Bookstein's deductions that any distribution function reducing to
or
will be resilient to
aggregation ambiguity is applied to performance odds. The Rasch model and three measurable
compositions result. TEAMs work as unions of perfect agreement doing best with easy problems.
PACKs work as collections of perfect DISagreements doing best with intermediate and hard problems.
CHAINs work as connections of IMperfect agreements doing better than TEAMs with hard problems.
Four problem/solution necessities for inference are reviewed: uncertainty met by probability, distortion
met by additivity, confusion met by separability and ambiguity met by divisibility. Connotations,
properties and applications of TEAM, PACK and CHAIN groups are ventured.
Why do some organizations succeed while others fail? Why do groups of a particular kind work well in some situations but poorly in others? The psychology of group organization is rich but qualitative (Freud, 1921). Questions about how groups work seem non-mathematical. Nevertheless, algebra undertaken to obtain hierarchically stable measurement leads to a mathematics of group productivity.
Groups are composed of subgroups, subgroups of elements, elements of parts. Aggregations separate into sub-aggregations and come together into super-aggregations. The "entities" we experience become understood and useful as we learn how to see down into their substructures and up into the compositions they construct.
We know from experience that the way group members work together affects group success. But we do not know how to measure these effects, nor how to calculate effective group organizations. Quantification requires models which measure group strength as functions of member strengths. These models must be hierarchically robust. They must maintain their metric across descending and ascending levels of composition. Can a mathematics be developed which defines the different ways group members might work together such that their individual measures can be combined mathematically to calculate an expected measure for the group?
"Composition analysis" is our name for the mathematics of how component measures combine to produce composite measures. Our deliberations will lead to measurement models which are infinitely divisible and, hence, inferentially stable. The models will compose and decompose smoothly from one level of aggregation to another. When the models work for groups of individuals, they will work for individuals within groups. When they work for individuals, they will work for parts within individuals. Although presented here as groups of persons, these models apply to groups of any kind: ideas, problems, cells.
Bookstein (see Appendix) shows that any distribution function that reduces to
or
will be indifferent to the intervals used for grouping and hence resilient to aggregation ambiguity.
Bookstein's functions specify the divisibility needed for composition analysis. They also specify the
arithmetic needed for quantitative comparisons. To relate these functions to the procedures of
measurement, we enlarge the + and x arithmetic inside functions G and H to include "procedural"
compositors
and
. These procedural "additions" and "multiplications" represent whatever
empirical procedures are discovered to operationalize measurement, as in "aligning sticks end-to-end" to
"add" length, and "piling bricks top-to-bottom" to "add" weight.
Two composition rules follow:
A Procedural Addition rule:
A Procedural Multiplication rule:
These rules compose and decompose ad infinitum, as in:
To discover the consequences for composition analysis, we will apply each rule to observation
probabilities. Observation probabilities are addressed we intend to use these rules on data in order to
estimate measures for empirical compositions. These probabilities will be expressed as odds because
0 < P < 1 is an awkward measure while 0 < [P/(1 - P)] < maintains equal ratios and log_{e} odds maintain
equal differences. Our application of Bookstein's functions to odds will determine what compositions
compositors
and
imply and hence what compositions are quantifiable.
Three compositions will result:
1. a TEAM union of perfect agreement,
2. a PACK collection of helpful DISagreements and
3. a CHAIN connection of IMperfect agreements.
We will deduce measurement models for these compositions, which, because of their divisibility, are indifferent to composition level and resilient to aggregation ambiguity. The resulting models will be the stable laws of composition analysis.
Finally we will place divisibility in the theory of inference which motivates these deductions and venture some interpretations of the three compositions.
In order to apply the composition rules, we need a stochastic measurement model with parameters that follow the rules of arithmetic and estimates that enable comparisons between strengths B_{n} and B_{m} of objects n and m which are invariant with respect to whatever relevant, but necessarily incidental, measuring agents are used to manifest the comparison.
Measurement means quantitative comparison. Quantitative comparison means differences or
ratios. Since odds are ratios, ratios are their comparison. The procedural comparison
of objects
n and m is:
Defining H as odds [P/(1 - P)] gets:
Estimation requires that strengths B_{n} and B_{m} be manifest by a relevant measuring agent i of difficulty D_{i}. Inferential stability requires that the comparison (B_{n} - B_{m}) be independent of task difficulty D_{i}.
