A new measurement in psychology has emerged from a confluence of scientific and social forces which are producing a revolution in social science methodology. We begin by reviewing how the semiotics of C.S.Peirce revise and enrich our interpretation of S.S. Stevens' four "kinds of measurement" into a creative dynamic for the evolution of one kind of useful measurement. Then we recall two turning points in social history which remind us of the antiquity and moral force of our need for stable measures. Next we review the psychometric and mathematical histories of measurement, show how the obstacles to inference shape our measurement practice and summarize Georg Rasch's contributions. This brings us to some applications of the "new" measurement models produced by Rasch's work. Finally we review some mistakes that the history of measurement can teach us to stop making.
Peirce Semiotics
Semiotics is the science of signs. It provides a developmental map of creative thought which shows how thinking progresses stepbystep from qualitative flashes of insight to the reproducible quantitative laws of process which are the tool and triumph of science. Table 1 outlines six levels of conscious signs through which science is reached (Buchler 1940, Sheriff 1994).
C.S.Peirce 1903 
Stevens 1939 
W.Kinston 1985 
6 STEPS TO SCIENCE 
5. possible ICON 
FANCY qualitative  
6. possible INDEX 
Entity real idea 
THOUGHT qualitative  
7. factual INDEX 
Nominal  Observable existent 
OBJECT qualitative 
8. possible SYMBOL 
Ordinal  Comparable quantity 
SCORE quantitative 
9. factual SYMBOL 
Interval Ratio 
Measurable unit 
MEASURE quantitative 
10. arguable SYMBOL 
Relatable process 
RELATION quantitative  
Peirce's complete set of signs mark out ten steps in the evolution of knowing (Sheriff 1994). The four earliest steps, however, precede the kind of awareness we ordinarily recognize as scientific so they are omitted here. Note how Peirce improves on Stevens in two ways: 1) Twice as many identifiable steps. 2) A clear sequence of connected thinking from the wildest qualitative hypothesis to the most objectively quantitatively measured process. 
The first level of "scientific" consciousness is private "fancy", a flash of thought, a wild hypothesis. The thought is ours alone. We may never rethink it. Peirce calls this first level of awareness a "possible icon". It is the seed of creativity.
Then some wild ideas acquire meaning. We think them again and encounter others who also think them. This is the step to Peirce's second level. A particular idea becomes more than a private fancy. It becomes something to return to, to talk about. It is still but a quality. We cannot point and say, "There it is." But we can think and talk it. Eddington (1946) and Kinston (1985) call this level an "entity". Peirce calls it a "possible index".
The next step up is an invention of a way to "see" what we mean. We nominate an "experience", a "thing", to serve as an instance of our "idea". This makes our idea observable and "real". Now our idea is a thing of the world. We can do more than think it. We can seek it. We can count it. We have reached the third level at which Stevens'(1946) first "kind of measurement", the "nominal", finally emerges. Eddington and Kinston call this level an "observable" and Peirce calls it a "factual index".
Stevens Revised
Peirce's first two steps to science precede Stevens' "nominal". This shows us that a nominal is not where science begins. A nominal is only reached as a consequence of an evolution of prior possible icons and indices. It is the successful development of these precursors which bequeath the nominal its potential merit as a qualitative thought with quantitative possibilities. We dream before we think. We think before we point. The scientific harvest which may follow depends on the fertility of our imagination and the cultivation of our thought.
Then, as we gather examples of our evolving entity, we discover preference. Some "things" please us. Others do not. The presence and absence of our entities acquire irresistible valence. We begin to count on more of their "goodness", less of their "badness". Our preferences propel us to a fourth level, a higher sign, a "comparable", corresponding to Steven's "ordinal" kind of measurement. We discover that entity nomination is not only a necessary step toward science, but that we are unable to nominate without preferring. Steven's calls this "ordinal measurement". Peirce calls it a "possible symbol". Our choices and pursuits of this latency are the crucible in which the tools of science are forged.
At this stage the subtle and decisive invention of abstract quantification begins to emerge. Valuing more or less of particular "things" begins to imply valuing more or less of a common "something" these "things" imply, a latent variable. We begin to abstract from counting apples to ideas of "appleness".
To count is to begin to measure. But counting concrete objects is ambiguous. Counts of real objects do not maintain the magnitude equivalence necessary to qualify as measures. When bricks do not approximate a single size, each piece requires individual attention, building is hard to plan, requires continuous readjustment and suffers awkward irregularity. It takes uniform bricks to regularize construction, to draw plans. It is the same in trade. How could my grocer sell me a mix of apples and oranges without the currency they share.
This earthy necessity propels us upward to a fifth level. We invent an even more abstract sign, the Eddington and Kinston "measurable", which reaches beyond a comparison of "more" or "less" to build an imaginary variable along which we can define and maintain an ideal of perfectly equal units which, while abstract in theory, are, nevertheless, approximable in practice. Stevens calls this "interval measurement". Peirce calls it a "factual symbol". It is the fundamental language of science. We will discover that its successful realization requires clear ideas as to the obstacles to inference and an understanding of their mathematical solutions.
Stevens distinguishes ratios from intervals. But logarithms of ratios are intervals. And exponentiated intervals are ratios. If we have one, we have the other. The only basis for a distinction would be a belief in "natural origins". But there are none. Look around. We put our origins wherever they work best: either end of a yardstick, where water (or alcohol) becomes solid, where molecular motion is extrapolated to cease or.... Our origins are convenient reference points for useful theories. When our theories change so do our origins. (Wright & Masters 1982 p.9)
Stevens' taxonomy stops here. Peirce and Kinston reach one step further to the level of Peirce's "arguable symbol" and Kinston's "relatable". This is where theories of process begin. What good is one variable, if it does not lead to another? And no, at last, a job for linear statistics emerges. To see what our variables imply about process, we plot them against one another. We analyze their variance, estimate regressions and model equations.
Implications for Practice
Peirce's semiotics specify six steps which we must climb through to reach our pride of science. A quick regression of raw data is always misleading. When we do not think and build our way through the five steps which precede statistical analysis with care and meaning, our "research" comes up empty. We can always get some "numbers". But they will not add up to reproducible meaning.
Mistaking Stevens' nominal/ordinal/interval/ratio taxonomy as four "kinds of measurement", each with its own "kind of statistics", has polarized social scientists into a qualitative/quantitative antipathy which cuts off communication, paralyses progress and overlooks the inevitability of a careful stepwise evolution beginning two steps before Stevens' nominal and going a step beyond his interval/ratio to reach, not two or four kinds of measurement but just one kind of science.
The moment we nominate a possible index as sufficiently interesting to observe, we experience a preference which elevates that nominal to an ordinal. Then as we replicate our observations we cannot resist counting them and using these counts as though they were measures. The final step from concrete ordinals to abstract intervals is forced upon us by the necessities of commerce. Just as we cannot trade without currency, we cannot reason quantitatively without linear measures.
Thus the discovery, invention and construction of scientific knowledge begins with 1) a wild hypothesis which gets thought into 2) a reproducible idea, which becomes realized in 3) an observable instance, to be interpreted as 4) a concrete ordinal which is carefully built into 5) an abstract measurable suitable, finally, for analyses in relation to other measurables and so to the construction of 6) promising theories.
