# How to prove that the chance of occurrence of event A increases in presence of event B, when both have a different Poisson distribution?

If we have the events:

• A is the number of rainfalls/month in a given period of time (rate of which is equal in each month)
• B is the number of hailstorms/month in a given period of time (rate of which increases in winters)

Can we statistically prove the probability of event A increases whenever the probability of event B increases? If yes, how?

EDIT: thank you for the valuable input. The reason I am interested in this question is because in EEG tracings for diagnosing epilepsy we look at "spikes" which are events occurring at more or less a constant rate (though the rate differs in awake and sleep, but I am only interested in spikes in sleep). Normally, the rate remains constant during sleep, but in few cases, it has been seen to increase in the presence of an event called "k-complex". It is hypothesized that these k-complexes actually increase the epileptogenisity and hence the "spikes". So if we have a poisson of "spikes during sleep" and "k-complexes" (rarer events during a specific sleep stage) during sleep, can we statistically prove that spikes increase when number of k-complexes are increased?

• One very basic way to argue this is through "confounding" - both A and B are counting things that are "caused by" clouds. Hence, in the abscence of data on cloud, they may appear to cause each other. Is your question "time based"? By this I mean your hypothesis is something like "hail today means more rain in the (near) future, on average." – probabilityislogic Feb 2 '14 at 13:20

If A is Poisson, you would speak of its rate of occurrence, rather than its probability.

If this is to be based on data, rather than a fully identified joint distribution, a simple approach is to condition on is observed values of B and the interest is if the observed counts of A tends to be higher when the count of B is higher.

So the question would be 'How would we see if the rate of occurrence of A increases when the occurrence of B is higher?'

You might do something simple, like look for a positive correlation (not necessarily linear - you might look at monotonic association via something like a nonparametric correlation, perhaps).

Another approach would be to do a Poisson regression (GLM) of the counts of A on counts of B (hopefully including any other known or likely important covariates); in some other cases, B might be treated as an exposure by including an offset in a Poisson regression (but I don't think that would be suitable for this particular example).

Here's an illustration with simulated data (in this case I know the model, because I created the data): As we see from the plot, the two are positively associated. Their Pearson correlation is:

 cor(x,y)
 0.8057106


Here I fit a GLM (in R) with identity link (more commonly, a log link would be used, but the model here is closer to the data generating model in this particular example). In this case, we fit $E(Y|X=x) = \beta_0+\beta_1 x$, which looks like a regression model, but with the GLM here the model takes account of the fact that the observations are conditionally Poisson. The command

summary(glm(y~x,family=poisson(link=identity)))


fits the model mentioned above, with output:

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)  13.9571     3.6826    3.79 0.000151 ***
x             0.9533     0.1806    5.28 1.29e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 42.907  on 39  degrees of freedom
Residual deviance: 15.284  on 38  degrees of freedom
AIC: 232.81


In this case, the fitted model reproduces the actual process used to create it fairly well; the true model is that $x$ was generated from a conditionally Poisson model (with mean that took two different values), and $y$ was equal to $x$ plus a 'background' Poisson process (with constant intensity).

The interpretation here is that when $x$ increases by $1$, on average the $y$ count increases by about $0.95$

The intercept term picked up the background process (which had population mean 13), and the slope term picked up the $x$ effect (which had population coefficient 1).

As gung suggests below, if you take daily rainfall and daily hail as Bernoulli (rained or not, hailed or not, for each day), then you can deal with probability rather than counts, and there are a variety of ways to model that. His suggestions in comments are a good way of looking at the problem (quite a bit more sophisticated than my suggestions here), and would get you closer to conditioning on estimated underlying probability per unit time rather than directly observed rate per unit time.

• This is the right advice. However, given the OP's description of the data (rainfalls in months), I was wondering if it might be possible to consider each day a Bernoulli trial, & use estimated probabilities as a latent variable to predict hail, perhaps w/ a SEM. OTOH, thinking of the physical processes behind these data, it it even possible to hail w/o rain? – gung - Reinstate Monica Feb 2 '14 at 5:06
• Thanks @gung it's definitely worth pointing out that the daily occurrence of rainfall would perhaps be better seen as Bernoulli rather than Poisson (and then the question's use of 'probability' fits better), so thanks for that. As for 'is it possible to hail without rain', in a practical sense a gauge will normally have water in it even if it was all frozen as it came down. I'm not sure how you could be sure none of it was liquid before it hit the ground rather than after - but perhaps I should leave that issue for people with more subject-expertise. – Glen_b -Reinstate Monica Feb 2 '14 at 5:29