Funnel plot for meta analysis of prevalence studies

[Shripada Rao]

Hi,
I am conducting a meta-analysis to estimate pooled prevalence of a disease using stata 18.0. Freeman Tukey double arcine transformation was used to stabilise variances. The pooled prevalence was 0.10 (CI: 0.05 to 0.16). However, when I generated funnel plot, the x axis shows the pooled prevalence to be 0.6, instead of 0.10. Any help to know the reason and fix this issue will be highly appreciated. Enclosed is the generated funnel plot.

[Tiago Pereira]

Hi, Shripada.

If your outcome is the arcsine transformed proportion, the horizontal axis will also be arcsine transformed. It is not supposed to be clinically interpretable. You can manually convert to proportion, but this can be a bit tedious.

All funnels plots using metric such as odds ratios, relative risk, hazard ratios and ratio of means will present the horizontal axis on the log-scales (their standard errors are for the log-estimates). This is not different in your case.

Importantly, you should plot the sample size on the vertical axis and the log odds on the horizontal axis - using simple two way graphs, and plotting the summary log-odds as a vertical line. This is the best approach to detect small-study effects or publication bias. The approach you used is not particularly useful to detect funnel plot asymmetry caused by publication bias.

[Shripada Rao]

Thank you Tiago,
The above figure is the default output on Stata 18.0, when I generated funnel plot for this outcome. I could not find the options to keep sample size on vertical axis. Moreover, since it was a prevalence study, not sure how to use log odds ratios on x axis.

[Tiago Pereira]

Hi, Shripada.

It is log-odds, not log-odds ratio. See an example below:

clear
set obs 10
gene id = _n
gene n = rbinomial(100,0.3)
gene N = round(runiform()*100)+100

*! proportions
gene p = n/N
*! log odds
gene logodds = ln(p/(1-p))
*! variance of log odds
gene var_logodds = (1 / (N * p)) + (1 / (N*(1 - p)))

*! summary effect from the fixed-effect IV meta-analysis
sum logodds [aw=1/var_logodds]
local fe_summary_es = r(mean)
*! funnel plot
twoway (scatter N logodds, xline(`fe_summary_es') )

[Shripada Rao]

Hi Tiago, thanks so much. It looks excellent. Would it be possible to use statistical tests such as Egger's for such data?

[Tiago Pereira]

Yes. You can apply Egger's test. If your main analysis used the arcsine transformation, you can also assess the funnel plot using it, as you did initially. However, it would be reassuring to plot N vs log-odds, just in case.

Reference to justify plotting N versus log-odds:
https://www.sciencedirect.com/scienc...95435614000869

[Shripada Rao]

Hi Tiago, thank you for all your help and the informative article. I tried plotting the N vs log odds. Please let me know if it is correct.

[Tiago Pereira]

Look good. However, without the data, it is not possible to check in detail.

[Shripada Rao]

Thank you Tiago, here is the excel sheet.

[Tiago Pereira]

It seems you added a "continuity" correction (e.g., 0.5) to avoid zero estimates. I am not sure how this approach performs in meta-analysis proportion, though. Besides that, overall, it is correct.

[Shripada Rao]

Hi Tiago, thank you very much for all the help. I have a query regarding the results of Egger's test. Hope you don't mind having a look.
The results of meta-analysis are slightly different for Freeman Tukey vs Logit transformation approach, which is probably due to differences in the approaches.
However, if Egger’s test is conducted after meta-analysis using Freeman Tukey approach, the Egger’s test p value is 0.325 which indicates absence of small study effects.
On the other hand, if Egger’s test is applied after conducting meta-analysis using the logit transformation approach, the Egger’s test p value is 0.0134, which indicates presence of small study effects.
I was not sure of the reason behind this important discrepancy. Thank you for the advice.

[Tiago Pereira]

Hard to tell. Egger's test may be associated with a very high type-I error rate when there is large statistical heterogeneity (e.g., many false positives when there is no publication bias/small-study effects, but proportions vary more than expected by change). There are recommendations that we should not that test it in meta-analysis of interventions when heterogeneity is high. I am not sure regarding your case, which is meta-analysis of prevalence.

[Shripada Rao]

Hi Tiago, Thank you very much for all the advice. Very kind of you.