Long before science or mathematics emerged as professions the commercial, architectural, political and moral necessities for abstract, exchangeable units of unchanging value were recognized and pursued. A fair weight of seven was a tenant of faith among seventh century Muslims. Muslim leaders were censured for using less "righteous" standards. (Sears, 1997)
Twelve centuries ago Caliph 'Umar b. 'Abd al'Aziz, ruled that:
The people of alKufa have been struck with trial, hardship, oppressive governments and wicked practices. The righteous law is justice and good conduct. I order you to take in taxes only the weight of seven. (Damascus, 723)
Seven centuries ago King John decreed that:
There shall be one measure of wine throughout Our kingdom, and one of ale, and one measure of corn, to wit, the London quarter, and one breadth of cloth, to wit, two ells within the selvages. As with measures so shall it be with weights. (Magna Carta, Runnymede, 1215)
Some say the crux of the French Revolution was outrage against unfair measures. The true origins of stable units for length, area, volume and weight were the necessities of commerce and politics. The steam engine is responsible for our measures of temperature and pressure.
Counting Events does NOT Produce Equal Units
The patriarch of educational measurement, Edward Thorndike, observed:
If one attempts to measure even so simple a thing as spelling, one is hampered by the fact that there exist no units in which to measure. One may arbitrarily make up a list of words and observe ability by the number spelled correctly. But if one examines such a list one is struck by the inequality of the units. All results based on the equality of any one word with any other are necessarily inaccurate. (Thorndike 1904, p.7)
Thorndike saw the irregularity in counting concrete events, however indicative they might seem (Engelhard 1991, 1994). One might observe signs of spelling. But simply counting would not measure spelling. The problem of entity ambiguity is ubiquitous in science, commerce and cooking. What is an apple? How many apples make a pie? How many little apples equal one big one? Why don't three apples always cost the same? With apples, we solve entity ambiguity by renouncing the concrete apple count and turning, instead, to abstract apple volume or weight. (Wright 1992, 1994)
Raw Scores are NOT Measures
Thorndike was not only aware of the "inequality of the units" counted but also of the nonlinearity of the "raw scores" counting produced. Raw scores are bound to begin at "none right" and end at "all right". But the linear measures we intend raw scores to imply have no boundaries. Figure 1 is a typical raw score to measure ogive. It shows how the monotonically increasing ogival exchange of one more right answer for a measure increment is steepest in the middle where items are dense and flat at the extremes of 0% and 100%. One more right answer implies the least measure increment near 50%, but an infinite increment at each extreme. In Figure 1 the measure distance along the horizontal axis which corresponds to a 10 percentile raw score increment from 88% to 98% up the vertical axis is 5 times greater than the measure distance corresponding to a 10 percentile raw score increment from 45% to 55%.
Number of Items 
Normal Test 
Uniform Test 
10  2.0  3.5 
25  4.6  4.5 
50  8.9  6.0 
100  17.6  8.0 
These calculations are explained on pages
143151 of Best Test Design (Wright & Stone
1979). The ratio for a normal test of L
items is: log_{e}{2(L1)/(L2)}/log_{e}{(L+2)/(L2)} 
Table 2 shows the numerical magnitude of this raw score bias against extreme measures for tests of normally distributed and uniformly distributed item difficulties. The tabled values are ratios formed by dividing the measure increment corresponding to one more right answer at the next to largest extreme step by the measure increment corresponding to one more right answer at the smallest central step. Even when item difficulties spread out uniformly in equal increments, the raw score bias against measure increments at the extremes of a 25 item test is a factor of 4.5. When the item difficulties of a 50 item test distribute normally, the bias against extreme measures is a factor of 8.9!
Raw score bias is not confined to dichotomous responses. The bias is just as severe for partial credits, rating scales and, of course, the infamous Likert Scale, the misuse of which pushed Thurstone's seminal 1920's work on how to construct linear measures from ordinal raw scores out of use.
This raw score bias in favor of central scores and against extreme scores means that any linear statistical method like analysis of variance, regression, generalizability, LISREL or factor analysis that misuses nonlinear raw scores or Likert scales as though they were linear measures will produce systematically distorted results. Like the nonlinear raw scores on which they are based, all results will be target biased and sample dependent and hence inferentially ambiguous. (Wright & Stone 1979 pp 49, Wright & Masters 1982 pp 2737, Wright & Linacre 1989) Little wonder that so much social "science" remains transient description of never to be reencountered situations, easy to doubt with almost any replication.
An obvious first law of measurement is:
Before applying linear statistical methods, use a measurement model to construct linear measures from your observed raw data.
There are many advantages to working with modelcontrolled linearization. Each measure is accompanied by a realistic estimate of its precision and a mean square residualfromexpectation evaluation of the extent to which the raw ordinal data from which the measure has been estimated fit the measurement model. When we now advance to graphing results and applying linear statistics to analyze relationships among variables, we not only have linear measures to work with, we also have estimates of their statistical precision and validity.
Thurstone's Measurement
Between 1925 and 1932 Louis Thurstone published 24 articles and a book on how to construct good measures.
Unidimensionality:
The measurement of any object or entity describes only one attribute of the object measured. This is a universal characteristic of all measurement. (Thurstone 1931, p.257)
Linearity:
The very idea of measurement implies a linear continuum of some sort such as length, price, volume, weight, age. When the idea of measurement is applied to scholastic achievement, for example, it is necessary to force the qualitative variations into a scholastic linear scale of some kind. (Thurstone & Chave 1929, p.11)
Abstraction:
The linear continuum which is implied in all measurement is always an abstraction...There is a popular fallacy that a unit of measurement is a thing  such as a piece of yardstick. This is not so. A unit of measurement is always a process of some kind...
Invariance:
... which can be repeated without modification in the different parts of the measurement continuum. (Thurstone 1931, p.257)
Samplefree item calibration:
The scale must transcend the group measured. One crucial test must be applied to our method of measuring attitudes before it can be accepted as valid. A measuring instrument must not be seriously affected in its measuring function by the object of measurement...Within the range of objects...intended, its function must be independent of the object of measurement. (Thurstone 1928, p.547)
Testfree person measurement:
It should be possible to omit several test questions at different levels of the scale without affecting the individual score ... It should not be required to submit every subject to the whole range of the scale. The starting point and the terminal point...should not directly affect the individual score. (Thurstone 1926, p.446)
Guttman's Scale
In 1944 Louis Guttman showed that the meaning of a raw score, including one produced by Likert scales, would be ambiguous unless the score defined a unique response pattern.
If a person endorses a more extreme statement, he should endorse all less extreme statements if the statements are to considered a scale...We shall call a set of items of common content a scale if [and only if] a person with a higher rank than another person is just as high or higher on every item than the other person. (Guttman 1950, p.62)
According to Guttman only data which approximate this kind of conjoint transitivity can produce unambiguous measures. A deterministic application of Guttman's "scale" is impossible. But his ideal of conjoint transitivity is the kernel of Norman Campbell's (1920) fundamental measurement. Notice the affinity between Guttman's "scalability" and Ronald Fisher's "sufficiency" (Fisher 1922). Both call for a statistic which exhausts the information to which it refers.
Although mathematics did not initiate the practice of measurement, it is the mathematics of measurement which provide the ultimate foundation for better practice and the final logic by which useful measurement evolves and thrives. As we review some of the ideas by which mathematicians and physicists built their theories, we will discover that, although each worked on his own ideas in his own way, their conclusions converge to a single formulation for measurement practice.
Concatenation
In 1920 physicist Norman Campbell deduced that the "fundamental" measurement on which physics was built required the possibility of explicit concatenation, like joining the ends of sticks to concatenate length or piling bricks to concatenate weight. Because psychological concatenation seemed impossible, Campbell concluded that there could be no fundamental measures in psychology.
Sufficiency
In 1920 Ronald Fisher, while applying his "likelihood" version of inverse probability to invent maximum likelihood estimation, discovered a statistic so "sufficient" that it exhausted from the data in hand all information concerning its modeled parameter. Fisher's discovery is much appreciated for its informational efficiency. But it has a concomitant property which is far more important to the construction of measurement. A statistic which exhausts all modelled information enables conditional formulations by which a value for each parameter can be estimated independently of the values of all other parameters in the model.
