Confidence Intervals: Closed or Open Intervals

Read post first! Are confidence intervals closed or open intervals?

  • Closed

    Votes: 1 33.3%
  • Open

    Votes: 1 33.3%
  • Unsure/Undecided

    Votes: 1 33.3%

  • Total voters
    3

ondansetron

TS Contributor
#1
I want to pose a question, mainly theoretical in nature, but applied responses are more than welcome. First, I'll define some terms.

An interval from a to b is said to be closed if it is the case that for any value x in the interval from a to b, a <= x <= b. That is, the interval is inclusive of the stated endpoints and is denoted [a,b]. Values can take on a or b.

An interval from a to b is said to be open if it is the case that for any value x in the interval from a to b, a < x < b. That is, the interval is NOT inclusive of the stated endpoints and is denoted (a,b). Values can become increasingly close to a and b, but may never reach a or b.

So, that being said, are confidence intervals open or closed intervals? Feel free to include and sources and absolutely include your rationale! Try to clarify if the argument is practical vs theoretical.

I'll start with two arguments, one for each side. Please critique!

Open- that is, the CI (a,b) does not include it's endpoints.
1. Alpha is the maximum acceptable probability of a Type I error. Therefore, in a two tailed test, +/-Z (alpha/2) is the smallest (in magnitude) value Z statistic for which we will reject Ho. Fail to Reject Ho occurs otherwise.
2. A (1-alpha)*100 CI is constructed by finding +/-Z(alpha/2) for the endpoints.
3. As noted before, these are also the minimum (in magnitude) values for which we Reject Ho.
4. Therefore, the CI must not include its endpoints, and is therefore an open interval. That is, for a test statistic equal to a or b we reject Ho.

Closed- that is, the CI [a,b] does include it's endpoints.
1. Acceptance region method wherein values exceeding (in magnitude) Z(alpha/2) lead to a rejection of Ho. Fail to Reject Ho occurs otherwise.
2. Therefore, the CI must include its endpoints, and is therefore a closed interval interval. That is, for a test statistic equal to a or b, we will fail to reject Ho.

I'm not being totally rigorous here, partly because I don't do math-type on the computer, but also because I'm sure many could run circles around me with more rigorous exposition, so I'll leave that to them.

Overall, I think a CI is an open interval!
 
Last edited:

hlsmith

Omega Contributor
#2
Yeah, there is probably always rounding error on 1.96 every time as well. It is too late to think too hard but I wonder if a huge data simulation can shine some light on it. Also, do you say < 0.05 or </=0.05?

So my answer would be whichever one includes 95% in the integral! So is this like how 12 noon with 0 seconds is AM, but 12 noon and 1 second is PM because the last hour is now closed and we are on to the next hour.

Odan, how is school going? Using many Cochrane reviews?

PS I will vote in the morning!
 

ondansetron

TS Contributor
#3
Yeah, there is probably always rounding error on 1.96 every time as well. It is too late to think too hard but I wonder if a huge data simulation can shine some light on it. Also, do you say < 0.05 or </=0.05?
I would omit rounding errors for the purpose of the theoretical argument but certainly it's a practical point. I use less than or equal to .05 (<=) since the definition I learned for alpha is the maximum probability of a Type I error. In other words, the researcher says, "In the long run, what is the frequency with which I am OK with making a Type I error?" Therefore, if the researcher says 5% (.05), then a p-value of .05 (truly .05) would be a p-value where one rejects Ho (or similarly if the z statistic was truly 1.96, would reject Ho). Practically speaking, <.05 and >.05 is what we would deal with nearly always as far as a calculated p-value goes. Regarding the "maximum probability of a Type I error," comment, you can see this in discrete examples where the true Type I error rate is no greater than alpha because the values of the discrete distribution are such that you will incorrectly reject Ho 3% of the time, for example.

So my answer would be whichever one includes 95% in the integral! So is this like how 12 noon with 0 seconds is AM, but 12 noon and 1 second is PM because the last hour is now closed and we are on to the next hour.
I agree when thinking about it from the perspective that a CI is a range of values to estimate the parameter and the interval is technically a set of "fail to reject Ho" values.

Odan, how is school going? Using many Cochrane reviews?
School is pretty busy, but it's going! Every now and then we see them!
 

Dason

Ambassador to the humans
#5
Closed. But an argument can be made either way.

But with that said anybody would be an idiot to disagree with me.