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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?

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    $\begingroup$ 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? $\endgroup$ – whuber Apr 16 '15 at 21:19
  • $\begingroup$ @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... .) $\endgroup$ – whuber Apr 16 '15 at 21:35
  • $\begingroup$ 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 $\endgroup$ – Kim Apr 16 '15 at 22:06
  • $\begingroup$ Did you notice that the midpoint of the two ends of the interval is the difference in means (-0.028)? $\endgroup$ – Glen_b Apr 17 '15 at 1:41
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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.

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  • $\begingroup$ @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. $\endgroup$ – rnso Apr 17 '15 at 15:41
  • $\begingroup$ The numbers have changed from the original post. They now make more sense. $\endgroup$ – Duncan Apr 17 '15 at 15:42
  • $\begingroup$ 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 $\endgroup$ – Kim Apr 17 '15 at 16:56
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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.

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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.

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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?

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