1. Measurement, Meaning and Morality
John Michael Linacre, Ph.D.
University of Sydney, Australia

2. I. Measurement Science  Measurement  Society ???
Treaty of the Meter

3. I. Measurement Science  Measurement  Society ???
Treaty of the Meter
Society  Measurement  Science !!!
Babylonian land survey BCE

4. I. Measurement Science  Measurement  Society ???
Treaty of the Meter
Society  Measurement  Science !!!
Babylonian land survey BCE
Measurement = Size or Quantity

5. I. Measurement Science  Measurement  Society ???
Treaty of the Meter
Society  Measurement  Science !!!
Babylonian land survey BCE
Measurement = Size or Quantity
Quantity = Numerable Amount

6. I. Measurement
Science  Measurement  Society ???
Treaty of the Meter
Society  Measurement  Science !!!
Babylonian land survey BCE
Measurement = Size or Quantity
Quantity = Numerable Amount
“Measurement is the imposition of the rules of arithmetic on the world around us.”

7. “One more unit means the same amount extra,
One Unit Extra …
“One more unit means the same amount extra,
no matter how much we already have.”

8. “One more unit means the same amount extra,
One Unit Extra …
“One more unit means the same amount extra,
no matter how much we already have.”
One more Orange = One more unit of Juice ???

9. “One more unit means the same amount extra,
One Unit Extra …
“One more unit means the same amount extra,
no matter how much we already have.”
One more Orange = One more unit of Juice ???
One more Orange = 59 cc. – 177 cc. of Juice
Texas Department of Agriculture

10. “One more unit means the same amount extra,
One Unit Extra …
“One more unit means the same amount extra,
no matter how much we already have.”
One more Orange = One more unit of Juice ???
One more Orange = 59 cc. – 177 cc. of Juice
Texas Department of Agriculture
So, we trade Oranges by
abstract arithmetical units of volume or weight!

11. Measures: Numbers aren’t enough ….
Earthquakes

12. Measures: Numbers aren’t enough ….
Earthquakes
???

13. Measures: Clarity requires additivity
Earthquakes
!!!

14. Educational and Psychological Additive Units?
“Measurement, in any true sense, is impossible in psychology, but their opinion might change if new facts were established”
Final Report BAAS, 1940
???

15. Educational and Psychological Additive Units?
“Measurement, in any true sense, is impossible in psychology, but their opinion might change if new facts were established”
Final Report BAAS, 1940
???
“the stars .... we would never by any means investigate their chemical composition”
Auguste Comte, 1842
???

16. Educational and Psychological Additive Units?
“Measurement, in any true sense, is impossible in psychology, but their opinion might change if new facts were established”
Final Report BAAS, 1940
???
“the stars .... we would never by any means investigate their chemical composition”
Auguste Comte, 1842
???
1859, Gustav Kirchoff - spectral analysis of Sun !!!

17. Old Facts of Measurement: Concatenation
Rod(A)
Rod(B)
= Rod(A+B)
N. Campbell: additivity requires rules of concatenation:
The length rule is: “Place rods end-to-end” !!!

18. Old Facts of Measurement: Concatenation
Rod(A)
Rod(B)
= Rod(A+B)
N. Campbell: additivity requires rules of concatenation:
The length rule is: “Place rods end-to-end” !!!
We need the psychological concatenation rule for:
Outcome(Bni)
Outcome(Bmi)
=Outcome(Bni+Bmi)
Person n
Person m
???
Item i

19. Conjecturing a rule based on probability …..
Let Pni be the probability of success of
person n on item i, inferred from data:
0  Pni  1

20. Conjecturing a rule based on probability …..
Let Pni be the probability of success of
person n on item i, inferred from data:
0  Pni  1
Commensurate with an infinite latent variable:
0  Pni / (1-Pni)  

21. Conjecturing a rule based on probability …..
Let Pni be the probability of success of
person n on item i, inferred from data:
0  Pni  1
Commensurate with an infinite latent variable:
0  Pni / (1-Pni)  
-   log(Pni / (1-Pni))  

22. Conjecturing a rule based on probability …..
Let Pni be the probability of success of
person n on item i, inferred from data:
0  Pni  1
Commensurate with an infinite latent variable:
0  Pni / (1-Pni)  
-   log(Pni / (1-Pni))  
Suppose: Outcome(Bni) = log(Pni / (1-Pni))
Outcome(Bmi) = log(Pmi / (1-Pmi))
Then: Outcome(Bni) + Outcome(Bmi) = …..

23. Revising a rule based on probability …..
Outcome(Bni) + Outcome(Bmi) =
log(Pni / (1-Pni)) + log(Pmi / (1-Pmi))
= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )

24. Revising a rule based on probability …..
Outcome(Bni) + Outcome(Bmi) =
log(Pni / (1-Pni)) + log(Pmi / (1-Pmi))
= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )
So, a revised concatenation rule: Let “Outcome” be
“the log-odds of coincident scored observations”

25. Revising a rule based on probability …..
Outcome(Bni) + Outcome(Bmi) =
log(Pni / (1-Pni)) + log(Pmi / (1-Pmi))
= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )
So, a revised concatenation rule: Let “Outcome” be
“the log-odds of coincident scored observations”
Due to self-coincidence, Outcome(Bni) is unchanged ….

