PS: I would be glad NOT to start a discussion about pros and cons of CIs, and about typical misconceptions and misuses and so on. In fact I think that the centering of the intervals is a cause fore several misconceptions about these intervals, but in either case key question remains: why are such intervals centered at all?

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strangely enough i never added the condition of centering to my CIs ... the only added condition is that this interval should have the smallest size under H0.

whether or not this results in a centered interval depends on H0 and this never bothered me much !

maybe i should reconsider this more seriously (but ignorance is bliss !) ...

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Your problem is with the interpretation of the relationship between CI and tests. While the formulas might run parallel, the purpose is different. There is no null hypothesis at all needed for the CI. The CI is rather a procedure that guarantees that according to some assumptions about the underlying probability, the true parameter will be included within the interval with a probability equal to the confidence level.

Whether the interval is symmetric or not is not of the utmost importance. But centering often results in the shortest CIs, which is a desirable property.

The connection to tests is that you will reject hypothesis that lie outside the interval. This is just a corollary of the definition of the CI. Since the true value is included with probability 'a' within the interval, the probability of rejecting the null being true is given by 1-a. This applies to any CI procedure.

Hope this helps.

José, I still have problems understanding this. You write:

(1) "while the formulas might run parallel, the purpose is different."

The interval was developed by Neyman in the context of testing, mathematically as the inverted rejection region. Am I wrong here?

(2) "The CI is rather a procedure that guarantees that according to some assumptions about the underlying probability, the true parameter will be included within the interval with a probability equal to the confidence level."

Yes, but this probability *refers* to a hypothesis. Placed around H0, and when H0 is true, 95% of the statistics will fall inside. Placing the interval around the maximum likelihood estimate defines this value as the H0 value. And this I don't understand (well, I actually do, but the reasons are not at all compatible with a frequentistic point of view!). A Frequentists can say: "Given H0 is at the maximum likelihood estimate, and given that this is true, then 95% of the statistics will fall inside this interval." But the interpretation of this interval is inverse, like "95% of such intervals include the TRUE value". I simply do not see and do not understand how this interpretation follows from the construction (via the sampling distribution!). The likelihood is a function of the data (that may be evaluated for any arbitrary selected hypothesis). It gives no information about the (hmmm... how to call it?) "credibility" of any of these hypotheses. But the CI seems suddenly to provide such a thing.

(3) " Since the true value is included with probability 'a' within the interval,[...]"

How can you say? If the 'a'-CI is aound H0, and H0 is true, then the probability is 'a' that the *statistic* will be in this interval. There is nothing said about the probability that H0 is true. This does not change when I define the MLE as H0.

Look at the formulae for CI. As you say they do not include anything about any H0. That proves conceptually they are independent of a particular null. They DO talk about a TRUE value of the parameter being estimated, but you never know the value of such parameter. If you do not know it, obviously you cannot test for it unless you had miracle powers.

This does not mean that CIs are not valuable for testing. The first thing you need to develop a classical test is a significance level, the probability of rejecting the null when it is true.

That is where the background assumptions about the underlying distribution are key, Those are needed to establish the distribution of the ESTIMATE around the true parameter, and serve you establish a probability for an estimate, given a sample size, being farther away from the true parameter that a certain quantity.

When you build the CI, you give a backward interpretation to this distribution of the ESTIMATE around the true parameter: you fix a probability and look how wide would be the estimate around the true parameter that you do not know. But you know your estimate. If you place such interval around the estimate that you have gotten you know it cannot be farther away from the true parameter but with probability equal to the significance level.

Difficult...

Thanks for your patience helping me!

This is still causing me headache: "If you place such interval around the estimate that you have gotten you know it cannot be farther away from the true parameter but with probability equal to the significance level."

I understand that a given proportion of such intervals is expected to include the true value. What I do not understand is how this argument can be inversed. I can not infere how likely the true value is in a particular, given CI. For instance, consider you repeated an experiment, so you actualy have two CIs for the same parameter. These CIs will have a different location and a different width. Let's take the case that interval A is quite narrow compared to interval B what is quite wide. From your explanation I understand that for both intervals the probability is equal that any of them contains the true value. I'd rather say that the wider interval B has a higher probability to include the true value! I think not both statements can be correct...

Another example: The two intervals are both narrow, and they do not intersect. How can one argue that each of these intervals has the same (high, say 95%) chance to include the true value?

