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Fisher's Exact Test01:08

Fisher's Exact Test

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Fisher's exact test is a statistical significance test widely used to analyze 2x2 contingency tables, particularly in situations where sample sizes are small. Unlike the chi-squared test, which approximates P-values and assumes minimum expected frequencies of at least five in each cell, Fisher's exact test calculates the exact probability (P-value) of observing the data or more extreme results under the null hypothesis. This feature makes it especially valuable when the assumptions of...
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Poisson Probability Distribution01:09

Poisson Probability Distribution

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A Poisson probability distribution is a discrete probability distribution. It gives the probability of a number of events occurring in a fixed interval of time or space if these events happen at a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on average, there are five words spelled incorrectly in 100 pages. The interval is 100 pages.
The...
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Testing a Claim about Population Proportion01:24

Testing a Claim about Population Proportion

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A complete procedure for testing a claim about a population proportion is provided here.
There are two methods of testing a claim about a population proportion: (1) Using the sample proportion from the data where a binomial distribution is approximated to the normal distribution and (2) Using the binomial probabilities calculated from the data.
The first method uses normal distribution as an approximation to the binomial distribution. The requirements are as follows: sample size is large...
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Bonferroni Test01:10

Bonferroni Test

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The Bonferroni test is a statistical test named after Carlo Emilio Bonferroni, an Italian mathematician best known for Bonferroni inequalities. This statistical test is a type of multiple comparison test to determine which means are different than the rest. Bonferroni test can minimize the Type 1 error by reducing the significance level alpha, which otherwise increases with sample pairs.
The means of different samples are first paired in all possible combinations.
The null hypothesis of the...
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Wald-Wolfowitz Runs Test II01:17

Wald-Wolfowitz Runs Test II

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The Wald-Wolfowitz runs test, commonly referred to as the runs test, is a nonparametric test used to assess the randomness of ordered data. The test evaluates the number of runs, which are consecutive sequences of similar elements within the data. If the number of runs is significantly higher or lower than expected, the data is considered non-random, indicating a detectable pattern or structure.
For binary data, runs are identified using symbols such as + and −, or equivalently, 1s and 0s. In...
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Test for Homogeneity01:23

Test for Homogeneity

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The goodness–of–fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to conclude whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence. The hypotheses for the test for homogeneity can...
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Related Experiment Video

Updated: Dec 29, 2025

Combined Immunofluorescence and DNA FISH on 3D-preserved Interphase Nuclei to Study Changes in 3D Nuclear Organization
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On powerful exact nonrandomized tests for the Poisson two-sample setting.

Stefan Wellek1,2

  • 1Department of Biostatistics, CIMH Mannheim, Mannheim Medical School of the University of Heidelberg, Mannheim, Germany.

Statistical Methods in Medical Research
|February 1, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces the Poisson-Boschloo test, a powerful nonrandomized method for comparing two Poisson distributions. It offers nearly optimal power and simplifies sample size calculations compared to existing randomized tests.

Keywords:
ConditioningPoisson distributionequivalenceincreased nominal levelnoninferioritysample-size calculationuniformly most powerful unbiased test

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Area of Science:

  • Statistics
  • Biostatistics
  • Hypothesis Testing

Background:

  • Comparing two independent Poisson distributions often involves testing the ratio of population means.
  • Existing conditional tests are optimally powerful only with randomized decisions at the boundary.
  • There is a need for nonrandomized, powerful tests in this domain.

Purpose of the Study:

  • To adapt Boschloo's method for constructing a powerful nonrandomized test for Poisson distributions.
  • To introduce the Poisson-Boschloo test and evaluate its performance.
  • To extend the methodology to two-sided equivalence testing.

Main Methods:

  • Adapted Boschloo's 1970 approach for binomial data to the Poisson case.
  • Developed the Poisson-Boschloo test based on a cutoff for the observed total number of events.
  • Analyzed power loss compared to the randomized Uniformly Most Powerful Unbiased (UMPU) test.
  • Extended the construction for two-sided equivalence testing.

Main Results:

  • The Poisson-Boschloo test achieves power nearly equivalent to the randomized UMPU test.
  • Sample size calculations for the Poisson-Boschloo test can use existing UMPU test procedures.
  • Approximate sample size calculation methods are rendered unnecessary.
  • The test construction successfully extends to two-sided equivalence testing.

Conclusions:

  • The Poisson-Boschloo test provides a powerful and practical nonrandomized alternative for comparing Poisson distributions.
  • It simplifies sample size determination, making it more accessible.
  • The methodology is adaptable for equivalence testing, offering similar advantages.