This follows because the functional presence of any parameter can be replaced by its sufficient statistic (Andersen 1977). Without this replacement each parameter estimation, each attempt to construct a generalizable measure, is forever foiled. The incidental distributions of the other parameters make every "measure" estimate situationspecific. Generality is destroyed.
This leads to a second law of measurement:
When a model employs parameters for which there are no sufficient statistics, that model cannot construct useful measurement because it cannot estimate its parameters independently of one another.
Divisibility
In 1924 Paul Levy (1937) proved that the construction of a law for probability distributions which is "stable" with respect to arbitrary decisions as to what is countable required infinitely divisible parameters (Feller 1950, 271). Levy's divisibility is logarithmically equivalent to conjoint additivity (Luce & Tukey 1964). Levy's conclusions were reenforced in 1932 when A.N.Kolmogorov (1950 pp.9 & 57) proved that independent parameter estimation also requires divisibility.
In 1992 Abe Bookstein reported his astonishment at the mathematical equivalence of every counting law he could find (1992, 1996). Provoked to decipher how this ubiquitous equivalence could have occurred, he discovered that the counting formulations were not only members of one simple mathematical family, but that they were surprisingly robust with respect to ambiguities of entity (which elements to count), aggregation (at what hierarchical level to count) and scope (for how long and how far to count). As he sought to understand the source of this remarkable robustness Bookstein discovered that the necessary and sufficient formulation was Levy's divisibility.
Additivity
American work on the mathematical foundations of measurement came to fruition with the proof by Duncan Luce and John Tukey (1964) that Campbell's concatenation was a physical realization of the mathematical law which is necessary for fundamental measurement.
The essential character of simultaneous conjoint measurement is described by an axiomatization for the comparison of effects of pairs formed from two specified kinds of "quantities"... Measurement on interval scales which have a common unit follows from these axioms.
A close relation exists between conjoint measurement and the establishment of response measures in a twoway table ...for which the "effects of columns" and the "effects of rows" are additive. Indeed the discovery of such measures...may be viewed as the discovery, via conjoint measurement, of fundamental measures of the row and column variables. (Luce & Tukey 1964, p.1)
Their conclusion writes a third law of measurement:
When no natural concatenation operation exists, one should try to discover a way to measure factors and responses such the "effects" of different factors are additive. (Luce & Tukey 1964, p.4)
The common measures by which we make life better are so familiar that we seldom think about "why" or "how" they work. A mathematical history of inference, however, takes us behind practice to the theoretical requirements which make measurement possible. Table 3 articulates an anatomy of inference into four obstacles which stand between raw data and the stable inference of measures they might imply.
OBSTACLES  SOLUTIONS  INVENTORS 
UNCERTAINTY have > want now > later statistic > parameter 
PROBABILITY binomial odds regular irregularity misfit detection 
Bernoulli 1713 Bayes 1764 Laplace 1774 Poisson 1837 
DISTORTION nonlinearity unequal intervals incommensurability 
ADDITIVITY linearity concatenation conjoint additivity 
Fechner 1860 Helmholtz 1887 N.Campbell 1920 Luce/Tukey 1964 
CONFUSION interdependence interaction confounding 
SEPARABILITY sufficiency invariance conjoint order 
Rasch 1958 R.A.Fisher 1920 Thurstone 1925 Guttman 1944 
AMBIGUITY of entity, interval and aggregation 
DIVISIBILITY independence stability reproducibility exchangeability 
Levy 1924 Kolmogorov 1932 Bookstein 1992 de Finetti 1931 
Uncertainty is our motivation for inference. The future is uncertain by definition. We have only the past by which to foresee. Our solution is to capture this uncertainty in a matrix of inverse probabilities which regularize the irregularities that interrupt the continuity between what seems certain now but must be uncertain later.
Distortion interferes with the transition from observation to conceptualization. Our ability to figure things out comes from our faculty to visualize. Our power of visualization evolves from the survival value of body navigation through the space in which we live. Our antidote to distortion is to represent our observations of experience in the bilinear form that makes them look like the mostly two dimensional space we see in front of us. To "see" what experience "means", we "map" it.
Confusion is caused by interdependencies. As we look for tomorrow's probabilities in yesterday's lessons, confusing interactions intrude. Our resolution of confusion is to simplify the complexity we experience into a few shrewdly crafted "dimensions". We define and measure our dimensions one at a time. Their authority is their utility. "Truths" may be unknowable. But when our inventions "work", we have proven them "useful". And when they continue to work, we come to believe in them and may even christen them "real" and "true".
Ambiguity, a fourth obstacle to inference, occurs because there is no objective way to determine exactly which operational definitions of conceptual entities are the "right" ones. As a result only models which are indifferent to level of composition are robust enough to survive the vicissitudes of entity ambiguity. Bookstein (1992) shows that this requires functions which embody divisibility or additivity as in:
Fortunately the mathematical solutions to Distortion, Confusion and Ambiguity are identical. The parameters that govern the probability of the data must appear in either an entirely divisible or an entirely additive form. No mixtures of divisions and additions, however, are admissible.
The Probability Solution
The application of Jacob Bernoulli's 1713 binomial distribution as an inverse probability for interpreting the implications of observed events (Thomas Bayes, 1764, Pierre Laplace, 1774 in Stigler 1986, pp. 6367, 99105) was a turning point in the mathematical history of measurement. Our interests reach beyond the data in hand to what these data might imply about future data, still unmet, but urgent to foresee.
The first problem of inference is how to predict values for these future data, which, by the meaning of "inference", are necessarily "missing". This meaning of "missing", of course, must include not only the future data to be inferred but also all possible past data which were lost or never collected. Since the purpose of inference is to estimate what future data might be like before they occur, methods which require complete data (i.e. cannot analyze present data in which some values are missing) cannot be methods of inference. This realization engenders a fourth law of measurement:
Any statistical method nominated to serve inference which requires complete data, by this requirement, disqualifies itself as an inferential method.
But, if what we want to know is "missing", how can we use the data in hand to make useful inferences about the "missing" data they might imply? Inverse probability reconceives our raw observations as a probable consequence of a relevant stochastic process with a useful formulation. The apparent determinism of formulae like F = MA depends on the prior construction of relatively precise measures of F and M (For a stochastic discussion of this formulation of the "multiplicative law of accelerations" in which M is realized by solid bodies and F is realized by instruments of force, see Rasch 1960, 110114). The first step from raw observation to inference is to identify the stochastic process by which an inverse probability can be defined. Bernoulli's binomial distribution is the simplest process. The compound Poisson is the stochastic parent of all such measuring distributions (Rasch 1960, 122).
The Mathematical Solution
The second problem of inference is to discover which mathematical models can determine the stochastic process in a way that enables a stable, ambiguity resilient estimation of the model's parameters from data in hand. At first glance, this step may look obscure. Table 3 suggests that its history followed many paths, traveled by many mathematicians and physicists. One might fear that there was no clear second step but only a variety of unconnected possibilities with seemingly different resolutions. Fortunately, reflection on the motivations for these paths and examination of their mathematics leads to a reassuring simplification. Although each path was motivated by a particular concern as to what inference must overcome to succeed, all solutions end up with the same simple, easy to understand and easy to use formulation. The mathematical function which governs the inferential stochastic process must specify parameters which are either infinitely divisible or conjointly additive i.e. separable.
What does this summarize to?
1. Measures are inferences,
2. Obtained by stochastic approximations,
3. Of one dimensional quantities,
4. Counted in abstract units,
5. Which are impervious to extraneous factors.
To meet these requirements,
Measurement must be an inference of values for infinitely divisible parameters which define the transition odds between observable increments of a theoretical variable.
