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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 09:55:50 -0500

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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