Talk:Confidence interval

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Inconsistency in interpretation?[edit]

In the “Practical example” section, just above the “Theoretical example” sections starts, we read these two contradictory statements.

First we read this:

One cannot say: "with probability (1 − α) the parameter μ lies in the confidence interval."

Then after a short amount of explanation, we find this:

That is why one can say: "with confidence level 100(1 − α)%, μ lies in the confidence interval."

To me, these two statements are effectively saying the same thing. Any comments? — Preceding unsigned comment added by (talk) 06:25, 26 July 2012 (UTC)

The statements are different, because confidence is not probability. When one says of a procedure to take sample data from a population and construct an interval estimate of a population parameter that there is a 95% confidence of the interval covering the parameter, one is making a statement about the procedure, not about the calculated interval. It means that if this procedure were repeated and lots of interval estimates calculated from lots of sample sets, then 95% of the calculated intervals would be expected to cover the population parameter. It does not mean that one can say of a particular sample set and its calculated interval that this one has a 95% chance of covering the parameter. Still less does it mean that there is a 95% probability of the parameter being within the interval, because in frequentist statistics one can only assign probability values to random variables: the parameter is either within the interval or it isn't - it is not a variable. To ask, what is the probability that the parameter is inside the interval? is a question that does make sense in Bayesian statistics, but that is a different concern: confidence intervals are fundamentally a frequentist concept. Confusing confidence with probability is very common, unfortunately, even within the published literature. Dezaxa (talk) 12:02, 26 July 2012 (UTC)

"95% probability" is correct (in the frequentist framework) if you're regarding the interval as random. But if you're talking about the observed interval, given a certain sample - as we usually are in applied statistics -, then it's "95% confidence" you should say, because the probability that the parameter is within a fixed interval is always either zero or one. FilipeS (talk) 12:18, 24 February 2013 (UTC)
Sorry, Filipe, the word "random" doesn't even make sense in your statement. And No, whether to a frequentist, a Bayesian, or anyone else: A confidence interval does not represent the probability of the unknown parameter's lying in the confidence interval.Daqu (talk) 19:39, 27 February 2013 (UTC)
Filipe is right. There is a lot of misunderstanding around confidence intervals. A clear distinction has to be made between the "random confidence interval", in fact consisting of two random variabels, being the endpoints of the interval, and the realisation of this interval, the observed confidence interval, consisting of tap fixed numbers, the endpoints. The confidence coefficient is the probabilkty for the random inerval to cover the (value of the) parameter.Nijdam (talk) 23:04, 27 February 2013 (UTC)
No, Nijdam. Since the actual parameter is unknown — and the parameters have no known probability distribution — there is no way for someone to construct a confidence interval that is known to have a specified probability of containing it.Daqu (talk) 04:01, 8 March 2013 (UTC)
No, Filipe and Nijdam are correct. Treating the observed data as random (and hence the end-points of the interval as random), the probability ( or, if you want to think of the parameter as in some sense random, the conditional probability) given the true value of the parameter that the interval covers the true value is equal to the confidence level. This is true for all possible values of the "true parameter". This is essentially the definition of a confidence interval, so there can be no mistaking it. Of course, the probability here is NOT the conditional probability given the observed data. (talk) 23:54, 8 March 2013 (UTC)

Confidence limits[edit]

Confidence limits redirects to this article, however, the term is not mentioned even once in the article. It would be nice to clarify the relationship between confidence intervals and confidence limits, or, if (as I assume) the limits are the endpoints of the interval, to state this explicitly in the article. Vegard (talk) 09:34, 25 September 2012 (UTC)

After all this time, this article contains totally erroneous statements about what a confidence interval is, and how it is computed[edit]

For example, in the introduction:

"The level of confidence of the confidence interval would indicate the probability that the confidence range captures this true population parameter given a distribution of samples."

Not at all. (And this is not the only place where this mistake is made in this article.)

This is *not* what a confidence interval means. That is surely the reason that the carefully written first paragraph does not mention anything implying that confidence level means the probability of the parameter lying in it.

It is essential that those editing this article fully understand the correct definition of the subject of this article -- or else that they step aside and make way for those who do.

Otherwise, this article will continue to misinform many thousands of readers over a period of more and more years.Daqu (talk) 19:29, 27 February 2013 (UTC)

Signing in to add a "me, too" here. The confidence interval specifically says ONLY that you can say with X% confidence that subsequent measurements will fall within this same range. It says nothing directly about whether the true value is close to the value you happened to measure. See [1] Cellocgw (talk) 16:06, 15 August 2013 (UTC)

When computing the CI of a mean (assuming a Gaussian population), the multiplier is a critical value from the t distribution. With very large n, this converges on 1.96 for 95% confidence. For smaller n, the multiplier is higher than 1.96. This article is simply wrong. HarveyMotulsky (talk) 14:58, 14 February 2014 (UTC)

Relation to significance testing[edit]

Unless I'm mistaken, an inconsistency seems to have crept in. The second paragraph contains the sentence:

If a corresponding hypothesis test is performed, the confidence level is the complement of respective level of significance, i.e. a 95% confidence interval reflects a significance level of 0.05.

while under the hypothesis testing section there is:

It is worth noting, that the confidence interval for a parameter is not the same as the acceptance region of a test for this parameter, as is sometimes thought. The confidence interval is part of the parameter space, whereas the acceptance region is part of the sample space. For the same reason the confidence level is not the same as the complementary probability of the level of significance.

