Metaregression on meta-analysis of proportion with metaprop and metafor

Question

[EDITED for more clarity]

I performed a meta-analysis of single proportion with logit transformation, which I made using the metaprop function, with a random intercept logistic regression model as the default option in the meta package.

Let's say for example that i want to meta-analyze the proportion of asymptomatic on the total number of patients, and I have a moderator:

moderator <- c(1,2,2,2.5,4)
m1 <- metaprop(c(10,20,30,40,50), c(100,300,400,330,240), comb.fixed=FALSE, prediction=TRUE)

So I have defined the meta-analysis object in m1.

To perform a meta-regression with the moderator, I use the TE of the meta-analysis, with the following code:

model1 <- rma(yi=m1$TE, sei=m1$seTE, method="ML", mods = ~ moderator)

Which gave me as result:

tau^2 (estimated amount of residual heterogeneity):     0.0000 (SE = 0.0224)
tau (square root of estimated tau^2 value):             0.0011
I^2 (residual heterogeneity / unaccounted variability): 0.00%
H^2 (unaccounted variability / sampling variability):   1.00
R^2 (amount of heterogeneity accounted for):            100.00%

Test for Residual Heterogeneity:
QE(df = 3) = 6.5458, p-val = 0.0879

Test of Moderators (coefficient 2):
QM(df = 1) = 26.2205, p-val < .0001

Model Results:

           estimate      se      zval    pval    ci.lb    ci.ub 
intrcpt     -3.2733  0.2623  -12.4772  <.0001  -3.7875  -2.7591  *** 
moderator    0.4735  0.0925    5.1206  <.0001   0.2922   0.6547  *** 

Two question are:

1. It is correct to conduct the meta-regression like this? it won't be better to use the transformed proportion in the meta-regression model?

For example with a code like this:

model2 <- rma(yi=transf.ilogit(m1$TE), sei=transf.ilogit(m1$seTE), method="ML", mods = ~ moderator)

which gave me totally different result:

Mixed-Effects Model (k = 5; tau^2 estimator: ML)

tau^2 (estimated amount of residual heterogeneity):     0 (SE = 0.1933)
tau (square root of estimated tau^2 value):             0
I^2 (residual heterogeneity / unaccounted variability): 0.00%
H^2 (unaccounted variability / sampling variability):   1.00
R^2 (amount of heterogeneity accounted for):            0.00%

Test for Residual Heterogeneity:
QE(df = 3) = 0.0128, p-val = 0.9996

Test of Moderators (coefficient 2):
QM(df = 1) = 0.0308, p-val = 0.8607

Model Results:

           estimate      se    zval    pval    ci.lb   ci.ub 
intrcpt      0.0114  0.6430  0.0177  0.9859  -1.2490  1.2717    
moderator    0.0444  0.2532  0.1754  0.8607  -0.4519  0.5407    

---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

2. If I exponentiate the coefficient from the first meta-regression model, I get an OR; see for example:

round(exp(coef(summary(model1))[-1,c("estimate", "ci.lb", "ci.ub")]), 2)

Which gave me the OR of 1.47 [95% CI 0.02-98.71]

is it correct to say that "for each unit increase of moderator, the odds of bein asymptomatic increase by 47%"?

3. Sticking with the model 1, I may need to plot the relationship between the moderator and the prevalence of asymptomatic disease. Following the instructions reported here: https://www.metafor-project.org/doku.php/plots:meta_analytic_scatterplot, and with a little tuning (since I have to transform the logit transformed proportions in the plot) i used this code:

preds <- predict(model1, newmods=c(1:4), transf=transf.ilogit)
plot(NA, NA, xlim=c(1,4), ylim=c(0,1), xlab="Moderator", ylab="Prevalence")
lines(1:4, preds$pred, col="black")
lines(1:4, preds$ci.lb, lty="dashed", col="black")
lines(1:4, preds$ci.ub, lty="dashed", col="black")
size <- 1 / sqrt(m1$seTE)
size <- size/10 / max(size)
symbols(moderator, transf.ilogit(m1$TE), circles=size, inches=FALSE, add=TRUE, bg="black")

Which gave me this plot:

Is that ok to represent in the plot the relationship between the moderator and the transformed proportion, even if the meta-regression was conducted with the untransformed one?

Answer 1

If you start out your analysis using a random intercepts logistic regression model, then I would suggest to stick to that framework also for your meta-regression analysis (your model1 does not use logistic regression, but uses the 'standard' inverse-normal model of the logit transformed proportions as outcome). The meta package actually uses rma.glmm() from the metafor package to fit logistic models, so your model m1 could be obtained directly with:

m2 <- rma.glmm(measure="PLO", xi=c(10,20,30,40,50), ni=c(100,300,400,330,240))
m2
predict(m2, transf=transf.ilogit)

And rma.glmm() can also be used to fit your meta-regression model with:

m3 <- rma.glmm(measure="PLO", xi=c(10,20,30,40,50), ni=c(100,300,400,330,240), mods = ~ moderator)
m3

The only change required in your code for the plot is size <- 1 / sqrt(m3$vi).