The Fundamental Measurement Model
In 1953 Georg Rasch (1960) found that the only way he could compare performances on different tests of oral reading was to apply the exponential additivity of Poisson's 1837 distribution (Stigler 1986, pp.182183) to data produced by a sample of students responding simultaneously to both tests. Rasch used the Poisson distribution as his model because it was the only distribution he could think of that enabled the equation of two tests to be entirely independent of the obviously arbitrary distribution of the reading abilities of the sample.
As Rasch worked out his solution to what became an unexpectedly successful test equating, he discovered that the mathematics of the probability process, the measurement model, must be restricted to formulations which produced sufficient statistics. Only when his parameters had sufficient statistics could these statistics replace and hence remove the unwanted person parameters from his estimation equations and so obtain estimates of his test parameters which were independent of the incidental values of the person parameters used in the model.
I never heard Rasch refer to Guttman's conjoint transitivity. Yet, as Rasch describes the necessities of his probability function, he defines a stochastic solution to the impossible requirement that data conform to a deterministic conjoint transitivity.
A person having a greater ability than another should have the greater probability of solving any item of the type in question, and similarly, one item being more difficult than another one means that for any person the probability of solving the second item correctly is the greater one. (Rasch 1960, p.117)
Rasch completes his argument for a measurement model on pages 117122 of his 1960 book. His "measuring function" on page 118 specifies the divisible definition of fundamental measurement for dichotomous observations as:
where P is the probability of a correct solution, f(P) is a function of P, still to be determined, b is a ratio measure of person ability and d is a ratio calibration of item difficulty.
Rasch explains this "measuring function" as an inverse probability "of a correct solution, which may be taken as the imagined outcome of an indefinitely long series of trials (p.118). Because "an additive system...is simpler than the original...multiplicative system," Rasch takes logarithms:
and transforms P into its logistic form:
Then Rasch asks: "Does there exist a function of the variable L which forms an additive system in parameters for person B and parameters for items D such that:
(pp.119120)
Finally, asking "whether the measuring function for a test, if it exists at all, is uniquely determined" Rasch proves that:
"is a measuring function for any positive values of C and A, if f_{0}(P) is", which "contains all the possible measuring functions which can be constructed from f_{0}(P). So that "By suitable choice of dimensions and units, i.e. of A and C for f(P), it is possible to make the b's and d's vary within any positive interval which may...be...convenient" (p.121).
Because of "the validity of a separability theorem (due to sufficiency):
It is possible to arrange the observational situation in such a way that from the responses of a number of persons to the set of items in question we may derive two sets of quantities, the distributions of which depend only on the item parameters, and only on the personal parameters, respectively. Furthermore the conditional distribution of the whole set of data for given values of the two sets of quantities does not depend on any of the parameters (p.122).
With respect to separability the choice of this model has been lucky. Had we for instance assumed the "NormalOgive Model" [as did Thurstone in the 1920's] with all s_{i} = 1  which numerically may be hard to distinguish from the logistic  then the separability theorem would have broken down. And the same would, in fact, happen for any other conformity model which is not equivalent  in the sense of f(P) = C{f_{0}(P)}^{A} to f(P) = b/d...as regards separability.
The possible distributions are...limited to rather simple types, but...lead to rather far reaching generalizations of the Poisson...process (p.122).
The Compound Poisson
Rasch (1960, 1961, 1963, 1968, 1969, 1977) shows that formulations in the compound Poisson family, such as Bernoulli's binomial, are not only sufficient for the construction of stable measurement, but that the "multiplicative Poisson" (Feller 1957, 270271) is the only mathematical solution to the formulation of an objective, sample and testfree measuring function. Andrich (1995, 1996) confirms that Rasch separability requires the Poisson distribution for estimating measures from discrete observations. Bookstein (1996) shows that the stochastic application of the divisibility necessary for ambiguity resilience requires the compound Poisson.
Conjoint Additivity and Rasch
Although conjoint additivity is acknowledged to be a decisive theoretical requirement for measurement, few psychologists realize that Rasch models are its practical realization. Rasch models construct conjoint additivity by applying inverse probability to empirical data and then testing these data for their goodnessoffit to this construction (Keats 1967, Fischer 1968, Perline, Wright, Wainer 1978).
The Rasch model is a special case of additive conjoint measurement... a fit of the Rasch model implies that the cancellation axiom (i.e. conjoint transitivity) will be satisfied...It then follows that items and persons are measured on an interval scale with a common unit. (Brogden 1977, p.633)
Rasch models are the only laws of quantification which define objective measurement, determine what is measurable, decide which data are useful and expose which data are not.
Our data can be obtained as responses to nominal categories like:
yes/no
right/wrong
present/absent
always/usually/sometimes/never
strongly agree/agree/disagree/strongly disagree
Nominal labels like these invite ordinal interpretation from more to less: more yesness, more rightness, more presence, more occurrence, more agreement. Almost without further thought, we take the asserted hierarchy for granted and imagine that our declared categories will inevitably implement a clear and stable ordinal scale. But whether our respondents actually use our categories in the ordered steps we intend cannot be left to imagination. We must find out by analyzing actual responses how respondents actually use our categories. When this is carefully done, it frequently emerges that an intended rating scale of five or six seemingly ordinal categories has actually functioned as a simple dichotomy.
Binomial transition odds like P_{nix}/P_{nix1} can implement the inferential possibilities of data collected in nominal categories interpreted as ordered steps. When we label our categories: x = 0,1,2,3,4,,, in their intended order, then each numerical label can count one more step up our ordered scale.
WRONG x = 0 
RIGHT x = 1 
STRONGLY DISAGREE x = 0 
DISAGREE x = 1 
AGREE x = 2 
STRONGLY AGREE x = 3 
We can connect each x to the situation for which we are trying to construct measures by subscribing x as x_{ni} so that it can stand for a rating recorded by (or for) a person n on an item i and then specify P_{nix} as the inverse probability that person n obtains x as their rating on item i.
The transition odds that the rating is x rather than x  1 become P_{nix}/P_{nix1}.
P_{nix1}  P_{nix}  
x  2  x  1  x  x + 1 
Then we can introduce a mathematical model which "explains" P_{nix} as the consequence of a conjointly additive function which specifies exactly how we want our parameters to govern P_{nix}.
Our parameters can be B_{n}, the location measure of person n on the continuum of reference; D_{i}, the location calibration of item i, the instrumental operationalization of this continuum and F_{x}, the gradient up the continuum of the transition from response category (x1) to category (x).
The necessary and sufficient measurement models are:
in which the symbol "==" means "by definition" rather than merely "equals".
The first model satisfies the Levy/Kolmogorov/Bookstein divisibility requirement. The second model satisfies the Campbell/Luce/Tukey conjoint additivity requirement. On the left of either formula a concrete datum x_{ni} is replaced by its abstract Bernoulli/Bayes/Laplace stochastic proxy P_{nix}.
In practice we apply the second formulation because its additivity produces parameter estimates in the linear form to which our eyes and feet are so accustomed. Our penchant for linearity is more than visual. Fechner (1860) showed that when we experience a ratio impact of light, sound or pain, our nervous system takes a logarithm so that we can see how the impact feels on a linear scale. Nor was Fechner the first to notice this neurology. Pythagorean tuning puts musical instruments out of tune at each key change. Tuning to notes which increase in frequency by equal ratios, however, produces an "equally tempered" scale of notes which sound equally spaced in any key. Pythagorean tuning is keydependent. Equally tempered tuning is keyfree. Bach's motive for writing "The WellTempered Clavier" was to demonstrate the validity of this 17th century discovery.
