PS28C Psychometrics

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PS28CPsychometrics

• Lecture 2
• What is Measurement and How Do We Do It?

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Measurement

• Measurement is one of the three foundations of
  psychometrics
• Measurement is
• The process of assigning numbers to represent a
  quantifiable difference between people
• Good measures faithfully reflect differences
  between people

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Hypothetical Constructs

• A hypothetical construct is a theoretical
  description of a some attribute
• Exist only as abstractions
• A collection of ideas describing the properties
  of an object or phenomena

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Hypothetical Constructs

• Can be simple or complex
• Simple constructs have one or two properties
  which are loosely associated
• Complex constructs have many properties which are
  more strongly associated

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Hypothetical Constructs

• Basic foundation of theories
• Hypothetical constructs inform the measurement
  process
• Provide descriptions of the properties that your
  measure should include

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Numbers as a Language

• Numbers and mathematics are a language describing
  the world in quantifiable terms
• Written language preceeded numbers
• Ancient Egyptians had first simple system of
  numbers

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Numbers as a Language

• Numbers evolved from pictographs
• Mathematics developed as system to combine
  numbers
• Numbers are like words while mathematics is the
  grammar
• Numbers are intimately tied to objects in the
  real world

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Numbers as a Language

• Numbers and mathematics reflect the sensed world
  but do not determine it
• Mathematical models are simply a shorthand way of
  describing behaviour

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A Controversy in Psychometrics

• Clash of two different world views in
  psychometrics
• Hypothetical constructs are abstractions outside
  of real objects
• Numbers and mathematics are reflections of real
  objects

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A Controversy in Psychometrics

• Ancient dispute between Plato and Aristotle
• Plato suggests that our senses are unreliable and
  we must focus on abstract properties
• Aristotle suggests that our senses are reliable
  and should be basis of science

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Measurement Theory

• Most critical factor is understanding how things
  differ
• Differences in quality versus quantity
• Measurement systems must correctly reflect the
  number of differences that truly exist

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Levels of Measurement

• Four properties need to be addressed in
  determining level of measurement
• Distinctiveness
• Ordering by magnitude
• Equality of intervals
• True zero point

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Levels of Measurement

• Four levels of measurement
• Nominal
• Ordinal
• Interval
• Ratio
• Properties and types of permissible
  transformations determines level of measurement

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Nominal Measures

• Unique numbers arbritarily assigned to distinct
  objects or observations
• Numbers reflect qualitative differences and not
  quantitative differences
• All types of mathematical changes are possible

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Ordinal Measures

• Numbers reflect rank ordering of objects
• Reflect quantitative distinctions in order of
  magnitude
• Any mathematical transformation possible as long
  as rank ordering preserved

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Interval Measures

• Assignment of numbers to observations reflect
  fact that differences occur at equally spaced
  intervals
• Possible to calculate and interpret ratios
  between intervals via linear transformations

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Interval Measures

• Permissible linear transformations follow this
  form
• Y aX b
• Where Y is new value
• a is a constant
• b is a constant, and
• X is the original value

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Interval Measures

• Example
• 3 Persons with differing leadership
• Person A 10
• Person B 20
• Person C 40
• B-A20-1010 C-B40-2020
• Ratio of Differences
• C-B/B-A 40-2020

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Interval Measures

• Y 2X 10
• Person A30
• Person B50
• Person C90
• B-A50-3020 C-B90-5040
• Ratio(C-B)/(B-A)40/202

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Ratio Measures

• Numbers assigned to observations reflect fact
  that ratios between observations are
  interpretable
• Have true zero point
• Only permissible form of transformation is
  multiplication by a constant

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Ratio Measures

• Permissible transformations take this form
• Y aX
• Where Y is the new scale value
• a is a constant, and
• X is the original scale value

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Ratio Measures

• Scores on a measure of ability to play tennis
• Person A9
• Person B3
• Ratio A/B9/33
• Apply transformation of Y5X
• Y5(9)45 Y5(3)15
• Ratio A/B45/153

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Levels of Measurement and Mathematics

• Level of measurement determines the how different
  observations can be combined
• At nominal level observations can not be added,
  subtracted, multiplied or divided
• At ordinal level observations can not be added,
  subtracted, multiplied or divided
• Rank ordering is only possible combination

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Levels of Measurement and Mathematics

• At interval level observations can be added and
  subtracted but not divided
• At ratio level observations can be added,
  subtracted, multiplied and divided