Questions tagged [poisson-distribution]

A discrete distribution defined on the non-negative integers that has the property that the mean is equal to the variance.

Filter by
Sorted by
Tagged with
34
votes
2answers
11k views

Is there a boxplot variant for Poisson distributed data?

I'd like to know if there is a boxplot variant adapted to Poisson distributed data (or possibly other distributions)? With a Gaussian distribution, whiskers placed at L = Q1 - 1.5 IQR and U = Q3 + 1....
7
votes
1answer
627 views

The Number of Exponential Summands in a Fixed Interval is Poisson

What is the most clever way to prove that the number of independent exponential summands in a fixed interval is distributed as a Poisson random variable? I can do it one way, but I would like to know ...
76
votes
1answer
86k views

How to interpret coefficients in a Poisson regression?

How can I interpret the main effects (coefficients for dummy-coded factor) in a Poisson regression? Assume the following example: ...
32
votes
4answers
33k views

Checking if two Poisson samples have the same mean

This is an elementary question, but I wasn't able to find the answer. I have two measurements: n1 events in time t1 and n2 events in time t2, both produced (say) by Poisson processes with possibly-...
88
votes
5answers
211k views

Relationship between poisson and exponential distribution

The waiting times for poisson distribution is an exponential distribution with parameter lambda. But I don't understand it. Poisson models the number of arrivals per unit of time for example. How is ...
64
votes
1answer
24k views

Why is the square root transformation recommended for count data?

It is often recommended to take the square root when you have count data. (For some examples on CV, see @HarveyMotulsky's answer here, or @whuber's answer here.) On the other hand, when fitting a ...
21
votes
4answers
19k views

How does a Poisson distribution work when modeling continuous data and does it result in information loss?

A co-worker is analyzing some biological data for her dissertation with some nasty Heteroscedasticity (figure below). She's analyzing it with a mixed model but is still having trouble with the ...
44
votes
2answers
28k views

Poisson regression to estimate relative risk for binary outcomes

Brief Summary Why is it more common for logistic regression (with odds ratios) to be used in cohort studies with binary outcomes, as opposed to Poisson regression (with relative risks)? Background ...
41
votes
4answers
114k views

How to calculate a confidence level for a Poisson distribution?

Would like to know how confident I can be in my $\lambda$. Anyone know of a way to set upper and lower confidence levels for a Poisson distribution? Observations ($n$) = 88 Sample mean ($\lambda$) =...
31
votes
1answer
9k views

Error metrics for cross-validating Poisson models

I'm cross validating a model that's trying to predict a count. If this was a binary classification problem, I'd calculate out-of-fold AUC, and if this was a regression problem I'd calculate out-of-...
21
votes
4answers
13k views

What is the distribution of the rounded down average of Poisson random variables?

If I have random variables $X_1,X_2,\ldots,X_n$ that are Poisson distributed with parameters $\lambda_1, \lambda_2,\ldots, \lambda_n$, what is the distribution of $Y=\left\lfloor\frac{\sum_{i=1}^n X_i}...
8
votes
1answer
3k views

How is a Poisson rate regression equal to a Poisson regression with corresponding offset term?

I do not understand the role of weights in "weighted Poisson regression". What exactly is being weighted? Is it the contribution of the observation to the log-likelihood of the model, or something ...
17
votes
2answers
11k views

Deriving the bivariate Poisson distribution

I've recently encountered the bivariate Poisson distribution, but I'm a little confused as to how it can be derived. The distribution is given by: $P(X = x, Y = y) = e^{-(\theta_{1}+\theta_{2}+\...
11
votes
3answers
10k views

Should I use an offset for my Poisson GLM?

I am carrying out research to look at differences in fish density and fish species richness when using two different underwater visual census methods. My data was originally count data but then ...
22
votes
9answers
9k views

Time series for count data, with counts < 20

I recently started working for a tuberculosis clinic. We meet periodically to discuss the number of TB cases we're currently treating, the number of tests administered, etc. I'd like to start ...
44
votes
8answers
59k views

How can I test if given samples are taken from a Poisson distribution?

I know of normality tests, but how do I test for "Poisson-ness"? I have sample of ~1000 non-negative integers, which I suspect are taken from a Poisson distribution, and I would like to test that.
33
votes
3answers
34k views

How do I interpret the 'correlations of fixed effects' in my glmer output?