Measuring Applied SelfCare
Here is an example of a Rasch analysis of 3128 administrations of the 8 program PECS^{©} Applied SelfCare LifeScale.
This scale asks about the success of eight selfcare programs:
1. BOWEL
2. URINARY
3. SKIN CARE
4. Health COGNIZANCE
5. Health ACTIVITY
6. Health EDUCATION
7. SAFETY Awareness
8. MEDICATION Knowledge
Nurses rate patients on their competence in each of these SelfCare programs according to a seven category scale (labeled 17 in this case). This scale rises from "Ineffective" at label x=1 through three levels of "dependence" at x=2,3,4 and two levels of "independence" at x=5,6 to "normal" at label x=7.
The data matrix we analyze has 3128 rows and 8 columns, a row for each patient n, and a column for each program i with cell entries for patient n on program i of x_{ni} recorded as an ordinal rating from 1 to 7. This matrix of 25,024 data points is analyzed to estimate the best possible:
1. 8 selfcare program calibrations to define a SelfCare construct,
2. 48 rating step gradients to define each program's rating scale structure,
3. 3,128 patient reports, with SelfCare measures when data permit.
All estimates are expressed in linear measures on the one common scale which marks out a single dimension of "SelfCare".Here are excerpts from a BIGSTEPS (Wright & Linacre 1997) analysis of these data. As you scan this example, do not worry about every detail. Relax and enjoy how this kind of analysis can reduce potentially complicated data into a few wellorganized tables and pictures.
ENTERED: 3128 PATIENTS ANALYZED: 2145 PATIENTS 8 PROGRAMS 56 CATEGORIES SUMMARY OF 2145 MEASURED (NONEXTREME) PATIENTS RAW MODEL INFIT OUTFIT SCORE COUNT MEASURE ERROR MNSQ MNSQ MEAN 26.2 7.5 41.56 4.94 .88 .89 S.D. 11.0 .9 22.09 1.35 1.06 1.10 REAL RMSE 5.80 ADJ.SD 21.32 SEPARATION 3.68 RELIABILITY .93 S.E. OF PATIENT MEAN .48 MAXIMUM EXTREME SCORE: 5 PATIENTS MINIMUM EXTREME SCORE: 94 PATIENTS LACKING RESPONSES: 884 PATIENTS VALID RESPONSES: 94.3% 
Output from BIGSTEPS (Wright, B.D. & Linacre, J.M. 1997) 
Summarizing Results
Table 4 summarizes the results of an 91% reduction of the raw data (from 3128 x 8 = 25024 ordinal observations to 2145 + 8 + 56 = 2209 linear measures) and documents the success of its reconstruction into one unidimensional measurement framework. The linear logits for this analysis were rescaled to center the eight test programs at 50 and mark out 10 units per logit. The resulting measures reach from 10 to 110 with 98% of the action between 0 and 90.
1. Of the 3128 patient records reported in Table 4:
2145 nonextreme records are measured on the SelfCare construct,
5 are rated at the high extreme of "normal" on all programs,
94 are rated at the low extreme of "ineffective" on all programs,
884 lack all but one or two ratings and are set aside.
2. The data for the 2145 measured patients are 94.3% complete.
3. The mean patient measure for the 2145 measured patients = 41.56.
4. The observed patient S.D. = 22.09 measure units.
5. Root Mean Square Measurement Error [RMSE] = 5.80 Measure Units.
6. Correction for measurement error and misfit adjusts the observed S.D. = 22.09 to the more realistic value of ADJ.S.D. = 21.32.
7. The separation ratio for distinguishing among these 2145 SelfCare measures is (ADJ.S.D.)/RMSE = 21.32/5.80 = 3.68.
8. The corresponding "test reliability" = (3.68)^{2}/[1+(3.68)^{2}] = .93.
PROGRAMS STATISTICS: MEASURE ORDER ENTRY RAW INFIT OUTFIT NUM SCORE COUNT MEASURE ERROR MNSQ MNSQ PTBIS HARDEST PROGRAM 8 5964 1866 57.4 .3 .96 .97 .82 KNOWS MEDICATIONS 6 5828 1832 57.1 .3 .66 .68 .89 HEALTH EDUCATION 5 6294 1800 52.1 .3 .72 .74 .87 HEALTH ACTIVITY 7 7458 2137 48.9 .3 .71 .78 .86 SAFETY AWARENESS 4 7413 2135 48.2 .3 .47 .49 .91 HEALTH COGNIZANCE 3 7290 2139 48.1 .3 1.22 1.29 .77 SKIN CARE PROGRAM* 1 7858 2141 44.6 .3 .82 .92 .83 BOWEL PROGRAM 2 8168 2135 43.7 .3 1.09 1.16 .80 URINARY PROGRAM EASIEST PROGRAM MEAN 7034. 2023. 50.0 .3 .83 .88 S.D. 829. 148. 4.8 .0 .23 .24 
* The SKIN CARE PROGRAM shows slight misfit with INFIT & OUTFIT = 1.22 & 1.29 
Output from BIGSTEPS (Wright, B.D. & Linacre, J.M. 1997) 
Program Calibrations
Table 5 lists the 8 program calibrations from which the right side of Figure 3 will be mapped. This table shows the programs in their order of difficulty to accomplish. (Note that HEALTH COGNIZANCE matches SKIN CARE and HEALTH EDUCATION matches KNOWS MEDICATIONS in difficulty.)
Listed for each selfcare program is a program calibration MEASURE, calibration standard ERROR and two mean square ratio [MNSQ] fit statistics by which the validity of that calibration can be judged. The MNSQ fit statistics are ratios of observed residual variance to expected residual variance. When observed variance is no more than expected, these fit statistics are near 1.00. Their values in Table 5 show that the only program afflicted with tangible calibration uncertainty is the SKIN CARE program with mean square residual ratios of observed to expected at 1.22 and 1.29.
This slight misfit suggests that skin care may interact with other variables like patient age, sex or impairment for some patients. Other pages of BIGSTEPS output identify exactly which patients manifest such interactions and bring out the rating pattern diagnostics most useful for understanding the ratings of these particular patients.
The analysis of the extent to which any particular set of data cooperates with a measurement model to define an intended variable and so to estimate useful measures along that variable is decisive in the construction of good measures. We cannot do justice to this important topic here. But an excellent place to find articulate explanations and applications is in the work of Richard Smith (Smith 1985, 1986, 1988, 1991, 1994, 1996).
Rating Scale Analysis
Figure 2, derived from a larger sample of patients rated on the same instrument, shows how BIGSTEPS analyzes the way each set of rating categories
works to obtain patient ratings on each program. The program shown is the BOWEL program.
