# How to interpret negative 95% confidence interval?

I have performed an unpaired t-test on two groups of data:

Group 1 (control): mean = 0.087 (n = 15) SD = 0.028

Group 2 (treatment): mean = 0.115 (n=12) SD = 0.042

The t-test revealed a non-significant difference between the groups with the 95% confidence interval being -0.056 to 0.00068

What does the negative confidence interval mean? I understand that I can't reject the null hypothesis because zero is contained within the CI range. But why is the range more "weighted" towards a negative value rather than a positive?

• The statistics you report are in conflict with one another. The estimated mean difference of $0.087- 0.12 = -0.033$ would ordinarily be exactly midway between the two confidence limits, but it is not. Please check your calculations. But regardless--since the difference in means is negative, why would it be in the least strange to have a negative confidence limit? How could that possibly be avoided and still make sense? – whuber Apr 16 '15 at 21:19
• @iacobus Sorry about the misunderstanding. Thanks for setting me straight. I'll restore your answer and delete the comment. (Initially I took you to be the OP and only discovered belatedly that you are not... .) – whuber Apr 16 '15 at 21:35
• Thank you very much for answering my question.Can I make any comments on the likelihood of the real effect size being zero, based on this CI? Sorry if this is an ignorant question but I am very new to all this – Kim Apr 16 '15 at 22:06
• Did you notice that the midpoint of the two ends of the interval is the difference in means (-0.028)? – Glen_b -Reinstate Monica Apr 17 '15 at 1:41

For a two-sample t-test (paired or unpaired), what you are looking at is the difference between the means of the two samples. The 95% confidence interval is providing a range that you are 95% confident the true difference in means falls in. Thus, the CI can include negative numbers, because the difference in means may be negative.

For a very basic example, let's say that your control group has a mean of $1$ and your treatment group has a mean of $2$. The difference between these will be $-1$. When you calculate the confidence interval for the true difference in means (not just the sample difference), it will be centered on $-1$. The confidence interval (whatever it is) will by definition fall more in the negative side than the positive side. However, if you reversed the calculation and did treatment-control instead, you would get a range falling more in the positive side. It would not affect your final conclusion.

[EDIT] The numbers in the question got updated, but I'll leave this comment here for future reference: In the numbers you give (as whuber points out in a comment), your confidence interval should center on $-0.033$ because that's the difference in your sample means. Because it doesn't, there's likely some error in your calculations somewhere.

• @whuber & @ Duncan: I am getting same value for (0.087-0.115) and ((-0.056)-0.00068)/2. Both are -0.028. Probably you did not take into account -ve value of -0.056. Please correct me if I am wrong. – rnso Apr 17 '15 at 15:41
• The numbers have changed from the original post. They now make more sense. – Duncan Apr 17 '15 at 15:42
• Yes, I re-calculated and amended the post. I am so sorry for any confusion. I am still unsure as to why a negative confidence interval suggests that "actually group 1 may be higher than group 2" as @rnso states below. Could someone be so kind as to explain this? Thank you – Kim Apr 17 '15 at 16:56

The range is "weighted" because the estimated differences in the means is not exactly zero. The CI of the difference is the point estimate +/- 1.96 * SE and it will only be symmetric about zero when the point estimate is zero.

There is no inconsistency in difference between means and center point of confidence intervals. Both are 0.028

In simple terms, a negative confidence interval in this setting means that although observation is that mean of group 2 is 0.028 higher than group 1, the 95% confidence interval suggest that actually group 1 may be higher than group 2.

Or does the CI definition change based on what we are using it for, as it only estimates our confidence at say 95% that the true value falls between the upper and lower limits of CI intervals

For example; In trying to figure out the effect of Y~ ßX + intercept, the 95% CI for X effect will give an estimate of true ß value (± 2.SE of ß)

Whereas in non inferiority study, the 95% CI is an estimate whether drug X (new) is non inferior than drug Y (baseline) by assigning a predefined margin M

Another is CI of means difference above

As such the negative or non negative values discussion have many meanings to it depending on how one uses 95% CI for?