A Review of Mathematical Methods in Sociometry

Abstract

Current mathematical and computer methods for the analysis of group and organizational structures of communication and evaluation are reviewed. Sociometric data collection, the detection of cliques and subgroups, sociometric index construction, and computer methods in sociometry are discussed. Clique detection methods are classified as either linkage methods, in which sociometric data are treated as a linear graph, or as distance methods, in which data are treated as a configuration of points in a space. Several linkage and distance analysis methods are discussed and compared. It is suggested that some of the methods of numerical taxonomy could be applied to the analysis of group and organizational structures.

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References

Alba, R.D. (1972) "A graph-theoretic definition of a sociometric clique." J. of Mathematical Sociology 3: 113-126.

--- ( 1971) "COMPLT: a program for analyzing sociometric data and clustering similarity matrices." Columbia University Bureau of Applied Social Research. (mimeo)

——— and M.P. Gutmann (1971) "SOCK: a sociometric analysis system." Columbia University Bureau of Applied Social Research. (mimeo)

Armor, D.J. and A.S. Couch (1971) The Data-Text Primer: An Introduction to Computerized Social Data Analysis Using the Data-Text System. New York : Free Press.

Augustson, J.G. and J. Minker (1970) "An analysis of some graph theoretical cluster techniques." J. of Assn. for Computing Machinery 17: 571-588.

Bain, R. (1943) "Sociometry and social measurement." Sociometry 6: 206-213.

Ball, G.H. and D.J. Hall (1965) ISODATA, A Novel Method of Data Analysis and Pattern Classification. Stanford: Stanford Research Institute.

Bavelas, A. (1950) "Communication patterns in task-oriented groups ." J. of Acoustical Society of America 57: 271-282.

——— (1948) "A mathematical model for group structures." Applied Anthropology 7: 16-30.

10.

Beaton, A.E. (1966) "An interbattery factor analytic approach to clique analysis." Sociometry 29: 135-145.

11.

Beauchamp, M.A. (1965) "An improved index of centrality." Behavioral Sci. 10: 161-163.

12.

Berge, C. (1962) The Theory of Graphs and its Applications. New York: John Wiley.

13.

Beum, C.O. and E.G. Brundage (1950) "A method for analyzing the sociomatrix." Sociometry 13: 141-145.

14.

Bjerstedt, A. (1956) Interpretations of Sociometric Choice Status. Copenhagen: Munksgaard.

15.

Bock, R.D. and S.Z. Husain (1950) "An adaptation of Holzinger's B-coefficients for the analysis of sociometric data." Sociometry 13: 146-153.

16.

Bonner, R.E. (1964) "On some clustering techniques." IBM J. of Research and Development: 22-33.

17.

Boorman, S.A. and C.C. Olivier (1973) "Metrics on spaces of finite trees." J. of Mathematical Psychology 10: 26-59.

18.

Borgatta, E.F. (1951) "A diagnostic note on the construction of sociograms and action diagrams." Group Psychotherapy 3: 300-308.

19.

——— and W. Stolz (1963) "A note on a computer program for rearrangement of matrices." Sociometry 26: 391-392.

20.

Boyle, R.P. (1969) "Algebraic systems for normal and hierarchical sociograms." Sociometry 32: 99-119.

21.

Cartwright, D.P. and F. Harary (1968) "On the coloring of signed graphs." Elemente der Mathematik 23: 85-89.

22.

——— (1956) "Structural balance: a generalization of Heider's theory." Psych. Rev. 63: 277-292.

23.

Cattell, R.B. and M.A. Coulter (1966) "Principles of behavioural taxonomy and the mathematical basis of the taxonome computer program." British J. of Mathematical and Stat. Psychology 19: 237-269.

24.

Cohen, S. and C.M. Vincent (1968) "An introduction to SPEAKEASY." Argonne National Laboratory, Illmois.

25.

Coleman, J.S. (1958) "Relational analysis: the study of social organizations with survey methods." Human Organization 17: 28-36.

26.

Coombs, C.H. (1964) A Theory of Data. New York: John Wiley.

27.

Cormack, R.M. (1971) "A review of classification." J. of Royal Stat. Society 134: 321-367.

28.

Criswell, J.H. (1949) "Sociometric concepts in personnel administration ." Sociometry 12: 287-300.

29.

Cronbach, L.J. and G.C. Gleser (1953) "Assessing similarity between profiles." Psych. Bull. 50: 456-473.

30.

Davis, J.A. (1970) "Clustering and hierarchy in interpersonal relations: testing two graph theoretical models on 742 sociomatrices." Amer. Soc. Rev. 35: 841-851.

31.

--- ( 1967) "Clustering and structural balance in graphs." Human Relations 20: 181-187.

