SIZE OF ANIMAL POPULATIONS 



195 



SOME FURTHER CONSIDERATIONS 

 An alternative derivation 



Formulae 3 to 7 were reached by Pearson by 

 means of Bales' theorem, which is objected to as 

 invalid by many mathematical statisticians (.Ken- 

 dall 1944, p. 176 et seq.)- Dr. S. Lee Crmnp has 

 suggested (private communication) that an esti- 

 mate of A'^ may be arrived at by other moans, as 

 follows. Drawing samples of fixed size n from a 

 population A'^ of which Tare marked, the probabil- 

 ity that, in a sample of n, f are marked is 



Pit)- 



(N-n)\nm{N-T)\ 



NW.iT-ty. (n-ty. {N-T-n+t)l 



,(10) 



whence 



in+l)(r+l) 



E 



(t+ 



^ytL^|=iv+i-P(0)(iv-r-7i). . .(11) 



where E ( ) denotes mathematical expectation and 

 P (0) is the probability of getting no tags in the 

 sample. 



This means that 



(n+DJT+l) 

 (t+l) 



1 



(12) 



is an estimate of A'^ biased by an amount P (0) 

 (N—T~n). If conditions arc such that a sample 

 of n with no marked individuals is very unlikely, 

 the bias is negligible. We may say that formula 12 

 is an efi'ectiveh' unbiased estimate of A'^. 



^There the numbers are all large, formula 12 

 reduces immediately to formula 1 or formula 6. 



Unfortunately, an estimate of the variance of 

 the estimate of A'^ given in formula 12 has not yet 

 been obtained. 



Chapman (1948) has considered the problem of 

 determining the value or values of A'^ which make 

 P (t), formula 10, a maximum. He found that the 



nT 

 maximum-likelihood estimate of A^ is — > or if that 



is fractional, the integer immediately below 



nT 



t ' 



Confidence limits on the population estimate 



The method of confidence intervals, due to 

 NejTnan (1934), may be employed to determine 

 the range of values within which we may expect A' 

 to lie. A discussion of the theory of confidence 



intervals is beyond the scope of this paper, and 

 reference is made to the original paper of Neyman 

 or to the discussion of Cranacr (1946, p. 507 et seq.) 

 or that of Kendall (1946, p. 62 et seq.). 



The confidcnco limits of the estimate of the 

 tag ratio in the population may be obtained as 

 follows (Cramer 1946, p. 515): 



Suppose wc have a population consistinK of a finite 

 number .V of individuals, .Vp of which pos-sess a certain at- 

 tribute A, while the remaining Xg = X— Np do not possess 

 A. It is now required to estimate the unknown proportion 

 p . . . Let us draw a random sample of n individuals 

 without replacement, and observe the numljer v of indi- 

 viduals in the sample possessing the attribute A. In 

 current text-books on probability, it is shown that we have 



Further the variable p*=- is approximately normally dis- 

 tributed, when n and N—n are large. Taking p* as an 

 estimate of p, we now assume as above that the error of ap- 

 proximation in the normal distribution can bo neglected. 

 The probability that p* lies between the limits pdb 



/A[-n pq ig 

 V A/-1 n 



then equal to e, where X has the same 



significance as in the preceding example. (Note: where X 

 was stated to be the lOOe % value of a normal deviate, and 

 € is the confidence level.) 



In Cramer's notation E ( ) denotes mathe- 

 matical expectation (or mean value) and D^ ( ) 

 denotes the variance. 



A' and n have the same meaning as in our earlier 



T 

 formulae, 1 to 12; p is equal to — > and y is equal to 



t in those formulae. 



For any given values of A^, n, and T we can cal- 

 culate the limits within which w* = - mav be ex- 



' n 



pected to fall for a given confidence level, «, by 

 the formula 



p±X 



iN-n pq 

 \N-1 ' n  



(13) 



where 



T 



Given values of n and T from an experiment, 

 we can, then, by formula 13 calculate for various 

 values of p, as ordinates, the limits within which 

 p*, the tag ratio of the sample, as abscissae, may 

 be expected to fall for a given value of the con- 

 fidcnco level e. The (turves connecting these 

 points will form the confidence limits corresponding 