The necessary and sufficient model is:
because task difficulty D_{i} cancels so that the n
m comparison maintains the same difference of
strengths regardless of which tasks are convenient to manifest these strengths (Rasch, 1960).
Applying procedural multiplication to success odds defines group success odds, when group
members work according to the procedural operator
as the following product of group member success odds:
The group composition specified by this first law of stable measurement can be seen by applying probabilities P_{n} and P_{m} to the outcomes possible when persons n and m work on a task according to the multiplication of their success odds. Figure 1 shows the two outcomes which occur in this composition.
m loses 0 |
m wins 1 | |
n loses 0 |
TEAM 00 loses |
Disagreement Absent |
n wins 1 |
Disagreement Absent |
TEAM 11 wins |
Agreement (11) wins or agreement (00) loses. Disagreements (10) and (01) are absent because they do not occur in the equation which defines TEAM composition. TEAMs work as unions of perfect agreement.
Applying Rasch odds to TEAM work with group strength represented by
gets:
Taking logs and generalizing to any size group defines N-member TEAM strength as:
The strengths of TEAM members, relative to task difficulty (B_{n} - D), add up to TEAM strength, relative to task difficulty (B_{T} - D). TEAMs are concatenations of relative strengths, accumulated in linear form.
Applying procedural addition to success odds defines group success odds, when group members
work according to the procedural operator
as the following addition of group member success
odds:
The group composition specified by this second law of stable measurement can be seen by applying probabilities P_{n} and P_{m} to the outcomes possible when persons n and m work on a task according to the addition of their success odds. Figure 2 shows the three outcomes which occur in this composition.
m loses 0 |
m wins 1 | |
n loses 0 |
PACK 00 loses |
PACK 0 1 wins |
n wins 1 |
PACK 1 0 wins |
Agreement Absent |
Helpful disagreements (10) and (01) win. Unhelpful disagreement (00) loses. Agreement (11) is absent because it does not occur in the equation which defines PACK composition. PACKs work as collections of perfect DISagreements.
Applying Rasch success odds to PACK work gets:
or
This is also a concatenation, but of absolute (not relative to problem difficulty) strengths, accumulated in exponential form.
Taking logs and extending to a group of any size defines N-member PACK strength B_{P} as:
with
Log_{e} (NW) is the amount PACK strength increases with PACK size N and member heterogeneity
W. W brings in member heterogeneity through member strength variance B^{2}, skew:
kurtosis:
Positive skew and kurtosis amplify the impact of stronger PACK members.
The homeostasis of most groups induces homogeneity. When heterogeneity emerges, members regroup toward homogeneity. As long as member strength variance B^{2} stays small, so that B<.3 for 1< W <1.1 or B<.5 for 1<W<1.2, then PACK strength can be modelled as:
The perfect disagreements of PACK members collect to benefit the PACK. As PACK size increases so does PACK strength. Unlike TEAMs, PACK strength is independent of task difficulty.
Applying procedural addition to failure odds defines group failure odds, when group members
work according to the procedural operator
as the following addition of group member failure
odds:
The group composition specified by this third law of stable measurement can be seen by applying probabilities P_{n} and P_{m} to the outcomes possible when persons n and m work on a task together according to the addition of their failure odds. Figure 3 shows the three outcomes which occur in this composition.
m loses 0 |
m wins 1 | |
n loses 0 |
More than one loss Absent |
CHAIN 0 1 loses |
n wins 1 |
CHAIN 1 0 loses |
CHAIN 11 wins |
Perfect agreement (11) wins. Disagreements (10) or (01) lose. Outcome (00) is absent because it does not occur in the equation that defines CHAIN composition. CHAINs work as connections of IMperfect agreements.
Applying Rasch failure odds to CHAIN work gets:
or
a concatenation of absolute weaknesses in exponential form.
Taking logs and extending to a group of any size defines N-member CHAIN strength B_{C} as:
with
which member homogeneity simplifies to:
The imperfect agreements of CHAIN members are connect against the danger of harmful disagreement. Like PACKs, CHAIN strength is independent of problem difficulty. Unlike PACKs, as CHAIN size increases, CHAIN strength decreases.
COMPARING COMPOSITIONS
To see the differences among TEAMs, PACKs and CHAINs consider the possibilities for groups of three in Figure 4 and for groups of any size in Figure 5.
