Confluent analysis

confluence analysis

A collection of methods of mathematical statistics relating to the analysis of a priori postulated functional relationships between quantitative (random or non-random) variables $ X ^ {(} 1) \dots X ^ {(} p) $ under conditions when it is not the variables $ X ^ {(} s) $ themselves that are observable, but rather random variables

$$ \tag{1 } \widetilde{X} {} _ {i} ^ {(} s) = \ X _ {i} ^ {(} s) + \epsilon _ {i} ^ {(} s) ,\ \ i = 1 \dots n , $$

where $ \epsilon _ {i} ^ {(} s) $ is the random error of measuring $ X _ {i} ^ {(} s) $ for $ X ^ {(} s) $ at the $ i $- th observation, and $ n $ is the number of observations. Here, the general form of the functional ( "structural" ) relationship being considered between the unobservables $ X ^ {(} 1) \dots X ^ {(} p) $ is assumed to be known. Part of confluence analysis is the construction of statistical estimators for the unknown values of the parameters occurring in the equations of these "structural" relationships, and also the construction of statistical tests for testing different hypotheses concerning the nature of the relationships.

The development of the theoretical and applied aspects of confluence analysis is carried out for structural relationships that are linear in form (or linearizable by means of a suitable transformation of the original variables). Within the framework of the linear model of confluence analysis, the a priori postulate of $ m $ linear relations between $ p $ variables, $ m < p $, can be stated as the assumption that there exist $ p - m $" common" factors $ Y ^ {(} 1) \dots Y ^ {(} p- m) $ such that

$$ \tag{2 } X ^ {(} s) = \lambda _ {s,1} Y ^ {(} 1) + \dots + \lambda _ {s,p-} m Y ^ {(} p- m) ,\ \ s = 1 \dots p , $$

where the matrix $ \Lambda = \| \lambda _ {sk} \| $, $ s = 1 \dots p $, $ k = 1 \dots p - m $, has rank $ p - m $. Parametrization of the model of confluence analysis in the form (1)–(2) enables one to state the fundamental problems in terms of statistical estimation of the unknown values of the parameters $ \lambda _ {s,k} $ and statistical testing of hypotheses associated with them. Formally, the model (1)–(2) looks the same as the model of factor analysis; however, the problems of confluence analysis and factor analysis have hardly anything in common: while the aim of confluence analysis is in the description of the structural relationships existing between variables $ X ^ {(} 1) \dots X ^ {(} p) $, the basic problem in factor analysis is the construction and interpretation of the common factors $ Y ^ {(} 1) \dots Y ^ {(} p- m) $. At the same time it is possible to speak about the contiguous nature of the problems of confluence analysis and regression analysis: Some partial schemes of confluence analysis can be imbedded into the framework of schemes of regression analysis (for example, if for the observations (1) it is required to show the single dependence on the variable $ X ^ {(} 1) $, measured with error, from the remaining variables that are measured without errors).

References

[1]	R. Frisch, "Statistical confluent analysis by means of complete regression systems" , Univ. Oslo Econom. Inst. (1934)
[2]	T.C. Koopmans, "Linear regression analysis of economic time series" , Publ. Netherl. Econom. Inst. , 20 , Netherlands Econom. Inst. (1937)
[3]	M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2 , Griffin (1983) pp. Chapt. 29
[4]	E. Malinveaud, "Statistical methods of econometrics" , North-Holland (1970) pp. Chapt. 10 (Translated from French)
[5]	S.A. Aivazyan, I.M. Bogdanovskii, Zavodsk. Labor. , 40 : 3 (1974) pp. 285–295

Comments

The methods originally proposed by R. Frisch [1], and indeed the name "confluence analysis" itself, have dropped out of use in Western literature (1988). They have been superceded by the methodology of K.G. Jøreskog [a1] for the analysis of linear structural relationships.

The words "structural" and "functional" refer to the cases when the common factors $ Y ^ {(} j) $ are random variables or unknown (non-random) quantities, respectively.

References

[a1]	K.G. Jøreskog, "Structural analysis of covariance and correlation matrices" Psychometrika , 41 (1978) pp. 443–477