Statistical Foundations: Measurement Scales. Psychology 790 Lecture #1 8/24/2006

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1 Statistical Foundations: Measurement Scales Psychology 790 Lecture #1 8/24/2006

2 Today s Lecture Measurement What is always assumed. What we can say when we assign numbers to phenomena. Implications for statistical procedures.

3 Measurement

4 Purpose and Definition of Measurement Measurement is the process of translating information from an empirical system into a numerical system. An empirical system contains observable (at some level) events that are of interest. A numerical system is a set of numbers that describe the observable events. The purpose of measurement is to quantify an attribute. The term attribute is used broadly.

5 More on Measurement Measurement is a process by which we assign numerical values to observable entities such that the numerical relations reflect a faithful representation of the empirical relationship among entities. It is important to recognize that the social sciences measurement can be very inaccurate. This is a premise that will be neglected throughout this course.

6 Quantification An important first-step in measurement is determining whether a variable is categorical or continuous. Why? This property of a variable determines how we quantify the variable, how we model its statistical behavior, and the way we analyze data regarding that variable.

7 Nominal Scale With categorical or nominal variables, people either belong to a category or not. Examples: country of origin biological sex (male or female) married vs. single Quantitative question: How many people belong to each category? What frequency distributions are useful for

8 Scales of Measurement: Nominal Scale Sometimes numbers are used to designate category membership. Example: Gender 1 = Female 2 = Male The numbers do not have numeric implications; they are simply convenient labels. What does it mean to say that a sample has an average gender of 1.45?

9 Frequency Distribution for Nominal Responses Because you are good students, all of you filled out the background questionnaire already. Here are your responses:

10 Bivariate Frequencies The two tables before were frequency distributions for the categorical variables of agreement with each question. We can also display these in bivariate fashion (aka contingency table, crosstabulation, etc ):

11 Continuous Variables With continuous variables, people vary in a graded way with respect to the property of interest. Examples: age intelligence marital discord Quantitative question: How much? or To what extent or degree?

12 Continuous Measurement, Abstractly Envision you have a set of entities O consisting of N distinct objects. We can label each object o 1, o 2,,o N. Take a pair of these objects: o i and o j. Suppose there is some property that pertains to each object: temperature, weight, length, age, intelligence, motivation,

13 More Abstract Measurement Each object has a certain amount of the property of interest. In principle, we could assign a number t(o i ), to each object i. We will let t( ) be the function that indicates the quantity of the property without error. In reality, t( ) may not be feasible Can you come up with a test (a t( ) ) that perfectly measures a person s intelligence?

14 So What Do We Have? When we make an attempt at quantifying a property of interest, we really are making a measurement rule, m(o i ), that assigns a number to a property. One such rule: we use the sum of the number of items correct to measure a persons intellect. The varying levels of assumptions regarding the measurement rules defines the level of measurement of our property Philosophically, we are defining a set of axioms which our measurement rule must satisfy in order to be considered a given level of measurement.

15 Scales of Measurement: Continuous Variables When we assign numbers to people (i.e., scale people) with respect to a continuous variable, those numbers represent something that is more meaningful than those used with nominal variables. Exactly what those numbers mean, and how they should be treated, however depends on the set of axioms satisfied by the measurement.

16 Levels of Measurement Commonly, we consider three differing levels of measurement (note: this distinction is debatable, and comes from Stevens, 1946): Ordinal Interval Ratio If you find yourself unable to sleep, try reading: Stevens, S.S. (1946). On the theory of scales of measurement. Science, 103,

17 Scales of Measurement: Ordinal Ordinal: Designates an ordering; quasi-ranking Does not assume that the intervals between numbers are equal. Example: finishing place in a race (first place, second place) 1st place 2nd place 3rd place 4th place 1 hour 2 hours 3 hours 4 hours 5 hours 6 hours 7 hours 8 hours

