MB 813 Multivariate Methods Carroll School of Management, Boston College

Notes on Measurement
Steve Borgatti, Boston College

What is Measurement?
• 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 different kinds of measurements, such as the familiar measurement scales of Stevens (nominal, ordinal, interval, ratio)

Examples of Mapping Relationships

• 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.
• 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.
• Meaningfulness also dictates the uniqueness of measurement -- what other sets of measurement could be seen as equally valid

Family of Measurement Scales

• 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 is measurement or something else entirely
• Only equality preserved. 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" in nominal measurement. There is just same or not.
• 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:
 Object M1 M2 M3 A 22 0 99 B 22 0 99 C 22 0 99 D 23 -1 98 E 24 67 12
• There is no simple normalization technique with nominal scales that lets you compare variables measured on different scales. What you have to do is 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 call into just one category of the column variable, you can establish a 1-1 mapping of one measurement to the other.

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 measurements are equally valid:
 Object M1 M2 M3 A 22 0 99 B 22 0 99 C 22 0 99 D 23 1 150 E 24 67 152
• To normalize an ordinal scale, you convert the values to rank order values, for example, normalizing each of the scales above would yield:
 Object M1* M2* M3* A 1 1 1 B 1 1 1 C 1 1 1 D 2 2 2 E 3 3 3
• By normalizing variables, you can see whether a set of measured variables are really measuring the same thing

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 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.
• the intervals between measured values of objects have ratio scale properties
• 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 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:
 Object M1 M2 M3 M3 A 22 32 220 230 B 22 32 220 230 C 22 32 220 230 D 23 33 230 240 E 24 34 240 250
• Ratios of values are not meaningful in interval 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. The contradiction indicates that ratios of interval measurements are not meaningful.
• It is often said that interval scales lack of zero point. That's kind of sloppy language. What it means 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 what is zero in one scale is another 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. Normalizing each of the variables above would yield:
 Object M1 M2 M3 M4 A -.75 -.75 -.75 -.75 B -.75 -.75 -.75 -.75 C -.75 -.75 -.75 -.75 D 0.50 0.50 0.50 0.50 E 1.75 1.75 1.75 1.75
• 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.
• 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.

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. The measurements M1, M2 and M3 are equally valid measures of given object property, but M4 is not measuring the same thing:
 Object M1 M2 M3 M4 A 22 220 11 12 B 22 220 11 12 C 22 220 11 12 D 23 230 11.5 13 E 24 240 12 14
• Ratio scales are said to have a defined zero point. This is because the admissible transformations (of the form Y = mX) do not include adding a constant, so no sliding of the scale up and down is permitted without changing the meaning of the values.
• To normalize a ratio scale, you perform a particular congruence 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). Normalizing each of the variables above would yield:
 Object M1 M2 M3 M4 A 0.44 0.44 0.44 0.43 B 0.44 0.44 0.44 0.43 C 0.44 0.44 0.44 0.43 D 0.45 0.45 0.45 0.46 E 0.47 0.47 0.47 0.50
• 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, from say exactly the same thing. The last column is different however, indicating that it measures something else.
• Note all 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.

• 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 scale 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:
 Object M1 M2 M3 A 22 12 11 B 22 12 11 C 22 12 11 D 23 13 11.5 E 25 15 12
• 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. Normalizing each of the variables above would yield:
 Object M1 M2 M3 A -0.8 -0.8 -0.3 B -0.8 -0.8 -0.3 C -0.8 -0.8 -0.3 D 0.2 0.2 0.2 E 2.2 2.2 0.7
• 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, from say exactly the same thing. The last column is different however, indicating that it measures something else.
• Note all 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.

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 scales exists.