Measurement levels

Mapping empirical to numerical relations

=> The mapping function delivers a distinct (homomorph) mapping of the empirical to numerical relations

Thus, each empirical relation corresponds to a numerical relation

• nominal scales

• ordinal scales

• interval scales

• ratio scales

• Simplest kind of ”measuring“

• Not really scaling, rather labelling – objects with the same attribute value get the same measurement value (equivalence relation)

• each transformation which assigns different numbers to objects with different attributes is allowed

• Examples – Category systems(ICD,DSM)

  • person with anxiety disorder is not better or worse than person with depression

  • => disorders are qualitatively different / distinct

• Scale orders objects along some continuum (order relation)

• Possible statements:

  • two objects are equal or different

  • larger or smaller relations

• Monotone transformations of the measurement values is allowed

• order remains intact

  • 1,2,3,4 or 8,9,10,11 => order relation invariant (unverändert)

• Example: soccer table

Differences can be interpreted

• 30–10 equals 80–60

• But: Does the difference in ability between grades 'very good' and 'good' equal the difference between 'fair' and 'failure‘?

  => The interval scale property of school grades can be doubted.

• You may argue that the numerical intervals are equal. However, it's important to remember that these intervals refer to the numerical relational system, not to the empirical relational system.

• In this example, the mapping function is incorrect because the relations of the empirical relative are not mapped properly

• There is no predefined measurement unit and no absolute but an arbitrary (!) yet meaningful zero point

• Example: Temperature scales

  • Celsius scale: centered on freezing point of water as zero point

  • Fahrenheit: Centered on the temperature produced by the mixing of equal weights of snow and common salt

• In addition: true zero point

  • unit not fixed

  • ratios can be interpreted

    • object a is twice as large as object b

• Example: physical scales (length, volume, Kelvin scale)

• Allows us to compare ratios between numbers

• Example,if you measured the time it takes 3 people to run a race, their times may be

  • 10 seconds (Racer A),

  • 15 seconds (Racer B) and

  • 20 seconds (Racer C)

    
    

    => You can say that it took Racer C twice as long as Racer A. Unlike the interval scale, the ratio scale has a true/natural zero value

Interval scale: Are ratios of scale values meaningful?

• No, because the zero point is arbitrary (but not meaningless)

• Illustration: Is the following statement justified when using a Celsius or Fahrenheit scale?: ”It is twice as hot as yesterday“

Celcius °

• First day: 10 degrees Celsius

• Second day: 20 degrees Celsius

• The second day was twice as hot (20°C / 10°C = 2) as the first day, right?

Fahrenheit

• We could also use the corresponding values of the Fahrenheit scale

• First day: 50 degrees Fahrenheit

• Second day: 68.

• It looks like the second day was 68°F / 50°F = 1.36 as hot as the first one, not twice as hot

Hint #1: Measurement as assignments of numbers to represent relations between objects.

Hint #2: Does 0% (i.e., number) correspond to zero importance (i.e., empirical property)? What‘s the natural zero point of “perceived importance“?

What is completely obscured here

• Using real numbers (as seen in previous examples) pertains to the numerical relational system

• However, nothing is necessarily implied regarding the empirical relational system, which concerns relationships between properties of sets of objects (such as perceived pain, attitude statements, intelligence, etc.).

Examples

• response time

• Factor scores (i.e., z-scores) from factor analysis (real number line)

Extraversion scale, expressed as z-scores; participants A, B, and C

• A: z-score = 0

• B: z-score = 1

• C: z-score = 2

  
  

  => Difference between A and C is twice the difference between B and C

  => However, does it follow that the difference in extraversion (i.e., empirical relational system) between A and C is twice the difference between B and C?

By merely assigning these numbers to objects, we have not at all established a mapping of the empirical to numerical relations

We just assume relations between these numbers to correspond to relations between degrees of extraversion