Essentials for Measurement

1. Essentials for Measurement

2. Basic requirements for measuring • The reduction of experience to a one dimensional abstraction. • More or less comparisons among persons and items. • The idea of linear magnitude inherent in positioning objects along a line. • A unit determined by a process which can be repeated without modification over the range of the variable.

3. Let’s consider weight • At some point, weight was constructed… why? • Is it one dimensional? • Can we make comparisons of more and less? • Does it have linear magnitude? (1 lb + 1 lb = 2 lbs?) • Do we have a process to determine weight which we can repeat without modification over the range of the variable?

4. Social science measures should follow the same criteria • Just like weight, height, time and temperature are measured with “universally” useful instruments, our task is to devise instruments to measure variables in the human sciences. • Psychometrics is often more about the “psycho” and less about the “metrics.” • Rasch modeling does not replace or supercede statistical analyses; it should precede it.

5. We start by searching for the possibility of order • “Amount” of an attribute in a person vs. “amount” in another person • “Amount” in an item vs. “amount” of that attribute in another item • Can we level items such that endorsing the next item indicates more of the attribute in the person?

6. The Rasch model is probabilistic • Guttman’s idea: • If you endorse an extreme statement, you will endorse ALL less extreme statements. This makes a scale. • With Rasch: • If you endorse an extreme statement, there is a good probability that you will endorse all less extreme statements.

7. Objectivity • Values should have similar meaning over time and place. • The measure (set of items) assigned to the construct must be independent of the person taking these items. • Does the weight of 1 pound on a scale depend on what a person is measuring? • Should the difficulty of an item depend on the distribution of abilities of persons responding to the item?

8. Conjoint Additivity • To be additive, units must be identical. • Are apples additive? • 1 Apple + 1 Apple = 2 Apples. • But 2 Apples are twice as much as 1 Apple only when the 2 Apples are perfectly identical. • Real apples are not! • Rasch measurement forms an equal interval linear scale, just like weight.

9. Conjoint Additivity • When any pair of measurements have been made with respect to the same origin on the same scale, the difference between them is obtained merely by subtraction. • Rasch measurement creates a single person/item yardstick with person “ability” (Bn) estimated in conjunction with item “difficulty” (Di). • Bn-Di > 0, Probability the person will answer “correctly” (Pxni)> .05. • Bn-Di < 0, Pxni < .05. • Bn-Di = 0, Pxn = .05.

10. “Fit” to the model • Fit statistics indicate where the principles of probabilistic conjoint measurement have been sufficiently satisfied to justify the claim that results can be used as a scale with interval measurement properties.

11. Rasch unit for “counting”: a logit • Logit: A Log-Odds Unit • Transformation of the raw score scale (ordinal) into an interval scale: • The raw score percentage is converted into its success-to-failure ratio • The logarithm of this score is taken • In this way, the bounded outcome of probabilities (ogive) is straightened.

12. What is a success-to-failure ratio?