Neither of these includes a citation. It would be helpful if we could have a clarification and an authoritative citation. Dezaxa (talk) 15:48, 4 June 2013 (UTC)

Example in the "Conceptual basis" section[edit]

How would people feel if I worked on a new example for the "conceptual basis" section of the article. Currently, the example involves a measured value of a percentage. I find this confusing since the confidence level is also stated as a percentage. Maybe an alternate example with a different type of measurement could be used? Sirsparksalot (talk) 20:47, 12 September 2013 (UTC)

I am going to ask this question under my new username and see what people think (I used to edit under Sirsparksalot, now I use JCMPC). Please provide comments because I think that this is an important issue for this page. It seems as though the 90%/95% example that is given keeps flipping back and forth in a mini editing war. Personally, I feel like some of the problem is that the example given is confusing since it involves polling, which has "measured" values reported as percentages. Maybe it's my own view, but I find it easy to mix up which value is referring to the CI and which is referring to the poll results. Could we use an alternate example to help reduce some of this confusion, perhaps something such as a length measurement? JCMPC (talk) 16:29, 22 November 2013 (UTC)

I agree, please do, and if you can think of an example that would be illustrated by the bar chart that's already to the right of this section then that would be even better :-) Mmitchell10 (talk) 15:21, 21 December 2013 (UTC)

Grumpiness, pedantry and overly long introductions[edit]

I'm approaching this page as a non-statistician and I appreciate the hard work gone into the careful crafted introduction on this page. However, as a reader I want to know from the first sentence of an article what this is and why I should care and it doesn't quite achieve this. My background is critical appraisal of medical journal articles and an accepted definition - however correct or otherwise - is "the range in which there is a 95% probability that the true value lies". I know that statisticians get a bee in their bonnet about use of the word probability in this context and get all uppity about referring to the population variable as a random statistic and to refer to this is the introduction just seems grumpy. My current favourite succint explanation is "the range within which we can be 95% confident that the true value for the population lies" [2] . I think this or something similar should be a first sentence as it makes the whole character of the article that much more approachable and understandable. This is a common flaw of most of the statistics article where too much effort is spent being "correct" before trying to make it understandable or engaging. Arfgab (talk) 13:48, 12 April 2014 (UTC)

The problem with saying that a CI is "the range in which there is a 95% probability that the true value lies" is that this is simply incorrect. You may well be right in saying that it is an accepted definition in some circles, but it is nevertheless just a common misconception. The article's opening paragraph is not being pedantic, it is just carefully worded to avoid a common error. Exchanging this for "the range within which we can be 95% confident that the true value for the population lies" is not much better, because the word confidence invites the reader to misinterpret it as probability. The critical point is that the 95% refers to the procedure and the random intervals it produces, not to the results of any particular data set. Dezaxa (talk) 21:03, 18 April 2014 (UTC)
This isn't the place to discuss what a CI is, but rather than saying 'this is simply incorrect', I think it would be fairer to say that this is something which is disputed. Mmitchell10 (talk) 12:03, 19 April 2014 (UTC)
Dezaxa is right to say 'simply incorrect', above. For a nice nice example of how the 75% confidence interval - constructed from particular observed data - can actually include the true parameter value with probability 1, see e.g. Ziheng Yang (2006), Computational Molecular Evolution, pp. 151-152.

The CI is such a commonly used statistic that I am amazed that there is no agreement as to what it means. The page is not much help for a non-statistician and apparently not even to a statistician. Really now, what is a CI? Pcolley (talk) 22:23, 28 April 2014 (UTC)

I agree it's amazing that there is so much confusion about this, especially because the answer is very clear and has been repeatedly stated by people above: a correctly specified 95% CI should contain the true value exactly 95% of the times if repeated experiments are made (frequentist CI). It is NOT, I repeat NOT !!! "the range in which there is a 95% probability that the true value lies", regardless how much people would like to think that. The latter is called a Bayesian credible / credibility interval and has its own wikipedia page Incidentally, the latter also gives the correct definition for a frequentist CI, although incomprehensively written otherwise. FlorianHartig (talk) 07:04, 12 June 2014 (UTC)

Cite error: There are <ref> tags on this page, but the references will not show without a {{reflist}} template (see the help page).