TEAM | PACK | CHAIN | ||
WIN | Agreement 111 |
Helpful Disagreement 100,010,001 |
Agreement 111 | |
LOSE |
Agreement 000 |
Disagreement 000 |
Harmful Disagreement 011,101,110 | |
ABSENT | 100,010,001 110,101,011 |
111 110,101,011 |
000 100,010,001 |
AGREE | DISAGREE | |
WINS | TEAM all 1's |
PACK a single 1 |
LOSES | TEAM all 0's |
CHAIN a single 0 |
TEAM | PACK | CHAIN | |
all 1's | WINS | *** | WINS |
both 0's & 1's |
*** *** |
One 1 WINS |
One 0 loses |
all 0's | loses | loses | *** |
TEAMs are united in perfect agreement. Win or lose, no disagreement can occur. PACKs and CHAINs distinguish disagreement, but conversely. PACKs win by a single winning disagreement. CHAINs lose by a single losing disagreement.
To help a TEAM, a member's strength must be stronger than problem difficulty. Members weaker than problem difficulty decrease TEAM strength. Adding to a PACK increases PACK strength. Adding to a CHAIN decreases CHAIN strength.
The measurement models for composition analysis in Figure 6 enable us to deduce which of these compositions works best against problems of different difficulties.
TEAMs:
B_{n} < D hurts
PACKs:
CHAINs:
TEAMs vs PACKs. When is one united TEAM agreement on what is best more effective than a collection of PACK disagreements?
Since
TEAMs do better than PACKs when average group strength is greater than problem difficulty by [(+log_{e} N)/(N - 1)]. This defines the upper curve in Figure 7.
TEAMs vs CHAINs. When is TEAM organization better than CHAIN organization?
Since
This is the lower curve in Figure 7.
To read Figure 7, find group size N = 4 on the horizontal axis. Go up to the upper curve and left to the vertical axis to read that a group of four must average half a logit more strength than problem difficulty to do better as a TEAM than a PACK.
When problem D is harder than
PACK disagreement is more
productive than TEAM agreement. As problem difficulty (D - B) increases, the value of TEAM work
declines. The turning point at which PACKs become better than TEAMs is always greater than zero.
Below (B - D) = (-log_{e} N)/(N - 1), a TEAM becomes the least productive group organization. Figure 8
formulates the relative strengths of TEAMs, PACKs and CHAINs.
WHEN: THEN:
Figure 9 uses Figure 7 to show how relationships between problem difficulty, group size and group organization can be used to design optimal work groups. The upper group of five, averaging .56 logits more able than their problem, should work best in TEAM agreement. The middle group of three, averaging only .18 logits more able than their problem, should work better in PACK disagreement. The bottom group of seven, averaging .64 less able than their problem, however, encounter an additional consideration. Optimal organization for this group depends on the cost/benefit balance between success and failure. When opportunity invites, PACK disagreements should be more productive. When danger looms, however, CHAIN commitment to maintain agreement may be safer.
VISUALIZING GROUP MIXTURES. When empirical measures B_{G} are estimated from group performance, we can see where each B_{G} fits on the line of TEAM, PACK and CHAIN compositions implied by its member measures {B_{n} for n = 1,N} by plotting B_{G} at:
in an XY-plot benchmarked by a TEAM, PACK, CHAIN line with intercept (A_{ND}, 0), slope one and composition reference points:
GENERALIZING THE MEASUREMENT MODEL
To expand probability
in Q, write its log
odds
Subscripting P and (1-P) to P_{1} and P_{0} for
leads to
a Rasch model for any
number of ordered steps: x = 1, 2, 3, , , m-1, m, ... , . This
model constructs additive conjoint measurement from data obtained
through any orderable categories: dichotomies, ratings, grades,
partial credits (indexing x_{i} and Q_{ix} to item i), comparisons,
ranks, counts, proportions, percents...
We can use this model to articulate a variety of frequently encountered facets. To represent a measure for person n, we introduce person parameter B_{n}. To produce an observable response x_{ni}, we provoke person n with item i designed to elicit manifestations of the intended variable. To calibrate item i, and so construct a quantitative definition of the variable, we introduce item parameter D_{i}. To calibrate the resistance against moving up in item i from category x-1 to x, we add item step parameter F_{ix}. With D_{i} and F_{ix} in place, we can estimate test-free person measures which, for data which follow the model, are stable with respect to item selection.
