Conduct Web experiments using PHP, Part 2
By Paul Meagher2005-03-18
Poisson sampling model
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The binomial distribution converges to the poisson distribution for large values of N and small values of p . While the binomial distribution is still conceptually applicable to simulating your results, the poisson distribution is more often used to represent rare-event distributions and behaves more reliably than my current binomial distribution implementation when supplied with extreme parameter settings (such as large N values and small p values).
The poisson distribution accepts only one parameter, a success-rate parameter called lambda (lambda). For a null effects poisson sampling model , you can efficiently simulate the effects of four independent poisson variables by instantiating the poisson distribution once with λ=0.045 and calling the RNG method four times (simulating 100 experimental trials per RNG call).
Listing 3. Simulating cell counts under the null effects poisson sampling model
<?php
require_once "config.php";
require_once PHPMATH . "/PDL/PoissonDistribution.php";
$lambda = 0.045;
$trials = 100;
$pois = new PoissonDistribution($lambda);
$cell1 = array_sum($pois->RNG($trials));
$cell2 = array_sum($pois->RNG($trials));
$cell3 = array_sum($pois->RNG($trials));
$cell4 = array_sum($pois->RNG($trials));
?>
The contingency table data appearing in Table 2 was generated by this script.
You can implement a poisson effects sampling model by instantiating your four poisson distributions with different lambda values to represent the differential effectiveness of your factors in eliciting a response.
Tutorial Pages:
» Categorical data analysis
» 2x2 contingency tables
» Sampling model
» Discrete probability distributions
» Binomial sampling model
» Poisson sampling model
» Envisioning your results
» Eliciting your prior distribution
» Model fitting with chi-square
» Null effects model
» Independence model
» Prior model
» DOE explorer
» Explorer output
» Conclusions
» Resources
First published by IBM developerWorks
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