282 Distribution of numbers of mutants in bacterial populations 



Maximal likelihood method: smaller counts 



In this section we describe the method of arriving at the maximal likelihood estimate of 

 m from an experiment falling within the scope of Tables 1 and 2; i.e. one in which the 

 majority of cultures have fewer than 64 mutants. 



p r is the probability of a culture having r mutants. The log likelihood of a set of N values 

 of r is (apart from irrelevant terms) 



L=S [log p r ], 



S denoting summation over the N experimental values of r. 

 The maximal likelihood value of m is that satisfying 



(m 7-1 m>\ 



SO that ^IrJrZSr m 



where t r = Zc,, r (^' { ~j ■ (55) 



Thus the maximal likelihood estimate of m is that satisfying 



S P-^ -0, (56) 



L Pr - 



t r has been computed for a range of values of m exactly as described earlier for p r , and 

 in Table 4 values of {t r —p r )/p r are listed for a range of values of m and for the same 

 grouped ranges of r as were used previously. 



The method of estimating m is therefore the following. A preliminary estimate of m is 

 obtained either by the median method or by equating e~ m to the proportion of cultures 

 without mutants. Table 4 is entered at the value of m nearest to this preliminary estimate, 

 and a value of {t r —p r )/p r read off for each of the N experimental values of r. The N values 

 are summed. The procedure is repeated for several adjacent values of m, and thence 



(graphically or otherwise) the value of m inferred which would make S - — — =0. 



The variance of this maximal likelihood estimate of m is given by the relation 



7T3p 

 Pr 



S here means summation over all values of /• from to infinity, and is to be distinguished 

 from S, meaning summation over the N experimental observations. 



