Measurement & Meaningfulness

1. Measurement & Meaningfulness Psych 818 - DeShon

2. Goal of Science • Discover & elucidate lawful relations among variables of interest • Mathematics provides the engine • The laws of Nature are written in the language of mathematics. - Galileo • The unreasonable effectiveness of mathematics • Hamming (1980)- The American Mathematical Monthly • Wigner (1960) - Comm. Pure Appl. Math

3. Quantification • The mathematics of science is oriented toward discovering relations among quantifiable characteristics of entities • Quantification implies amount and amount is typically represented using numbers • Numbers are complex!

4. Measurement • Measurement of some attribute of a set of things is the process of assigning numbers to the things in such a way that relationships of the numbers 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 - Sarle (1987)

5. Numbers • We take them for granted… • Huge effort and revolutions in our representation of mathematics were required to end up where we are • In the beginning, • Natural numbers (counting) • Whole numbers (includes zero) • zero was originally used as a place holder • Rational numbers (ratio of two integers) • Integers • Incorporates negative numbers (16th century)

6. Numbers • Imaginary numbers - sqrt(-1) • Complex numbers • Our focus is on Real numbers • Any number that may be expressed by an infinite decimal representation • Continuous, infinitely long number line • Negative, Positive, or zero

7. Properties of Real Numbers • Real number system has 4 parts • A set consisting of the Real numbers • An order relation, > • An addition function, + • A multiplication function, ∙

8. Properties of Real Numbers • 12 Axioms – for all x, y, & z • Associative • (x + y) + z = x + (y + z) • (x ∙ y) ∙ z = x ∙ (y ∙ z) • Commutative • x + y = y + x • x ∙ y = y ∙ x • Distributive • x ∙ (y + z) = x ∙ y + x ∙ z

9. Properties of Real Numbers • Additive Identity • 0+x = x • Additive inverse • x + y = 0 • Multiplicative identity • 1 ∙ x = x • Multiplicative Inverse • x ∙ y = 1

10. Properties of Real Numbers • Trichotomy • For any real numbers x, y, exactly one of the following three statements is true: x=y, x>y, or y>x. • Transitivity • For any real numbers x, y, z, if one has x > y and y > z, then one necessarily has x > z

11. Properties of Real Numbers • Additive compatibility • If x, y, z are real numbers, and x > y, then x + z > y + z • Multiplicative compatibility • If x, y, z are real numbers, and x > y and z > 0, then xz > yz.

12. Quantity • Quantification requires that the underlying construct is quantifiable • The amount of the construct is quantifiable in a manner consistent with the real numbers • Different from quality or category • Well structured

13. Meaningfulness • “The hallmark of a meaningless proposition is that its truth-value depends on what scale or coordinate system is employed, whereas meaningful propositions have truth-value independent of the choice of representation” • Mundy, B. (1986). On the general theory of meaningful representation. Synthese, 67, 391-437.

14. Meaningfulness • A statement using scales is meaningful if its truth or falsity is unchanged when all scales in the statement are transformed by admissible transformations - Roberts (1987) • If a statement, S (I am shorter than the empire state building), is true for one scale of height (e.g., inches) it is true for all scales of height obtained by multiplying by a positive constant.

15. Meaningfulness • When we measure something, the resulting numbers are often 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 • Statistical conclusions should not depend on these arbitrary decisions, because we could have made the decisions differently • The statistical analysis should say something about reality and not an arbitrary choice regarding meters or feet • If a given statement may be either true or false depending on arbitrary choices, then that statement is not demonstrably meaningful

16. Meaningfulness • Example: • A common and meaningless statement made by a weather forecaster “it’s twice as warm today as yesterday because it was 40 degrees Fahrenheit today but only 20 degrees yesterday • Works for Fahrenheit • What about a different temperature scale (i.e., Celcius)? • 30f = -1.11c • 60f = 15.55c • The relationship 'twice-as' applies only to the numbers, not the attribute being measured (temperature).

17. Meaningfulness • Example: Panel Tasting • 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 just want to draw conclusions about the measurements (the 1-to-5 ratings), then measurement doesn’t matter • Just use a t-test on the mean ratings across foods • But if we want to draw conclusions about flavor, then we must consider how flavor relates to the ratings! • Measurement is critical here!

18. Meaningfulness • Panel Tasting example (cont.)… • Ideally: • The ratings should be linear functions of the flavors • Same slope for each judge. • If so, ANOVA can be used to make inferences about mean goodness-of-flavors • If the judges have different slopes relating ratings to flavor, or if the functions are not linear, then this ANOVA will not allow us to make inferences about mean goodness-of-flavor.

19. Meaningfulness • Perhaps we can only be sure that the ratings are monotone increasing functions of flavor. • Here, 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.

