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Soft Modelling by Latent Variables: The Non-Linear Iterative Partial Least Squares (NIPALS) Approach

Published online by Cambridge University Press:  05 September 2017

Herman Wold
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Abstract

The NIPALS approach is applied to the ‘soft’ type of model that has come to the fore in sociology and other social sciences in the last five or ten years, namely path models that involve latent variables which serve as proxies for blocks of directly observed variables. Such models are seen as hybrids of the ‘hard’ models of econometrics where all variables are directly observed (path models in the form of simultaneous equations systems) and the ‘soft’ models of psychology where the human mind is described in terms of latent variables and their directly observed indicators. For hybrid models that involve one or two latent variables the NIPALS approach has been developed in [38], [41] and [42]. The present paper extends the NIPALS approach to path models with three or more latent variables. Each new latent variable brings a rapid increase in the pluralism of possible model designs, and new problems arise in the parameter estimation of the models. Iterative procedures are given for the point estimation of the parameters. With a view to cases when the iterative estimation does not converge, a device of range estimation is developed, where high profile versus low profile estimates give ranges for the parameter estimates.

Type
Part III — Statistical Theory
Information
Journal of Applied Probability , Volume 12 , Issue S1 , 1975 , pp. 117 - 142
Copyright
Copyright © 1975 Applied Probability Trust 

