st: RE: META with confidence intervals

 From "Steichen, Thomas" To 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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