32.

--- and S. Leinhart (1971) "The structure of positive interpersonal relations " in Volume 2 of J. Berger (ed.) Sociological Theories in Progress. Boston: Houghton Mifflin.

33.

Davis, J.A. and C.F. Warnath (1957) "Reliability, validity, and stability of a sociometric rating scale." J. of Social Psychology 45: 111-121.

34.

Dunnington, M.J. (1957) "Investigation of areas of disagreement in sociometric measurment of preschool children." Child Development 28:93-102.

35.

Edwards, A.F. and L.L. Cavalli-Sforza (1965) "A method for cluster analysis ." Biometrics 21: 362-375.

36.

Eng, E. and R.L. French (1948) "The determination of sociometric status." Sociometry 11: 368-371.

37.

Festinger, L. , S. Schachter , and K. Back (1950) Social Pressures in Informal Groups: A Study of Human Factors in Housing. Stanford: Univ. Press.

38.

Fishbein, M. (1965) "Prediction of interpersonal preferences and group member satisfaction from estimated attitudes." J. of Personality and Social Psychology 1: 663-667.

39.

——— (1963) "An investigation of the relationships between beliefs about an object and the attitude toward that object." Human Relations 16: 223-239.

40.

Flament, C. (1963) Applications of Graph Theory to Group Structure . Englewood Cliffs, N.J.: Prentice-Hall .

41.

Ford, L.R. and D.R. Fulkerson (1962) Flows in Networks. Princeton, N.J.: Princeton Univ. Press.

42.

Forsyth, E. and L. Katz (1946) "A matrix approach to the analysis of sociometric data: preliminary report." Sociometry 9:340-347.

43.

Foster, C.C. and W.J. Horvath (1971) "A study of a large sociogram III: reciprocal choice probabilities as a measure of social distance." Behavioral Sci. 16: 429-435.

44.

Foster, C.C. , A. Rapoport , and C.J. Orwant (1963) "A study of a large sociogram II: elimination of free parameters." Behavioral Sci. 8: 56-65.

45.

Gardner, E.F. and G.G. Thompson (1956) Social Relations and Morale in Small Groups. New York: Appleton-Century-Crofts.

46.

Glanzer, M. and R. Glaser (1959) "Techniques for the study of group structure and behavior: I. Analysis of Structure." Psych. Bull. 56: 317-332.

47.

Gleason, T.C. (1971) "DIP: a directed graph processor." University of Michigan Institute for Social Research.

48.

——— (1969) "Multidimensional scaling of sociometric data." University of Michigan Institute for Social Research.

49.

——— and D. Cartwright (1967) "A note on a matrix criterion for unique colorability of a signed graph." Psychometrika 32: 291-296.

50.

Green, P.E. and F.J. Carmone 1970) Multidimensional Scaling and Related Techniques in Marketing Analysis. Boston: Allyn & Bacon .

51.

Greenwood, M. and G.U. Yule (1920) "An inquiry into the nature of frequency distributions representative of multiple happenings." J. of Royal Stat. Society 83: 255-279.

52.

Guimaraes, L.L. (1968) "Matrix multiplication in the study of interpersonal communication." M.A. thesis. Michigan State University .

53.

Harary, F. (1969) Graph Theory. Reading, Mass .: Addision-Wesley.

54.

--- ( 1954) "On the notion of balance of a signed graph." Michigan Mathematical J. 2: 143-146.

55.

——— and I.C. Ross (1957) "A procedure for clique detection using the group matrix." Sociometry 20: 205-215.

56.

Harary, F. , R.Z. Norman and D.P. Cartwright (1965) Structural Models: An Introduction to the Theory of Directed Graphs. New York: John Wiley.

57.

Heider, F. (1946) "Attitudes and cognitive organization." J. of Psychology 21: 107-112.

58.

Holland, P.W. and S. Leinhart (1973) "The structural implications of measurement error in sociometry." J. of Mathematical Sociology 3: 85-111.

59.

——— (1970) "A method for detecting structure in sociometric data." Amer. J. of Sociology 76: 492-513.

60.

Hubbell, C.H. (1965) "An input-output approach to clique identification ." Sociometry 28: 377-399.

61.

International Business Machines Corporation ( 1970) "APL/360 user's manual." White Plains, New York.

62.

Inter-University Consortium for Political Research (1971) OSIRIS II User's Manual. Ann Arbor, Mich.: Institute for Social Research.

63.

Jennings, H.H. (1950) Leadership and Isolation. New York: Longmans, Green.

64.

Johnson, S.C. (1967) "Hierarchical clustering schemes." Psychometrika 32: 241-254.

65.

Katz, L. (1953) "A new status index derived from sociometric analysis ." Psychometrika 18: 39-43.

