there are 2 basic ways.

the 'hard' way is to use the variance-covariance matrix of the coefficients.

note that you have to do this at the level of the coefficients then exponentiate.

call the current coeff of the new reference level betanr.

call the current coeff of the nth group betan.

the new coeff of the nth group relative to the new reference level is

betannr=betan-betanr.

its variance is v(betan)+v(betanr)-2*cv(betan, betanr).  you then take the square root to get the standard error and do the usual computations to get the confidence interval.

the easy way is to make up an alternative set of indicators with your desired new reference level as the reference level and rerun the analysis.  if you are using software that automatically decides which class is the reference,  and you don't want to program indicators yourself, you might have to make up a new variable in which you scramble the classes so the desired reference level is in the 'right' place.

If you have  the data, do Ellen suggested by change the reference group. If not, you may do following: Suppose we just do for three groups, for more than 3 groups, the process is the same.

n11, n12 for yes/no for group 1,

 n21, n22 for yes/no for group 2,

n31, n32 for yes/no for group 3,

If you know all of them, it would be easy, but I think you should find the information for reference group, say n11, n12, define n1=n11+n12.

Hence you can get d=1/n11-1/n1

Variance( Ln(RR(1,2)))=1/n11+1/n21-1/n1-1/n2=d+1/n21-1/n2

From your data 2nd  group  :   RR      0.98(0.78-1.25) 2nd  group 

 Variance( Ln(RR(1,2)))=(ln(1.25)-ln(0.78))/3.92= 0.119092

Then 1/n21-1/n2=0.119092-d

Variance( Ln(RR(1,3)))=1/n11+1/n31-1/n1-1/n3=d+1/n31-1/n3

From your data 3rd  group  :   RR      .82(0.75-1.29) 3rd  group 

 Variance( Ln(RR(1,2)))=(ln(1.29)-ln(0.75))/3.92= 0.136951

Then 1/n31-1/n3= 0.136951-d

RR for (2end vs 3)=0.98/0.82=1.195

Variance( Ln(RR(2,3))) = 0.136951-d+0.119092-d=0.136951 +0.119092-2d

The only thing you need to do is to find, the number (yes/no) for your reference group.

Many studies use the logistic regression, the OR is reported, you can do the same thing. VAR(ln(OR))=1/n11+1/n12+1/n21+1/n22

If there are multiple logistic regression used by some studies, you still can use the above approach, but minor differences may be founded for re-run the model by change the reference group as the highest one.

Thank you both for answering my question. I did not have the origianl data, and I just have the RR and 95% CI for each categories. Hence, I think Li's answer is more suitable to solve my question. I want to know whether the approach can use if the effect size is hazard ratio(HR) from COX model?

HR is usually handle the time to event data, the time is continuous variable. Hazard is defined as f(t)/S(t), f is density, S is the survival distribution S=1-F, F is CDF. However we can use it to handle contingency table. In such a case, if no other factor considered, we can use Kaplan_Meier method. Then the similar approach for HR or OR as I gave would be used for HR, the step is:

1.

As part of the Kaplan-Meier calculations, compute the number of observed events (deaths, usually) in each group (Oa, and Ob), and the number of expected events assuming a null hypothesis of no difference in survival (Ea and Eb).

2.

The hazard ratio then is:

HR= (Oa/Ea)/(Ob/Eb)

3.

The standard error of the natural logarithm of the hazard ratio is: sqrt(1/Ea + 1/Eb)

4.

The lower 95% confidence limit of the hazard ratio equals:

exp( (Oa-Ea)/V - 1.96*sqrt(1/Ea + 1/Eb))

5.

The upper 95% confidence limit equals:

exp( (Oa-Ea)/V + 1.96*sqrt(1/Ea + 1/Eb))

if a is reference group, b and c are two level of treatment group, you just replace b by c in 1)-5).

Then the HR(a,b)/HR(a,c) =HR(c,b). If you know, Ea, you can estimate Var (ln(HR(c,b)) by using 95% CI of HR(a,b) and HR(a,c)

PS: V is total variance. Because you have the results, HR and 95% CI for B vs A and C vs A,  you want find HR B vs C, you don't need know V.

I would suggest the Orsini method: Article From floated to conventional confidence intervals for the re...

You can recalculate the RR/HR by using a different reference group (for example the lowest physical activity level) with data easily accessible from the published article.