What are these 2.5% and 97.5%? I didn't understand what is written in the documentation. Anyone could explain to me in a more practical terms?
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4$\begingroup$ The help page, under "Value," states "A matrix (or vector) with columns giving lower and upper confidence limits for each parameter. These will be labelled as (1-level)/2 and 1 - (1-level)/2 in % (by default 2.5% and 97.5%)." Which aspect(s) of this need explaining? $\endgroup$– whuber ♦Commented Jun 16, 2020 at 17:33
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$\begingroup$ @whuber these labels. I've heard about a confidence interval having 95% of confidence,in this case, [11.79, 14.27] of 95% chance to my variable value be there. I don't know how to interpret these two numbers: 2.5% and 97.5% $\endgroup$– user45523Commented Jun 16, 2020 at 17:39
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$\begingroup$ Please see stats.stackexchange.com/search?q=two-sided+confidence. $\endgroup$– whuber ♦Commented Jun 16, 2020 at 18:16
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$\begingroup$ @whuber Maybe I should study more statistics, where can I go to study deeper into confidence intervals? Which branch of statistics? I will search for a book about this $\endgroup$– user45523Commented Jun 16, 2020 at 18:35
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$\begingroup$ The relevant concepts are covered in all introductory stats textbooks, so you will do well to consult your favorite before you look any further afield. $\endgroup$– whuber ♦Commented Jun 16, 2020 at 21:42
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1 Answer
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A 95% confidence interval (CI) is in fact an algorithm with the following property:
Suppose you re-run an experiment many, many times (from sampling from the population over measuring your data to fitting your model) and estimate the same parameter of interest with an associated 95% CI. Then, because of noise in your data, you will get a different parameter estimate and a different 95% CI each time. But 95% of these different CIs will cover the true parameter value.
(Yes, this is unintuitive, and frequently gotten wrong. Here is a recent discussion.)
Now, of course a 95% CI is not unique. It consists of two end points. You could in each case calculate a lower 1% limit (such that when you repeat the experiment many times, the true parameter value will be lower than this 1% limit in only 1% of cases) and an upper 96% limit (i.e., out of your many experiments, the true parameter value will be higher than this limit in 4% of cases). The two together will yield an interval that covers the true parameter value in 95% of cases, so this is a 95% CI.
However, symmetry is appealing. So the "standard" 95% CI consists of a lower 2.5% limit and an upper 97.5% limit. When you repeat your experiment many times, the true parameter value will be below the CI in 2.5% of cases and above it in another 2.5% of cases - and the CI will cover it in 95% of cases.
The numbers you see are the lower (2.5%) and upper (97.5%) limits of this standard symmetric CI for your three estimated parameters in the single run of the experiment you are analyzing right now.
The Wikipedia page is somewhat confusing, but there is a lot of material there.
answered Jun 16, 2020 at 19:35
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$\begingroup$ Great answer, thank you very much. So if I understood well, these numbers refer to how percentage of the data is bellow of these limits. So, we have 2.5% of the value bellow 11.79 and 97.5% of the value bellow 14.27. Is that correct? $\endgroup$ Commented Jun 16, 2020 at 21:26
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$\begingroup$ These two comments appear to be at cross purposes: it is not the case that the confidence values refer to "percentage of the data." They refer to the sampling distribution of the parameter estimate. The two concepts are vastly different. $\endgroup$– whuber ♦Commented Jun 17, 2020 at 14:23
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1$\begingroup$ @whuber: you are right, I misread the OP's comment, thanks for pointing this out. $\endgroup$ Commented Jun 17, 2020 at 14:27
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1$\begingroup$ @user45523: no, the CIs have very little to do with how the data are distributed. They pertain to the distribution of the parameter estimate. (For instance: collect more and more data from the same distribution and estimate the mean. Its CI will get narrower and narrower and therefore contain less and less of your observed data.) $\endgroup$ Commented Jun 17, 2020 at 14:29