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I fit the model y = a + b * x. And 95% CI for estimates of a and b are (a1, a2), (b1, b2), respectively.

If we have a new observation x0, then the estimated response is a_hat + b_hat * x0. And the 95% CI is (a1 + b1 * x0, a2 + b2 * x0)

Case 1, now given y = 0, I want to estimate the responding x. I guess the confidence interval for x is (a1 / b2, a2 / b1). Am I right?

Case 2, if we fit a model y = a + b * x + c * z + d * x * z , a_hat<0, b_hat,c_hat,d_hat>0, Given y = 0, how to calculate the confidence interval for x + z?

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    $\begingroup$ For these questions to be answerable you need to supply more information, including (at a minimum) the probabilistic terms in each model and the method by which the parameters are being estimated. $\endgroup$
    – whuber ♦
    Jan 18, 2014 at 22:53
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    $\begingroup$ For OLS regression, it is not the case that the 95% CI for $\hat y$ is $(a_l + b_lx_0,\ a_u + b_ux_0)$. $\endgroup$
    – gung - Reinstate Monica
    Jan 18, 2014 at 23:32
  • $\begingroup$ Yea. We still need error term. I think this problem is looking for a calibration interval. $\endgroup$
    – OMG
    Jan 18, 2014 at 23:37
  • $\begingroup$ What is a "calibration interval" in this context? $\endgroup$
    – whuber ♦
    Jan 18, 2014 at 23:38
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    $\begingroup$ @whuber: in this context, calibration is inverse prediction. See p145 of Seber&Lee Linear Regression Analysis (2ed) $\endgroup$
    – kjetil b halvorsen ♦
    Nov 23, 2018 at 18:17

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You seem to be asking about inverse prediction, also known as calibration. In short, if you have a linear regression model $$ Y=\beta_0+\beta_1 x +\epsilon $$ and you estimate it based on some selected values $x_1, x_2, \dotsc, x_n$. Then, you want to use the estimated model (and estimated parameters $\hat{\beta_0},\hat{\beta_1}$ to find the value of $x$ that will give some choosen value of $Y$, say $y_0$, and a confidence interval for that $x$.

This is discussed in Seber and Lee, page 145, for instance, with many references. There is a dedicated R package on CRAN ,investr. And it has been discussed on this site Error bars, linear regression and "standard deviation" for point

(I will try to come back to post a fuller account)

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