BOWEL PROGRAM RATING CATEGORY PROBABILITIES P R 1.0 Ineffective Normal O 111 7777 B 11 77 A 11 77 B .8 1 Independent 7 I 1 7 L 1 7 I 1 Dependent 44 7 T .6 1 44 4 7 Y 1 4 4 7 .5 1 2222 3333 4 4 7 O * 233 * 4 7 F .4 21 32 43 4 7 2 1 3 2 4 3 4 7 R 2 1 3 2 4 3 5*6*666 E 22 * * 3 55 6* 6 S .2 2 31 42 3 5 6 *55 66 P 2 3 1 4 2 5* 6 7 4 5 6 O 222 33 1*4 2 5 3* 7 4 5 666 N 222 33 44 11 *** 667*3 44555 6666 S .0 ********************************************************* E 25 5 15 35 55 75 95 115 MEASURE 3 1 1 1 11112222222234222241333431311 1 2 6 4 4 31671212716708664379546145138996 7 1 2 3 6 PATIENT 6 55216473961979258068811527561823331476839 9 9 3 2 Q S M S Q BOWEL PROGRAM RATING STEP CALIBRATIONS CATEGORY STEP OBSERVED STEP STEP EXPECTED SCORE MEASURES LABEL VALUE COUNT MEASURE ERROR STEP.5 AT STEP STEP+.5 Ineffective 1 618 NONE ( 6 ) 13 2 2 1049 7 .6 13 13 22 Dependent 3 1794 21 .4 22 30 40 4 4 2503 38 .3 40 50 58 Independent 5 860 66 .4 58 63 67 6 6 727 66 .4 67 72 78 Normal 7 1010 70 .5 78 ( 84 ) mode mean 
Output from BIGSTEPS (Wright, B.D. & Linacre, J.M. 1997) 
The plot in the top half of Figure 2 shows how the probabilities of category use, 1 to 7, move to the right across the variable from low measures below 20 to high measures above +110. Each rating category in turn, from left to right, shows a distinct region of greatest probability, except categories 5 and 6 which were underused in these data. This discovery raises a question about the utility of trying to distinguish between categories 5 and 6 when rating the BOWEL program. If raters do not use categories 5 and 6 to distinguish successive levels of BOWEL program success in a reliable way, it could be more efficient to combine 5 and 6 into one category of "independence" for rating success on this program. This rescoring might be mistaken as "throwing away data". But experience with this kind of analysis has shown that rescoring to combine indistinct adjacent categories usually increases the reliability of measurement.
The table at the bottom of Figure 2 gives a statistical summary of how these ratings performed. Successive columns show OBSERVED COUNTs, STEP MEASUREs and EXPECTED SCORE MEASURES at each rating level from 1 to 7. In the OBSERVED COUNT column we see that the use of categories 5 and 6, at counts of 860 and 727, fall below the use of category 4, at a count of 2503. This is why the probability curves for those two categories do not surface in the plot above. Rescoring these categories together produces a combined count of 1587 which fits right into a unimodal distribution of observed counts from 618 through 1048, 1794, 2503, 860+727=1587 to 1010.
Making a MAP
To expedite interpretation of this SelfCare construct, the distribution of patient measures and the program calibrations from Table 5 are plotted on the construct MAP in Figure 3. The left column benchmarks the linear units of the measurement framework from 0 to 90 units.
MEASURE PATIENTS RATING.APPLIED SELFCARE 90 . 6.KNOWS MEDICATIONS . .# Q 6.HEALTH ACTIVITY . 6.SAFETY AWARENESS 80 .## 6.SKIN CARE PROGRAM .### 6.BOWEL PROGRAM .## 6.URINARY PROGRAM .##### 70 .### .##### ######### ######### S 60 .######## 4.KNOWS MEDICATIONS .########## .#### 4.HEALTH ACTIVITY .########## 4.SAFETY AWARENESS 50 .##### 4.SKIN CARE PROGRAM .####### 4.BOWEL PROGRAM ##### 4.URINARY PROGRAM ######## M 40 .############ 3.KNOWS MEDICATIONS .########### .######### 3.HEALTH ACTIVITY .###### 3.SAFETY AWARENESS 30 .###### 3.SKIN CARE PROGRAM .####### 3.BOWEL PROGRAM .####### 3.URINARY PROGRAM .###### 20 .##### S .###### .##### .#### 1.KNOWS MEDICATIONS 10 .#### .# 1.HEALTH ACTIVITY . 1.SAFETY AWARENESS .## 1.SKIN CARE PROGRAM 0 .# 1.BOWEL PROGRAM .### Q 1.URINARY PROGRAM 
Output from BIGSTEPS (Wright, B.D. & Linacre, J.M. 1997) 
The second column plots the frequency distribution of the 2096 patients measuring between 0 and 90. M marks the mean patient measure. The two S's mark plus and minus one standard deviation. The two Q's mark plus and minus two standard deviations.
Finally the right column of the MAP shows the calibration locations of the six most distinct selfcare programs. The relative positions of these programs provide define this SelfCare construct. The programs are shown at four of their rating seven levels:
1 marking "Ineffective".
3 and 4 marking two levels of "Dependence" and
6 marking "Independence".
The MAP shows how this definition of SelfCare moves up from ratings of ineffectiveness in the 0 to 10 measure region, through successive levels of dependence in the 25 to 60 measure region to ratings of independence in the 75 to 90 measure region.
The program difficulty hierarchy within each level begins with the URINARY and BOWEL programs which are the easiest programs to succeed on and moves up through SKIN CARE, SAFETY and ACTIVITY to reach MEDICATIONS which is the hardest to succeed on.
The clinical implications of this hierarchy are that SelfCare education has the best chance of success when it begins at the easy end with URINARY and BOWEL programs and does not attempt the more challenging ACTIVITY and MEDICATIONS programs until the mastery of the easier programs is well underway.
Diagnostic KEYs
MAPs and numerical summaries of program and category performance are essential to the analysis of the measurement quality of a test. Our ultimate concern, however, is the measurement and diagnosis of each patient. This brings us to the two diagnostic KEYs illustrated in Figure 4.


Output from BIGSTEPS (Wright, B.D. & Linacre, J.M. 1997) 
After the construct MAP of Figure 2, KEYs like those in Figure 4 are the second most useful outcome of a BIGSTEPS analysis. Figure 4 shows two patients who received the same rating totals of 36 raw score points, but who differ decisively in the best estimates of their overall SelfCare measures and in the diagnostic implications of their different SelfCare rating patterns.
The typical (i.e. fitting) patient at the top of Figure 4 measures at 60 SelfCare units which puts them at the 70th percentile among a normative group of 8561 patients. This patient is verging on SelfCare independence but can still benefit from additional help with the hardest programs, MEDICATION and EDUCATION.
The atypical (i.e. misfitting) patient at the bottom of Figure 4 has the same raw score. But, as soon as we draw a vertical line through their ratings of 5's and 6's on the easiest six programs, the implications of their pattern of ratings are quite different. Now we see that when their totally "ineffective" ratings of 1's on MEDICATION and EDUCATION are set aside, they measure at 73 units and the 85th percentile.
This patient is well into independence on everything except EDUCATION and MEDICATION. But, for these two programs, their training must go back to the beginning and start over again at the earliest levels. The KEY record makes it obvious that their SelfCare curriculum can ease up on everything except EDUCATION and MEDICATION.
Notice how easily we become aware of the idiosyncratic aspects of a patient's status when a welldefined frame of reference has been constructed. The KEYs make it clear that the two patients in Figure 4 should not be treated the same just because they happen to have the same raw score of 36. On the contrary, when we have a frame of reference we learn more about each patient than just the overall measure their score may imply. We learn how they got their score? And, when their pattern of ratings contains values which, because of the structure of our measurement frame of reference, are unexpected, we can identify and respond to these explicit expressions of the needs of those particular patients.
Methodological Summary
1. Raw scores are not measures. Raw scores are biased against offtarget performance, test and sample dependent and nonlinear.
2. Inverse probability explained by conjointly additive parameters enables the construction of clear measures from raw ratings.
3. Measures enable MAPs to define a variable in a onepage, easy to grasp graphical report of the construct our analysis has realized as a measurable variable.
4. Measures enable KEYs which apply the variable individually to each person to bring out their personal particulars in an easy to work from one page report.
The history of social science measurement not only teaches us what to do in order to succeed as scientists. It also shows us what NOT to do.
