Russlands Bankerott

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Monograph

Identifikator:: 869807978

URN:: urn:nbn:de:zbw-retromon-400

Document type:: Monograph

Title:: Russlands Bankerott

Place of publication:: Berlin-Charlottenburg

Publisher:: Plutus Verlag

Year of publication:: 1906

Scope:: 1 Online-Ressource ([5] Blatt, 125 Seiten)

Digitisation:: 2017

Collection:: Economics Books

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Document type:: Monograph

Structure type:: Chapter

Title:: I. Zur Warnung.

Collection:: Economics Books

XIL—PARTIAL CORRELATION, orl lation giving the correlation between X; and X, or other pair of variables when the remaining variables X, . . . . X, are kept constant, or when changes in these variables are corrected or allowed for, so far as this may be done with a linear equation. For examples of such generalised regression-equations the student may turn to the illustrations worked out below (pp. 239-247). 3. With this explanatory introduction, we may now proceed to the algebraic theory of such generalised regression-equations and of multiple correlation in general. It will first, however, be as well to revert briefly to the case of two variables. In Chapter IX, to obtain the greatest possible simplicity of treatment, the value of the coefficient »=p/o 0, was deduced on the special assump- tion that the means of all arrays were strictly collinear, and the meaning of the coefficient in the more general case was sub- sequently investigated. Such a process is not conveniently applicable when a number of variables are to be taken into account, and the problem has to be faced directly: i.e. required, to determine the coefficients and constant term, if any, in a regression-equation, so as to make the sum of the squares of the errors of estimate a minimum. We will take this problem first for the case of two variables, introducing a notation that can be conveniently adapted to more. Let us take the arithmetic means of the variables as origins of measurement, and let z;, , denote deviations of the two variables from their respective means. Then it is required to determine a, and &,, in the re- gression-equation z=, +b, kL 58) so as to make Z(z, -a,+b,,.,)% for all associated pairs of deviations x, and a, the least possible. Put more briefly, if we write N.S o=3(x, ~ a. + b;5.2,)% . . (9) so that s,, is the root-mean-square value of the errors of estimate in using regression-equation (a) (¢f. Chap. IX. § 14), it is required to make s;, a minimum. Suppose any value whatever to be assigned to b,,, and a series of values of a, to be tried, s,, being calculated for each. Evidently s,, would be very large for values of a; that erred greatly either in excess or defect of the best value (for the given value of &,,), and would continuously decrease as this best value was approached ; the value of s, , could never become negative, though possibly, but exceptionally, zero. If therefore the values of s,, were plotted to the values of a; on a diagram, a curve would be obtained more or less like that of fiz. 44. The best value of a, for which s,, attained its aq

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