# Confidence interval and confidence region

Could you please tell me what is the difference between confidence interval and confidence region in the following sense?

For example, we have s multiple linear regression model. For individual confidence intervals, we use $t$-statistics to find individual confidence intervals for regression parameters but I found in many books that when authors write confidence intervals based on $F$ distribution,they call it confidence region not the intervals.

What does it mean by region and how does it relate to confidence intervals, specifically in view of multiple regression?

As @NBrouwer says, a confidence interval is for an individual variable, so it is a (one-dimensional) interval. This is the case for e.g. the confidence interval for an individual regression coefficent.

However, if you build 'confidence intervals' for more than one variable at a time, i.e. for a multivariate parameter $\beta=(\beta_0, \beta_1, \dots \beta_n)$ then you get a region in an (n+1)-dimensional space. In two dimensions this could be a rectangular region, an ellips, or another shape.

The Bonferroni correction e.g. requires you (for the two variable case) for a 0.95 confidence level to construct two intervals - one for each dimension - using a 0.975 confidence level. The confidence region then looks like $\{(x,y) | \bar{x}_L \le x \le \bar{x}_H \& \bar{y}_L \le y \le \bar{y}_H \}$ which defines a rectangular region (the 'bar' means a fixed value and the subscripts L and H mean Low and High). Such a rectangular region could be seen as a (two-dimensional) interval.

For confidence 'intervals' based on statistics like a $\chi^2$-statistic (in 2 dimensions) your region will be an ellips (see also How to find the maximum axis of ellipsoid given the covariance matrix? and Chi-Square-Test: Why is the chi-squared test a one-tailed test? - where the equation for an ellips can be seen in the definition of the $X^2$).

Similar for the F-statistic; you build an interval for all the coefficients jointly, so it is a multidimensional region.

So a one-dimensional confidence region is a confidence interval. Rectangular confidence regions might be called confidence intervals, depending on your definition of an (multi-dimensional) interval. Or ''confidence region'' is the more general name, ''confidence inteval'' is a special case of a ''confidence region''.

Confidence intervals are usually for individual parameters in a model or individual predictions from a model; both would be considered point estimates. Confidence regions usually cover the complete range of data that went into the model, and incorporate both uncertainty in the parameter estimates and prediction error.

• The conflation of "data" with "parameters" in this post is troubling because it could confuse many people. The subsequent conflation of "confidence" and "prediction" risks further confusion. Do you think you could edit the post to fix these problems?
– whuber
Feb 21, 2015 at 20:31