Python Programming for Biology: Bioinformatics and Beyond

Tailed tests and p-values

A simple test to perform with a probability distribution is to evaluate the likelihood that a

value  was  generated  by  the  distribution,  which  is  a  matter  of  knowing  the  mathematical

form  of  the  distribution.  Although  this  may  be  helpful  for  some  discrete  probability

distributions  (as  illustrated  in  the  previous  chapter),  as  the  number  of  possible  outcomes

becomes larger the probability of specific outcomes gets ever smaller. If we are trying to

test  whether  an  observation  was  generated  by  a  probability  distribution,  it  often  doesn’t

help  to  know  that  a  particular  observed  outcome  is  itself  highly  unlikely.  This  notion  is

taken to the extreme for continuous probability distributions where the probability that any

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single,  precise  real  number  value  was  generated  by  a  probability  distribution  is  zero,

because technically there are an infinite number of values that could have arisen. Hence,

rather  than  thinking  about  the  likelihood  of  a  single  value  we  instead  tend  to  test  the

probability of a value falling within a given range. Mathematically we can formulate this

as an integral of the probability mass function (introduced in Chapter 21), i.e. the area of a

region under the curve defining the probabilities for each value (see

Figure 22.3

).