¹ The American Science of Politics (London 1959), 247–48.Google Scholar

² Hoffmann's, “International Relations: The Long Road to Theory,” World Politics, XI (April 1959), 346–77CrossRefGoogle Scholar, is perhaps the most significant recent combination of theoretical insight and methodological confusion regarding nonadditivities in international relations research. The above quotation is taken from a revised version of his “Long Road to Theory” article: Hoffmann, Stanley, ed., Contemporary Theory in International Relations (Englewood Cliffs 1960), 45.Google Scholar See also pp. 40–44, 46–52.

³ “The Sociological Study of Conflict,” in International Sociological Association, The Nature of Conflict: Studies on the Sociological Aspects of International Tensions (Paris 1957)Google Scholar, chap. 1, partly reprinted in Hoffmann, Contemporary Theory, 124–36. The quotation above is taken from p. 133 of the Hoffmann reader. See also Hoffmann's “Long Road to Theory,” 364.

Rapoport, Anatol has made the same point, with more homely examples, in the glossary of his Strategy and Conscience (New York 1964), 308.Google Scholar “Additivity” is there defined as “a property possessed by measurable quantities whose magnitudes can be meaningfully added when the quantities are combined. For example, the combined weight of Cow A and Cow B equals the sum of two weights. However, the utilities of objects or events are not necessarily additive. The utility of a cup of coffee in which an ounce of salt has been dissolved need not equal [for most people it will be a good deal less than] the utility of a cup of coffee plus die utility of an ounce of salt.”

⁴ Besides the enormous statistical literature on random errors and statistical significance, it might be worth citing just a few of the more imaginative works on nonrandom errors. In decreasing degrees of mathematical sophistication are Frisch, Ragnar, Statistical Confluence Analysis by Means of Complete Regression Systems (Oslo 1934)Google Scholar; Blalock, Hubert M. Jr., Causal Inferences in Nonexperimental Research (Chapel Hill 1964)Google Scholar, esp. chap. 5; and Naroll, Raoul, Data Quality Control (Glencoe 1962).Google Scholar

⁵ Hoffmann, Contemporary Theory, 45.

⁶ Although many measurement problems can thus be “solved” by using lower “levels of measurement” with higher levels of precision, there remain much more serious issues known technically as problems of measurement reliability (reproducibility) and validity (truthfulness). For a helpful discussion of these problems, the reader is referred to the social science literature on research methods, e.g., Festinger, Leon and Katz, Daniel, eds., Research Methods in the Behavioral Sciences (New York 1953).Google Scholar

⁷ “Long Road to Theory,” 364.

⁸ Contemporary Theory, 42. See also pp. 174–84. The sensible inner quotations in this passage are taken from Cohen, Morris R. and Nagel, Ernest, An Introduction to Logic and Scientific Method (New York 1934), 266–67.Google Scholar

⁹ For those who enjoy historical or genetic arguments, etymologically the word “statistics” comes from a medieval Latin term for “state"; in the eighteenth century it acquired somewhat similar meanings in both British and German intellectual circles, and was defined by one Baron J. F. von Bielfield as “the science that teaches us what is the political arrangement of all the modern states of the known world.” See Yule, G. Udny and Kendall, M. G., An Introduction to the Theory of Statistics, 14th ed. (New York 1957)Google Scholar, xvi-xix; and Lazarsfeld's, Paul F. “Notes on the History of Quantification in Sociology: Trends, Sources and Problems,” in Woolf, Harry, ed., Quantification: A History of Meaning of Measurement in the Natural and Social Sciences (Indianapolis 1961), 147–203.Google Scholar

