XIL—PARTIAL CORRELATION. 
which is the value found by the previous indirect method of 
Chapter IX. From the fact that &,, is determined so as to 
make the value of 3(a, — b,,7,)? the least possible, the method 
of determination is sometimes called the method of least squares. 
Evidently all the remaining results of Chapter IX. follow from 
this, and notably we have for 01.9 the minimum value of s,,, 
the standard-deviation of errors of estimate 
of =0(1 -mp?) : . (a) 
4. Now apply the same method to the regression-equation 
for m variables. Writing the equation in terms of deviations, 
it follows from reasoning precisely similar to that given above 
that no constant term need be entered on the right-hand 
side. For the partial regression-coefficients (the coefficients of 
the z’s on the right) a special notation will be used in order 
that the exact position of each coefficient may be rendered quite 
definite. The first subscript affixed to the letter 4 (which will 
always be used to denote a regression) will be the subscript of 
the z on the left (the dependent variable), and the second will 
be the subscript of the « to which it is attached ; these may 
be called the primary subscripts. After the primary subscripts, 
and separated from them by a point, are placed the subscripts 
of all the remaining variables on the right-hand side as secondary 
subscripts. The regression-equation will therefore be written 
in the form 
Ty =bos,  n-Tg+byun,, n.23+ sre Oinos... m1" Zn i {) 
The order in which the secondary subscripts are written is, 
it should be noted, quite indifferent, but the order of the 
primary subscripts is material; eg. b,, _ , and boys. 
denote quite distinct coefficients, x, being the dependent variable 
in the first case and ro in the second. A coefficient with » 
secondary subscripts may be termed a regression of the pth order. 
The regressions b,,, by1s byg bg, ete., in the case of two variables 
may be regarded as of order zero, and may be termed total as 
distinct from partial regressions. 
5. In the case of two variables, the correlation-coefficient Tyg 
may be regarded as defined by the equation 
719 = (b19.07)% 
We shall generalise this equation in the form 
Tam... .0=0un... inban,.. in : +) 
This is at present a pure definition of a new symbol, and it 
remains to be shown that ry, may really be regarded as, 
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