66.

——— and J.H. Powell (1960) "Probability distributions of random variables associated with a structure of the sample space of sociometric investigations ," pp. 309-317 in J. L. Moreno (ed.) The Sociometry Reader. New York: Free Press.

67.

Kruskal, J.B. (1964a) "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis." Psychometrika 29: 1-27.

68.

--- ( 1964b) "Nonmetric multidimensional scaling: a numerical method ." Psychometrika 29: 115-130.

69.

Leinhart, S. (1971) "SOCPAC I: a FORTRAN IV program for structural analysis of sociometric data." Behavioral Sci. 16: 515.

70.

Leontief, W.W. (1941) The Structure of the American Economy, 1919-1929 . Cambridge, Mass.: Harvard Univ. Press.

71.

Lerner, M.J. (1965) "The effect of responsibility and choice on a partner's attractiveness following failure." J. of Personality 33: 178-187.

72.

Lin, N. ( 1970) "Measuring prestige as a property of group structure." Johns Hopkins University. (mimeo)

73.

Lindzey, G. and D. Byrne (1968) "Measurement of social choice and interpersonal attractiveness," pp. 452-525 in Volume 2 of G. Lindzey and E. Aronson (eds.) The Handbook of Social Psychology. Reading, Mass.: Addison-Wesley.

74.

Lingwood, D.A. (1969) "Interpersonal communication, research productivity, and invisible colleges." Ph.D. dissertation. Stanford University .

75.

Lorr, M. (1968) "A review and classification of typological procedures ." Presented at the meetings of the American Psychological Association, San Francisco.

76.

——— and B.K. Radhakrishnan (1967) "A comparision of two methods of cluster analysis." Educ. and Psych. Measurement 27: 47-53.

77.

Luce, R.D. (1950) "Connectivity and generalized cliques in sociometric group structure." Psychometrika 14: 169-190.

78.

——— and A.D. Perry (1949) "A method of matrix analysis of group structure ." Psychometrika 14: 95-116.

79.

MacRae, D., Jr. (1960) "Direct factor analysis of sociometric data." Sociometry 23: 360-371.

80.

McKinney, J.C. (1948) "An educational application of a two-dimensional sociometric test." Sociometry 11: 356-367.

81.

McQuitty, L.L. (1963) "Rank order typal analysis." Educ. and Psych. Measurement 23.

82.

——— (1957) "Elementary linkage analysis for isolating orthogonal and oblique types and typal relevancies." Educ. and Psych. Measurement 17: 207-229.

83.

--- and J.A. Clark (1968) "Clusters from iterative, intercolumnar correlational analysis." Educ. and Psych. Measurement 28: 211-238.

84.

Michener, C.D. and R.R. Sokal (1957) "A quantitative approach to a problem in classification ." Evolution 11: 130-162.

85.

Moreno, J.L. (1934) Who Shall Survive? Washington, D.C.: Nervous and Mental Disease Publishing Company .

86.

——— H. H. Jennings , J. H. Criswell , L. Katz , R. R. Blake , J. S. Mouton , M. E. Bonney , M. L. Northway , C. P. Loomis , C. Proctor , R. Tagiuri , and N. Nehnevajsna [eds.] (1960) The Sociometry Reader. New York: Free Press.

87.

Mouton, J.S. , R.R. Blake , and B. Fruchter (1960a) "The reliability of sociometric measures," pp. 320-361 in J. L. Moreno et al. (eds.) The Sociometry Reader. New York: Free Press.

88.

--- ( 1960b) "The validity of sociometric responses," pp. 362-387 in J. L. Moreno et al. (eds.) The Sociometry Reader. New York: Free Press.

89.

Mulligan, G.D. and D.G. Corneil (1972) "Corrections to Bierstone's algorithm for generating cliques." J. of Assn. for Computing Machinery 19: 244-247.

90.

Needham, R.M. (1961) "The theory of clumps, II." Cambridge Language Research Unit Report M. L. 139, Cambridge, England.

91.

Newcomb, T.M. (1961) The Acquaintance Process. New York: Holt, Rinehart & Winston.

92.

Nie, N.H. , D.H. Bent , and C.H. Hull (1970) SPSS: Statistical Package for the Social Sciences . New York: McGraw-Hill.

93.

Norman, R.Z. and F.S. Roberts (1972) "A derivation of a measure of relative balance for social structures and a characterization of extensive ratio systems." J. of Mathematical Psychology 9: 66-91.

94.

Norman, W.T. (1967) "Cluster analysis: a diametric rationale and procedure ." University of Michigan. (mimeo)

95.

Northway, M.L. (1940) "A method for depicting social relationships obtained by sociometric testing." Sociometry 3: 144-150.