Do NOT Use Raw Scores As Though They Were Measures
Some social scientists still believe that misusing raw scores as measures does no harm. They are unaware of the consequences for their work of the raw score bias against extreme scores. Some believe they can construct measures by decomposing raw score matrices with some kind of factor analysis. There is an incomplete relation between measurement construction and factor analysis, but factor analysis does not construct measures (Smith 1996, Wright 1996, especially Table 5). All supposedly useful results from raw score analyses are spoiled by their nonlinearity, extreme score bias and sample dependency (Embretson 1998).
Do NOT Use Models that Fail to Converge
Among those who have seen their way beyond raw scores to "Item Response Theory" [IRT] there is a selfdestructive misunderstanding concerning the measurement necessity for conjoint additivity. These adventurers cannot resist the blandishments of Birnbaum's (1968) "2P" and "3P" IRT models:
These nonadditive models are mistaken as improvements over a "1P" Rasch model because they bring in an item scaling parameter A_{i} to estimate a "discrimination" and a lower asymptote parameter C_{i} to estimate a "guessing" level for each item.
Here is what happens when these models are applied:
Item discriminations "increase without limit." Person abilities "increase or decrease without limit." (Lord 1968, pp.10151016)
Even for data generated to fit the 3PL (3PL, three parameter logistic model) model exactly, "only item difficulty [the Rasch parameter] is satisfactorily recovered by [the 3P computer program] LOGIST." (Lord 1975 p.13) "If restraints are not imposed, the estimated value of discrimination is likely to increase without limit." (Lord 1975 p.14) "Left to itself, maximum likelihood estimation procedures would produce unacceptable values of guessing." (Lord 1975 p.16)
During "estimation in the two and three parameter models...the item parameter estimates drift out of bounds." (Swaminathan 1983, p.34)
"Bias [in person measures] is significant when ability estimates are obtained from estimated item parameters...And, in spite of the fact that the calibration and crossvalidation samples are the same for each setting, the bias differs by test." (Stocking 1989, p.18)
"Running LOGIST to complete convergence allows too much movement away from the good starting values." (Stocking 1989, p.25)."
The reason why 2P and 3P IRT models do not converge is clear in Birnbaum's (1968 pp. 421422) estimation equations:
These equations are intended to iterate reciprocally to convergence. When the first equation is applied to a person with a correct response x_{i} = 1 on an item with discrimination a_{i} > 1, their ability estimate is increased by the factor a_{i}. When the second equation is applied, the same person response x_{i} = 1 is multiplied by their increased ability estimate which further increases discrimination estimate a_{i}. The presence of response x_{i} = 1 in both equations produces a feedback which escalates the estimates for item discrimination a_{i} and person measure to infinity.
Do NOT Use Models that Fail to Minimize Residuals
The sine qua non of a statistical model is its success at predicting its data. The simplest evaluation of success is the mean square residual between each piece of data x and its modeled expectation E_{x}, as in the mean of (x  E_{x})^{2} over all observed values of x. Ordinarily, the more parameters a model uses, the smaller the mean square residual must become. Otherwise why add more parameters. Should we encounter a parameter the addition of which increases our mean square residuals, we have exposed a parameter that does not belong in our model.
Hambleton and Martois (1983) used LOGIST to analyze 18 sets of data twice, first with a 1 item parameter Rasch model and second with a 3 item parameter Birnbaum model. In 12 of their 18 experiments, much to their surprise, two less item parameters, i.e. the Rasch model, produced smaller mean square residuals than the 3 item parameter model. In the six data sets where this did not happen, the tests were unusually difficult for the students. As a result, attempting to estimate "guessing" parameters reduced residuals slightly more than the Rasch model without a guessing constant.
Had a single apriori guessing constant been set at a reasonable value like C = .25 for all items and the data reanalyzed with a 1P Rasch model so modified, Hambleton and Martois would have discovered that one wellchosen apriori guessing constant did a better job than attempting to estimate a full set of item specific guessing parameters. When we encounter a situation in which the addition of a parameter makes things worse, we have proven to ourselves that the parameter in question does not belong in our model.
Do NOT Destroy Additivity
The additivity of the parameters of a model can be tested by attempting to separate parameters for independent estimation by subtraction.
Letting G_{ni} = log_{e}[P_{ni}/(1P_{ni})] stand for the log_{e} odds side of the measurement model for a dichotomy, consider the Rasch equations:
whenG_{ni} = B_{n}  D_{i}G_{mi} = B_{m}  D_{i}G_{nj} = B_{n}  D_{j}
thenG_{ni}  G_{mi} = B_{n}  B_{m} and D_{i} drops out of consideration.
andG_{ni}  G_{nj} = D_{j}  D_{i} and B_{n} drops out of consideration.
Now consider the corresponding 2P equations:
whenG_{ni} = A_{i}(B_{n}  D_{i})G_{mi} = A_{i}(B_{m}  D_{i})G_{nj} = A_{j}(B_{n}  D_{j})
thenG_{ni}  G_{mi} = A_{i}(B_{m}  B_{n}) and we are stuck with A_{i}
andG_{ni}  G_{nj} = B_{n}(A_{i}  A_{j}) + A_{i}D_{i}  A_{j}D_{j} and we are stuck with B_{n}
When we add an item discrimination parameter we cannot separate A and B in order to estimate them independently.
But Parameter Additivity is NOT Enough
Additive parameters can be asserted to govern a monotonic probability function over an infinite range, yet fail to construct fundamental measurement. The stochastic formulation of G_{ni} is also decisive.
Consider Goldstein (1980):
and Samejima (1997):
two stochastic models for G_{ni} which appear to specify conjoint additivity, but do not construct fundamental measurement.
Not only does neither model provides sufficient statistics for B and D, but both models fail to construct unique measures. To see this, reverse the direction of the latent variable and focus on person deficiency (B_{n}), item easiness (D_{i}) and task failure (1P_{ni}).
Rasch:
becomes
in which nothing changes but direction. However,
Goldstein:
becomes
which does NOT equal log_{e}[log_{e}(P_{ni})] unless [logP_{ni}][log_{e}(1P_{ni})] = 1
and
Samejima:
becomes
which does NOT equal
making Samejima's model a Rasch model.
For Goldstein and Samejima, merely measuring from the other end of the ruler produces a second set of measures which are incommensurable with the first. The mere assertion of additivity on one side of a model is not enough. To produce fundamental measurement, the whole model must reproduce itself.
Do NOT Destroy Construct Stability
Finally there is a substantive illogic in attempting to define a construct with item characteristic curves [ICC] which cross because their slopes differ due to differing discriminations or their asymptotes differ due to differing guessing parameters. Crossing curves cause the hierarchy of relative item difficulty to change at every ability level. This destroys the variable's criterion definition.
WOODCOCK READING MASTERY TESTS  
DIFFICULTY  SAMPLE TASK  
Mastery Scale  Grade Scale 50% Mastery  
25 41 58 70 86 101 114 124 143 159 174 192 211 240 
1.1 1.3 1.4 1.5 1.7 1.8 2.0 2.2 2.8 3.3 4.1 5.7 9.3 12.9 
is red . . . . . down black away . . . . . cold drink . . . . . shallow through octopus . . . . allowable hinderance equestrian . . . . heterogeneous 
. A . . B . C . . D . . E  
FIXED ITEM POSITIONS DEFINE VARIABLE 
Table 6 shows a word recognition ruler built by Woodcock (1974). The left column marks the Mastery Scale "inches" which measure equal intervals of word recognition mastery. The center column gives norms from 1st to 12th Grade. The right column lists some of the words which define this construct. "Red", a short easy word, is recognized at 1st Grade. When we finally reach "heterogeneous", we see that it takes a 12th Grader to recognize it. Woodcock's ruler implements a continuous construct which can be used to make detailed wordbyword interpretations of children's wordrecognition measures.
