ON TESTS FOR LINEARITY OF REGRESSION 
IN FREQUENCY DISTRIBUTIONS. 
By JOHN BLAKEMAN, M.Sc, B.A. 
In considering any frequency distribution of two characters, say x and y, one 
of the things to be determined is whether the regression curve o\ y ow. x is linear 
or non-linear. 
In his memoir " On the general theory of skew correlation and non-linear 
regression*" Karl Pearson has given the test for linearity of regression expressed 
in terms of the correlation coefficient r and the correlation ratio 77. 
It is there shown that, while 97 is by definition an essentially positive quantity 
r may be either positive or negative ; that rj- and r'^ are both quantities which 
can only vary between the limits 0 and 1 and moreover 'rf is never less than ; 
finally, that the necessary and sufficient condition for linear regression is rf — = 0. 
It has been remarked that r may be positive or negative, but it is obvious that 
whenever r works out negative we can, by changing the direction of either one 
of the axes of reference, make r positive. Thus, in all frequency distributions, the 
axes may be so drawn that r shall be positive. There is therefore no loss of 
generality in assuming r positive and, for simplicity of statement, in what follows 
r will always be taken as positive. 
77 
With this convention we see that, if we write f = t?^ — ^ = — r, ■07 = loge - , 
the test for linear regression may be written indiscriminately as f =0, or ^ = 0, or 
«7 = 0, and in many other equivalent ways. 
Now, in any actual distribution, we shall not expect to find the quantities 
^, 37 absolutely zero but, even when the regression of 7/ on a; is linear, we shall 
expect the values of ^, ^, in- to be influenced in the usual way by the fact that our 
material is only a random sample of the general population. 
Thus, as in all statistical tests, having found ^, or ^, or 117, to have certain 
positive values, we want to know if these values are such as might arise from 
* Drapers' Company Research Memoirs. Biometric Series ii. Dulau and Co. 
