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I understand the basics of confidence intervals, the central limit theorem, etc, to be able to know things like given N samples of random variable, we're 68/95/99.7 percent sure the variable is within these two values.

I'm wondering about how this generalizes to cases where we're trying to derive a relationship between two variables. Ie, if we're trying to measure how some variable Y depends on a variable X, one thing I guess we could do is set X to 1, then get a 95% confidence interval for X = 1. Then set X += 0.1 and do that again, and again again etc, to derive a 95% "confidence relationship", or something.

This seems kind of sloppy though. Is there some other statistical mechanism designed for doing things like this (ie, finding a relationship between X and Y that we are 95% sure is the right relationship).

edit: Some more info on what I'm looking for:

Say I have a bunch of (x,y) pairs. I'm trying to express the y's as a function of x. x values might repeat. I also don't necessarily care about expressing the relationship algebraically. So, the naive thing to do would be to average over the ys for every possible x value we have data for, and plot that. (so if the dataset was {(1,2), (1,3), (2, 2)}, I'd plot the points {(1,2.5), (2,2)}).

This totally disregards data about how much y data we got for each x value, though. It also seems to be ignoring the fact that we suspect the data to be correlated in some way. Like if our data points were {(1 + 1e-10, 4), (1, 17)}, my naive approach would plot {(1,17), (1 + 1e-10, 4)}. But considering how close 1 and 1e-10 are, we'd think that the values 4 and 17 should be close to each other. So maybe something like {(1, 10.5), (1 + 1e-10, 10.5)} would be more accurate.

Or, in another sense, if our dataset was {(1,16), (1+1e-10,16.2}), the fact that we got two values so close to 16 both times should seem to "inspire our confidence". But since the x values are technically different we won't notice this.

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    $\begingroup$ You can get confidence intervals on a correlation, and you don't do it in the way that you imagine. There is a very simple introduction at onlinestatbook.com/2/estimation/correlation_ci.html The key is to use Fisher's z transformation, which is an atanh scale. $\endgroup$
    – Nick Cox
    May 28 '13 at 0:40
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    $\begingroup$ Important: a confidence interval IS NOT the interval in which there is a 67/95/99.7 percent probability of finding the parameter of interest! Another easy way of finding confidence intervals is to perform sub-sampling (e.g. bootstraping). $\endgroup$
    – Néstor
    May 28 '13 at 1:29
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    $\begingroup$ @Néstor raises a good point, but it may be hard to understand. If so, this thread: why does a 95% CI not imply a 95% chance of containing the mean? may help you. $\endgroup$ May 28 '13 at 2:24
  • $\begingroup$ @NickCox so a lot of that seemed a bit over my head, but I got the impression it was working off a model of linear correlation, right? what if I'm not expecting the values to be correlated linearly? $\endgroup$
    – Alec
    May 28 '13 at 5:30
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    $\begingroup$ You have edited your post. It does seem as if you don't want a confidence interval for a correlation coefficient. I can't follow what you do want. There is going to be a persistent communication problem if you are using terms like correlation and confidence in your own informal senses, when there are technical uses of the same words. $\endgroup$
    – Nick Cox
    May 28 '13 at 7:47
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The word you missed is regression. You have $(x,y)$ pairs, and wants to "express the y's as a function of x". That is regression, and usual regression procedures will take care of "x values might repeat".

You also say "I also don't necessarily care about expressing the relationship algebraically". That could mean that you want something like nonlinear regression or nonlinear smoothing (maybe splines). Without more context it is difficult to say!

Some posts you could look into: Practical description of LOESS and smoothing splines?

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