I have a collection of several sets of two arrays X and Y, where the x and y values can be 0, 1 or 2. I calculate the real Mutual Information between X and Y. Then I shuffle X and Y and calculate again a Mutual Information (MI) of such random case. I repeat the process many times, so for each instance of the collection I end up with one real value of Mutual Information and a distribution of MI values from a null model (random). I want to now how significant is the real value but also be able to compare such significance with other instances of the collection. If the underlying distribution for the null model were gaussian, calculating z-score like
zscore = (MI - mu) / std,
where mu and std are respectively the mean and standard deviation for the null model distribution of MI, would be my solution. But how I can do this when the underlying distribution is a Poisson?
If you could provide any cite I would appreciate.
Thank you very much!
Variance of a Poisson random variable is equal to its expectation. When modelling a Poisson outcome using generalized linear model the canonical link function is a natural logarithm. If there is a difference in baseline variance / expectation between observations in such a model we would use a log transformed baseline as an offset. In this sense it might be meaningfull to compare ratios of the real MI and the mean of the null model MI distribution for each instance (relative MI?) or alternatively log of this ratio which would be a difference between log real MI and log mean MI of the null distribution. When comparing ratios value of 1 corresponds to real MI being exactly equal to expected value under the null. Values above 1 mean real MI is larger, and values below 1 that it is smaller than expected under the null distribution. If you take a logarithm of this ratio, your "standardized" values will be somewhat similar to the standardized values under the normal distribution where 0 means exactly equal to the expected value, positive number reflects real MI larger than expected and negative value reflects real MI smaller than expected under the null distribution.
Variance of a Poisson random variable is equal to its expectation. When modelling a Poisson outcome using generalized linear model the canonical link function is a natural logarithm. If there is a difference in baseline variance / expectation between observations in such a model we would use a log transformed baseline as an offset. In this sense it might be meaningfull to compare ratios of the real MI and the mean of the null model MI distribution for each instance (relative MI?) or alternatively log of this ratio which would be a difference between log real MI and log mean MI of the null distribution. When comparing ratios value of 1 corresponds to real MI being exactly equal to expected value under the null. Values above 1 mean real MI is larger, and values below 1 that it is smaller than expected under the null distribution. If you take a logarithm of this ratio, your "standardized" values will be somewhat similar to the standardized values under the normal distribution where 0 means exactly equal to the expected value, positive number reflects real MI larger than expected and negative value reflects real MI smaller than expected under the null distribution.
If you have the simulation results you have a good estimate of the sampling distribution of the MI under the null hypothesis. That's all you need, and there is no need to standardize anything.
The p-value for your observed MI is the fraction of simulated MI values that are "more extreme" (larger?) than the observed value. Note that two-sided tests are not well defined when the sampling distribution is not symmetric.
Integer valued random variables are usually scaled with the so-called "thinning" (introduced by Renyi). When applied to Poisson distributions, it simply scales the parameter. You can find the basic properties and definitions in connection with the "law of small numbers".
But it is probably of more theoretical than practical use.
Mario, would you please present exact formulas applied to evaluate the quantities of your interest? Or, an article where they are defined?
Sincerely
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