1. Levels of Measurement
Where were we: ah, yes! Representational measurement … NOW …
(Numerical) Representations of a measurement structure are not *usually unique.
*Except for absolute measurement… not normally a problem for Soc Scs
This is the starting point of Classical Measurement Theory’s “Levels of Measurement” … How unique are these values then?
ANSWER: Original values can be transformed so long as they preserve the original structure properties…

2. Levels of Measurement: TRANSFORMATIONS
 “Permissible ( = Mandatory!) Transformations
Are those that can be made to the original values and not change defining structure/properties
Usually a CLASS of transformation

3. Levels of Measurement
Once a level is assigned, and the consequent permissible transformations decided, this sets limits to what you may legitimately do within these constraints …
“Appropriate Statistics”
aggregative measures which only perform permissible operations on the values
E.g. no arithmetic ( etc) on ordinal values, so Median rather than Mean
A contested characteristic, as we shall see …

4. Levels of Measurement
An hierarchical arrangement of common scale types were termed “Levels of Measurement “
associated mainly with Stevens
exist in various versions, most common is “NOIR”
Defined by:
Basic operations (between objects w.r.t. property)
Structure-preserving transformation/s (main criterion of “level”)
Appropriate statistics

5. Absolute M Ratio Interval NM Ordinal Nominal (inc dichot.) none All
Name
Permissible Transformatn
Preserves
Direct Data
Aggregate measures
Absolute
none
All
(true 0) -- M
Ratio
similarity
(a/b)
(Unit change)
Counts, rates, distances
Interval
linear
(a-b)
..& origin
Rating
PM Correln. 
Covariance  NM
Ordinal
monotonic
order
Ranking
Kendall’s 
G & K’s 
Nominal
(inc dichot.)
isotonic
categories
Sorting
-sq based
Pairbonds

6. Criticisms of Steven’s LoM
Subject to considerable criticism:
Many scale types lie outside a simple hierarchy (e.g. POSET, qv)
Single Levels disguise heterogeneity (esp. types of ordering) (q.v)
Appropriate statistics can be arbitrary (and even “inappropriate”:
S.D. is not linear in data values
Spearman’s rho uses intervals for ordinal data
No inbuilt “tests” of compliance
Hence susceptible to “measurement by fiat”
especially Metric-Non-metric [Ordinal-Interval] difference
and therefore most crucial “barrier to quantification” overcome also by fiat?
… and therefore based on researcher’s judgment (or advocacy)?

7. Monotonic Transformations (ordinal data)

8. Types of Order: Weak Partial Strict

9. On what grounds did they criticise?
Measurement:
On what grounds did they criticise?
Models make sets of assumptions (e.g. well-behaved error) necessary to obtain a solution, which may or may not actually hold
“we buy information by making assumptions
… and scales themselves are models “ CH Coombs
Metric models additionally assume equality of unit intervals and use that in deriving results.
If a model does not hold for a set of data, and we cannot justify metric assumptions, we have no way of knowing whether that failure is due simply to “pseudo-quantification”.

10. Measurement: BUT SURELY … Measurement … for what?
What are measured variables to be used for?
Main motivation to measure (and raise measurement status of variables) arose because of desire to use powerful Multivariate models of analysis …
Which require[d] “metric” (interval level, or higher ) variables …
n.b. Before development of
non-metric methods,
categorical models,
iterative estimation needing computing

11. Measurement:
By contrast, most social science data are (at least on Rep. Criteria) non-metric.
So? Quantification = (raising LoM) is needed. How?
(1) Treat “interval-looking” variables AS IF interval
Rigorists (scornfully) cried foul: “Pseudo-quantification“
Representationalists (snootily) termed it “Index Measurement” (contrast to RepM)
Both condemned it.

12. Measurement: (2a) Make weaker (non-metric) assumptions
The so-called “Non-metric revolution” associated with U of Michigan, and esp. Clyde Coombs
Make a virtue of ordinality and build models on this basis
Not a huge success
e.g. fully non-metric FA which sought only to recover rank-order of co-ordinates on a factor
Non-metric Unfolding, which was not robust against error nor fully solvable in absence of all data rank-orderings (I-scales)

13. Measurement:
(2b) UNTIL the advent of Roger Shepard: <trumpets off-stage> Showed that
“non-metric constraints, if imposed in sufficient number guarantee a metric result” – and produced (iterative) computer procedure to do it!
…. More of this anon tomorrow, when we consider another response …
(3) The Ubiquitous Rating Scale …
MEANWHILE … a Coombsian Model of Measurement Practices …
“Read, mark, learn and inwardly digest”!

14. Measurement: