266 



FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



F. PROBIT ANALYSIS 



111 the section on reproduction, we dealt with 

 two lots of data which can be analyzed by probit 

 transformation. These data consist of samples 

 of yellowtail that are used to determine (1) length 

 of the fish at maturity and (2) the date of spawn- 

 ing. Since the technique of probit analysis is not 

 commonly employed in fishery research, yet it has 

 been thoroughly tested, an explanation of its use 

 in this study is in order. 



Probit analysis has been used almost exclu- 

 sively in analyzing the results of biological assay 

 of chemicals tested on experimental animals, al- 

 though psychophysicists have used closely related 

 methods. It is the most thoroughly developed 

 method known for the analysis of quanta! (all or 

 nothing) response data, such as occurs in tests of 

 a chemical in which different concentrations cause 

 varying proportions of the experimental animals 

 to die. Developed largely from the studies of C. I. 

 Bliss, probit analysis was brought to its most 

 definitive form by Finney (1952), on whose work 

 this discussion is based. 



Our yellowtail data may be considered as ana- 

 logous to such doseage-response data. In deter- 

 mining length of the fish at maturity, the state of 

 maturity or immaturity is the quantal response 

 to the stimulus of growth. For the description of 

 the spawning season, the females are ripe or spent 

 in varying proportions as they are stimulated by 

 the vernal change in environment. 



The probit of a proportion P is defined as the 

 abscissa which corresponds to a probability P in 

 a normal distribution with mean 5 and variance 

 1 ; in symbols the probit of P is Y where 



V27T J-< 



'"du 



The transformation from percentage to probit 

 changes the usual sigmoid curve of percentage 

 response against stimulus to a straight line of the 

 type 



Y=a+bX 



in which Y is the probit and X is the stimulus. 



In the analysis of bioassay results, the typical 

 distribution curve of dosage X is decidely skewed 

 with a long tail on the right caused by the high 

 tolerance of a few animals (usually insects) . Such 

 a curve can usually be normalized by transforma- 



tion to common logarithms, and this has become 

 standard practice in bioassay. In our spawning 

 data, however, we have no evidence that such a 

 transformation is necessary. A satisfactory fit is 

 obtained by using the measures of time and length 

 directly. 



Probit regression lines may be fitted by eye 

 if there is little scatter of the points and an ac- 

 curate measure of the precision of the estimates 

 is not needed. Such a procedure is easy and rapid, 

 but it requires familiarity with the data and ex- 

 pected results. The arithmetic method of fitting 

 is, unfortunately, rather laborious, because a solu- 

 tion of maximum likelihood is required. This re- 

 sults from the increasing variance as the propor- 

 tion P approaches or 1. The values of the 

 probit Y must be weighted according to the ex- 

 pected Y and also according to the number of 

 observations used in obtaining the proportion P. 

 The expected Y is obtained from the eye-fitted 

 line and the weighting coefficients have been tabu- 

 lated by Fisher and Yates (1948, table 11). 



In our analysis of the spawning period, the 

 computations for the regressions of percentage of 

 spent fish against the date for the female yellow- 

 tail have been made as indicated in table F-l, 



Table F-l. 



Date 



-Probit analysis of the spawning period 

 of yellowtail, in 1943 



Apr. 20 



Apr. 20 



Apr. 20 



Apr. 20_... 



Apr. 27 



Apr. 27.... 

 May 4, 6. . 



May 7 



May 17.... 

 May 18.... 

 June 3, 7, 3 



June 9 



June 16 



June 23 



June 28 



June 29 



July 4 



Work- 

 ins 



probit 



3.719 

 3.415 

 3.701 

 4.186 

 4.142 

 4.220 

 4.047 

 4.300 

 4.859 

 5. 154 



5. 544 

 5.374 

 6.334 



6. Ill 

 7.002 

 7.052 

 7.421 



SUMMARY OP REGRESSION COMPUTATIONS 



1 .r = day of the year minus 100. 

 ! From' table 40, p. 217. 