When person n responds directly to item i, producing response x_{ni}, we can collect x_{ni}'s over persons and items and construct person measures on the item-defined variable. But, when persons are observed through performances which are not self-scoring, then we need a rater j to obtain rating x_{nij} of person n's performance on item i. But we know that even the best trained raters vary in the way they use rating scales. To calibrate raters, we add rater parameter C_{j}. With C_{j} in place, we can estimate rater-free, as well as test-free, person measures which, for data that fit, will be stable with respect to rater selection as well as item selection.
As comprehension of the measurement context grows, we can add more facets, a task parameter A_{k} for the difficulty of the task on which person n's performance is rated by rater j on item i to produce x_{nijk} and so on.
In order to obtain inferential stability [Fisher sufficiency (1920), Thurstone invariance (1925), a stochastic Guttman scale (1944), Rasch objectivity (1960), and Luce and Tukey conjoint additivity (1964)] we need only combine these parameters additively into a many-facet model (Linacre, 1989) such as:
where B_{n} is the person parameter, D_{i} is the item parameter, C_{j} is the rater parameter, A_{k} is the task parameter and F_{ix} is the item step parameter.
Compositions can be studied in any facet of a many-facet model. Consider:
rewritten for x = 0,1 to simplify presentation. The measurement models for TEAMs of animate elements, persons and raters, and for BLOCKs of inanimate elements, items and tasks, are listed in Figure 10.
Group Type Measurement Model
Person TEAM n = 1,...,N
Item BLOCK i = 1,...,L
Rater TEAM j = 1,...,M
Task BLOCK k = 1,...,H
For TEAM and BLOCK measures to increase with group size, the average measure of the grouped facet must exceed:
The PACK and CHAIN formulations in Figure 11 are simpler. For PACKs and CHAINs the levels of other facets do not matter. More persons make person PACKs stronger, but person CHAINs weaker. More items, raters or tasks make PACKs easier to satisfy, but CHAINs more difficult.
PACKs CHAINs
Persons
Items
Raters
Tasks
Four problems interfere with inference:
Uncertainty is the motivation for inference. We have only the past by which to infer the uncertain future. Our solution is to contain uncertainty in probability distributions which regularize the irregularities that disrupt connections between what seems certain now but must be uncertain later.
Distortion interferes with the transition from data collection to meaning representation. Our ability to figure out comes from our faculty to visualize. Visualization evolved from the survival value of safe body navigation. Our solution to distortion is to represent data in bilinear forms that make the data look like the space in front of us. To "see" what experience "means", we "map" it.
Confusion is caused by interdependency. As we look for tomorrow's probabilities in yesterday's lessons, interactions intrude and confuse us. Our solution is to force the complexities of experience into few enough invented "dimensions" to make room for clear thinking. The authority of these fictions is their utility. We will never know their "truth". But, when our fictions "work", they are usually useful.
The logic we use to control confusion is enforced singularity. We investigate the possibilities for, define and measure one dimension at a time. The necessary mathematics is parameter separability. Models which introduce putative "causes" as separately estimable parameters are the founding laws of quantification. They define measurement. They determine what is measurable. They decide which data are useful, and which are not.
Ambiguity is the fourth problem for inference. We control hierarchical ambiguity by using measurement models which embody divisibility.
Bookstein's functions:
for resilience to aggregation ambiguity contain the divisibility necessary to stabilize quantitative inference (Feller, 1966). They also contain the parameter separability and linearity necessary to alleviate confusion and distortion. Models which follow from Bookstein's functions implement:
1. the concatenation and conjoint additivity which Norman Campbell (1920) and Luce and Tukey (1964) require for fundamental measurement,
2. the exponential linearity which Ronald Fisher (1920) requires for estimation sufficiency and
3. the parameter separability which Thurstone (1925) and Rasch (1960) requires for objectivity.
The measurable compositions are TEAMs, PACKs and CHAINs. The measurement models necessary and sufficient for quantitative composition analysis are linear mixtures of the Rasch models for measuring these compositions. Figure 12 summarizes the problems of inference and their current.
