Handout
Measurement Theory

This is a simplified account of formal representational measurement theory.

Measurement is the assignment of numbers to objects in such a way that physical relationships and operations among the objects correspond to arithmetic relationships and operations among the numbers.

• Mapping of (a property of) objects to numbers. If we call this mapping F, then the measured value of object x is f(x)
  • e.g., the height of person A can be written f(A), and the height of someone twice as tall as A might be written 2*f(A)
• The mapping is like an isomorphism in the sense that certain relationships among the objects are also mapped to mathematical relationships among the numbers.
• Since not all relationships among the objects will correspond to relationships among the measured values, measurement is a kind of model -- there is an implicit theory about what is important and what is not
• Exactly which relationships among objects are preserved as arithmetic relationships among measured values is what defines and differentiates different kinds of measurements, such as the familiar measurement scales of Stevens (nominal, ordinal, interval, ratio)

The case of measuring weight

• When we measure weight, two objects that balance each other at each distance on a balance beam are assigned the same measured value (i.e., f(x) = f(y), and if two objects have the same measured value (f(x)=f(y), they balance each other -- mirror image
• If objects X and Y are placed together on the left side of a balance scale and they perfectly balance an object Z on the right side, then it will be the case that f(x)+f(y) = f(z) -- the combining operation is mirrored by arithmetic addition
• Suppose an object weighs 5 pounds. Then five of these objects will weight 5*5 = 25. So multiplication of measured scores by a constant maps to the physical operation of putting that many items on the scale
• But not all mathematical relationships among measured values have a counterpart in physical operations. For example, the breakdown of an object's weight into its prime multipliers doesn't say anything about the objects themselves. Similarly, if the weight of one object happens to be the log of the weight of another, this does not imply any special relationship among the objects

Meaningfulness

• Scales of measurement are distinguished from each other by which arithmetic operations are meaningful (because they map to physical properties) and which are not.
• IMPORTANT NOTE: Scales of measurement are independent of the properties they measure. I can choose to measure mass using a ratio scale (which is what we usually do) or an interval scale, or an ordinal scale ... etc.
  • so it is not that "temperature is interval" and "weight is ratio". Temperature can be measured on any scale, and is commonly measured on both ratio and interval scales
• Meaningfulness also dictates the uniqueness of measurement -- what other sets of measurements could be seen as equally valid

Family of Measurement Scales

• Ideally, we define a new measurement scale whenever needed. That is, for a given research need, we would defined a measurement system that assigns values to objects in such a way that certain relationships among the objects are captured by relationships among the measured values, but others are not. Which relations are captured is referred to as "properties  preserved" in the discussion below.

• However, there are 6 scales that are frequently used and well-studied. These are shown in Figure 1.

• For pedagogical reasons, the following discussion is ordered a little differently

Nominal Scale Measurement

• Also known as classification and categorical measurement. Some controversy over whether classification should be considered measurement or something else entirely
• Equality property. Only the equality property is preserved by nominal measurement. A system of measurement in which only equality of numbers has meaning. If we measure weight with a nominal scale, then if f(A) = f(B) we can be sure that A and B weigh the same, but if f(A) = 12 and f(B) = 13, we can't tell whether B weighs more than A -- there is no "more than" relation in nominal measurement. There is just same or not same. In this sense, when applied to attributes like mass, nominal measurement is a very weak kind of measurement because it fails to tell us things we regard as important, such as which object is heavier.
• Uniqueness. Nominal measurement is characterized by a tremendous lack of uniqueness: many alternative measurements equally valid. Suppose A weighs same as B and C, D weighs twice as much as A, and E weighs 3 times as much as D. Then any of the following measurements are equally valid:

• To compare two nominal variables that may be measured using different scales, you want to "normalize" the values so you can see how well they correspond to each other. There is no simple normalization technique to do this, but it can be done.
  • One approach: Construct a contingency table to cross-classify observations of one variable against the other. Then, if all observations falling into a given category or the row variable fall into just one category of the column variable, you can establish a 1-1 mapping of one measurement to the other. i.e., in variable A, a "2" corresponds to a "92" means in variable B
  • More sophisticated approach: use a technique called correspondence analysis (particularly a variant called optimal scaling) to work out a set of scores that maximize correspondence between the two variables

Ordinal Scale

• preserves 2 properties: equality and ordinality
• Equality property (like Nominal scaling): If we measure weight with an ordinal scale, then if f(A) = f(B) we can be sure that A and B weigh the same, and vice-versa
• Ordinality property:  If we measure weight with an ordinal scale, then if f(A) > f(B), we can be sure that A weighs more than B. But we don't know how much more.
• Uniqueness. Ordinal scales are unique up to a monotone transformation. A monotone transformation T is one that assigns new values such that if f(x) > f(y) in the original scale, then T(f(x)) > T(f(y)) in the newly transformed scale.
• The following ordinal measurements of the same attribute of objects of A through E are equally valid:

