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

John Michael Linacre, Ph.D.

University of Sydney, Australia

Measurement, Meaningand Morality

Science Measurement Society ???Treaty of the Meter - 1875

I. Measurement

Science Measurement Society ???Treaty of the Meter - 1875

Society Measurement Science !!!

Babylonian land survey - 1500 BCE

I. Measurement

Science Measurement Society ???Treaty of the Meter - 1875

Measurement = Size or Quantity

Society Measurement Science !!!

Babylonian land survey - 1500 BCE

I. Measurement

Science Measurement Society ???Treaty of the Meter - 1875

Measurement = Size or Quantity

Quantity = Numerable Amount

Society Measurement Science !!!

Babylonian land survey - 1500 BCE

I. Measurement

Science Measurement Society ???Treaty of the Meter - 1875

Measurement = Size or Quantity

Quantity = Numerable Amount

“Measurement is the imposition of the rules of arithmetic on the world

around us.”

Society Measurement Science !!!

Babylonian land survey - 1500 BCE

I. Measurement

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

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 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 JuiceTexas Department of Agriculture

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 JuiceTexas Department of Agriculture

So, we trade Oranges by abstract arithmetical units of volume or weight!

One Unit Extra …

EarthquakesMeasures: Numbers aren’t enough ….

Earthquakes

???

Measures: Numbers aren’t enough ….

Earthquakes

!!!

Measures: Clarity requires additivity

“Measurement, in any true sense, is impossible in psychology, but their opinion

might change if new facts were established” Final Report BAAS, 1940 ???

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

???

???

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 !!!

???

???

Educational and Psychological Additive Units?

Rod(A)Rod(B)

= Rod(A+B)

N. Campbell: additivity requires rules of concatenation:The length rule is: “Place rods end-to-end” !!!

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 nPerson m

Item i ???

Old Facts of Measurement: Concatenation

Let Pni be the probability of success of person n on item i, inferred from data:

0 Pni 1

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)

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))

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) = …..

Conjecturing a rule based on probability …..

Outcome(Bni) + Outcome(Bmi) = log(Pni / (1-Pni)) + log(Pmi / (1-Pmi))

= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )

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”

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 ….

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”

Outcome (Bni+Bmi) = log( Probability of coinciding on success / Probability of coinciding on failure )

= log( (Pni * Pmi) / ((1-Pni)*(1-Pmi)) )

Due to self-coincidence, Outcome(Bni) is unchanged ….

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”

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) !!!

Due to self-coincidence, Outcome(Bni) is unchanged ….

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”

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!!

Due to self-coincidence, Outcome(Bni) is unchanged ….

Revising a rule based on probability …..

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 nPerson m

Item i !!!

Old Facts of Measurement: Concatenation

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

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:

!!!

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.

!!!

The simple dichotomous Rasch model …..

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”.

Least Fatigued

Person

Most FatiguedPerson

Level of DifficultyHow demanding/fatiguing is the

activity?

Equal

Intervals

More Demanding (Fatiguing)

Activity

Less Demanding (Fatiguing) Activity

Level of FatigueHow fatigued is the person?

1 2 3 4 5 6 7 8 9 10 11

Getting out of bed

Bathe & dress Sports activity

Yard work

Mallinson (2001)

WRAT3 Item Map and Absolute Scale

Measures must be:

III. Measurement, Meaning and Morality

ideal, but practicalrigorous, but accommodating

demanding, but forgiving

quantitative, but qualitative

forward-looking, but faithful to the past

fair, even-handed, honest

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)

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).

!!!

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 …. ???

In Politics: the bad …

“Sixty thousand measures of weight in France before the Revolution of 1789. ... 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)

In Religion:

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

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.

John Michael Linacre, Ph.D.

University of Sydney, Australia

Measurement, Meaningand Morality