¹⁰ Hoffmann, Contemporary Theory, 176. Similar calls for contextually valid definitions of foreign policy attitudes are Richard C. Snyder's plea for “concepts and theories which are situationally referred” (italics in the original) and Cantril and Free's development of a Self-Anchoring Striving Scale. See Snyder, , “Some Recent Trends in International Relations Theory and Research,” in Ranney, Austin, ed., Essays on the Behavioral Study of Politics (Urbana 1962), 129Google Scholar; and Cantril, Hadley and Free, L. A., “Hopes and Fears for Self and Country,” American Behavioral Scientist, VI, Supplement (October 1962).Google Scholar An exciting and comprehensive characterization of conflict contexts is Leiserson, Michael A., “Coalitions in Politics,” unpubl. diss., Yale, 1965.Google Scholar

Problems of aggregating dissimilar cases have of course been discussed in other social sciences. In particular see Naroll, Raoul and D'Andrade, Roy, “Two Further Solutions to Galton's Problem,” American Anthropologist, LXV (October 1963), 1053–67CrossRefGoogle Scholar; Orcutt, GuyGreenberger, Korbel, Rivlin, , Microanalysis of Socioeconomic Systems: A Simulation Study (New York 1961)Google Scholar; Thurstone, Louis, Multiple Factor Analysis (Chicago 1947)Google Scholar, especially the chapter on “the effects of selection.”

¹¹ That scientists lean toward the opposite belief is indicated by the popular doctrine that indicators are interchangeable. See Horwitz, Hortense and Smith, Elias (pseudonyms for a weightier intelligence), “The Interchangeability of Socio-Economic Indices,” in Lazarsfeld, Paul F. and Rosenberg, Morris, eds., The Language of Social Research (Glencoe 1955), 73–77Google Scholar; and Deutsch, Karl W., “Social Mobilization and Political Development,” American Political Science Review, LV (September 1961), 493–514.CrossRefGoogle Scholar The opposite doctrine, that indicators are variable, not interchangeable, is basic to Lazarsfeld's latent class analysis, wherein classes are subsets within which indicators are independent of each other. The procedure is designed to explain bivariate correlations and three variable interactions as due to differences among such subsets. See, for instance, Lazarsfeld, , “A Conceptual Introduction to Latent Structure Analysis,” in Lazarsfeld, , ed., Mathematical Thinking in the Social Sciences (Glencoe 1955), 341–87.Google Scholar

¹² Mathematically less sophisticated than latent structure analysis, factor analysis has long been an important methodology of psychometric research; it has more recently been used to conceptualize dimensions of national character, economic development, and international conflict. See Thurstone, Multiple Factor Analysis; Cattell, R. B., “The Dimensions of Culture Patterns of Factorization of National Characters,” Journal of Abnormal and Social Psychology, XLIV (1949), 443–69CrossRefGoogle Scholar; Berry, Brian J. L., “Patterns of Economic Development,” in Ginsburg, Norton, ed., Atlas of Economic Development (Chicago 1961)Google Scholar; Rummel, R. J., “Dimensions of Conflict Behavior Within and Between Nations,” General Systems, VIII (1963), 1–50Google Scholar; Alker, H. R. Jr., “Dimensions of Conflict in the General Assembly,” American Political Science Review, LVIII (September 1964), 642–57.CrossRefGoogle Scholar More general applications include Gregg, Phillip and Banks, Arthur S., “Dimensions of Political Systems: Factor Analysis of A Cross-Polity Survey,” American Political Science Review, LIX (September 1965), 602–14CrossRefGoogle Scholar; Bruce M. Russett, “Delineating International Regions,” in J. David Singer, ed., Insights and Indicators in World Politics (forthcoming); and R. J. Rummel and others, Dimensions of Nations (forthcoming).