PROBLEMS | SOLUTIONS | PARENTS |
UNCERTAINTY have want now later statistic parameter |
PROBABILITY distribution regular irregularity misfit detection |
Bernoulli 1713 De Moivre 1733 Laplace 1774 Poisson 1837 |
DISTORTION non-linearity unequal intervals incommensurability |
ADDITIVITY linearity arithmetic concatenation |
Luce/Tukey 1964 Fechner 1860 Helmholtz 1887 N.Campbell 1920 |
CONFUSION interdependence interaction confounding |
SEPARABILITY sufficiency invariance conjoint order |
Rasch 1960 R.A.Fisher 1920 Thurstone 1925 Guttman 1944 |
AMBIGUITY arbitrary grouping ambiguous hierarchy |
DIVISIBILITY independence stability reproducibility exchangeability |
Kolmogorov 1932 Levy 1924 Bookstein 1992 de Finetti 1931 |
For Bernoulli, De Moivre, Laplace and Poisson see Stigler (1986). For Kolmogorov and Levy see Feller (1966).
The prevalence, history and logic of the addition and multiplication rules establish Rasch measurement models as the necessary and sufficient foundations for measurement. Models which contradict the inferential necessities of: probability, linearity, separability and divisibility, cannot survive the vicissitudes of practice. Only data which can be understood and organized to fit a Rasch model can be useful for constructing measures.
Mathematics leads to three reference compositions which empirical composites must mix to be measurable. We can use group member measures to calculate TEAM, PACK and CHAIN expectations. We can use these expectations and empirical group measures to study TEAM/PACK/CHAIN mixtures. So much for mathematics. What can TEAMs, PACKs and CHAINs say about everyday life? How might we bring these mathematical ideas to practice as useful formulations for better living? Can these abstractions help us manage our infinitely complex experiences with living compositions, hierarchies of functioning, families of ideas and tasks? Can we construct maps by which to "see" how the compositions of which we are, by which we think and within which we live, might be better worked? Figure 13 lists some connotations which TEAMs, PACKs and CHAINs bring to mind. Figure 14 lists some properties which they imply. We end with some stories in which these compositions might participate.
SAFE SURE |
DANGEROUS UNSURE | |
AGREE | TEAM government formality convention |
CHAIN survival security discretion |
DISAGREE | PACK science opportunity invention |
chaos anarchy |
TEAM | PACK | CHAIN | |
WIN | virtue satisfaction justice |
pride triumph progress |
safety relief security |
LOSE | guilt indignation worry |
shame frustration disappointment |
fear recrimination despair |
Football. When a TEAM of players huddle to call a play, win or lose, they intend to act united. Should one of them err, he will hurt the TEAM. TEAM success is jeopardized by weak links in its CHAIN of players.
Lost Keys. What is the best way to look for a lost key? Should we all agree to look in the same place? Or, should we all agree to disagree as to where to look and spread out?. Each in a different place has the better chance of success. PACK work is the way to look for lost keys.
Mountain Climbing. Climbers rope for safety. As one climbs, everyone else hangs on. Then, should a climber slip, his anchored mates may be able to save him. When, however, a supposedly anchored mate is not hanging on or moves out of turn, then all may fall. CHAIN work is the way to climb mountains.
Cops and Robbers. When a crime is reported, the perpetrator is often unknown. Solving the problem begins hard. PACKs of detective TEAMs fan out to search of suspects. As evidence accumulates, however, deciding who's guilty becomes easier. The PACK of TEAMs converges in their solutions to one TEAM agreement and detains the most likely suspect.
Should the suspect go to trial, judgement will depend on a jury TEAM decision. But, if a contrary jurist holds out, the jury TEAM may become a failing CHAIN.
TEAM | PACK | CHAIN |
unite | collect | connect |
consolidate | accumulate | protect |
evaluate | explore | preserve |
unify | discover | secure |
agree | attack | defend |
uphold ground |
gain ground |
guard ground |
capitalize consensus |
optimize difference |
survive together |
play safe | take chance | hang on |
smug secure |
daring hopeful |
cautious worried |
virtue disapproval |
pride shame |
safety danger |
usual events |
rare events |
dangerous events |
easy problems |
hard problems |
risky problems |
successful jury |
missing key |
mountain climbing |
A Common Source of Misapprehension. A weak shooter, in solitude, misses repeatedly. But then, in sudden company, is seen to hit on what is now his Nth try. His PACK ability: B_{PN} = B + log_{e} N => B_{P1} = B' + log_{e} 1 = B', when only his finally successful shot is seen, will appear to be the ability B' of a stronger shooter who hits on his first try.
