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Monte Carlo
Distribution-Free Methods: | Nonparametrics | Bootstrap | Jackknife | Monte Carlo
description | simple example | how it works | caveats
Description and a simple example: Suppose you observe plants in a plot and would like to evaluate whether they are clumped, dispersed, or randomly located. You could use Monte Carlo simulations in this case to compare test statistics generated from your data with similar test statistics generated from a random model (Manly 1991).
From your actual data, you might calculate the distance of each plant from its nearest neighbor, for example. This is your test statistic. To find out if it is unusually high or low, you can define a random model for the placement of plants on the plot. If the plot is 1 meter squared, you might select random numbers from 0 to 1 to define the position of plants on the simulated plots. Next, calculate the distance to the nearest neighbor for all these simulated plants. Repeat this process many times until you have many values for your test statistic and can define its distribution. At this point, you can evaluate whether your observed test statistic is unusual compared to the distribution of test statistics generated from the simulated plots of plants.
How the method works: Once you define the underlying model, new samples are randomly selected from that model. There are no restrictions on the test statistics, you can calculate any value or measure you like. Once you have many (typically hundreds) of values from the simulated data, you can compare the value from your real data and determine if it is unusually high or low. You can use the percentage of values that are higher (or lower) as the p-value. For example, if 4 out of 1000 values are more extreme than yours, you can say that your results are significant at p < 0.005
Assumptions/limitations: The primary assumption for this method is that you have correctly specified the null model from which the random samples are drawn. If the null model you specify is too general, the test may be trivial and not provide a realistic evaluation of the test statistic from your real sample. If the model is too specific, the results from the random samples will look very much like your real data.