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st: RE: META with confidence intervals


From   "Steichen, Thomas" <STEICHT@rjrt.com>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: META with confidence intervals
Date   Tue, 29 Oct 2002 15:35:35 -0500

In my earlier reply to Paul O'Brien I slipped in a typographical
error... Charles Poole (forwarded via Jay Kaufman) privately 
pointed it out for me, thus this follow-up.

I claimed: 
   Data that follows log symmetry has the characteristic 
   that the following are all equal:
 
       rr/ll = (ul-ll)/2  = ul/rr

This is not true.

A correct form (and the one I used in my calculations) is:

   rr/ll = exp((ln(ul)-ln(ll))/2) = ul/rr

Charles pointed out that a simpler form is:

   rr/ll = sqrt(ul/ll) = ul/rr 

(I once knew this...)

My apologies to Paul for the error and my thanks to Charles
for the correction and to Jay for forwarding Charles' comment.

Tom Steichen

 
> Paul O'Brien writes: 
>  
> > I am combining two studies:
> > 
> > Study       RR    LCI   UCI
> > Study 1   0.7   0.1   8.2 
> > Study 2   0.6   0.1   6.4 
> > 
> > With the command: 
> > 
> > . meta rr ll ul, ci eform gr(f) print id(study)  
> > 
> > However, the confidence intervals listed in the print are 
> > different from what I have entered:
> > 
> > Meta-analysis (exponential form) 
> >        |  Pooled      95% CI         Asymptotic      No. of 
> > Method |     Est   Lower  Upper  z_value  p_value   studies 
> > -------+---------------------------------------------------- 
> > Fixed  |      0.645   0.142   2.927   -0.568    0.570      2 
> > Random |   0.645  0.142   2.927   -0.568    0.570 
> > Test for heterogeneity: Q=  0.010 on 1 degrees of freedom (p= 0.921) 
> > Moment-based estimate of between studies variance =  0.000 
> >           |      Weights      Study       95% CI 
> >     Study |   Fixed Random    Est   Lower   Upper 
> > ----------+---------------------------------------- 
> > Study 1 |    0.79    0.79    0.70    0.08    6.34 
> > Study 2|    0.89    0.89    0.60    0.08    4.80
> > 
> > What is the problem?
> 
> The problem is that your input data do not follow the expected
> ratios for log-based confidence intervals (probably because
> too few digits were retained).  -meta- uses your input CI to
> compute the standard error (se), assuming log symmetry, then 
> later recalculates the proper log-symmetric CI endpoints about 
> the point estimate using this standard error.
> 
> Data that follows log symmetry has the characteristic that the 
> following are all equal:
> 
>   rr/ll = (ul-ll)/2  = ul/rr
> 
> For your input data I get:
> 
>            rr/ll = (ul-ll)/2 = ul/rr
> study 1      7        9.06     11.71
> study 2      6          8      10.67
> 
> For the (rounded) recalculated values I get:
> 
> study 1     8.75      8.90      9.06
> study 2     7.50      7.75      8.00
> 
> These values are not exactly equal because the two-digit
> representation of the recalculated ll, .08, is not
> accurate enough.
> 
> For a more accurate value, note that -meta- uses the 
> following calculation to get the se:
> 
>  se = ( ln(ul) - ln(ll) ) / 2 / z  
> 
>   (where z is an appropriate Normal value)
> 
> For your study 1 data this generates:
> 
>  se = (  ln(8.2)  -   ln(.1)   ) / 2 / 1.96
>     = ( 2.1041342 - -2.3025851 ) / 2 / 1.96
>     = 4.4067192 / 2 / 1.96
>     = 1.1241631
> 
> Later, -meta- spits back the recalculated CI endpoints as:
> 
>  ll = exp( ln(rr) - z * se )
>  ul = exp( ln(rr) + z * se )
> 
> Or, for study 1:
> 
>  ll = exp( ln(rr) - z * se )
>     = exp( ln(.7) - 1.96 * 1.1241631 )
>     = exp( -.35667494 - 1.96 * 1.1241631 )
>     = exp( -2.5600346 )
>     = .07730206  (displayed as .08)
> 
>  ul = exp( ln(rr) + z * se)
>     = exp( ln(.7) + 1.96 * 1.1241631 )
>     = exp( -.35667494 + 1.96 * 1.1241631 )
>     = exp( 1.8466847 )
>     = 6.3387699  (displayed as 6.34)
> 
> Thus, using the exact ll and ul in the ratio calculations:
> 
>   rr/ll = (ul-ll)/2  = ul/rr = 9.06
> 
> 
> This suggests to me that more digits are required from your
> original data in order to properly meta-analyze the data.
> 
> Tom
> 
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