The accuracy of four approximations to noncentralF

The accuracy of the two-moment, three-moment, square root, and cube root approximations to the noncentralF distribution was assessed using 7,920 entries from Tiku’s (1967) power tables. Tiku’s tables list exact values of β for α= .005?.05, ν₁ = 1?12, ν₂ = 2?120, and ?= 0.5?3.0. Analysis of the errors showed generally satisfactory performance for all four approximations. The three-moment approximation was most accurate, registering a maximum error of only .009. The other three approximations had maximum errors of ±.02, except for the square root approximation at ν₂ = 2, where maximum errors of .05 occur. Approximation error increased with decreases in ν₁ and, less consistently, with increases in ν2. Error was nonmonotonically related to ?. A second investigation explored the accuracy of the approximations at values of αranging from .10 to .90. All four approximations degraded substantially in this situation, with maximum errors ranging from ?.09 to .05. If the analysis is restricted to cases where ν₁ τ; 1 and ν₂ τ; 2, maximum errors drop to roughly ±.03. We conclude that the approximations perform reasonably well for small αand moderately well for larger values, if certain restrictions are imposed. From a computational standpoint, however, there is little advantage to using approximate as opposed to exact methods unless exact values ofF _α are known in advance.     

Echolocation in the blinded rat

The difference threshold for detection of barriers at different distances using a modified method of limits was found, for four blinded rats, to be 9 in. White noise 20 dB above ambient level reduced discrimination performance to chance, indicating that auditory cues were involved in the discrimination. A second determination, utilizing the method of constant stimuli, confirmed the difference threshold to be 9 in. and reduced individual S variation by further training. Evidence with a black and a white barrier indicated the possibility that infrared heat cues do not enter into the discrimination.     

A study of position habits induced by reward and “frustration”

Predictions from Maier's theory of “frustration”-instigated behaviour have been tested in an experimental situation differing significantly from that in which the theory was propounded yet containing the central element of “frustration”—the insoluble problem.

A water discrimination unit was employed in which the performance of rats would be observed during attacks on insoluble problems, position problems or symbol problems.

Two groups, each containing ten Wistar albino rats, served as subjects. The research design consisted of the following phases: preliminary training, development of position responses, exposure to a symbol-reward problem with 50 per cent, punishment and exposure to a symbol-reward problem with 100 per cent, punishment. The design differed for the two groups only at the phase in which the position responses were established. During this phase one group was exposed to a position-reward problem and the other to an insoluble problem.

Position responses were established as frequently under position-“frustration” (position stereotypes) as under position-reward (position habits) conditions. Position stereotypes were more rigid—more resistant to extinction—than position habits under conditions of 50 per cent, punishment. Position stereotypes were as readily extinguished under 100 per cent, punishment as were position habits under 30 per cent, punishment.

The first two observations conform to predictions made from Maier's theory. The third does not. That is to say, not all situations containing the basic elements of “frustration” give rise to stereotyped behaviour patterns which are as rigid or “fixated” as Maier's theory would predict. It is a reasonable hypothesis that the characteristics of stereotyped responses established in certain “frustration” situations may be described adequately in terms of conventional learning principles without the necessity of resorting to a distinction between “goal-motivated” and “frustration-instigated” behaviour.     

Effect of CER on DRL responding

The effect of CER on DRL 16 was studied in four rats. All Ss showed complete CER suppression after five CER trials, together with some unconditioned post-shock suppression. This post-shock suppression showed complete recovery in all Ss after 10 days of five CER trials per day, but complete CER suppression continued throughout.     

Indefinite Divisibility

Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefinite extensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch.