# [R] poisson mean hypothesis

Peter Dalgaard p.dalgaard at biostat.ku.dk
Mon Sep 12 16:50:42 CEST 2005

```Thomas Lumley <tlumley at u.washington.edu> writes:

> Use ppois(x,lambda), which gives P(X<=x) for mean=lambda.
>
> Eg: lower one-sided test for observing no events with mean of 3.4
> > ppois(0,3.4)
>  0.03337327
>
> upper one-sided test for observing 8 events with a mean of 3.4 (need the
> -1 to include 8 in the rejection region)
>
> > ppois(8-1,3.4,lower.tail=FALSE)
>  0.02307394
>
> If you want an exact confidence interval there is a formula involving the
> quantiles of the gamma distribution (ie the qgamma() function) that I
> can't remember off hand. It might even be Garwood's formula.

Or you can clone the procedure in binom.test. In fact, using
binom.test with a sufficiently large n is a rather decent "cheat":

> binom.test(5,1e6)\$conf * 1e6

  1.623488 11.668293
attr(,"conf.level")
 0.95

> ppois(4,1.623488)
 0.975
> ppois(5,11.668293)
 0.02500060

The qgamma-based interval would seem to be from

> qgamma(.025, 5)
 1.623486

to

> qgamma(1 - .025, 5 + 1)
 11.66833

Notice that this is obtained as the intersection of the two one-sided
intervals, each at half the nominal alpha level. There are other
possible definitions (e.g. invert the two sided test), but this one
has the advantage of being computable.

--
O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
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~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)                  FAX: (+45) 35327907

```