96.

Nosanchuk, T.A. (1963) "A comparison of several sociometric partitioning techniques." Sociometry 26: 112-124.

97.

Ore, O. ( 1963) Graphs and Their Uses. New York: Random House.

98.

Parker-Rhodes, A.F. (1961) "Contributions to the theory of clumps, I." Cambridge Language Research Unit Report M. L. 138, Cambridge , England.

99.

Peay, E.R., Jr. (1970) "Extensions of clusterability to quantitative data with an application to the cognition of political attitudes." Ph.D. dissertation. University of Michigan.

100.

Pepinsky, H.B. , L. Siegel , and E.L. Vanatta (1952) "The criterion in counseling: a group participation scale." J. of Abnormal and Social Psychology 47: 415-419.

101.

Peterson, R.J. , S.S. Komorita , and H.C. Quay (1964) "Determinants of sociometric choices." J. of Social Psychology 62: 65-75.

102.

Phillips, D.P. and R.H. Conviser (1972) "Measuring the structure and boundary properties of groups: some uses of information theory." Sociometry 35: 235-254.

103.

Proctor, C.H. (1969) "Analyzing pair and point data on social relationships, attitudes and background characteristics of Costa Rican census bureau employees ." Proceedings of the Social Statistics Section of the American Statistical Association 457-465.

104.

——— (1967) "The variance of an estimate of linkage density from a simple random sample of graph nodes." Proceedings of the Social Statistics Section of the American Statistical Association 342-343.

105.

——— (1960) "A sociometric method for distinguishing social structures," pp. 308-315 in R. N. Adams and J. J. Preiss (eds.) Human Organization Research . Homewood, III.: Dorsey.

106.

--- and C.P. Loomis (1951) "Analysis of sociometric data," pp. 561-585 in M. Jahoda et al. (eds.) Research Methods in Social Relations, with Especial Reference to Prejudice. New York: Dryden.

107.

Purdy, G. (1973) "Finding the cliques of a graph." University of Illinois Center for Advanced Computation Technical Memorandum .

108.

Rapoport, A. and W.J. Horvath (1961) "A study of a large sociogram." Behavioral Sci. 6: 279-291.

109.

Riffenburgh, R.H. (1966) "A method of sociometric identification on the basis of multiple measurements." Sociometry 29: 280-290.

110.

Roistacher, R.C. (1972) "Peer nominations, clique structures, and exploratory behavior in boys at four junior high schools." Ph.D. dissertation. University of Michigan.

111.

--- ( 1971) "Peer nominations of exploratory behavior." State University of New York at Buffalo Studies in Psychotherapy and Behavioral Change 2.

112.

Saunders, D.R. and H. Schucman (1962) "Syndrome analysis: an efficient procedure for isolating meaningful subgroups in a nonrandom sample of a population." Presented at the Third Annual Meeting of the Psychonomic Society , St. Louis.

113.

Sawrey, W.L. , L. Keller , and J.J. Conger (1960) "An objective method of grouping profiles by distance functions and its relation to factor analysis." Educ. and Psych. Measurement 20: 651-674.

114.

Shepard, R.N. (1972) "A taxonomy of some principal types of data and of multidimensional methods for their analysis," in Volume 1 of R. N. Shepard , A. K. Romney , and M. B. Nerlove (eds.) Multidimensional Scaling: Theory and Applications in the Behavioral Sciences. New York: Seminar.

115.

--- ( 1972) A.K. Romney , and S.B. NERLOVE Multidimensional Scaling : Theory and Applications in the Behavioral Sciences. New York: Seminar.

116.

Smucker, O. (1949) "Near-sociometric analysis as a basis for guidance ." Sociometry 12: 326-340.

117.

Sokal, R.R. and P.H.A. Sneath (1963) Principles of Numerical Taxonomy. San Francisco: W. H. Freeman.

118.

Sonquist, J.A. (1970) Multivariate Model Building: The Validation of a Search Strategy. Ann Arbor, Mich.: Institute for Social Research.

119.

Spilerman, S. (1966) "Structural analysis and the generation of sociograms ." Behavioral Sci. 11: 312-318.

120.

Taylor, M. (1969) "Influence structures." Sociometry 32: 490-502.

121.

Thorndyke, R.L. (1953) "Who belongs in the family?" Psychometrika 18: 267-276.

122.

Torgerson, W.S. (1958) Theories and Methods of Scaling. Reading, Mass.: Addision-Wesley.

123.

Van Valkenburg, M.E. (1964) Network Analysis. Englewood Cliffs, N.J.: Prentice-Hall.

124.

Wright, B. and M.S. Evitts (1961) "Direct factor analysis in sociometry." Sociometry 24: 82-98.