Cancer Clustering Phenomenon Whitney Lamm November 30, 2004

Outline What is a Cancer Cluster? Scientific and Public definitions The One-dimensional and D-Dimensional Cases Expected logarithmic growth rates Purpose Collecting the simulated data MatLab program Matrices of 1’s and 0’s Results Conclusions Further Research Questions

What is a Cancer Cluster? The Public’s definition The public usually thinks of a cancer cluster as a large group of the population in a certain area of the country that are diagnosed with cancer due to some type of industrial pollution, usually caused by a chemical contamination. The Epidemiologists define a Cancer Cluster as “a geographic area, time period, or group of people with a greater than expected number of cases of cancer”

One and D-dimensional Cases The expected growth rate for the longest run of heads grows logarithmically For the case of D-dimensions, more specifically two dimensions, the expected growth rate is logarithmic, but more closely resembles the following

One and D-dimensional Cases In both the one dimensional and d-dimensional cases, the limiting distribution is a Gumbel or Type I extreme value distribution.

Purpose Find a way to simulate cancer clusters in two-dimensions Does the simulated data grow logarithmically? How does changing the probability of susceptibility affect the simulated cancer clusters? Estimate the expected cancer cluster size for the population of the United States.

Collecting Simulated Data What is the best way to simulate data for the two dimensional case of cancer clusters? MatLab program creates a random n by n matrix composed of 1’s and 0’s finds largest blocks of 1’s within larger n by n matrix, these blocks represent simulated cancer clusters

Collecting Simulated Data

Results For 100 trials, with probability .5. For 100 trials of Matrix size 1,000.

Results (continued)

Results (continued)

Results (continued)

Conclusions Does the maximum cluster size grow logarithmically as n increases? Yes, but much slower than we would expect for the cases of the common and base two log, but a more accurate fit to the square root logarithmic expression. How does decreasing the probability affect the maximum block size? The block size decreases as the probability decreases.

Conclusions (continued) Can we predict the expected cluster size for the population of the US? The second method of approximation was found to be more accurate for the simulated data, so we will assume that the expected block size for the population of the US would be between a 3 by 3 matrix and a 4 by 4 matrix.

Further Research Questions How close is the distribution of the two-dimensional simulations to a Gumbel distribution (Type I extreme value distribution)? Increase the number of trials and/or the sample size Change the probabilities for different regions of the simulations What would occur?

Are there any questions? Thank You for your time! Are there any questions?