The null hypothesis is that there should be no difference between the proportion (number)
of subjects who can achieve transitivity but not relative thinking, and the proportion of
subjects who do not achieve transitivity but do achieve relative thinking. Put another
way, we could say that the number of subjects performing discrepantly in each direction
on two tasks should not differ significantly from a chance distribution. A one-sided
(directional) test would be appropriate here if the authors were, in advance, looking for
discrepancies in a given direction.
Worked ExampleData from the study by Blasingame and McManis (1977) is used to illustrate
computational details for the binomial test when π=0.5, (sample proportion P= Q=0.5)
and n<25. The exact probability of obtaining values as extreme or more extreme than the
observed values is evaluated using the binomial equation, see equation 4.1 Binomial
Probability in Chapter 4.
p=nCr×pr×(1−p)n−r
Binomial
probability
The probability of obtaining 4 or fewer discrepant performance ratings is given by the
sum of the 5 probabilities:
p 0 discrepant observations=17!/(0!×17!)×0.5^0 ×0.5^17 =0.0000076
p 1 discrepant observations=17!/(1!×16!)×0.5^1 ×0.5^16 =0.0001297
p 2 discrepant observations=17!/(2!×15!)×0.5^2 ×0.5^15 =0.0010376
p 3 discrepant observations=17!/(3!×14!)×0.5^3 ×0.5^14 =0.0051880
p 4 discrepant observations=17!/(4!×13!)×0.5^4 ×0.5^13 =0.0181579
which equals 0.0245 allowing for rounding error. To avoid tedious calculations binomial
tables can be referred to or the SAS function PROBBNML can be used (see computer
analysis).
InterpretationThe null hypothesis of no difference in proportions is rejected at the 5 per cent level. In
the original paper the authors reported that there was a significant difference in
performance between relative thinking and transitivity, p=0.025. The authors interpreted
these findings as indicating a sequential development of relative thinking and transitivity,
in that order. (In a situation where a two-sided test would be appropriate, then the
calculated probability, using the above procedure, would simply be multiplied by two.)
Computer AnalysisBinomial calculations can be accomplished with ease using the PROBBNML function in
SAS. Functions are a valuable feature of SAS because they allow a variable to be defined
which is equal to a kind of built-in expression in the SAS language. There are nearly 150
different functions which fall into different types, for example, probability, arithmetic,
Inferences involving binomial and nominal count data 175