74 MASS. EXPERIMENT STATION BULLETIN 264 



The simple correlation coefficient, calculated by the ordinary method, is a 

 satisfactory measure only when the two characteristics being considered show 

 true linear regression. Regression is said to be linear when the actual mean 

 values of the dependent variable in each array of the independent variable 

 lie in a straight line. For example, if the mean length of molt period for each 

 of the eight hatches consistently increases or decreases at a constant rate, 

 molt period shows linear regression on hatching date. 



Each correlation table presented in this report was tested for linearity of 

 regression first by the X^ method of Pearson* (1924). This test may be ap- 

 plied to different types of distributions to determine whether departures from 

 expectation are within the limits of normal samples. In this report a large 

 value of X- indicates that deviations in length of molt period lie outside the 

 normal correlation surface and that the correlation ratio rather than the 

 correlation coefficient must be used to express the degree of correlation. The 

 correlation ratio has also been calculated and the difference between the cor- 

 relation coefficient and the correlation ratio tested for significance by Blake- 

 man's (1905) method. 



In biological work the significance of a simple correlation coefficient can- 

 not safely be assumed to depend upon its absolute magnitude in all cases. 

 This is true particularly when correlating inherited characters where there 

 may or may not be crossing over. Interrelations between characters also 

 complicate the situation and require the use of partial correlation methods. 

 For purposes of selection, however, the absolute magnitude of the correlation 

 coefficient is probably the most significant. 



Among the measures of significance of correlation the method of Ezekiel 

 (1929) is probably the most satisfactory. This method assimies that the 

 stjuare of the correlation coefficient measures the proportion of variance in 

 the dependent variable that is associated with the independent variable. In 

 other words, if r ^ .30 then r" = .09, and selection based on the independent 

 characteristic may be expected to modify the dependent characteristic to the 

 extent of 9 per cent in the desired direction. On this basis a simple correla- 

 tion coefficient of less than .30 is of little value as a guide in selection. 



Environmental Factors 



i. Hatching Date versus Levgth of Molt Period. 



The hatching dates of the birds used in these studies extended over a 

 rather limited period of time, namely the 49-day period beginning March 25 

 and ending near the middle of May. Hatching dates were kept constant from 

 year to year. Since date of hatching has a significant effect upon rate of 

 growth (Hays and Sanborn, 1929) as well as upon several characteristics 



*Pearson has suggested that the X- method, when applied to regression lines (Slut- 

 sky, 1913) gives the best fit when the theoretical standard deviation of arrays, which 

 is equal to the standard deviation of the population multiplied by the square root of 

 1 — r^ is used as a constant. Pearson also suggests that the theoretical frequency of 

 arrays on a Gaussian distribution might also improve the fit. Since the numbers used 

 in the studies reported in this paper are rather large, it is believed that the actual 

 frequency of arrays will closely approach the theoretical in most cases. The method of 

 Slutsky (1913) for calculating the value of X^ for each array is used, except that the 

 theoretical standard deviation is used in all cases. 