• To normalize an ordinal scale, you convert the values to rank order values, for example, normalizing each of the scales above would yield:

• By normalizing variables, you can see whether a set of measured variables are really measuring the same thing. i.e., you take away numerical differences that are arbitrary (due to different measurement properties) and leave only the differences that reflect differences in the underlying property being measured.
• Note: we tend to use an asterisk after a variable name to indicate the normalized version of the variable

Presence-Absence Scales

• Not recognized by formal measurement theory, but occurs often enough to be discussed
• Often we "measure" whether a trait is present or not. For example, for a new species we might record whether it has fur or not, is warm-blooded, gives birth to live young, etc. By convention we use a value of 1 to indicate presence of a trait and 0 to indicate absence.
• Presence/absence variables can be seen as as a kind of ordinal measurement. They have the ordinal property that 1 indicates more of some attribute than a 0. On they other hand, the underlying attribute might not be something that makes sense to think about in terms of "more of". You might think of it as classification: either it is a dog (1) or it isn't (0). In this case we might think of presence absence scales as nominal.

Interval Scale

• preserves 3 properties: equality, ordinality, and interval ratios
• Equality property (like Nominal scaling): If we measure weight with an interval scale, then if f(A) = f(B) we can be sure that A and B weigh the same, and vice-versa
• Ordinality property:  If we measure weight with an interval scale, then if f(A) > f(B), we can be sure that A weighs more than B.
• Difference or interval Ratios property. Suppose we are measuring weight. If f(A)-f(B) = 10, and the difference between B and C is 20, then we know the difference in mass between B and C is greater than the difference in mass between A and B (A......B..........C), and in fact we know that the difference between B and C is twice as much as between A and C.
  • the intervals between measured values of objects have ratio scale properties but the scale values themselves do not
• Uniqueness. Interval scales are unique up to a linear transformation (Y = mX+b). In other words, if you measure a set of objects on an interval scale, and then multiply and/or add a constant to each value, the resulting values are equally as valid as the original values. This is because the ratios of the intervals between the numbers are not affected by linear transformations. The following measurements are equally valid:

• Ratios of values are not meaningful in interval scales (nor in ordinal or nominal scales). Consider asking whether the temperature in one city is twice as hot as in another. Measure temperature in Fahrenheit. City A is 80 degrees and City B is 40 degrees. So it looks like A is twice as hot as B. But suppose instead we measure temperature in centigrade. City A is 28 deg, and City B is 4 degrees. So now it looks like A is 7 times as hot as B. This contradicts our previous result of twice as hot. Yet centigrade and fahrenheit are both valid measurements. In fact, one is just a linear transformation of the other:  F = 9/5C + 32. (and C = 5/9F - 17.78) The contradiction indicates that ratios of interval measurements are not meaningful.
• It is often said that interval scales lack a zero point. This is kind of sloppy language, but what it is intended to mean is that the value of zero has no special meaning in an interval scale -- it is just a value one unit above -1 and two units below +2. The constant "b" in the linear transformation allows you to slide the scale up and down so that what is zero in one scale is a different value in a different, equally valid, interval scaling of the same object property.
• To normalize an interval scale, you perform a linear transformation that creates a normalized version of the variable with the property that the mean is zero and the standard deviation is one. This linear transformation is called standardizing or reducing to z-scores.
  • to standardize a variable, first subtract its mean, then divide by the std dev. i.e., if x is a variable, then the standardized version of x, called x*, is given by this equation:  x*(i) = (x(i) - m(x))/s(x), where x(i) is the ith value of variable x, m(x) is the mean of x, and s(x) is the standard deviation of x
  • Normalizing each of the variables above would yield:

• Note that all the values are the same -- this indicates that all four columns are just linear transformations of each other and therefore, from an interval scaling point of view, say exactly the same thing. The four variable measure the same attribute of the objects, using different interval scales
• Note all also that the standardized values can be interpreted as deviations from the mean. D is just slightly above the mean of all objects on this variable, while E is quite a bit higher than the mean. When an attribute is normally distributed, most of the standardized measured values will be near 0, and about 95% of values will be between -2 and +2, and only about 2.5% will be as large or larger than 2.

