Essentials for Measurement

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Essentials for Measurement
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Basic requirements for measuring

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

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Lets 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.

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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?

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The Rasch model is probabilistic

• Guttmans 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.

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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?

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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.

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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 gt 0, Probability the person will answer
  correctly (Pxni)gt .05.
• Bn-Di lt 0, Pxni lt .05.
• Bn-Di 0, Pxn .05.

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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.

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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.

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What is a success-to-failure ratio?