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References

[1] Adelman, I. and Morris, C. T. (1967) Society, Politics, and Economic Development. The Johns Hopkins Press, Baltimore. (Revised ed. 1971.) Google Scholar
[2] Adelman, I. and Morris, C. T. (1973) Economic Growth and Social Equity in Developing Countries. Stanford University Press, California.Google Scholar
[3] Ågren, A. (1972) Extensions of the Fix-Point Method. Theory and Applications. Doctoral Thesis, University of Uppsala, Sweden.Google Scholar
[4] Areskoug, B. (1974) Computer program MIMIC. Department of Statistics, University of Göteborg, Sweden.Google Scholar
[5] Basmann, R. L. (1957) A generalized classical method of linear estimation of coefficients in a structural equation. Econometrica 25, 77–83.CrossRefGoogle Scholar
[6] Bergstrom, R. (1974) Studies in the Estimation of Interdependent Systems, Especially the Fix-Point and Iterative Instrumental Variables Methods. Doctoral Thesis, University of Uppsala, Sweden.Google Scholar
[7] Blalock, H. M. Jr. (1971a) Causal models involving unmeasured variables in stimulus-response situations. In Causal Models in the Social Sciences, pp. 335–347. Aldine, Atherton, Chicago.Google Scholar
[8] Blalock, H. M. Jr., ed. (1971b) Causal Models in the Social Sciences. Aldine, Atherton, Chicago.Google Scholar
[9] Bodin, L. (1974) Recursive Fix-Point Estimation. Theory and Applications. Doctoral Thesis, University of Uppsala, Sweden.Google Scholar
[10] Brown, T. M. (1959) Simplified full maximum likelihood and comparative structural estimates. Econometrica 27, 638–653.Google Scholar
[11] Christoffersson, A. (1970) The One Component Model with Incomplete Data. Doctoral Thesis, University of Uppsala, Sweden.Google Scholar
[12] Coleman, J. S. (1973) Effects of school learning: The IEA findings. Harvard-IEA Conference, Teachers College, Harvard University, 12–17 November 1973.Google Scholar
[13] Costner, H. L., ed. (1971) Sociological Methodology 1971. Jossey-Bass, San Francisco.Google Scholar
[14] Friedman, M. (1957) A Theory of the Consumption Function. Princeton University Press, Princeton, New Jersey.Google Scholar
[15] Gini, C. (1921) Sull'interpolazione di una retta quando i valori della variabile indipendente sono affetti da errori accidentali. Metron 1, 63–82.Google Scholar
[16] Hauser, R. M. (1968) Family, School and Neighborhood Factors in Educational Performances in a Metropolitan School System. Unpublished Doctoral Dissertation, University of Michigan.Google Scholar
[17] Hauser, R. M. (1969) On ‘social participation and social status’. American Sociological Review 34, 587–611.Google Scholar
[18] Hauser, R. M. and Goldberger, A. S. (1971) The treatment of unobservable variables in path analysis. In Sociological Methodology 1971, pp. 87–117. Jossey-Bass, San Francisco.Google Scholar
[19] Jöreskog, K. G. (1973) Analysis of covariance structures. In Multivariate Analysis III, pp. 263–285. Academic Press, New York.Google Scholar
[20] Jöreskog, K. G. and Goldberger, A. S. (1973) Estimation of a model with multiple indicators and multiple causes of a single latent variable. Research Report 73–14, Department of Statistics, Uppsala University, Sweden.Google Scholar
[21] Kettenring, J. R. (1971) Canonical analysis of several sets of variables. Biometrika 58, 443–451.CrossRefGoogle Scholar
[22] Krishnaiah, P. R., ed. (1966) Multivariate Analysis. Academic Press, New York.Google Scholar
[23] Krishnaiah, P. R., ed. (1973) Multivariate Analysis III. Academic Press, New York.Google Scholar
[24] Lyttkens, E. (1966) On the fix-point property of Wold's iterative estimation method for principal components. In Multivariate Analysis, pp. 335–350. Academic Press, New York.Google Scholar
[25] Lyttkens, E. (1970) Some asymptotic formulas for standard errors and small-sample bias of FP estimates. In Interdependent Systems. Structure and Estimation, Chapter 3.9. North-Holland Publishers, Amsterdam.Google Scholar
[26] Lyttkens, E. (1973) The fix-point method for estimating interdependent systems with the underlying model specification. J. R. Statist. Soc. B 136, 353–394.Google Scholar
[27] Lyttkens, E. and Wold, H., eds. (1973) FP (Fix-Point), IIV (Iterative Instrumental Variables) and related approaches to ID (Interdependent) systems. Theory and applications. European Meeting of the Econometric Society, Oslo, August 28–31, 1973.Google Scholar
[28] Miller, R. G. (1974) The jack-knife—a review. Biometrika 61, 1–15.Google Scholar
[29] Mosbaek, E. J. and Wold, H. with contributions by Lyttkens, E., Ågren, A. and Bodin, L. (1970) Interdependent Systems. Structure and Estimation. North-Holland Publishers, Amsterdam.Google Scholar
[30] Pearson, K. (1901) On lines and planes of closest fit to systems of points in space. Philosophical Magazine (6) 2,559–572.Google Scholar
[31] Rao, C. R. (1964) The use and interpretation of principal component analysis in applied research. Sankhya A 26, 329–358.Google Scholar
[32] Theil, H. (1953) Estimation and simultaneous correlation in complete equation systems. Central Planning Bureau, The Hague.Google Scholar
[33] Tukey, J. (1958) Bias and confidence in not-quite large samples. Abstract. Ann. Math. Statist. 29, p. 614.Google Scholar
[34] Van Der Waerden, B. L. (1957) Mathematische Statistik. Springer, Berlin.Google Scholar
[35] Wold, H. (1965) Toward a verdict on macroeconomic simultaneous equations. Scripta Varia 28, 115–166. Pontifical Academy of Sciences, Vatican City.Google Scholar
[36] Wold, H. (1966a) Non-linear estimation by iterative least squares procedures. In Research Papers in Statistics, pp. 411–444. Festschrift for Neyman, J., ed. David, F. N. Wiley, New York.Google Scholar
[37] Wold, H. (1966b) Estimation of principal components and related models by iterative least squares. In Multivariate Analysis, pp. 391–420. Academic Press, New York.Google Scholar
[38] Wold, H. (1973a) Non-linear iterative partial least squares (NIPALS) modelling. Some current developments. In Multivariate Analysis III, pp. 383–407. Academic Press, New York.CrossRefGoogle Scholar
[39] Wold, H. (1973b) A panorama of OLS and NIPALS modelling. See [27], Chapter 2.Google Scholar
[40] Wold, H. (1974a) Path models with latent variables that have several indicators. To be published in Quantitative Sociology, ed. Blalock, H. M. Jr. Seminar Press, San Francisco.Google Scholar
[41] Wold, H. (1974b) Causal flows with latent variables. Partings of the ways in the light of NIPALS modelling. European Economic Review 5, 67–86.Google Scholar
[42] Wold, H. (1974c) Path models with one or two latent variate aggregates. Research Report 1974, 6, Department of Statistics, University of Göteborg, Sweden.Google Scholar
[43] Wold, H. and Lyttkens, E. eds. (1969) Non-linear iterative partial least squares (NIPALS) estimation procedures. Bull. Internat. Statist. Inst. 43, 29–51.Google Scholar
[44] Wold, H., Lyttkens, E. and Areskoug, B. (1974) The convergence of NIPALS estimation procedures for six path models with one or two latent variables. Manuscript in preparation.Google Scholar
[45] Wold, S. (1974a) A theoretical foundation of extrathermodynamic relationships (linear free energy relationships). Chemica Scripta 5, 97–106.Google Scholar
[46] Wold, S. (1974b) Pattern cognition and recognition (cluster analysis) based on disjoint principal components models. Technical Report No. 357, Department of Statistics, University of Wisconsin.Google Scholar