Ratio Scale

• Preserves 4 properties: equality, ordinality, interval ratios, and value ratios
• Equality property (like Nominal scaling): If we measure mass with a ratio scale, then if f(A) = f(B) we can be sure that A and B weigh the same, and vice-versa
• Ordinality property:  If we measure mass with a ratio scale, then if f(A) > f(B), we can be sure that A weighs more than B.
• Interval ratios property. If f(A)-f(B) = 10, and the difference between B and C is 20, then we know the difference in mass between B and C is greater than the difference in mass between A and B (A......B..........C), and in fact we know that the difference between B and C is twice as much as between A and C.
• Value Ratios property. If we measure mass with a ratio scale, then if A weighs twice as much as B, then f(A) = 2*f(B) and vice versa. I.e., ratios of the measured values correspond to ratios of the actual properties being measured.
• Uniqueness. Ratio scales are unique up to a congruence or proportionality transformation (Y = mX). In other words, if you measure a set of objects on a ratio scale, and then multiply each value by a constant, the resulting values are equally as valid as the original values. This is because the ratios of the intervals between the numbers are not affected by congruence transformations.
• In the table below, the measurements M1, M2 and M3 are equally valid measures of given object property, but M4 is not measuring the same thing. You can tell because the m1(E)/m1(A) = m2(E)/m2(A) = m3(E)/m3(A) = 1.0909 but m4(E)/m4(A) = 1.008.

• Ratio scales are said to have a defined zero point. This is because the admissible transformations (of the form Y = mX) do not include an additive constant (the "b" in the interval-scale transformation formula), so no sliding of the scale up and down is permitted without changing the meaning of the values.
  • in a sense (but don't take this too far), once you choose a meaningful zero point, there is no difference between a ratio scale and an interval scale. The Kelvin scale for measuring temperature is an illustration. To can get from Kelvin to Centigrade and Fahrenheit by simple linear transformations, as is always true in interval scale measurement. But because the 0 point of the Kelvin scale is the absence of heat, it is less arbitrary than centigrade (centered on freezing point of water) and fahrenheit (centered on who knows what). And when zero corresponds to the absence of something, the ratios of the scale values have meaning. in Kelvin, something that is 373K is 1.37 times hotter than something at 273K
• To normalize a ratio scale, you perform a particular "congruence" or "similarity" transformation that creates a normalized version of the variable with the property that the length of the vector is 1 (i.e., the Euclidean or L2 norm equals 1.0). In other words, to normalize a ratio-scaled variable, we divide each value of the variable by the square root of the sum of squares of all the original values. Normalizing each of the variables above would yield:

• Note that all the values except the last column are the same -- this indicates that the first three columns are just rescalings (in a ratio sense) of each other and therefore, measure exactly the same thing. The last column is different however, indicating that it measures something else.
• Note also that other ways of normalizing accomplish the same goal of making different measurements comparable. So we could just divide each column by the column sum, creating a new variable whose values add to 1. This allows interpretation of the rescaled values as proportions or shares of the whole. This is not the usual way but it works fine.

Difference Scale (aka Additive Scale)

• Note that "additive scales" is a term used for several different things. better to call these "difference scales".
• Preserves 4 properties: equality, ordinality, interval ratios, and interval equalities
• Equality property (like Nominal scaling): If we measure mass with a ratio scale, then if
  f(A) = f(B) we can be sure that A and B weigh the same, and vice-versa
• Ordinality property:  If we measure mass with a ratio scale, then if f(A) > f(B), we can be sure that A weighs more than B.
• Interval ratios property. If f(A)-f(B) = 10, and the difference between B and C is 20, then we know the difference in mass between B and C is twice as much as between A and C.
• Interval equalities property. If we measure mass with an additive scale, then if the difference between A and B is 10 units using one scale, then the difference is 10 units using any valid scale.
• Uniqueness. Additive scales are unique up to a "translation" transformation (Y = X + b).  In other words, if you measure a set of objects on an additive scale, and then add a constant to each value, the resulting values are equally as valid as the original values. This is because the intervals between values are not affected by translation transformations. The measurements M1 and M2 are equally valid measures of given object property, but M3 is not measuring the same thing:

• Additive scales are said to lack a defined zero point. This is because the admissible transformations (of the form Y = X +b) effectively allow sliding the scale up and down without changing the meaning of the values -- it is only the gaps between the values that matter.
• To normalize an additive scale, you perform a particular translation transformation that creates a normalized version of the variable with the property that the mean of the transformed vector is 0. To do this, we just subtract the mean of the original values (also called "centering"). Normalizing each of the variables above would yield:

• Note that all the values except the last column are the same -- this indicates that the first two columns are just rescalings (in a difference scale sense) of each other and therefore, say exactly the same thing. The last column is different however, indicating that it measures something else.
• Note also that other ways of normalizing accomplish the same goal of making different measurements comparable. So we could just subtract the column sum from each value

Absolute Scale

• preserves all properties discussed above.
• Uniqueness. Absolute scales are unique up to an identity transformation (Y = X). In other words, they are completely unique and no (non-trivial) transformation of the numbers is permissible.
• As a result of their uniqueness, no normalization of absolute-scaled variables is needed (nor exists).