Figure 5 shows the relative locations of Rasch item calibrations for five words from Woodcock's ruler. It does not matter whether you are a 1st, 2nd or 3rd Grader, "red", "away", "drink", "octopus" and "equestrian" remain in the same order of experienced difficulty, at the same relative spacing. This ruler works the same way and defines the same variable for every child whatever their grade. It obeys the Magna Carta.
To obtain the construct stability manifest in Figure 5 we need the kind of item response curves which follow from the standard definition of fundamental measurement. Figure 6 shows that these Rasch curves do not cross. In fact, when we transform the vertical axis of these curves into logodds instead of probabilities, the curves become parallel straight lines, thus demonstrating their conjoint additivity.
Figure 7, in contrast, shows five 3P Birnbaum curves for the same data. These five curves have different slopes and different asymptotes. There is no sign of conjoint additivity.
Figure 8 shows the construct destruction produced by the crossing curves of Figure 7. For a 1st Grader, "red" is said to be easier than "away" which is easier than "drink" which is easier than "octopus". But for a 3rd Grader the order of item difficulty is different. Now it is "away" rather than "red" that is easier. "Red" has become even harder than "drink"! And "octopus" is now almost as easy to recognize as "red", instead of being up near "equestrian". What is the criterion definition of this variable? What construct is defined? The definition is different at every ability level. There is no construct! No ruler! No Magna Carta!
Much as we might be intrigued by the complexity of the Birnbaum 3P curves in Figure 7, we cannot use them to construct measures. To construct measures we require orderly, cooperating, noncrossing curves like the Rasch curves in Figure 6. This means that we must take the trouble to collect and refine data so that they serve this clearly defined purpose, so that they approximate a stochastic Guttman scale.
When we go to market, we eschew rotten fruit. When we make a salad, we demand fresh lettuce. We have a recipe for what we want. We select our ingredients to follow. It is the same with making measures. We must think when we select and prepare our data for analysis. It is foolish to swallow whatever comes. Our data must be directed to building a structure like the one in Figures 5 and 6  one ruler for everyone, everywhere, every time  so we can achieve a useful, stable construct definition like Woodcock's wordrecognition ruler.
The lamentable history of the Birnbaum model is a cautionary tale of myopic promiscuity. Guessing is celebrated as an item asset. Discrimination is saluted as a useful scoring weight. Crossed item characteristic curves are swallowed without hesitation.
Models which build fundamental measurement are more choosy. They recognize guessing, not as an item asset but, as an unreliable person liability. They identify variation in discrimination as a symptom of item bias and multidimensionality (Masters, 1988). Instead of parameterizing discrimination and guessing and then forgetting them, models for fundamental measurement analyze the data for statistical symptoms of misfit including variations in discrimination and guessing, identify their item and person sources and weigh their impact on measurement quality.
In practice, guessing is easy to minimize. All one has to do is to test on target. Should a few lucky guesses crop up, it will not be items that produce them. The place to look for lucky guessing is among lucky guessers. The most efficient and fairest way to deal with lucky guessing, when it does occur, is to detect it, to measure the advantage it affords the lucky guesser, in case that matters, and, finally, to decide what is the most reasonable thing to do with the improbably successful responses that lucky guessing chanced on.
In the history of science there are vast differences between gerrymandering models to give best local descriptions of transient data and searching, instead, for better data that brings inferentially stable meaning to parameter estimates. It is the search for better data which begets discovery. The only way discovery can emerge is as an unexpected discrepancy from an otherwise stable frame of reference. When we study data misfit, we discover new things about what we are measuring and how people tell us about it. These discoveries strengthen and evolve our constructs as well as our ability to measure them.
The social history of stable units for fair trade makes clear that when units are unequal, when they vary from time to time and place to place, it is not only unfair, it is immoral. So too with the misuse of necessarily unequal and so unfair raw score units.
The purpose of measurement is inference. No measurement model which fails to meet the requirements for inference of probability, additivity, separability and divisibility can survive actual practice. Physics and Biology do not use and would never consider using models with intersecting trace lines. We measure to inform our plans for what to do next. If our measures are unreliable, if our units vary in unknown ways, our plans will go wrong and, eventually, our work will perish.
This simple point is sometimes belittled. It is not negotiable. It is vital and decisive! We will never build a useful, let along moral, social science until we stop deluding ourselves by analyzing local concrete ordinal raw scores as though they were general abstract linear measures.
Thurstone's "New" Measures
Thurstone outlined his necessities for a "new" measurement in the 1920's:
1. Measures must be linear, so that arithmetic can be done with them.
2. Item calibrations must not depend on whose responses they were estimated from  must be samplefree.
3. Person measures must not depend on which items they were estimated from  must be testfree.
4. Missing data must not matter.
5. The method must be easy to apply.
The mathematics needed to satisfy Thurstone's necessities and to make them practical were latent in Campbell's 1920 concatenation, Fisher's 1920 sufficiency and the divisibility of Levy and Kolmogorov and realized by Rasch's 1953 Poisson measurement model.
Rasch's "New" Models
The implications of Rasch's discovery have taken many years to reach practice (Wright 1968, 1977, 1984, Masters & Wright 1984). Even today there are social scientists who do not understand or benefit from what Campbell, Levy, Kolmogorov, Fisher and Rasch have proven (Wright 1992). Yet, despite this hesitation to use fundamental measurement models to transform raw scores into linear measures so that subsequent statistical analysis can become fruitful, there have been many successful applications (Wilson 1992, 1994, Fisher & Wright 1994, Engelhard & Wilson 1996, Wilson, Engelhard & Draney 1997).
Infinitely divisible (conjointly additive) models for the inverse probabilities of observed data enable:
1. The conjoint additivity which Norman Campbell (1920) and Luce and Tukey (1964) require for fundamental measurement (Brogden 1977, Perline, Wright & Wainer 1979, Wright 1985, 1988).
2. The exponential linearity which Ronald Fisher (1922) requires for estimation sufficiency (Andersen 1977, Wright 1989).
3. The parameter separability which Louis Thurstone (1925) and Rasch (1960) require for objectivity (Wright & Linacre 1989).
Measurement is an inference of values for infinitely divisible parameters which define the transition odds between observable increments of a theoretical variable.
No other formulation can construct results which a rational scientist, engineer, business man, tailor or cook would be willing to use as measures. Only data which can be understood and organized to fit this model can be useful for constructing measures. When data cannot be made to fit this model, the inevitable conclusion is that those data are inadequate and must be reconsidered (Wright 1977).
Rasch's models are applicable to every imaginable raw observation: dichotomies, rating scales, partial credits, binomial and Poisson counts (Andrich 1978, Masters & Wright 1984) in every reasonable observational situation including ratings faceted to: persons, items, judges and tasks (Linacre, 1989).
Computer programs which apply Rasch models have been in circulation for 30 years (Wright & Panchapakesan, 1969). Today, convenient, easy to use software for applying Rasch's "measuring functions" is readily available (Wright & Linacre 1997, Linacre & Wright 1997). Today it is easy for any social scientist to use computer programs like these to take the decisive step from unavoidably ambiguous concrete raw observations to welldefined abstract linear measures with realistic estimates of precision and explicit quality control. Today there is no methodological reason why social science cannot become as stable, as reproducible and as useful as physics.
Benjamin D. Wright
MESA Psychometric Laboratory
This paper was published as: Wright B.D. (1999) Fundamental measurement for psychology. In S.E. Embretson & S.L. Hershberger (Eds.), The new rules of measurement: What every educator and psychologist should know. Hillsdale, NJ: Lawrence Erlbaum Associates.
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Jan. 25  Feb. 22, 2019, Fri.Fri.  Online workshop: Practical Rasch Measurement  Core Topics (E. Smith, Winsteps), www.statistics.com 
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