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Suppose you don't know the estimates of a linear regression model (b0, b1 are unknown) and you are given a 95% prediction interval at x = 6 to be [5,15].

Then, since prediction intervals are numerically symmetric you find the midpoint and this is your Y^hat at that given value of x.

(15 + 5) / 2 = 10.

So here, Y^hat = 10.

Is the statistically correct or can you not find the estimated Y-value of the regression line using a prediction interval?

Sorry if this doesn't make sense, I am a bit confused.

Thanks.

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  • $\begingroup$ You're logic is right. For linear regression, prediction intervals involve adding and subtracting the same quantity from the prediction. The prediction is hence the mid point of the prediction interval end points. Note, this is only true for linear regression. $\endgroup$ Mar 22, 2021 at 16:34

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The prediction interval is

$$ \hat{y}_{h} \pm t_{(1-\alpha / 2, n-2)} \times \sqrt{M S E \times\left(1+\frac{1}{n}+\frac{\left(x_{h}-\bar{x}\right)^{2}}{\sum\left(x_{i}-\bar{x}\right)^{2}}\right)}$$

for a simple linear regression. The mean of this quantity is $\hat{y}_h$. Thus, you can simply take the average of the prediction interval points and recover the prediction.

This is not true generally speaking for other models however. Prediction intervals integrate over the uncertainty in the parameters and the likelihood, and not all likelihoods are symmetric about their mean (e.g. the gamma).

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I would say that in 98% of cases, this works fine.

If you are interested in the 2% cases where it doesn't, read on.

  • This presupposes that the 95% prediction interval is a central one, consisting of the 2.5% and the 97.5% quantile predictions. This is often an unspoken assumption, but of course, you could also have a 95% prediction interval that consists of the 4% and the 99% quantile predictions. Or PI that is not defined by two quantiles, but is specified as the shortest 95% PI. Brehmer & Gneiting (2020) are a nice reference.
  • This also presupposes that your PI is calculated using a symmetric distribution. This is again usually satisfied: in OLS, we use a $t$ distribution. But this does not hold for PIs for Generalized Linear Models (GLMs), e.g. if you are using a gamma distribution.
  • Finally, this presupposes that "the estimated $y$ value" is the conditional expectation. This is often what you want. But not always. You may be interested in the conditional median, or the conditional (-1)-median. It depends on your loss function (Kolassa, 2020).

However, I reiterate that my caveats are usually academic.

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