¹³ Developed and underdeveloped “regions,” containing twenty-eight and seventythree nations respectively, are distinguished on the basis of per capita gross national products, in Russett, Bruce M. and others, World Handbook of Political and Social Indicators (New Haven 1964).Google Scholar

¹⁴ I have discussed these issue dimensions most fully in Part I of Alker, and Russett, , World Politics in the General Assembly (New Haven 1965).Google Scholar The procedure of looking first at the regional, then the universal, and finally the residual (i.e., the regionaluniversal) factor loadings resembles one suggested by Tukey, John W.; see his “The Future of Data Analysis,” Annals of Mathematical Statistics, XXXIII (March 1962), 60–61.Google Scholar He refers to multiple response data for which a small number of descriptive variables exist. He considers (a) regressing response data on the descriptive variables and then (b) factor analyzing residual responses (unaccounted for by the regression) as conveying more meaning than a one-step factor analysis: “Consider a study of … children's personalities, as revealed in their use of words. … [If sexes are known,] elimination of the additive effect of sex would almost surely lead to more meaningful ‘factors,’ and eliminating of reading speed as well … might lead us to even closer grips with essentials.” Tukey's “residual factor analysis” procedure is clearly a valuable supplement to the regional factor analysis procedure, especially as a way of uncovering contextually adjusted relationships of universal validity.

¹⁵ In the seventy-roll-call factor analysis reported in Alker, “Dimensions of Conflict,” votes 4, 8, and 10 (but not vote I) all loaded heavily on a “UN supranationalism” factor. A detailed examination of the appropriateness of the supranationalism label for this particular UN factor is presented in Alker, , “Supranationalism in the United Nations,” Peace Research Society: Papers, III (1966), 197–212.Google Scholar

¹⁶ In the more detailed analyses referred to in the previous footnotes, votes 3 and 6 loaded on a “cold-war factor” and votes 2 and 7 formed part of a “Moslem/selfdetermination” dimension. For interpretive purposes the rather high loading of Chinese representation on the self-determination factor and the rather low loading of the “West Irian self-determination” resolution on the same voting component suggest both the possibilities and limits of the “self-determination” tendency in UN voting.

¹⁷ Regarding UN votes, which are themselves a relatively homogeneous body of data, it should be noted that “East-West,” “North-South,” “self-determination,” “cold-war” and “Moslem issues” factors were all identified in the larger studies referred to previously. In this sense the analysis above has helped confirm the earlier larger universal analysis by not uncovering any new factors. By way of contrast, using essentially the same regional factor analysis procedure as that discussed here, Eric Allardt has found several significant regional factors not found in a universal analysis. After noting in the case of noninterchangeable indicators that using the same operational indicator for the same theoretical concept would lead to erroneous results when applied within different subgroups or different areas, Allardt distinguishes “traditional” from “emerging” radicalism because unemployment is his best ecological indicator of insecurity (and Communist voting) in backward areas, while housing problems serve best to index insecurity in the cities. See his “Patterns of Class Conflict and Working Class Consciousness in Finnish Politics,” in Allardt, and Littunen, Y., Cleavages, Ideologies, and Party Systems (Helsinki 1964).Google Scholar

¹⁸ Using roll-call terminology, the factor model assumed by almost all methods of factor analysis is

The vote of any nation i on roll call j (V_ji) is assumed to result from a sum of underlying national scores on K general factors (F_ki's), each national factor score being weighted by the appropriate factor loading (a_jk, the loading of roll call j on factor k). Factor loadings are assumed not to depend on i, the country concerned. U_ji is the “uniqueness” of nation i's position on roll call j, the part of V_ji not explainable in general factor terms. See Harmon, Henry H., Modern Factor Analysis (Chicago 1960)Google Scholar, chap. 2 for further details.

¹⁹ A more elaborate statistical procedure for comparing matrices of factor loadings derived from the same variables is described in Ahmavaara, Yrjo, “On the Unified Factor Theory of Mind,” Annales Akademiae Scientiarum Fennicae, Serial B, No. 160 (Helsinki 1957).Google Scholar

As Thurstone suggests, any such comparison should allow the same factors to account for differing variances in each subpopulation. One of the reasons why Thurstone urges carefully selecting the variables to be included in a factor analysis is that background or experimental variables (in our example, the contextual variable of economic development) influencing particular observed variables can cause them to load on different factors, thus making factor interpretation extremely hazardous.

