# Why are p-values probabilities rather than likelihoods?

The p-value is the probability of obtaining test results at least as extreme as the results actually observed during the test, assuming that the null hypothesis is correct.

Why is the p-value a probability rather than a likelihood function?

• I think that a p-value is more like a cumulative probability, whereas a likelihood is the probability of a single point (for discrete case) and P(X in (t,t+dt)) (for a continuous case). – M. Austin Jan 18 at 12:28
• @M.Austin a likelihood (in the statistical definition) is not a probability. It is a function of parameter(s) given an observation. It may be non-negative and not guaranteed to sum to 1. As a simple example consider a Bernoulli(p). If $x=0$ observed integrating over $0<p<1$ gives you $1/2$ and not 1. – Lucas Roberts Jan 18 at 20:15

For example, suppose you think that a statistic comes from a Poisson distribution. You could write the probability as $$P_{\lambda}(x)$$. That is, you're treating $$\lambda$$ as a fixed parameter that gives rise to a function of the variable $$x$$. Or you could write it as $$P(x,\lambda)$$. This notation treats $$x$$ and $$\lambda$$ as both being variables, and the probability being a function of the two. Or you could write it as $$P_x(\lambda)$$. This notation treats $$x$$ as a fixed parameter, and the probability being a function of $$\lambda$$.