Let us talk about a particular CI around some estimate for a particular sample. It either includes the true value or it does not. It is 1 or 0. There is no probability of it including the true value of , say, 0.4. It is either 0 or 1. The problem is you don't know it. That is why, in order to understand the CI, you should not be thinking about a particular realization. You have to think ex-ante. Before obtaining the sample.

What the CI construction procedure guarantees is that if you build intervals that way, the probability of them including the true value is given by the confidence level. That is why I started talking about PROCEDURES

Your problem in the last examples is comparing different realizations ex-post.

Width is connected to precision. The fact that the interval is narrow is just telling you that the estimate is more precise and you do not need to build such a wide interval in order to guarantee a given probability of having the true value: That happens because you have more information on the whereabouts of the true value than when the interval is wider.

I reccomend you to check java applets such as http://www.ruf.rice.edu/~lane/stat_sim/conf_interval/

You will see how intervals for different realizations might overlap or not,, that does not matter. The only thing you care about is that intervals built that way include the true value with a certain probability. If you increase sample size intervals are all narrower since you have a more precise estimate.

The CI is not centred around H0 and is not required to be. The 100x(1-alpha)% CI is the set of all values of the population mean that would not be rejected at the alpha level IF adopted as the value under the null. For a t-test it is the solution of all values of t_alpha/2

Thank you Stehpen. This is a good way to look at the construction of the CI, what then automatically will be centered around the sample estimate. And it makes the equivalency obvious between test and CI.

José, I somehow feel that we go in circles. I absolutely agree that any particular CI does or does not contain the true value (we just don't know), and that from a frequentist's point-of-view there is no probability accociated with that.

I further agree with "the probability of them including the true value is given by the confidence level" - the "them" means that we are looking at the CI as a random variable (what is the case by definition), and for a random variable, probabilities can be assigned.

Actually I agree to everything you wrote...

The problem starts we we do have a realization, a particular sample and a particular CI. Then it is often taken as the one particular interval that indicates this particular range in what we have a given (say 95%) chance to find the true estimate. This conclusion is wrong - as far as I understand. And if this is wrong, what is the benefit of providing such an interval? A particular interval still does not tell me where the true value is, and it does not even tell me were it likely is. And I think this would already be a non-frequentist interpretation, because "likely" means that we assign proabilities to different parameter values, and this shoul be logically impossible. Actually, even considering a point-estimate as the most likely value of the parameter is a wrong conclusion (based on frequentistic reasoning). It rather is the parameter value for which the data is most likely. If not everything was seriousely wrong what I learned so far, this cannot simply be inverted. Even more: the parameter value can not have a probability assigned (because it is not a random variable; it is just something we don't know). So a statement like "the population parameter is most likely close to the sample estimate" is nonsense (philosophically! not so practically, I know). Similarily, claiming that the CI is a range of the most likely values of the parameter is also nonsense. So why should such an interval be constructed around an estimate? (*)

(*) I'd say: to see all values of H0 that could be rejected at the given level of confidence - but this again seems non-sensical since the H0 should be specified a-priori anyway.

I agree wtih your last starred comment and, from a practical point of view, that is the way I see CIs. They provide me an idea of the measuring rod I am using.

Note that your arguments about not liking CI would also apply to point estimates: why give a point estimate if I know it is not the true value anyway?

Returning to the point, if we build a SME (structural model equation) that fits dataset, then the SME function must be able to estimate: what is the mean for any interval taken from (0;1) cummulative populations? My results are that the mean U(point i to point i+1) is always between the values of variables predicted by good models and that the must frequent situation is that those means are not centered. If that is the case, then we should not speak of Confidence Intervals, CI, but instead of Confident SEM that are able to represent datasets properly. And "properly" means here that it keeps good isomorphism and fitting with original dataset, even if not exact due to the random factors mentioned by Jochen and to our initial ignorance about corresponding frequences. Thanks Jochen for the question, emilio

I agree with José Antonio Ortega but want to add a few things for emphasis.

1. Confidence intervals need not be symmetric about an estimator. The usual confedence intervals for chi squared random variables or a mean proportion are examples.

2. The end points of the interval are random variables under repeated sampling.

3. The confidence interval does not depend on H_0. For example, if the confidence interval is [.013, .471], then both H_0: theta=0 and H'_0: theta=.01 are rejected.