Negative confidence intervals in regression

#1
Hi all,

So I'm really stumped. Basically I have a negative standardised coefficient value of -0.35, and a 95% CI equals (-0.47, -0.23). I'm having trouble interpreting the interval because it's negative.

I'd have thought that, the lower and upper bounds would be reversed. Because if the coefficient is closer to -1 that'd mean it's a stronger effect. However, here, in the upper bound, it's saying that at best beta could be -0.23, which is closer to no effect (0). But it makes more sense for it to be -47, because that indicates a stronger effect. It doesn't make sense saying "at worst beta could be -.47" because that value would indicate the variable is a stronger predictor of the independent variable, not a weaker one.

Am I making sense here?

Many thanks!
 
#2
Similarly, regarding confidence intervals on semi partial correlations, when you square the lower and upper bounds, you get higher % of variance explained in the lower bound. Don't you usually want a higher percentage of variance explained? However, the upper bound is again, closer to 0, which is not what you want when it comes to effect size and significance.

If the number is positive however (and this goes for the b value in my question above), then you have a larger number in the UB and lower in the LB which is what you want.

Perhaps the answer to all this is simple -- you just reverse the bounds?

I am using Xeci if that helps.
 
#3
I think I may have found the answer to my question, after nearly 2 hours

When the values in a CI are <1, the lower and upper bounds become inverted? Is that correct?



Many thanks!
 

BGM

TS Contributor
#4
The CI itself does not have any problem.

It depends on which parameter you are interested in - if you are constructing the confidence interval for \( \beta \) then the result you obtain is correct. However if you are interested in other, say \( |\beta| \) then you obviously would like to obtain a different interval.
 
#5
Hi BGM,

Thanks for your answer, but it still doesn't make it clear why, in the lower bound, there is a larger number tending towards -1, which is the reverse that you get for non-negative numbers.

It is for the beta value. However, if -.47 is in the lower bound, this is saying that at worst -.47 could be the beta value, but, it is preferable to have -.47 because this is a stronger predictive effect. It makes no sense to me to say "at best, beta could be -0.23" when -.47 is a stronger effect.

Many thanks!
 

BGM

TS Contributor
#6
I am not sure how to make you feel better about this. Mathematically speaking we have \( -2 < -1 < 0 \) but if you consider the magnitude only you have \( |-2| > |-1| > 0 \)

So, perhaps you just want to construct the CI for \( |\beta| \) instead; but remind that the confidence limits of \( |\beta| \) are not the absolute value of the confidence limits of \( \beta \)
 
#7
Hi there,

Thanks for your reply! What is the meaning of the bracketed |\beta| and \beta? I know that's a very newbie questions but you've used the brackets | | a few times but unsure what they mean!

Many thanks.
 
#9
Okay great, so I'll just place a CI around the absolute value, which reverses the bounds, and place a negative sign on them.

Thanks!
 

BGM

TS Contributor
#10
That is the reason why I have to put down the notice in the previous post - you cannot simply reverse the limits and put a positive sign if you are constructing the CI for the absolute value of parameter.

To see this, note that

\( 0 < L(x) < |\beta| < U(x) \)

\( \iff -U(x) < \beta < -L(x) < 0 \text{~~or~~} 0 < L(x) < \beta < U(x) \)

Therefore

\( \Pr\{0 < L(X) < |\beta| < U(X)\} = 1 - \alpha \)

\( \Rightarrow \Pr\{-U(X) < \beta < -L(X) < 0\} + \Pr\{0 < L(X) < \beta < U(X)\} = 1 - \alpha \)

So unless one of the term is zero (that means the confidence limits are always positive/negative, which is true for parameters with domain in one sign only), the confidence intervals for \( \beta \) and \( |\beta| \) are different.

In your case, if you just reverse the confidence limits and put a positive sign over there, the confidence interval form should have a greater coverage probability for \( |\beta| \).
 
#11
Hi,

Thanks for your help I appreciate it. However, I don't understand - I'm not that advanced in math and don't know what all the notation means. I need it to be in layman's terms, but thanks for your help regardless.


What I did in XECI was put .50 for the b value instead of the observed -.50, and as a result got a CI on \beta with the larger effect in the upper bound, and placed a negative sign on the bounds....

I just need to know how to interpret a negative confidence interval, eg (-.37, -.20) for \beta. Because the larger effect is in the lower bound, it makes no sense for me to say "an effect at worst a \beta value of -.37" it makes more sense -.37 would be in the upper bounds because it's a stronger effect, as it would be if b was an absolute value.

Thanks, but I understand if you can't dumb your answers down for me :)
 

CB

Super Moderator
#12
Hi all,

So I'm really stumped. Basically I have a negative standardised coefficient value of -0.35, and a 95% CI equals (-0.47, -0.23). I'm having trouble interpreting the interval because it's negative.

I'd have thought that, the lower and upper bounds would be reversed. Because if the coefficient is closer to -1 that'd mean it's a stronger effect. However, here, in the upper bound, it's saying that at best beta could be -0.23, which is closer to no effect (0). But it makes more sense for it to be -47, because that indicates a stronger effect. It doesn't make sense saying "at worst beta could be -.47" because that value would indicate the variable is a stronger predictor of the independent variable, not a weaker one.

Am I making sense here?

Many thanks!
I wonder if the problem here is the language that you're trying to use to describe the confidence interval. Why are you using words like "best" and "worst"? These are subjective (almost moral) evaluations - whether they make sense depends on what you mean by "best" and "worst". The problem might disappear if you use different language in your interpretation. See the WP page on confidence intervals for ideas.
 
#13
Yes, I only just worked that out - I was being too literal with the best and worst interpretation, I think it's all good now.

Thank you so much for your help!