194 ECOLOGY AND LIFE HISTORY OF THE COMMON FROG 



air temperature is closely correlated with mean water temperature and 

 the effect of the approximation is to lower the correlation, not to raise 

 it. Of course, we ought to use the best measures available, but if there 

 are no water temperatures on record, we can make shift with what we 

 can get, confident that if we had the better measure, our conclusions 

 would be even more certain, and very unlikely to create a relation 

 where none exists. 



The number of degrees of freedom in tliis work is doubtful. It is 

 certainly not as high as 2,734, the number of variables in the multiple 

 regression equation, if only because the number of meteorological 

 stations was less than the number of phenological stations, so that the 

 weather records were used for several observations in the same year. 

 Moreover, weather records from widely separated parts of the country 

 are almost independent, but those from adjacent areas are certainly not. 

 This difficulty has been evaded by the scale of values in Table 8. A 

 correlation coefficient of 0-2 is significant with only 100 pairs of 

 variates, and there are certainly far more degrees of freedom than that. 

 In fact, in such a large scheme, questions of statistical significance play 

 a far less important part than in the small scale experiments and 

 observations usually encountered. 



It has sometimes been pointed out that multiple regression equations 

 are never complete, because we have no means of knowing when we 

 have included all relevant factors, and if, as usually happens, there are 

 correlations between all of them, we can never know that our co- 

 efficients are correct. There is, philosophically, no doubt about all 

 this, but there are three practical points, one mathematical and the 

 others logical, that are insufficiently stressed in the argument. It is 

 probably rather rare for the ecology of an animal to be dominated 

 by one factor or a very few. These are the easy cases to recognize, 

 and our present knowledge is probably overweighted, so that the 

 numerous cases in which multiple causation is at work have not been 

 understood. Now, whenever a number of roughly equal causes are 

 operating, the individual partial correlation coefficients must be weak, 

 even if together they provide a complete description. Weak co- 

 efficients have little influence on each other, as can be seen from the 

 following formula for calculating a partial correlation coefficient from 

 three simple ones — 



'"■' " V(i - rj}{i - r,3^) 



