BIOLOGY OF THE ATLANTIC MACKEREL 201 



The outstanding peculiarity in the survival curve is, of course, the abrupt 

 change of level and slope at the age of 40 days, or length of 10 mm. To investigate 

 the possibility that this might have been due to the mathematical effect of a fluctua- 

 tion in growth rate, rather than a fluctuation in mortality rate, let it be assumed that 

 the mortality rate through and beyond this period was constant, and compute the 

 changes in growth rate required to fit this hypothesis. The resulting new values for 

 growth rate, in terms of days required to grow one mm. in length, are as follows: 



Millimeters: -Dai" Millimeters — Continued. Day 



9 3.04 13 15 



10 --- .80 14 .18 



11 .38 15 09 



12 . 24 



Thus, this hypothesis would require growth at an ever-accelerating rate from 10 

 mm. on, such that less than a day would be occupied in growing from a length of 10 

 to a length of 15 mm., and by that time growth would be at the rate of 10 mm. per 

 day. Clearly this hypothesis is untenable, for such high growth rates are not only 

 absurd per se, but also inconsistent with the distributions of lengths of larvae taken 

 on successive cruises; and it may be concluded that the outstanding peculiarity in 

 the mortality curve cannot have resulted from a fluctuation in growth rate. This 

 demonstration, having proved that it requires striking changes in growth rate to 

 produce material effects on the survival curve, indicates also that errors of the order 

 of magnitude which likely exist in the determination of growth would not materially 

 affect the determination of mortality rates. 



Thus far attention has been centered on the possible elements of selective error 

 or bias connected either with collection of the material or the subsequent mathemati- 

 cal treatment. There remains the question of the effect of random variability. This 

 could not alter the level or the trend of the survival curve, for random variability 

 would produce empirical values that tend to deviate equally above and below the 

 true values, so that the sole effect would be on the scatter of points, or, in other words, 

 the relative reliability of fit by any lines expressing their trends. This is readily 

 investigated by conventional statistical methods. 



Because the points in the curve obviously lie along straight lines over consider- 

 able segments, such lines have been fitted, by the method of least squares, to various 

 combinations of segments. Since our interest hes principally in the mortality rates 

 expressed by the slopes of the lines, attention may be focussed on the 6 value, or 

 regression coefficient, in the equation: 22 



y=a-\-bx 



which describes these lines. The standard deviation s of the regression coefficient b 

 may be estimated by the formula 



_ S(y-D 2 

 tin 1 — 2 



To investigate the reliability of the slopes of the lines for various segments of the 



diagram, one may calculate 



, Jb-Pi/S(x-x)* 



8 



and find, from published tables, the probability, P, that any other slope /3 might 

 result from sampling the same universe. Being interested in knowing the limits of 



» The symbols given In this and following equations are those used by Fisher (1932). 



