Confidence limits for the Poisson
Confidence interval calculation for Poisson rates.
cipoisson(k, time = 1, p = 0.95, method = c("exact", "anscombe"))k |
Number of successes |
time |
Total time on trial |
p |
Probability level for the (two-sided) interval |
method |
The method for computing the interval. |
The likelihood method is based on equation 10.10 of Feller, which relates poisson probabilities to tail area of the gamma distribution. The Anscombe approximation is based on the fact that sqrt(k + 3/8) has a nearly constant variance of 1/4, along with a continuity correction.
There are many other proposed intervals: Patil and Kulkarni list and
evaluate 19 different suggestions from the literature!. The exact
intervals can be overly broad for very small values of k, many of
the other approaches try to shrink the lengths, with varying success.
a vector, matrix, or array.
If both k and time are single values the result is a
vector of length 2 containing the lower an upper limits.
If either or both are vectors the result is a matrix with two columns.
If k is a matrix or array, the result will be an array with one
more dimension; in this case the dimensions and dimnames (if any) of
k are preserved.
F.J. Anscombe (1949). Transformations of Poisson, binomial and negative-binomial data. Biometrika, 35:246-254.
W.F. Feller (1950). An Introduction to Probability Theory and its Applications, Volume 1, Chapter 6, Wiley.
V. V. Patil and H.F. Kulkarni (2012). Comparison of confidence intervals for the poisson mean: some new aspects. Revstat 10:211-227.
cipoisson(4) # 95\% confidence limit # lower upper # 1.089865 10.24153 ppois(4, 10.24153) #chance of seeing 4 or fewer events with large rate # [1] 0.02500096 1-ppois(3, 1.08986) #chance of seeing 4 or more, with a small rate # [1] 0.02499961
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