## Teams, Packs and Chains

In the 1920's Physicist Norman Campbell considered linear measurement of a social science variable to be impossible because he thought people could not be concatenated in any way similar to placing sticks end-to-end to measure length. More recently we have come to realize that people are always being "concatenated", i.e., accumulated to increase quantity according to specific rules. Teams, Packs and Chains are three methods of social concatenation.

Teams:
Teams work as unions of perfect agreements. When Team members are given a task to perform, they must all agree on what to do (right or wrong). For a two-member team, helpful agreement (11) wins, but unhelpful agreement (00) loses. Disagreements (10) and (01) are absent because they are eliminated by the rules that define Team behavior. Let Team concatenation be indicated by x.

If Pni is the probability of success of person n on item i, and Pmi is the probability of success of person m on item i, then their Team odds of success, under Rasch model conditions, is:

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

= exp (Bn+Bm -(Di+Di))

Take logs and generalize to N Team members. Define N-member Team strength as BT. Then

BT - Di = sum (Bn - Di)

The strengths of individual Team members relative to task difficulty (Bn - Di) add up to Team strength relative to task difficulty, (BT - Di). Teams are concatenations of relative strengths, accumulated in linear form. Adding a member more able than the task increases Team strength. Adding a member less able than the task decreases Team strength and weakens the Team. More is often not better, as has been frequently observed about committees.

Packs:
Packs work as collections of helpful disagreements. When Pack members are given a task to perform, the first success wins for the Pack. If no member is successful, the Pack loses. Search- parties work this way. For a two-member Pack, helpful disagreements (10) and (01) win. Unhelpful agreement (00) loses. Helpful agreement (11) is absent because it is eliminated by the rules that define Pack behavior. Let Pack concatenation be indicated by +.

If Pni is the probability of success of person n on item i, and Pmi is the probability of success of person m on item i, then their Pack odds of success, under Rasch model conditions, is:

Pn+m/(1-Pn+m) = ( Pni(1-Pmi) + (1-Pni)Pmi ) / (1-Pni)(1-Pmi) exp(Bn+m) = exp(Bn) + exp(Bm)

a concatenation of absolute strengths accumulated in exponential form, which does not depend on Di.

When Bn and Bm are similar, this is approximated by:

exp(Bn+m) = 2 exp ((Bn + Bm)/2)

Take logs and extend to N similarly able Pack members. Define N- member Pack strength as BP. Then

BP = sum (Bn/N) + loge(N)

Pack composition is a collection because the helpful disagreements of Pack members collect to benefit the Pack. As N rises so must BP. The more the better. Unlike Teams, the strength of a Pack is independent of task difficulty.

Chains:
Chain composition is a connection of commitments against harmful disagreement. Member actions are linked by mutual reliance on not disagreeing. An ability to maintain complete agreement is Chain strength. The harm that can result from a single disagreement is Chain weakness.

When the Chain members are given a task to perform, the first failure loses for the Chain. If there is no failing member, the Chain wins. For a two-member Chain, helpful agreement (11) wins. Lack of agreement (10) and (01) loses. Unhelpful agreement (00) is absent because it is eliminated by the rules that define Chain behavior. Let Chain concatenation be indicated by *.

If Pni is the probability of success of person n on item i, and Pmi is the probability of success of person m, then their Chain odds of success, under Rasch model conditions, is:

Pn*m/(1-Pn*m) = Pni*Pmi /( Pni(1-Pmi) + (1-Pni)Pmi )

exp(-Bn*m) = exp(-Bn) + exp (-Bm)

a concatenation of weaknesses accumulated in exponential form, which does not depend on Di.

When Bn and Bm are similar this is approximated by:

exp(Bn*m) = 0.5 * exp((Bn+Bm)/2)

Take logs and extend to N similarly able Chain members. Define N-member Chain strength as BC. Then

BC = sum (Bn/N) - loge(N)

Adding another member or link to a Chain always decreases Chain strength.

Comparison:
The Figure depicts the effectiveness of the three types of concatenation as group size (of equally able members) increases for a range of problem difficulties.

Example: Imagine there are 10 people, all of ability -1 logit relative to the task to be performed:
Team ability = BT = 10 * -1 = -10 logits
Pack ability = BP = 10 * (-1 / 10) + loge(10) = 1.3 logits
Chain ability = BC = 10 * (-1 / 10) - loge(10) = -3.3 logits
so that: BP > BC > BT

Teams are most effective when a problem is very easy (at the top), otherwise Packs are most effective. Chains are least effective, except when a problem is very hard (at the bottom). Then Teams are least effective. A mob is a Team that chooses the wrong solution to a difficult social problem.

On a practical note, productive Japanese-style consensus- building, i.e., Team-work, requires that each difficult task be divided into several easy tasks, so that there is a high probability that each group member, acting alone, would have made the consensus-decision.

Teams, packs and chains. Wright BD. … Rasch Measurement Transactions, 1995, 9:2 p.432

Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
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Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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