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Related Concept Videos

Binomial Probability Distribution01:15

Binomial Probability Distribution

A binomial distribution is a probability distribution for a procedure with a fixed number of trials, where each trial can have only two outcomes.
The outcomes of a binomial experiment fit a binomial probability distribution. A statistical experiment can be classified as a binomial experiment if the following conditions are met:
There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
There are only two possible outcomes,...
Poisson Probability Distribution01:09

Poisson Probability Distribution

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...
Random Variables01:09

Random Variables

A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
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Binomial Expansion Using Pascal's Triangle01:30

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Expanding a binomial expression such as (a + b)n results in a predictable sequence of terms that can be systematically derived using Pascal’s Triangle. This triangular array of numbers plays a central role in understanding and computing the coefficients of binomial expansions.Pascal’s Triangle is constructed such that each row corresponds to the coefficients of a binomial raised to a power. The topmost row, known as the zeroth row, corresponds to (a + b)0, and each successive row gives the...
Wald-Wolfowitz Runs Test I01:17

Wald-Wolfowitz Runs Test I

The Wald-Wolfowitz test, also known as the runs test, is a nonparametric statistical test used to assess the randomness of a sequence of two different types of elements (e.g., positive/negative values, successes/failures). It examines whether the order of the elements in a sequence is random or if there is a pattern or trend present. This nonparametric test applies to any ordered data despite the population and sample data distribution, even if a higher sample size is available.
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Testing a Claim about Population Proportion

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Related Experiment Video

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A Real-world What-Where-When Memory Test
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Published on: May 16, 2017

On models for binomial data with random numbers of trials.

W Scott Comulada1, Robert E Weiss

  • 1UCLA Center for Community Health, 10920 Wilshire Boulevard, Suite 350, Los Angeles, California 90024-6543, USA. scomulad@ucla.edu

Biometrics
|August 11, 2007
PubMed
Summary

This study introduces Bayesian multivariate Poisson models to analyze binomial outcomes where trial counts vary. This approach enhances understanding of success probabilities and correlated failures in complex studies.

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Area of Science:

  • Biostatistics
  • Statistical Modeling

Background:

  • Binomial outcomes involve successes (s) and failures (f) in independent trials (n).
  • In many studies, trial counts (n) are random variables, not fixed by design.
  • Standard logistic regression may not fully capture insights from joint modeling of (s, f) or handle n=0 observations.

Purpose of the Study:

  • To propose Bayesian multivariate Poisson models for analyzing bivariate binomial responses (s, f).
  • To extend these models for longitudinal and multivariate longitudinal binomial data.
  • To provide a framework that incorporates correlated successes and failures and handles varying trial counts.

Main Methods:

  • Development of Bayesian multivariate Poisson models for the bivariate response (s, f).
  • Incorporation of random effects to model correlation between s and f.
  • Extension to longitudinal and multivariate longitudinal data structures.

Main Results:

  • The proposed models offer a flexible approach to joint modeling of successes and failures.
  • The methodology accommodates situations where the number of trials (n) is a random variable.
  • The models can provide deeper insights into success probabilities (pi) influenced by covariates, including cases with n=0.

Conclusions:

  • Bayesian multivariate Poisson models offer a robust alternative for analyzing binomial data, especially when trial counts vary.
  • This approach enhances the understanding of factors influencing success probabilities and the correlation between successes and failures.
  • The methodology is applicable to diverse fields, including teratology and public health interventions.