# Interpreting multilevel meta analysis coefficients and modelling the sigmoidal association

I am doing meta-analysis using the metafor package and I have the following data structure(used this example from Wolfgang's answer):

I have only one observation within each group. I have used the three-level model to fit the logit transformed proportions and have used the mean age as the predictor. Following is the code and output:

res <- rma.mv(yi, vi, mods = ~ mean.age, random = ~ 1 | study/group, data=dat)

Multivariate Meta-Analysis Model (k = 14; method: REML)

Variance Components:

estim    sqrt  nlvls  fixed       factor
sigma^2.1  0.0000  0.0000     10     no        study
sigma^2.2  0.3752  0.6125     14     no  study/group

Test for Residual Heterogeneity:
QE(df = 12) = 39.8526, p-val < .0001

Test of Moderators (coefficient 2):
QM(df = 1) = 1.0301, p-val = 0.3101

Model Results:

estimate      se     zval    pval    ci.lb   ci.ub
intrcpt     0.3130  1.2710   0.2463  0.8055  -2.1781  2.8041
mean.age   -0.0494  0.0486  -1.0149  0.3101  -0.1447  0.0460

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


Can anyone please help me with the interpretation of the coefficient of the mean age variable? My understanding is that the average log odds of passing decrease by 0.05 with the unit increase in the mean age. In my data, the proportions that I have for each group within each study are assumed to have a sigmoidal association with the quantitative variable in the model so I was wondering if the multilevel model would be able to model this association correctly?
Thanks!

Your interpretation of the coefficient is correct.

For a little tutorial on how to model non-linear associations in meta-regression using polynomial and restricted cubic spline models, see here:

http://www.metafor-project.org/doku.php/tips:non_linear_meta_regression

The same can be done with rma.mv() models. You wouldn't want to do this with so little data, but in essence you could use:

res <- rma.mv(yi, vi, mods = ~ rcs(mean.age, 4), random = ~ 1 | study/group, data=dat)

• Thank you so much @Wolfgang for your answer! So, the argument mods = ~ rcs(mean.age, 4) in the multilevel model will assume a sigmoidal relation between the proportions and mean.age, correct? and just to give you an idea regarding my data, I have on average 2 groups within each study(7 studies) and only observation in each group. Do you think that is reasonable? The objective of my analysis is to assume a sigmoidal association between proportions and the quantitative variable and also, to account for correlation between groups within each study. Thanks! May 10, 2020 at 1:27
• @frances123 restricted cubic splines are not, in general, sigmoids. May 10, 2020 at 13:32
• Thank you @mdewey for the clarification! So, can we implement a logistic regression model for a multilevel meta-analytic data using the metafor package? Thanks! May 11, 2020 at 20:30
• The rma.glmm() function fits logistic mixed-effects models, but it doesn't allow for specification of the random effects structure yourself. You might as well go straight to lme4::lmer() then. May 12, 2020 at 8:31
• Roughly speaking, yes, but models with more complex random-effects structures (like in this example) don't just have weights, but an entire weight matrix. May 16, 2020 at 9:47