I have the following output: ...
32
votes
4answers
16k views

How do I fit a multilevel model for over-dispersed poisson outcomes?

I want to fit a multilevel GLMM with a Poisson distribution (with over-dispersion) using R. At the moment I am using lme4 but I noticed that recently the ...
17
votes
2answers
26k views

Why is the probability zero for any given value of a normal distribution?

I noticed that in the Normal distribution, the probability $P(x=c)$ equals zero, while for the Poisson distribution, it will not equal zero when $c$ is a non-negative integer. My question is: Does ...
5
votes
1answer
1k views

Scaling vs Offsetting in Quasi-Poisson GLM

I recently modeled insurance claim frequency assuming a Quasi-Poisson distribution in R. My frequency dependent variable was calculated, in advance of modeling, as the number of claims per underlying ...
18
votes
2answers
11k views

How is it possible that Poisson GLM accepts non-integer numbers?

I am really stunned by the fact that the Poisson GLM accepts non-integer numbers! Look: Data (contents of data.txt): ...
17
votes
4answers
2k views

Strategy for deciding appropriate model for count data

What is the appropriate strategy for deciding which model to use with count data? I have count data that i need to model as a multilevel model and it was recommended to me (on this site) that the ...
26
votes
3answers
24k views

What is the appropriate model for underdispersed count data?

I am trying to model count data in R that is apparently underdispersed (Dispersion Parameter ~ .40). This is probably why a glm with ...
14
votes
1answer
4k views

How does one show that there is no unbiased estimator of $\lambda^{-1}$ for a Poisson distribution with mean $\lambda$?

Suppose that $ X_{0},X_{1},\ldots,X_{n} $ are i.i.d. random variables that follow the Poisson distribution with mean $ \lambda $. How can I prove that there is no unbiased estimator of the quantity $ \...
6
votes
2answers
4k views

Should the predictor variables be normally distributed for Poisson glm?

This is probably a really basic question, but it's the first time I've created a model that defines Poisson as its error family. In setting up my variables to make the model, should I be concerned ...
8
votes
1answer
12k views

Interpreting coefficients for Poisson regression

I don't understand how to interpret the coefficient from a Poisson regression relative to the coefficient from an OLS regression. Suppose I have time series data, my left-hand side variable is number ...
5
votes
1answer
238 views

Distribution of successes of poisson process followed by a binomial distribution

I have been stuck with a problem for a couple of days regarding the distribution of outcomes from a two-stage process. Specifically, what is the distribution of the number of successes in a poisson ...
30
votes
4answers
83k views

Difference between binomial, negative binomial and Poisson regression

I am looking for some information about the difference between binomial, negative binomial and Poisson regression and for which situations are these regression best fitted. Are there any tests I can ...
34
votes
1answer
10k views

Does Cox Regression have an underlying Poisson distribution?

Our small team was having a discussion and got stuck. Does anyone know whether Cox regression has an underlying Poisson distribution. We had a debate that maybe Cox regression with constant time at ...
23
votes
3answers
8k views

Poisson regression vs. log-count least-squares regression?

A Poisson regression is a GLM with a log-link function. An alternative way to model non-normally distributed count data is to preprocess by taking the log (or rather, log(1+count) to handle 0's). If ...
18
votes
2answers
12k views

When someone says residual deviance/df should ~ 1 for a Poisson model, how approximate is approximate?

I've often seen the advice for checking whether or not a Poisson model fit is over-dispersed involving dividing the residual deviance by the degrees of freedom. The resulting ratio should be "...
14
votes
7answers
4k views

Intuitively understand why the Poisson distribution is the limiting case of the binomial distribution

In "Data Analysis" by D. S. Sivia, there is a derivation of the Poisson distribution, from the binomial distribution. They argue that the Poisson distribution is the limiting case of the binomial ...
10
votes
1answer
6k views

How do you use the EM algorithm to calculate MLEs for a latent variable formulation of a zero inflated Poisson model?

