Why does the confidence interval change in poisson.test() and binom.test() based on the alternative hypothesis?

Question

For example:

poisson.test(x = 170,T = 70,r = 3,alternative = "two.sided",conf.level = .95)

    Exact Poisson test

data:  170 time base: 70
number of events = 170, time base = 70, p-value = 0.005157
alternative hypothesis: true event rate is not equal to 3
95 percent confidence interval:
 2.077216 2.822335
sample estimates:
event rate 
  2.428571 

gives me the answer I would expect for a 95% CI. But if I change the alternate to a one-tail test:

poisson.test(x = 170,T = 70,r = 3,alternative = "less",conf.level = .95)

    Exact Poisson test

data:  170 time base: 70
number of events = 170, time base = 70, p-value = 0.002502
alternative hypothesis: true event rate is less than 3
95 percent confidence interval:
 0.000000 2.758037
sample estimates:
event rate 
  2.428571 

We can see that the CI has changed drastically when I changed only the alternative hypothesis. Something similar happens in binom.test() as well. In the Biometrika article cited in the function (https://doi-org.colorado.idm.oclc.org/10.1093/biomet/26.4.404) I don't see any mention of changing the limits based on the alternative, and I can't think of any reason you would want to.

Am I missing something here?

Answer 1

One-sided hypothesis testing, such as in your second example

alternative hypothesis: true event rate is less than 3

result in a one-sided confidence interval. One-sided confidence intervals are maybe unusual, so seems unfamiliar. For a discussion see for instance Inverting a hypothesis test: nitpicky detail

One-sided confidence intervals follows naturally when constructing confidence intervals (CI) by inverting a hypothesis test, see Inverting a hypothesis test: nitpicky detail. The idea is to include in the CI parameter values that cannot be rejected using the test. But the rejection region for a test will be different for one-sided and two-sided tests, so the resulting CI must be different. For details see the above linked post.

A more detailed discussion is at Intuition for inverting one sided hypothesis tests