# Joint Confidence Interval

Given the following model:

${\rm Wage}_i=\beta_0+\beta_1 {\rm Married}_i+\beta_2{\rm Female}_i+\beta_3 {\rm Married}_i \times {\rm Female}_i + \varepsilon_i$,

I am interested in finding $\beta_1+\beta_3$ for different quantiles of the dependent variable (i.e. on average, how much more/less does married females make compared to single females for a given quantile). I already have the results of the quantile regression model, and I calculated $\beta_1+\beta_3$ for all quantile levels.

However, I am trying to find a confidence interval for $\beta_1+\beta_3$ to determine whether this quantity is statically significant for a given quantile level. One thing that came to mind is the Bonferroni joint confidence interval. Using this method I was able to find a lower bound for the joint confidence interval for each of $\beta_1$ and $\beta_3$, but I am not sure how to find a confidence interval for the sum of the two covariates.

Would it be correct to sum the two confidence intervals that we got (I didn't find evidence that support this approach)? Is there a way to determine such a confidence interval? Appreciate your help. Thanks.

• This is a question about predictions based on linear combinations. You will need to base your CI on the standard error of the combination, which is a function of the standard errors of the estimates, but also of the covariance of the estimates.. May 1 '14 at 23:21

Uh... let's see. . . this is just from vague memory, but I think that in your case: $s_{\beta_{1}+\beta_{3}} = \sqrt{s^{2}_{\beta_{1}} + s^{2}_{\beta_{3}} + 2cov_{\beta_{1}\beta_{3}}}$

I am a bit worried that for discrete changes, like marriage and gender, the partial derivatives do not give the right effect, which is inherited by their sum. With a continuous variable this seems more sensible, because the coefficient gives you the change in the outcome for an infinitesimal change in x, which is unlikely to shift a person to a different quantile. With discrete changes, I am more suspicious of that, and the interpretation is a bit more delicate. However, I am not aware of an alternative way to proceed.

Also, I am not sure "Joint CI" is the right title since you are not looking for that, and the bonferroni tag seems irrelevant.

A general way to get simultaneous confidence intervals, that assumes normality of the $\beta$s, is implemented in the R rms package. You can fit with the rms Rq function for quantile regression, then pass the resulting fit object to the contrast.rms function where you specify all the contrasts of interest and specify conf.type='simultaneous'.

Once you have calculated the standard error from Alexis' formula, you should be able to retrieve the confidence interval doing:

CI = 1.96 +/- SE

However, it is not clear whether it works for linear regression, as suggested in the CV post

Reproduce a confidence interval of linear regression in excel

• (1) The post you refer to concerns a spreadsheet mistake, not a problem with linear regression, and it's about prediction limits rather than confidence limits. (2) I didn't notice this the first time through, but did you see that the question concerns a quantile regression? It's not conventional linear regression. Even so, @Alexis is correct: given an estimated covariance matrix for the coefficient estimates, an approximate CI can be constructed for any linear combination of the coefficients.
– whuber
Jun 27 '14 at 19:02
• No I did not. Thanks for your comment, it is helpfull for me because I want to compute a joint confidence interval as well :). It works provided the 2 parameters I want to calculate the joint CI for are linearly related, isn't it? (I mean, even if is in a non-linear model)? Jun 27 '14 at 19:06
• That's a great question. If you are not doing least squares regression, the parameter estimates might not be linearly related, but even so their covariance matrix can be used as one way to construct joint CIs. This is justified for Maximum Likelihood estimation by an appeal to its asymptotic properties, so a sizable amount of data can be necessary.
– whuber
Jun 27 '14 at 19:10
• I am using the Levenberg Marquardt algorithm in Mathematica, I think it is using least square. I can try to fit my model with ML, data amount will not be an issue. Thanks! Jun 28 '14 at 22:37

I know the answer is found at this link. which says: If you want simultaneous confidence intervals for both the intercept and slope, using the Bonferroni method with joint confidence level α, set the level equal to 1 – α / 2...

• This is an approximation only. It often works well for intercept and slope in ordinary regression because frequently the intercept and slope estimates are uncorrelated or only weakly correlated. When it comes to two arbitrary coefficient estimates, though, as in the question, it is important to account for their correlation. A comment to the question by @Alexis points this out.
– whuber
Dec 12 '17 at 14:26