the first stick is longer than the second. Thus a relationship among the numbers (greater than) corresponds to a relationship among the sticks (longer than). If we lay two sticks end-to-end in a straight line and measure their combined length, then the number we assign to the concatenated sticks will equal the sum of the numbers assigned to the individual sticks (within measurement error). Thus another relationship among the numbers (addition) corresponds to a relationship among the sticks (concatenation). These relationships among the sticks must be empirically verified for the measurements to be valid.

Why should I care about measurement theory?
Measurement theory helps us to avoid making meaningless statements. A typical example of such a meaningless statement is the claim by the weatherman on the local TV station that it was twice as warm today as yesterday because it was 40 degrees Fahrenheit today but only 20 degrees yesterday. This statement is meaningless because one measurement (40) is twice the other measurement (20) only in certain arbitrary scales of measurement, such as Fahrenheit. The relationship 'twice-as' applies only to the numbers, not the attribute being measured (temperature).
When we measure something, the resulting numbers are usually, to some degree, arbitrary. We choose to use a 1 to 5 rating scale instead of a -2 to 2 scale. We choose to use Fahrenheit instead of Celsius. We choose to use miles per gallon instead of gallons per mile. The conclusions of a statistical analysis should not depend on these arbitrary decisions, because we could have made the decisions differently. We want the statistical analysis to say something about reality, not simply about our whims regarding meters or feet. If a given statement may be either true or false depending on arbitrary, unspecified choices, then that statement is logically meaningless.

Suppose we have a rating scale where several judges rate the goodness of flavor of several foods on a 1 to 5 scale. If we want to draw conclusions about the measurements, i.e. the 1-to-5 ratings, then we need not be concerned about measurement theory. For example, if we want to test the hypothesis that the foods have equal mean ratings, we might do a two-way ANOVA on the ratings.

But if we want to draw conclusions about flavor, then we must consider how flavor relates to the ratings, and that is where measurement theory comes in. Ideally, we would want the ratings to be linear functions of the flavors with the same slope for each judge; if so, the ANOVA can be used to make inferences about mean goodness-of-flavors, providing we can justify all the appropriate statistical assumptions. But if the judges have different slopes relating ratings to flavor, or if these functions are not linear, then this ANOVA will not allow us to make inferences about mean goodness-of-flavor. Note that this issue is not about statistical interaction; even if there is no evidence of interaction in the ratings, the judges may have different functions relating ratings to flavor.

We need to consider what information we have about the functions relating ratings to flavor for each judge. Perhaps the only thing we are sure of is that the ratings are monotone increasing functions of flavor. In this case, we would want to use a statistical analysis that is valid no matter what the particular monotone increasing functions are. One way to do this is to choose an analysis that yields invariant results no matter what monotone increasing functions the judges happen to use, such as a Friedman test. The study of such invariances is a major concern of measurement theory.

However, no measurement theorist would claim that measurement theory provides a complete solution to such problems. In particular, measurement theory generally does not take random measurement error into account, and if such errors are an important aspect of the measurement process, then additional methods, such as latent variable models, are called for. There is no clear boundary between measurement theory and statistical theory; for example, a Rasch model is both a measurement model and a statistical model.

What are permissible transformations?
Permissible transformations are transformations of a scale of measurement that preserve the relevant relationships of the measurement process. Permissible is a technical term; use of this term does not imply that other transformations are prohibited for data analysis any more than use of the term normal for probability distributions implies that other distributions are pathological. If Stevens had used the term mandatory rather than permissible, a lot of confusion might have been avoided.
In the example of measuring sticks, changing the unit of measurement (say, from centimeters to inches) multiplies the measurements by a constant factor. This multiplication does not alter the correspondence of the relationships 'greater than' and 'longer than', nor the correspondence of addition and concatenation. Hence, change of units is a permissible transformation with respect to these relationships.

What are levels of measurement?
There are different levels of measurement that involve different properties (relations and operations) of the numbers or symbols that constitute the measurements. Associated with each level of measurement is a set of permissible transformations. The most commonly discussed levels of measurement are as follows:

Nominal:
Two things are assigned the same symbol if they have the same value of the attribute.
Permissible transformations are any one-to-one or many-to-one transformation, although a many-to-one transformation loses information.
Examples: numbering of football players; numbers assigned to religions in alphabetical order, e.g. atheist=1, Buddhist=2, Christian=3, etc.
Ordinal:
Things are assigned numbers such that the order of the numbers reflects an order relation defined on the attribute. Two things x and y with attribute values a(x) and a(y) are assigned numbers m(x) and m(y) such that if m(x) > m(y), then a(x) > a(y).
Permissible transformations are any monotone increasing transformation, although a transformation that is not strictly increasing loses information.
Examples: Moh's scale for hardness of minerals; grades for academic performance (A, B, C, ...); blood sedimentation rate as a measure of intensity of pathology.
Interval:
Things are assigned numbers such that differences between the numbers reflect differences of the attribute. If m(x) - m(y) > m(u) - m(v), then a(x) - a(y) > a(u) - a(v).
Permissible transformations are any affine transformation t(m) = c * m + d, where c and d are constants; another way of saying this is that the origin and unit of measurement are arbitrary.
Examples: temperature in degrees Fahrenheit or Celsius; calendar date.
Log-interval:
Things are assigned numbers such that ratios between the numbers reflect ratios of the attribute. If m(x) / m(y) > m(u) / m(v), then a(x) / a(y) > a(u) / a(v).
Permissible transformations are any power transformation t(m) = c * m ** d, where c and d are constants.
Examples: density (mass/volume); fuel efficiency in mpg.
Ratio:
Things are assigned numbers such that differences and ratios between the numbers reflect differences and ratios of the attribute.
Permissible transformations are any linear (similarity) transformation t(m) = c * m, where c is a constant; another way of saying this is that the unit of measurement is arbitrary.
Examples: Length in centimeters; duration in seconds; temperature in degrees Kelvin.
Absolute:
Things are assigned numbers such that all properties of the numbers reflect analogous properties of the attribute.
The only permissible transformation is the identity transformation.
Examples: number of children in a family, probability.
These measurement levels form a partial order based on the sets of permissible transformations:

Weaker <-----------------------------------> Stronger

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