The zero inflated Poisson regression model is defined for a sample $(y_1,\ldots,y_n)$ by $$ Y_i = \begin{cases} 0 & \text{with probability} \ p_i+(1-p_i)e^{-\lambda_i}\\ k & \text{with ...
4
votes
1answer
3k views

UMVUE of $e^{-\lambda}$ from poisson distribution

Let $X_1,\ldots,X_n \sim$ Poisson$(\lambda)$. I wish to find UMVUE of $e^{-\lambda}$. Here $\overline{X}_n$ is complete and sufficient for $\lambda$ (hence for $e^{-\lambda}$??). Define $Y_i=1$ if $...
8
votes
1answer
3k views

Resolving heteroscedasticity in Poisson GLMM

I have long-term collection data, and I'd like to test, whether the number of animals collected is influenced by weather effects. My model looks like below: ...
6
votes
2answers
1k views

Can I treat a counting response variable as a continuous variable and run OLS?

What I learned from book is predicting counting is different from predicting a continuous variable. For example, if we want to predict how many mails a person get per day, we may use Poisson ...
3
votes
1answer
4k views

What is a statistical significance test for two Poisson distributions?

Say that I have two Poisson distributions. These were modelling after count data. How would I determine statistical significance between these two distributions? That is, how would I determine ...
5
votes
1answer
1k views

Determining confidence interval with one observation (for Poisson distribution)

When you make only one measurement, how do you determine 95% confidence interval for the Poisson distribution? Let's say that I make one measurement of a quantity $x$ and obtain $x=10$. Also it is ...
3
votes
2answers
221 views

Is there a probabilistic (not analytical) argument for why the sum of independent Poissons is Poisson?

I was just wondering if there was a simple probabilistic argument for why a Poisson random variable with parameter $\lambda_1$ plus a Poisson random variable with parameter $\lambda_2$ (both ...
3
votes
1answer
510 views

poisson glm to observe whether effects of artificial light on the number of bat passes in each location were significant

I am a university student looking for help with analysis for my dissertation, I do not have the best knowledge of R studios or statistics in general so please bear with me! I would like to perform a ...
1
vote
1answer
669 views

Poisson regression - dealing with widely different population sizes

I have a dataset of counts (responses to a marketing campaign), collected at the zipcode level. I am trying to use a poisson regression to determine underlying response rate of each zipcode. How do ...
29
votes
2answers
66k views

How to know if a data follows a Poisson Distribution in R?

I am an undergrad student and I have a project for my probability class. Basically, I have a dataset about the hurricanes that impacted my country for a series of years. In my probability Book, (...
6
votes
1answer
11k views

Conditional Distribution of Poisson Variables, given $\sum X_i$

An exercise asks me to show that if $X_1,X_2,\ldots,X_n$ is a random sample from a Poisson distribution with parameter $\theta$, the conditional distribution of $X_1,X_2,\ldots, X_{n-1}$, given $Y=\...
8
votes
2answers
10k views

Difference between shifted distribution and zero-truncated distribution

How would one derive the density or pmf of a distribution that is shifted to the right ? For example, a Poisson distribution that is truncated at zero , is shifted to the right . And it only takes ...
13
votes
1answer
18k views

Fitting a Poisson GLM in R - issues with rates vs. counts

I'm currently working on a project involving GLMs (and eventually GAMs) of some count data over time. Normally I'd do this in SAS, but I'm trying to move to R, and having...issues. When I fit a GLM ...
11
votes
1answer
5k views

OLS vs. Poisson GLM with identity link

My question reveals my poor understanding of Poisson regression and GLMs in general. Here's some fake data to illustrate my question: ...
9
votes
1answer
974 views

Mixing and dividing point processes

At the following figure at left side two realizations of point processes with different density (intensity) $\lambda_1$ and $\lambda_2$ is being mixed matching the center of the belonging areas to ...
11
votes
3answers
6k views

The weighted sum of two independent Poisson random variables

Using wikipedia I found a way to calculate the probability mass function resulting from the sum of two Poisson random variables. However, I think that the approach I have is wrong. Let $X_1, X_2$ be ...
11
votes
2answers
3k views

Is the Poisson distribution stable and are there inversion formulas for the MGF?

First, I have a question about whether the Poisson distribution is "stable" or not. Very naively (and I'm not too sure about "stable" distributions), I worked out the distribution of a linear ...
13
votes
2answers
6k views

poisson vs logistic regression

I have a cohort of patients with different length of follow-up. So far I´m disregarding the time aspect and just need to model a binary outcome-disease/no disease. I usually do logistic regression in ...

1
2 3 4 5 6