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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