18 Ordinal Quasi-Axioms Suppose we have a measurement procedure that gives a number m(o i ) to any object o i and also gives a number m(o j ) to any object o j. We say that this is measurement at the ordinal level if the following statements are true: 1. m(o i ) m(o j ) implies that t(o i ) t(o j ) 2. m(o i ) > m(o j ) implies that t(o i ) > t(o j )

19 Ordinal Example Consider the following mapping of behavior: we assign placements for four runners. 1st place 2nd place 3rd place 4th place m(o 1 )= 1 m(o 2 )= 2 m(o 3 )= 3 m(o 4 )= 3 1 hour 2 hours 3 hours 4 hours 5 hours 6 hours 7 hours 8 hours t(o i )

20 Ordinal Information Most scales constructed in the social sciences measure properties ordinally. Taking the mean of a set of ordinal numbers is unjustified: What does it mean to say someone s average finish was 1.45? Statistically, it may be more beneficial to treat ordinal numbers as nominal testing certain assumptions before making a leap to ordinal statistics Although doing this would render the rest of the class meaningless.

21 Scales of Measurement: Interval Interval: designates an equal-interval ordering. The distance between, for example, a 1 and a 2 is the same as the distance between a 4 and a 5. Example: Some think that intelligence tests are assumed to use an interval metric. This is a debatable distinction it depends on other measurement conditions being satisfied.

22 Interval Level Quasi-Axioms Suppose we have a measurement procedure that gives a number m(o i ) to any object o i and also gives a number m(o j ) to any object o j. We say that this is measurement at the interval level if the following statements are true: 1. m(o i ) m(o j ) implies that t(o i ) t(o j ) 2. m(o i ) > m(o j ) implies that t(o i ) > t(o j ) 3. For any object o i, t(o i ) = x if and only if m(o i ) = ax + b

23 Interval Explanation At the interval level, we assume that the measurement number m(o i ) is some linear function of the true magnitude x. We can make stronger inferences about objects measured at the interval level than we can about objects measured ordinally. For instance we can talk about the distance between two objects. Imagine: m( oi ) m( o j ) This implies that the true difference is four units (a is a scaling constant): t( o t( o i ) j ) 4 4 a

24 Interval Information Examples of interval level measurements include the year date, and temperature measured in Fahrenheit or Celcius scales. A note about temperature, though. Temperature is defined as random microscopic motions of the atomic and subatomic constituents of matter. Fahrenheit and Celcius provide a m(o i ) for an object i that relates ordinally to atomic conditions (certainly not linearly).

25 Scales of Measurement: Ratio Ratio: designates an equal-interval ordering with a true zero point (i.e., the zero implies an absence of the thing being measured). Example: number of speeding tickets a person has had: 0 quite literally means none. a person who has had 4 tickets has had twice as many as someone who has had 2.

26 Ratio Level Quasi-Axioms Suppose we have a measurement procedure that gives a number m(o i ) to any object o i and also gives a number m(o j ) to any object o j. We say that this is measurement at the ratio level if the following statements are true: 1. m(o i ) m(o j ) implies that t(o i ) t(o j ) 2. m(o i ) > m(o j ) implies that t(o i ) > t(o j ) 3. For any object o i, t(o i ) = x if and only if m(o i ) = ax + b 4. For any object o i, t(o i ) = x if and only if m(o i ) = ax

27 Ratio Level Information Ratio scales can relate the differences in the property of interest by comparison of magnitudes: m( oi ) m( o ) j t( oi ) t( o ) Examples of ratio scales: temperature (in Kelvin units), time, counts. j

28 Wrapping Up Measurement is something often taken for granted. Most statistical procedures in this course will involve interval or better measures. Commonly, this is the practice used in the social sciences. It may not be entirely correct, but is approximate.

29 Next Time We will cover probability Be sure to go to lab tonight (5pm, Room 4 Fraser Hall). Please be sure to complete the homework assignment by tomorrow.

Measurement and Measurement Scales

Measurement and Measurement Scales Measurement is the foundation of any scientific investigation Everything we do begins with the measurement of whatever it is we want to study Definition: measurement