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.
![](../images/measur1.gif) 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 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:
Object |
M1 |
M2 |
M3 |
A |
22 |
0 |
99 |
B |
22 |
0 |
99 |
C |
22 |
0 |
99 |
D |
23 |
-1 |
98 |
E |
24 |
67 |
12 |
- 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:
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. 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:
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
(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:
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. 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.
Object |
M1 |
M2 |
M3 |
M4 |
A |
22 |
220 |
11 |
122 |
B |
22 |
220 |
11 |
122 |
C |
22 |
220 |
11 |
122 |
D |
23 |
230 |
11.5 |
123 |
E |
24 |
240 |
12 |
124 |
- 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:
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, 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:
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. To do this,
we just subtract the mean of the original values (also called
"centering"). 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 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).
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