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I've read through the many excellent discussions on the site regarding interpretation of confidence intervals and prediction intervals, but one concept is still a bit puzzling:

Consider the OLS framework and we've obtained the fitted model $\hat y = X\hat\beta$. We're given a $x^*$ and asked to predict its response. We compute $x^{*T}\hat\beta$ and, as a bonus, we also provide a 95% prediction interval around our prediction, a la Obtaining a formula for prediction limits in a linear model. Let's call this prediction interval PI.

Now, which of the following (or neither) is the correct interpretation of PI?

  1. For $x^*$ in particular, $y(x^*)$ lies within PI with 95% probability.
  2. If we're given a large number of $x$s, this procedure to compute PIs will cover the true responses 95% of the time.

From @gung's wording in Linear regression prediction interval, it seems like the former is true (although I could very well be misinterpreting.) Interpretation 1 seems counterintuitive to me (in the sense that we're drawing Bayesian conclusions from frequentist analysis), but if it's correct, is it because we're predicting the realization of a random variable vs. estimating a parameter?

(Edit) Bonus question: Suppose we knew what the true $\beta$ is, i.e. the process generating the data, then would we be able talk about probabilities regarding any particular prediction, since we're just looking at $\epsilon$?

My latest attempt at this: we can "conceptually decompose" (using the word very loosely) a prediction interval into two parts: (A) a confidence interval around the predicted mean response, and (B) a collection of intervals which are just quantile ranges of the error term. (B) we can make probabilistic statements on, conditional on knowing the true predicted mean, but as a whole, we can only treat prediction intervals as frequentist CIs around predicted values. Is this somewhat correct?

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  • $\begingroup$ The answer I wrote at stats.stackexchange.com/a/26704 implies that something like (2) is the case (according to laws of large numbers) but definitely not (1). $\endgroup$ – whuber May 19 '14 at 15:02
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First, on the use of the word probability, frequentists don't have a problem with using the word probability when predicting something where the random piece has not taken place yet. We don't like the word probability for a confidence interval because the true parameter is not changing (we are assuming it is a fixed, though unknown, value) and the interval is fixed because it is based on data that we have already collected. For example if our data comes from a random sample of adult male humans and x is their height and y is their weight and we fit the general regression model then we don't use probability when talking about the confidence intervals. But if I want to talk about what is the probability of a 65 inch tall male chosen at random from all 65 inch tall males having a weight within a certain interval, then it is fine to use probability in that context (because the random selection has not yet been made, so probability makes sense).

So I would say that the answer to the bonus question is "Yes". If we knew enough information, then we could compute the probability of seeing a y value within an interval (or find an interval with the desired probability).

For your statement labeled "1." I would say that it is OK if you use a word like "approximate" when talking about the interval or probability. Like you mention in the bonus question, we can decompose the uncertainty into a piece about the center of the prediction and a piece about the randomness around the true mean. When we combine these to cover all our uncertainty (and assuming we have the model/normality correct) we have an interval that will tend to be too wide (though can be too narrow as well), so the probability of a new randomly chosen point falling into the prediction interval is not going to be exactly 95%. You can see this by simulation. Start with a known regression model with all the parameters known. Choose a sample (across many x values) from this relationship, fit a regression, and compute the prediction interval(s). Now generate a large number of new data points from the true model again and compare them to the prediction intervals. I did this a few times using the following R code:

x <- 1:25
y <- 5 + 3*x + rnorm(25, 0, 5)
plot(x,y)

fit <- lm(y~x)
tmp <- predict(fit, data.frame(x=1:25), interval='prediction')

sapply( 1:25, function(x){ 
    y <- rnorm(10000, 5+3*x, 5)
    mean( tmp[x,2] <= y & y <= tmp[x,3] )
})

I ran the above code a few times (around 10, but I did not keep careful count) and most of the time the proportion of new values falling in the intervals ranged in the 96% to 98% range. I did have one case where the estimated standard deviation was very low that the proportions were in the 93% to 94% range, but all the rest were above 95%. So I would be happy with your statement 1 with the change to "approximately 95%" (assuming all the assumptions are true, or close enough to be covered in the approximately).

Similarly, statement 2 needs an "approximately" or similar, because to cover our uncertainty we are capturing on average more than the 95%.

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The second is better. The first depends on what other information is known.

Using a random example, it is true that "95% of intervals (at 95% confidence) would include the true mean of [insert variable]".

On the other hand, if a result is obviously counter-intuitive, we cannot assert (1).

E.g., "my significance test at 95% confidence shows that height and weight are negatively correlated". Well that's obviously false, and we cannot say that there is a "95% probability that it is true". There is in fact, taking into consideration prior knowledge, a very small probability that it is true. It is, however, valid to say that "95% of such tests would have yielded a correct result."

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    $\begingroup$ This answer seems to discuss confidence intervals rather than prediction intervals. $\endgroup$ – whuber May 19 '14 at 15:03
  • $\begingroup$ @whuber The same principle applies. We are essentially dealing with confidence intervals for a certain variable (the "predicted" variable). $\endgroup$ – user45240 May 19 '14 at 15:06
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    $\begingroup$ There is an important distinction between a fixed value (like a parameter) and the value of a random variable. Moreover, the heart of the present question gets to this distinction: what can be said about the probability of that ("future") random outcome? It therefore appears inadequate--and possibly misleading--to treat this question as being one merely about the meaning of confidence. $\endgroup$ – whuber May 19 '14 at 15:11
  • $\begingroup$ @whuber The statement (2) in the post still does not imply statement (1). As in my example, a prediction which went against obvious intuition/background knowledge would not imply that future outcomes have 95% chance of falling in the PI. It is true that the process, 95% of the time, would give PI's containing the future outcome. But it is sometimes possible to detect when this has or has not happened. $\endgroup$ – user45240 May 19 '14 at 15:16
  • $\begingroup$ You are right, but if I'm reading your comment correctly I suspect it misses the point. The issue is not the fact that (by design) a PI has only a 95% chance of covering the future value or that additional data (or intuition) could give more information. The matter before us concerns whether a PI can be interpreted in terms of a conditional probability for the future value (based on the regression values). That indeed is the interpretation of a Bayes PI, as the O.P. notes, but it is invalid for a frequentist PI. $\endgroup$ – whuber May 19 '14 at 15:23

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