The ideally homogeneous set of data would be a variety of separate indicator variables not influencing each other but themselves caused in a linear additive fashion by factors, which may themselves be correlated. In such a case, factor analysis may be validly interpreted as offering a causal explanation of the manifest phenomena. I suspect that these assumptions are more approximately valid concerning the opinions and attitudes of frequently interacting individuals (e.g., the General Assembly) than for large (and useful) agglomerations of ecological and/or judgmental data such as the World Handbook of Political and Social Indicators or Banks, Arthur S. and Textor, Robert B., A Cross-Polity Survey (Cambridge, Mass., 1963).Google Scholar

For further discussion of these points see Blalock, Causal Inferences, 162–71; Cattell, R. B. and Dickman, K., “A Dynamic Model of Physical Influences Demonstrating the Necessity of Oblique Simple Structure, Psychological Bulletin, LIX (September 1962), 389–400CrossRefGoogle Scholar; Wright, Sewall, “The Interpretation of Multivariate Systems,” in Kempthorne, Oscar and others, Statistics and Mathematics in Biology (Ames, Iowa, 1954), 11–33.Google Scholar

²⁰ See Thurstone; and Cattell, R. B., Factor Analysis (New York 1952).Google Scholar

²¹ It is possible formally to summarize such a procedure in a single equation. Using the notation of equation (1), nonadditive factor loadings b_jk(r), where r indicates the region of data being discussed, can be defined by

For each value of r, the resid_ji(r) terms are the significant or large entries in the matrix of residual factor loadings. This way of thinking is basically nonadditive in that it denies the possibility of additively breaking a variety of indicators into a smaller set of conceptual variables without further breaking the data down into smaller subpopulations. It should also be mentioned that other, more radical departures from additive models are possible, such as Guttman's smallest space analysis and Gibson's latent profile analysis. See, Louis Guttman, “A General Nonmetric Technique for Finding the Smallest Euclidean Space for a Configuration of Points,” Psychometrika (forthcoming); and Gibson, W. A., “Three Multivariate Models: Factor Analysis, Latent Structure Analysis, and Latent Profile Analysis,” Psychometrika, XXIV (September 1959), 229–52.CrossRefGoogle Scholar

²² See footnote 2. In Political Community (Princeton 1957)Google Scholar, Deutsch et al. are quite explicit on how their background conditions combine. They distinguish among conditions that have been always or probably present in successful integrations (certainly no dismemberment of international reality); they rank integration methods in the order of apparent decisiveness; and they pay particular attention to disintegrating conditions likely to produce system changes.

²³ “The Integrative Process: Guidelines for Analysis of the Bases of Political Community,” in Deutsch, Jacob, Teune, Toscano, , and Wheaton, , The Integration of Political Communities (Philadelphia 1964), 3.Google Scholar

²⁴ Relevant literature on nonadditive explanations and the equivalent concept of statistical interaction includes Lazarsfeld, , “The Algebra of Dichotomous Systems” in Solomon, Herbert, ed., Item Analysis and Prediction (Stanford 1961)Google Scholar; Blalock, Hubert M. Jr., Social Statistics (New York 1960)Google Scholar, chap. 20, “Covariance Analysis”; Blalock, , “Theory Building and the Statistical Concept of Interaction,” American Sociological Review, XXX (June 1965), 374–80CrossRefGoogle Scholar; Anscombe, F. J. and Tukey, John W., “The Examination and Analysis of Residuals,” Technometrics, V (May 1963), 141–60CrossRefGoogle Scholar; and Coleman, James S., An Introduction to Mathematical Sociology (New York 1964)Google Scholar, esp. chap. 6.