20. Meaningfulness • Example: Fuel efficiency and vehicle cost • Could measure fuel efficiency as the distance (miles) that can be traveled on a gallon of gas (miles/gallon) • Or… • the gallons of gas required to travel a certain distance (gallons/mile) • How does the choice of the measure of fuel efficiency relate to the cost of the vehicle?

21. Meaningfulness

22. Meaningfulness • Run a linear model for both ways of measuring fuel efficiency • lm(cost~mpg) • lm(cost~gpm) Call: lm(formula = cost ~ mpg) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 22979.9 2224.1 10.332 5.39e-09 *** mpg 279.2 110.3 2.532 0.0209 * Residual standard error: 2015 on 18 degrees of freedom Multiple R-Squared: 0.2627 Call: lm(formula = cost ~ gpm) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 32076 1916 16.74 2.02e-12 *** gpm -67108 34770 -1.93 0.0695 . Residual standard error: 2136 on 18 degrees of freedom Multiple R-Squared: 0.1715

23. Permissible Transformation • Permissible transformations are transformations of a scale of measurement that preserve the relevant relationships of the measurement process • Example: Measuring length • Transform the unit of measurement from centimeters to inches  cm x 0.39 = in • measurements are multiplied by a constant factor. • This transformation does not alter the correspondence of the relationships 'greater than' and 'longer than', nor the correspondence of addition and concatenation. • Therefore, change of units is a permissible transformation with respect to these relationships

24. A bit of Measurement History • Extensive Measurement • Based on addition rule for combination • Ex: distance, mass, time • Derived Measurement: • Created by mathematical operations using functions of extensive measures • Ex: velocity = distance / time (speed of light meters per second) , Newtonian constant of gravitation, etc.

25. A bit of Measurement History • Campbell (1920; 1940) argued that psychological measurement was impossible • Prerequisite of measurement is some form of empirical quantification that can be experimentally accepted or rejected • Only known form of such quantification that satisfies the axioms of extensive measurement is the binary operation of concatenation • Psychology has no extensive measurement

26. History of Measurement • Stevens (1951) argued that this perspective on measurement was too narrow • Various levels or types of measurement

27. Meaningfulness • Stevens (1946) response • His Logic… • Measurement is the assignment of numerals to objects or events according to rules • Different assignment rules result in different kinds of scales and measurement • e.g., nominal, ordinal, interval, ratio

28. Meaningfulness • Stevens (cont)… • Meaningful inference requires: • Making explicit the various rules for assigning numbers (i.e., scales) • Making explicit the mathematical properties of the resulting scales (permissible transformations) • Making explicit the statistical operations that are applicable to measurements made with the different scale types

29. Scales of Measurement • Nominal • Requires the ability to determine equality • Two things are assigned the same symbol if they have the same value of the attribute. • Permissible transformations are any one-to-one (1s become 5s) or many-to-one transformation, although a many-to-one transformation loses information • Permissible statistics: frequency, mode • Examples: • numbering of football players; • numbers assigned to religions in alphabetical order, e.g. atheist=1, Buddhist=2, Christian=3, etc

30. Scales of Measurement • Ordinal • Requires the ability to determine order (greater than relations) • 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) • Examples: • Moh's scale for hardness of minerals • grades for academic performance (A, B, C, ...) • Race finishing place (1st, 2nd, 3rd)

31. Scales of Measurement • Ordinal (cont.) • Permissible transformations are any monotone increasing transformation, although a transformation that is not strictly increasing loses information • Permissible statistics • Median, Percentiles, Range, Rank-order correlation • Other statistics yielding same result for all monotonic transformations (order preserving)

32. Scales of Measurement • Interval • Requires the ability to determine the equality of differences • 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) • Examples: • Temperature in degrees Fahrenheit or Celsius • Calendar date

33. Scales of Measurement • Interval (cont.) • 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 (zero point) and unit of measurement are arbitrary • Permissible statistics • mean, standard deviation, product moment correlation

34. Scales of Measurement • 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 * md, where c and d are constants • Examples: • Density (mass/volume) • Fuel efficiency (miles/gallon)

35. Scales of Measurement • 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

36. Scales of Measurement • 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

37. Scales of Measurement • The lines separating these classifications is fuzzy • What about binary variables? • one-to-one transformations, monotone transformations, and affine transformations are identical • For a binary variable, you can't do anything with a one-to-one transformation that you can't do with an affine transformation • Hence binary variables are at least at the interval level

38. Does it Matter? • Parametric vs. non-parametric statistical tests • Example: t-test (Baker et al, 1966) “Strong statistics are more than adequate to cope with weak measurements”

39. Transformations • What does this perspective say about transformations?

40. Summing up Statistics Measurement • Statistics addresses the connection between inference and data • Measurement theory addresses the connection between data and reality • Both statistical theory and measurement theory are necessary to make reasonable inferences about reality Reality Data Inference