58 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



if h i? £i — £.2 — £ = — h , there is no response ; 

 if — h > £i — £ 2 — £ , response R 2 is given. 



If p(£)d£ is the probability that £ has a value in the range £ to 

 £ + d£ , then the probability that response R x occurs is obtained by 

 integrating p(£) with respect to £ from minus infinity to (e^ — e 2 — h). 

 This becomes evident when we see that if £ is any value less than 

 £1 — £ 2 — h , response R x is produced. If we let f\ be the probability 

 of response R x , P 2 the probability of response R 2 , and P the prob- 

 ability of neither response, and if we define 



then we may write 



P(x) = f* p(£)d£, (2) 



J -"50 



P^ = P(e 1 -B 2 -h), (3) 



P 2 = P(-e 1 + e 2 -h), (4) 



Po = l-P*-P z . (5) 



If Si = S 2 , response Ri may be considered the correct response 

 and R 2 the wrong response. In this case P r = P c , the probability of 

 a correct response, and P 2 = P w , the probability of a wrong re- 

 sponse. Any failure to respond, or any response other than cor- 

 rect or wrong such as "equal," "doubtful," could be included in the 

 proportion to be identified with P • It is commonly the case that when 

 a categorical judgment is required, so that either R x or R 2 is made 

 at each trial, the subject must lower his criteria for judgment. We 

 may interpret this with reference to the structure studied by assum- 

 ing a lowered threshold. For this case we set h = , whence P = 

 and Pi + P 2 = 1. Thus from a knowledge of only the standard error 

 of the probability distribution, one is able to calculate the probabil- 

 ities of the various responses to any given pair of stimuli when the 

 judgments are categorical; the additional parameter h enters when 

 "doubtful" judgments are allowed. 



Since complete symmetry has been assumed, it follows that 

 P x = P 2 for Si = S 2 . In general, this is not true for the observed pro- 

 portions. The amount by which the observed proportions differ from 

 equality is a measure of the bias of the subject. The simplest inter- 

 pretation of the bias is that the afferent thresholds are not exactly 

 equal so that the mean values £ x and £ 2 are not equal for S x = S 2 , but 

 £x(£) = s 2 (S) + x . Although x. will depend on Si, we shall not 

 consider this any further, preferring rather to incorporate # into £ , 

 so that p(£) has a mean value of x instead of zero. Thus modified, 

 the mechanism may be applied to the experimental data by F. M. 

 Urban shown in Table 1 (Urban, 1908). These are the average re- 



