For Poisson distribution, there are many different ways for
calculating the confidence interval. The paper by Patil and Kulkarni discusses
19 different ways to calculate a confidence interval for the mean of a Poisson
distribution.

The most commonly used method is the normal
approximation (for large sample size) and the exact method (for small sample
size)

**Normal Approximation:**

For Poisson, the mean and the variance are both lambda (λ ).

The standard error is calculated as: sqrt(λ /n) where λ is Poisson mean and n is sample size or total exposure (total person years, total time observed,…)

The standard error is calculated as: sqrt(λ /n) where λ is Poisson mean and n is sample size or total exposure (total person years, total time observed,…)

The confidence interval can be
calculated as:

λ ±z(α/2)*sqrt(λ/n).

The 95-percent confidence interval is calculated
as: λ
±1.96*sqrt(λ/n).

The 99-percent confidence interval is calculated as: λ ±2.58*sqrt(λ/n).

**EXACT method:**

Refer to the following paper for the description of this method:

- Garwood, F. 1936. Fiducial limits for the Poisson distribution. Biometrika. 28(3/4):437-442
- Ulm K. A simple method to calculate the confidence interval of a standardized mortality ratio. American Journal of Epidemiology 1990;131(2):373-375.

The confidence interval for event X is calculated as:

(qchisq(α/2, 2*x)/2, qchisq(1-α/2, 2*(x+1))/2 )

Where x is the number of events occurred under Poisson
distribution.

In order to calculate the exact confidence interval for
Poisson mean, the obtained confidence interval for the number of events need to
be converted to the confidence interval for Poisson mean.

Here are two examples from the internet:

Example 1:

Would like to know how confident I can be in my λ. Anyone know of a way to set upper and lower confidence levels for a Poisson distribution?

- Observations (n) = 88
what would the 95% confidence look like for this?

- Sample mean (λ) = 47.18182

With Normal Approximation, the 95% confidence interval is
calculated as:

47.18182 +/- 1.96* sqrt(47.18182/88)

This gives 45.7467, 48.617

With Exact method, we first need to calculate x (# of
events):

X = n * λ
= 88 * 48.18182 = 4152

The compute the 95% confidence interval for X = 4152. This
will give the 95% confidence interval for X as (4026.66, 4280.25)

The 95% confidence interval for mean (λ)
is therefore:

lower bound = 4026.66
/ 88 = 45.7575

upper bound = 4280.25 /88 = 48.6392

Say that 14 events are observed in 200 people studied for 1 year and 100 people studies for 2 years. Calculate the 95% confidence interval for Poisson mean

In
this example, the number of events (X) is given, the Poisson rate (λ) or mean needs to be calculated.

First step is to calculate the person year:

The
person time at risk is 200 + 100 x 2 = 400 person years

The poisson rate / poisson mean (λ)
is :

- Events observed = 14
- Time at risk of event = 400
- Poisson (e.g. incidence) rate estimate = 14/400 = 0.035

Normal
Approximation: 95% confidence interval is calculated as:

0.035 +/- 1.96* sqrt(0.035/400)

This will give the 95% confidence interval of (0.0167, 0.0533)

Exact approach: Calculate the 95% confidence interval for the number of events
(X) using:

(qchisq(0.025, 2*x)/2, qchisq(0.975, 2*(x+1))/2 )

and the result is: [7.65, 23.49]

Exact
95% confidence interval for Poisson mean is:

Lower bound = 7.65
/ 400 =0.019135 for lower bound and

Upper bound =
23.49 / 400 = 0.058724 for upper bound

We will then say the Poisson
mean is 0.035 with 95% confidence interval of (0.019, 0.059).

The following SAS programs can illustrate the calculations above:

data normal;

input lambda n ;

lower = lambda - probit(0.975)*sqrt(lambda/n);

upper = lambda + probit(0.975)*sqrt(lambda/n);

datalines;

47.18182 88

0.035 400

;

proc print data=normal;

title 'Normal Approximation for 95% confidence interval for Poisson mean';

run;

data exact;

input X;

lower = quantile('CHISQ',.025,2*x)/2;

upper = quantile('CHISQ',.975,2*(x+1))/2;

datalines;

4152

14

;

proc print data=exact;

title 'Exact method for 95% confidence interval for Poisson mean';

run;