A strong shooter, in solitude, hits repeatedly. But then, in sudden company, is seen to miss on what is now her Nth try. Her CHAIN ability: B_{CN} = B - log_{e} N => B_{C1} = B" - log_{e} 1 = B" , when only her finally unsuccessful shot is seen, will appear to be the ability B" of a weaker shooter who misses on her first try.
Solving Problems. When problems are easy, TEAMing ideas into one course of action should work best. When problems are hard, however, putting every egg in a single basket may not be as productive as deploying a PACK of diverse undertakings. When a mistake is fatal, however, then PACK diversity risks CHAIN weakness.
MESA Memorandum 67, 1994
Benjamin D. Wright
MESA Psychometric Laboratory
Bookstein, A. (1992). Informetric Distributions, Parts I and II, Journal of the American Society for Information Science, 41(5):368-88.
Campbell, N.R. (1920). Physics: The elements. London: Cambridge University Press.
de Finetti, B. (1931). Funzione caratteristica di un fenomeno aleatorio. Atti dell R. Academia Nazionale dei Lincei, Serie 6. Memorie, Classe di Scienze Fisiche, Mathematice e Naturale, 4, 251-99. [added 2005, courtesy of George Karabatsos]
Fechner, G.T. (1860). Elemente der psychophysik. Leipzig: Breitkopf & Hartel. [Translation: Adler, H.E. (1966). Elements of Psychophysics. New York: Holt, Rinehart & Winston.].
Feller, W. (1966). An introduction to probability theory and its applications, Volume II. New York: John Wiley.
Fisher, R.A. (1920). A mathematical examination of the methods of determining the accuracy of an observation by the mean error and by the mean square error. Monthly Notices of the Royal Astronomical Society,(53),758-770.
Freud, S. (1921). Group psychology and the analysis of the ego.
New York: Norton.
Guttman, L. (1944). A basis for scaling quantitative data.
American Sociological Review,(9),139-150.
Helmholtz, H.V. (1887). Zahlen und Messen erkenntnis-theoretisch betrachet. Philosophische Aufsatze Eduard Zeller gewidmet. Leipzig. [Translation: Bryan, C.L. (1930). Counting and measuring. Princeton: van Nostrand.].
Linacre, J.M. (1989). Many-facet Rasch measurement.
Chicago: MESA Press.
Luce, R.D. & Tukey, J.W. (1964). Simultaneous conjoint measurement. Journal of Mathematical Psychology,(1),1-27.
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. [Danish Institute of Educational Research 1960, University of Chicago Press 1980, MESA Press 1993] Chicago: MESA Press.
Stigler, S.M. (1986). The history of statistics. Cambridge: Harvard University Press.
Thurstone, L.L. (1925). A method of scaling psychological and educational tests. Journal of Educational Psychology,(16), 433-451.
A. Bookstein, University of Chicago
The concern of this appendix is our ability to observe and describe regularities in the face of ambiguity. Since much of Social Science data are based on ill-defined concepts, this ability has serious practical implications.
Policy decisions are often based on assumptions about how certain characteristics are distributed over an affected population. Such assumptions tend to be expressed in terms of functions describing statistical distributions. Although these functions critically influence our decisions, they are usually created ad hoc, with little or no theoretic support. We are interested in situations in which the particular values entering the function could be quite different, given a plausible redefinition of the concepts being probed. In such situations, it is reasonable to demand of the functions involved, that reasonable redefinition of key concepts not result in new functions that change the decisions being made.
We examined one case in which counts were described in terms of an unknown function, f(x) (Bookstein, 1992). We had a population of items, each with an associated latent parameter, x, indicating its potential for producing some yield over a period of time. The number of items with values of x between x_{0} and x_{0} + is given by f(x)*. It is convenient to let f(x) = A*h(x), with h(x) defined so h(1)=1.
The function, h(x) is unknown, but we would like the form that it takes not to depend on the size of the interval chosen. Demanding this constraint led to the condition that h(x) must obey: h(xy)=h(x)h(y). This functional constraint in many cases determines the function itself. For example, for h(x) a "smooth" function, only the form h(x)= A/x^{} is permitted.