²⁵ Cutright's, Phillips “National Political Development: Social and Economic Correlates” in Polsby, Nelson W.Dentler, Robert A., and Smith, Paul A., Politics and Social Life (Boston 1963), 569–82Google Scholar, has been an influential example of this approach. Another example is Russett, Bruce M., “Inequality and Instability: The Relation of Land Tenure to Politics,” World Politics, XVI (April 1964), 442–54.CrossRefGoogle Scholar

²⁶ The supranationalism and caucusing group variables have been operationalized in a slightly different manner from those reported in Alker, “Dimensions of Conflict in the General Assembly.” Western Big Three trade as a percentage of a nation's total is measured according to the same specifications. See Alker and Russett, World Politics in the General Assembly, for more details.

²⁷ For a discussion of mixed and purely multiplicative explanatory models geared to comparative and international politics, see Alker and Russett, “Multifactor Explanations of Social Change,” in the World Handbook Blalock's “Theory Building and the Statistical Concept of Interaction” is especially exciting because it derives “mixed” models in a deductive manner by multiplying variables that are themselves linear combinations of prior influences.

For qualitative contextual generalizations similarly susceptible to logical and mathematical formalization see Deutsch, Karl W. and Merritt, Richard, “Effects of Events on National and International Images,” in Kelman, Herbert C., ed., Internettional Behavior (New York 1965), 130–87Google Scholar; and James N. Rosenau, “Pre-Theories and Theories of Foreign Policy,” in R. B. Farrell, ed., Approaches to Comparative Politics and International Relations (Evanston, forthcoming).

²⁸ Taking logarithms of both sides of equation (4) shows us that coefficients b, c, and d are elasticity coefficients indicating the percentage rate of change in supranationalist voting (Y) associated with percentage changes in V, W, and X respectively:

Note that equation (5) is linear and additive, and may be subjected to ordinary least-square regression estimation procedures!

²⁹ Hoffmann, Contemporary Theory, 177.

³⁰ For example, Nagel, Ernest, “Wholes, Sums, and Organic Unities,” in Lerner, Daniel, ed., Parts and Wholes (Glencoe 1963), 135–55Google Scholar, makes the following relevant remarks: “… It is often claimed that a functional whole cannot be properly analyzed from an ‘additive point of view’; that is, the characteristic modes of functioning of its constituents must be studied in situ, and the structure of activities of the whole cannot be inferred from properties displayed by its constituents in isolation from the whole.… [The] mere fact that a system is a structure of dynamically interrelated parts does not suffice, by itself, to prove that the laws of such a system cannot be reduced to some theory developed initially for certain assumed constituents of the system.”

³¹ Relevant book-length treatments of the causal inference problem include Frisch, Statistical Confluence Analysis; Tinbergen, , Statistical Testing of Business Cycle Theories II: Business Cycles in the United States of America 1919–32 (Geneva 1939)Google Scholar; Maclver, R. D., Social Causation (Boston 1944)Google Scholar; Ando, Albert, Fisher, Franklin M., and Simon, Herbert A., Essays on the Structure of Social Science Models (Cambridge, Mass., 1963)Google Scholar; Blalock, Causal Inferences; and Wold, Herman, ed., Econometric Model Building: Essays on the Causal Chain Approach (Amsterdam 1964).Google Scholar

³² Frisch, 5–6; see also pp. 48, 86–87, and 121ff. for the use of causal language.

³³ Identifiability criteria are readably discussed in Valavanis, Stefan, Econometrics (New York 1959)Google Scholar, chap. 6, which should be read in conjunction with Koopman's early article on the identification problem cited therein. This problem is a very real one in nonrecursive causal systems involving simultaneous “feedback” and various logical interdependencies. In such interdependent or reciprocal systems, not to be further discussed here, each equation cannot be given an explicit hierarchical causal position, nor can its coefficients be estimated without knowledge of all other causal relations. For comparisons of the recursive and nonrecursive modeling approaches see Wold, , “Ends and Means in Econometric Model Building,” in Grenander, Ulf, ed., Probability and Statistics (New York 1959), 355–434Google Scholar; and Alker, “Causal Inference in Political Analysis,” in Joseph Bernd, ed., Mathematical Applications in Political Science, 2nd Series (Dallas, forthcoming). In technical language the possibility that any systematic theory of reality is hopelessly underidentified (e.g., contains too few distinct equations and too many unknowns) corresponds quite closely to the Crick Hoffmann view, shared by many humanists, that everything depends on everything else.

