# One sided testing of two Poisson distributions?

• Gerenuk
In summary: If you know the variance then you know the mean. The z-test is a good approximation to the t-test as the sample size increases. However, it is not a good approximation when the variances are not known (in which case you would use an analysis of variance).
Gerenuk
I want to test if one Poisson distributed result a is large than another one b.
I don't know much about statistics, but I understood the Wiki article about testing normal distribution however they need the number of samples there.

Basically I measure two Poisson distributed variables, I get two values and want to know the probability that one is larger than the other.

Can someone give my a quick reference (online or good book), where I can find my problem as close as possible?

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You can compute the difference between the two Poisson means and see whether the difference is significantly different from zero under the Skellam distribution.

Or (with a large enough sample) you can assume that the normal distribution will be a reasonable approximation.

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EnumaElish said:
You can compute the difference between the two Poisson means and see whether the difference is significantly different from zero under the Skellam distribution.

Or (with a large enough sample) you can assume that the normal distribution will be a reasonable approximation.

I think I just about know what to do with the Skellam. But it has a funny Bessel function.
I have both means approx. 500.
What would be the method with the normal distribution?

If you know the true variances, the z test.

If you are using computed variances, then technically you should use t test with equal or unequal variances, as the case may be. As the sample size increases, the z-test becomes a good approximation to the t-test (e.g. for n > 40).

EnumaElish said:
If you know the true variances, the z test.

If you are using computed variances, then technically you should use t test with equal or unequal variances, as the case may be. As the sample size increases, the z-test becomes a good approximation to the t-test (e.g. for n > 40).

I tried to look through these tests, but I'm not sure what to take. I only know that I measured a single value
x and single value y. Both are supposed to be Poisson (so I expect x+- sqrt(x) and y+-sqrt(y)).

In this case I'm not sure how interpret "computed variance" or "sample size".
I know about mathematics, but not of the formalities of statistics :(

I found the following (attachment) in
"An improved approximate two-sample poisson test" (M.D.Huffman)

Just to make sure I got it right and plug in the right values:
I use $\alpha=0.05, p=0.90$ as sensible values?
I look up z in a table? (i.e. $z_{0.95}=1.65, z_{0.90}=1.28$)
I estimate $\varrho$ from initial measurements.
Should I use equal counting time $d=1$ for best results?
By equation (4) I will find how long I have to measure...

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Gerenuk said:
I tried to look through these tests, but I'm not sure what to take. I only know that I measured a single value
x and single value y. Both are supposed to be Poisson (so I expect x+- sqrt(x) and y+-sqrt(y)).

In this case I'm not sure how interpret "computed variance" or "sample size".
I know about mathematics, but not of the formalities of statistics :(
In a z-test you are assumed to know both means and variances. Poisson is a one-parameter distribution (say k) where mean is a function of k, and variance is also a function of k; so you can derive both means and both variances if you know the k parameter for each of the distributions. Put differently, if you know the mean then you know the variance.

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## 1. What is one sided testing of two Poisson distributions?

One sided testing of two Poisson distributions is a statistical method used to compare two populations of count data, typically represented by Poisson distributions. It is used to determine if there is a significant difference between the means of the two populations, in a specific direction.

## 2. How is one sided testing of two Poisson distributions different from two sided testing?

In one sided testing, the researcher has a specific hypothesis about the direction of the difference between the two populations (e.g. one is larger than the other). In two sided testing, the researcher is interested in whether there is any difference between the two populations, regardless of direction.

## 3. What is the null hypothesis in one sided testing of two Poisson distributions?

The null hypothesis in one sided testing of two Poisson distributions is that there is no difference between the means of the two populations. This means that any observed difference is due to chance or random sampling variability.

## 4. What statistical test is used for one sided testing of two Poisson distributions?

The most commonly used test for one sided testing of two Poisson distributions is the one-sided Poisson test. This test calculates the probability of observing the data, or more extreme data, if the null hypothesis is true. If this probability is small enough (typically less than 0.05), the null hypothesis is rejected in favor of the alternative hypothesis.

## 5. What are some practical applications of one sided testing of two Poisson distributions?

One sided testing of two Poisson distributions is commonly used in fields such as healthcare, finance, and social sciences to compare the effectiveness of two treatments or programs, to determine if there is a significant difference in customer satisfaction between two companies, or to study the impact of a particular intervention or policy. It can also be used in quality control to assess whether a new process or product is better than the old one.

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