The functional constraint, h(xy)=h(x)h(y), also resulted from examining a wide range of other ambiguities in counting. Similar requirements occur in other contexts. In an interesting and important example, Shannon, in defining the properties of a measure of information, first considers the uncertainty of which of M equally likely events will occur. He argues that, if this is given by a function f(M), this function must obey f(MN) = f(M) + f(N). A discussion of the consequences of this assumption is found in Ash (1965).
In the example of information, a transition is made between discrete counts and continuous variables, the probabilities of events. But the constraint also plays a critical role in number theory, where a number of key "number theoretic" functions have a similar property (though it is usually assumed that the values corresponding to M and N are relatively prime.) For example, the function (n) giving the number of integer divisors of an integer n satisfies this condition. An excellent treatment of number theoretic functions from this point of view may be found in Stewart (1952).
Thus we find that the constraint is both strong, in the sense of determining the form of the functions satisfying it, and widespread. Both information and counts are central to much of Social Science policy making. It is the purpose of this appendix to show that other types of commonly occurring constraints are simply related to the one given above.
OTHER FORMS
The previous section defined a key relation that it is attractive for our functions to obey h(xy)=h(x)h(y). Given such a function, we can define other functions with interesting functional properties. For example, k(x) = log_{e}(h(exp(x))). Using the properties of log_{e}, h, and exp, it is easy to see that k obeys k(x+y) = k(x) + k(y). Similarly, given such a function k(x), we could define a function h(x) = exp(k(log_{e}(x))) which obeys the initial condition.
EVALUATING k(x)
We have some freedom in evaluating k(x), but not much. We can now list some consequences of the constraint.
1. We immediately see that, if k(x) satisfies the additivity condition, so does A*k(x), for any constant A. This allows us to choose k for which: k(1) = 1.
2. But if so, we can evaluate k(1/m), for any integer m, 1= k(1) = k({1/m + 1/m + ... + 1/m}^{m}) = m*k(1/m), so, k(1/m) = 1/m.
3. Similarly, k(mx) = m*k(x), so given any positive rational number, m/n, we have, k(m/n)= mk(1/n) = m/n.
4. Since k(x) = k(x+0) = k(x) + k(0), we must necessarily have, k(0) =0.
5. Also, since 0 = k(0) = k(x + (-x) ) = k(x) + k(-x), we can conclude that, k(-x) = -k(x), for arbitrary x.
Thus we see that the additivity condition strictly imposes values that k can take on the rational numbers, a set dense in the real number line. If k(x) is smooth, then, in general, we must have k(x) = Ax.
We can make a stronger statement: If k(x) is monotonic, say monotonically increasing, in any interval I, no matter how small, then it must be continuous in that interval and throughout its range. Thus, k(x) = Ax. For suppose k(x) is monotonic in a small interval including the irrational value x_{0}. Then we can find rational numbers r_{1} and r_{2}, also in I, for which r_{1} < x_{0} < r_{2}. Thus, we have k(r_{1}) < k(x_{0}) < k(r_{2}). This is true even if we choose a sequence of r_{1} and r_{2} increasingly close to x_{0}. For such values, k(r_{1}) = r_{1} and k(r_{2}) = r_{2} both approach x_{0}, so k(x_{0}) must itself equal x_{0}. Thus, at least in I, k(x) = x, for irrational as well as rational x.
But now consider any x. Certainly k(x) = k(r+(x-r)) =
r+k(x-r), for r a rational number near enough to x that there
exist values x_{1} and x_{2}, both in I for which x_{1} - x_{2} = x - r. Then, k(x-r) = k(x_{1} - x_{2}) = k(x_{1}) - k(x_{2}). But in this interval, we saw k(x)=x. Thus we have k(x) = r + x_{1} - x_{2} = r + (x - r) = x, as it was intended to prove.
Ash, R. (1965) Information Theory, New York: Wiley.
Bookstein, A. (1992) Informetric Distributions, Parts I and II, Journal of the American Society for Information Science, 41(5):368-88.
Stewart, B.M. (1952) Theory of Numbers, New York: MacMillan.
DERIVATION for pages 5 and 7 of COMPOSITION ANALYSIS by
Benjamin Drake Wright
Derivation of PACK and CHAIN measure approximations from their exponential (ratio) parameter definition.
PACK definition:
where
so that
Since
PACK measure becomes
CHAIN definition:
so that
Since
CHAIN measure becomes
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