³⁴ Much more detailed discussions of partially isolable subsystems in which even the above assumptions are not precisely correct are contained in Ando and others, Essays on the Structure of Social Science Models, and in Wold, “Ends and Means.” In a brilliant article drawing on this research, “The Architecture of Complexity,” General Systems, X (1965), 63–76Google Scholar, Herbert Simon argues that in complex systems made up of a large number of parts that interact in a nonsimple way, “the whole is more than the sum of the parts, not in an ultimate, metaphysical sense, but in the important pragmatic sense that, given the properties of the parts and the laws of their interaction, it is not a trivial matter to infer the properties of the whole. In the face of complexity, an in-principle reductionist may be at the same time a pragmatic holist” (pp. 63–64).

He goes on to argue that near-decomposability of many systemic interdependencies into hierarchical orderings of systems and subsystems means that we need not despair about never being able to unravel the web of causes involved. As a generalization of the notion of near-decomposability he suggests an “empty world hypothesis”—that “most things are only weakly connected with most other things; for a tolerable description of reality only a tiny fraction of all possible interactions needs to be taken into account” (p. 73).

³⁵ The plausibility of both models derives from my previous paper “Dimensions of Conflict in the General Assembly”—alliances in particular were found to be of decisive importance. Relating Soviet aid, trade, and alliances to UN voting in a single regression equation, as I did in that article, is clearly inappropriate from the point of view of confluence analysis because of their high intercorrelations (which violate the predictions of Model 2).

³⁶ Once causal interrelations have been made plausible, the relative importance of the actual coefficients in each equation of such a recursive model can be estimated by multiple regression techniques, or derived from a more fundamental theory about national behavior. Using this approach would be a good example of how statistical models (like the linear additive ones of factor and regression analysis) use a small number of statistical degrees of freedom to estimate or measure how environmental influences and human actions underlie voting behavior.

Even more flexibility would be allowed if multiplicative interaction terms among trade and alliances were introduced. The resulting nonrecursive equations could still be used, however, to derive the predictions that r and r₁₂, should be zero if the model and related assumptions are correct.

³⁷ Economists and simulation specialists have been the most imaginative quantitative researchers on nonadditivities in international relations. See for example “Confluence Analyses” by Richard Stone, Tinbergen, and Wold; also Guetzkow, Harold, “Structured Programs and Their Relation to Free Activity Within the Inter-Nation Simulation,” in Guetzkow, Alger, Brody, Noel, , and Snyder, , Simulation in International Relations: Developments for Research and Teaching (Englewood Cliffs 1963), 103–49.Google Scholar

³⁸ True international behavior seen from a sociological perspective may be thought of as that interaction which remains after national characteristics have been taken into account. Karl Deutsch and I. R. Savage, for example, have measured international proximities in transactional terms essentially using the idea of statistical interaction. For a readable review and extension of their approach see Goodman, Leo, “A Short Computer Program for the Analysis of Transaction Flows,” Behavioral Science, IX (April 1964), 176–86.CrossRefGoogle Scholar In more subjective and rationalistic terms Bernard might have noted the “superadditivity” postulate of “essential” bargaining games: such games assume that one adds more to the value of a coalition by joining it than by staying outside. In this regard see the papers by L. S. Shapley discussed in Luce, R. D. and Raiffa, H., Games and Decisions (New York 1957)Google Scholar, chaps. 6, 11.