Transcript Essentials for Measurement

Essentials for Measurement
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.
Let’s consider weight
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At some point, weight was constructed… why?
1) Is it one dimensional?
2) Can we make comparisons of more and less?
3) Does it have linear magnitude?
(1 lb + 1 lb = 2 lbs?)
4) Do we have a process to determine weight
which we can repeat without modification over
the range of the variable?
Social science measures should
follow the same criteria
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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.
We start by searching for the
possibility of order
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“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?
The Rasch model is probabilistic
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Guttman’s idea:
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If you endorse an extreme statement, you
will endorse ALL less extreme statements.
This makes a scale.
With Rasch:
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If you endorse an extreme statement, there
is a good probability that you will endorse all
less extreme statements.
Objectivity
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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.
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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?
Conjoint Additivity
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To be additive, units must be identical.
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Are apples additive?
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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.
Conjoint Additivity
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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).
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Bn-Di > 0, Probability the person will answer “correctly”
(Pxni)> .05.
Bn-Di < 0, Pxni < .05.
Bn-Di = 0, Pxn = .05.
“Fit” to the model
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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.
Rasch unit for “counting”: a logit
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Logit: A Log-Odds Unit
Transformation of the raw score scale (ordinal) into an
interval scale:
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The raw score percentage is converted into its successto-failure ratio
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The logarithm of this score is taken
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In this way, the bounded outcome of probabilities
(ogive) is straightened.
What is a success-to-failure ratio?
Person |Item
|
1
|1234576980
|---------12 +1111110101
1 +1111111000
27 +1111111000
4 +1111010100
33 +1111010001
25 +1110101000
35 +1110000000