# Interpret confidence interval upper and lower in linear regression

I'm struggling to understand the output of a linear regression analysis, specifically with respect to the Confidence Interval Lower and Upper values. I understand all of the other values, but I can't seem to relate the CI values reported to my 'very general' understanding of CI's.

My specific output is shown below. Any insight into what these numbers indicate would be very helpful.

• If you were to repeat your experiment 99 more times, 95 of them would yield coefficients within the CI. – Jay Schyler Raadt Dec 23 '17 at 23:59
• @Jay That's a common misunderstanding of a CI--it's definitely not true, not even as an expectation or asymptotic result. The problem is that you don't even know whether the true coefficients are covered by the CI of your data. If they aren't, then the rate at which future experiments yield coefficients within this CI is going to be much less than you think. – whuber Dec 24 '17 at 0:33
• @whuber you're right, I assumed the coefficients were empirically sound. They could be used as a prior probability though – Jay Schyler Raadt Dec 24 '17 at 0:43

The 95% in your 95% CI (assuming it is a 95% CI) refers to its long run "coverage" rate of the parameter you're estimating--in this case, regression intercepts and slopes. Kristoffer Magnusson has a nice intuitive visualization of this property of CIs which is often misinterpreted, as @whuber points out above. In essence, if you were to repeatedly draw random samples from the same population of the same size and fit the same model in each, 95% of CIs would contain the population value. If you're looking at Magnusson's visualization, watch the coverage rate over time (i.e., how many randomly sampled CIs from a simulated distribution actually contain the population mean), and you'll see that it begins to approach the level of confidence you specify in the long run.

For example, with your CI for sd_qty, you can't be sure (e.g. you cannot be "95% confident") that the true value of its slope is between -0.034 and .011, but you can be confident that if you were to re-fit this model with 100 random samples of the same size, from the same population, approximately 95/100 of the CIs for sd_qty would contain the population value for that slope.

In terms of interpreting the CI for the purposes of null-hypothesis significance testing, you look to see whether the expected null value is within the CI (for slopes, this expected null value is often [but doesn't have to be] 0); if it is not, you can reject the null hypothesis at the corresponding level of $\alpha$ (e.g., $\alpha$ = .05 for a 95% CI)--your 95% CI for ROC_DSRS_5 does not contain 0, for example, so we could reject the null for this slope. Otherwise, if the expected null value falls within the CI, you fail to reject the null as the interval suggests the estimated value [or one more extreme] would not be all that unusual if the null were true, such as in the case for the 95% CI of sd_qty, which straddles 0.

I understand all of the other values, but I can't seem to relate the CI values reported to my 'very general' understanding of CI's

For normal distributions, a 95% confidence interval for the mean is (eg 95% of the area under the normal curve is within) +/- 1.96 standard errors from the mean - so:

95% CI = coefficient +/- 1.96 * Std. Err.


Using the example posted, for intercept:

0.090447 +/- 1.96 * 0.014862606 =
CI Lower ~ 0.061317
CI Upper ~ 0.119577

• Although this describes how the CI might be computed, it doesn't provide any interpretation or explanation. Perhaps you could elaborate on what the results mean? – whuber Jan 26 '18 at 14:02