26. Revising a rule based on probability …..
Outcome(Bni) + Outcome(Bmi) =
log(Pni / (1-Pni)) + log(Pmi / (1-Pmi))
= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )
So, a revised concatenation rule: Let “Outcome” be
“the log-odds of coincident scored observations”
Due to self-coincidence, Outcome(Bni) is unchanged ….
Outcome (Bni+Bmi) = log( Probability of coinciding on success / Probability of coinciding on failure )
= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )

27. Revising a rule based on probability …..
Outcome(Bni) + Outcome(Bmi) =
log(Pni / (1-Pni)) + log(Pmi / (1-Pmi))
= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )
So, a revised concatenation rule: Let “Outcome” be
“the log-odds of coincident scored observations”
Due to self-coincidence, Outcome(Bni) is unchanged ….
Outcome (Bni+Bmi) = log( Probability of coinciding on success / Probability of coinciding on failure )
= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )
= Outcome(Bni) + Outcome(Bmi) !!!

28. Revising a rule based on probability …..
Outcome(Bni) + Outcome(Bmi) =
log(Pni / (1-Pni)) + log(Pmi / (1-Pmi))
= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )
So, a revised concatenation rule: Let “Outcome” be
“the log-odds of coincident scored observations”
Due to self-coincidence, Outcome(Bni) is unchanged ….
Outcome (Bni+Bmi) = log( Probability of coinciding on success / Probability of coinciding on failure )
= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )
= Outcome(Bni) + Outcome(Bmi) !!!
A concatenation rule for people taking test items!!

29. Old Facts of Measurement: Concatenation
Rod(A)
Rod(B)
= Rod(A+B)
N. Campbell: additivity requires rules of concatenation:
We have the psychological concatenation rule for:
Outcome(Bni)
Outcome(Bmi)
=Outcome(Bni+Bmi)
Person n
Person m
!!!
Item i

30. The “Final Report” was not the last word ….
A Glorious “New Fact” !!!
The “Final Report” was not the last word ….
There can be measurement in Education, Psychology and the Social Sciences equally as valid and rugged as in Physics!
But Social Science measurement requires painstaking effort, exactly as in Physics

31. The simple dichotomous Rasch model …..
Outcome(Bni) = log(Pni / (1-Pni))
Define: Outcome(Bni) = Bn - Di
Bn - Di = log(Pni / (1-Pni))
which can be rewritten as the Rasch model:
!!!

32. The simple dichotomous Rasch model …..
Outcome(Bni) = log(Pni / (1-Pni))
Define: Outcome(Bni) = Bn - Di
Bn - Di = log(Pni / (1-Pni))
which can be rewritten as the Rasch model:
!!!
Finding: the Rasch model is a concatenation rule, a “new fact” (Rasch, 1953), operationalizing “true” measurement for education and psychology.

33. II. Measurement and Meaning
Mohammed ibn-Musa al-Khowarizmi (830 CE, “Algorithm”)
Book: Hisab al-jabr w'al muqabala (“Algebra”)
“Calculating by restoring and comparing”
What we want to find out is “the thing”,
a value on a (latent) variable:
= shay’ (Arabic) heard as xay (Spanish)  x (English)
So meaning is expressed as measures on an x-axis representing amounts of the latent variable “thing”.

34. How demanding/fatiguing is the activity? How fatigued is the person?
Mallinson (2001)
Level of Difficulty
How demanding/fatiguing is the activity?
More Demanding (Fatiguing) Activity
Less Demanding (Fatiguing) Activity
Yard work
Getting out of bed
Sports activity
Bathe & dress 
Equal Intervals 1 2 3 4 5 6 7 8 9 10 11
Level of Fatigue
How fatigued is the person?
Least Fatigued
Person
Most Fatigued

35. WRAT3 Item Map and Absolute Scale

36. III. Measurement, Meaning and Morality
Measures must be:
ideal, but practical
rigorous, but accommodating
demanding, but forgiving
quantitative, but qualitative
forward-looking, but faithful to the past
fair, even-handed, honest

37. In Science:
“The development of [physical] metrology … shows that the same principle is being fulfilled in physics. The accord of ethical and physical principles was first noted by Sir Arthur Eddington, when in 1920 he chose the words from The Book of Deuteronomy* as an epigraph to the chapter on Weyl’s unified theory in his Space, Time and Gravitation We obviously live in the world where the fundamental principles of ethics and physics agree with each other.” !!!
(Tomilin, 1999)

38. In Politics: the good …
!!!
“Chau conferred great gifts, and the good were enriched. ... He carefully attended to the weights and measures, … and the good government of the kingdom took its course.” (Confucius, The Analects, 20. ca. 500 BCE).

39. In Politics: the good …
!!!
“Chau conferred great gifts, and the good were enriched. ... He carefully attended to the weights and measures, … and the good government of the kingdom took its course.” (Confucius, The Analects, 20. ca. 500 BCE).
But what if …. ???

40. In Politics: the bad …
“Sixty thousand measures of weight in France before the Revolution of the falsification of standards by the feudal land owners and the distrust, justified or not, of the peasants. A common demand was to unify weights and measures - not to avoid paying feudal dues but to assure an honest amount payable. The rallying cry: un roi, une loi, un poids, et une mesure (one king, one law, one weight, and one measure) was a slogan of equality and centralization, … one that the Revolution furthered.”
(Kennedy, 1989)

41. In Religion: “And O my people! give just measure and weight”
The Qur’an, The Prophet Hud, 11:85, ca. 600 CE

42. Measurement, meaning and morality - honestly, they work together.
In Religion:
“And O my people! give just measure and weight”
The Qur’an, The Prophet Hud, 11:85, ca. 600 CE
Surely this injunction applies equally to
educational, psychological and physical measures.
Measurement, meaning and morality
- honestly, they work together.

43. Measurement, Meaning and Morality
John Michael Linacre, Ph.D.
University of Sydney, Australia