Poisson Hypothesis Tests

  • Poisson Hypothesis Tests are integral to statistics as they model the number of times an event occurs in a fixed time, space or volume.
  • The Poisson distribution has a parameter λ (lambda), which represents the mean number of successful events in a given length of time.
  • A crucial assumption of this hypothesis test is that these events occur independently of each other - the happening of one event does not alter the probability of the next.
  • The Poisson distribution has a unique property that its mean (λ) is also its variance.
  • A one-sample Poisson test would assess if the observed results significantly deviate from what the expected results are under a Poisson model.
  • Null hypotheses (H0) and alternative hypotheses (H1) are central to these tests. The null hypothesis assumes that the difference is due to chance, whereas the alternative hypothesis assumes that there’s a statistically significant difference.
  • The test statistic for the Poisson Hypothesis Tests is the sum of (Observed-Expected)^2 / Expected over all categories. If this value is less than the critical value, the null hypothesis is accepted.
  • A two-tailed test is generally applied when it is unknown whether the true parameter will be greater or lesser than the nominal value.
  • A one-tailed test should be implemented when it can be assumed that the true value will be less/more than the nominal.
  • It is also worth noting that if the sample size is large (generally, if λ > 15), the Poisson distribution can be approximated to a normal distribution.
  • P-values also play a significant role. The probability that, if the null hypothesis were true, we would get the current statistics or one more extreme, is represented by the P-value. If P < 0.05, the result is considered statistically significant, meaning the null hypothesis can be rejected.
  • Like all statistical tests, Poisson Hypothesis Tests also have potential for type I and type II errors. Type I is rejecting the null hypothesis when it’s true (false positive), and type II is failing to reject the null hypothesis when it’s false (false negative).
  • Lastly, the power of a test is the probability it correctly rejects the null hypothesis when the null hypothesis is false. This measures the effectiveness of the test.