[R-meta] testing for and visualizing correlation between dependent traits in bivariate meta-analysis

[Sigurd Einum]

Thank you for this James! It wasn’t clear to me what you did with the transformations in your first reply, but now I get it. Log-transforming the estimates and using the delta method approximation for the variances/covariances seems to do the trick, and I no longer get the extreme cook’s distance values that Wolfgang pointed out (they are now all < 0.3).

Greatly appreciated!

Best,
Sigurd



From: James Pustejovsky <jepusto using gmail.com>
Sent: Friday, November 8, 2024 5:19 PM
To: Sigurd Einum <sigurd.einum using ntnu.no>
Cc: Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer using maastrichtuniversity.nl>; R Special Interest Group for Meta-Analysis <r-sig-meta-analysis using r-project.org>
Subject: Re: [R-meta] testing for and visualizing correlation between dependent traits in bivariate meta-analysis

Hi Sigurd,

Some comments inline below.

James

On Thu, Nov 7, 2024 at 1:52 AM Sigurd Einum <sigurd.einum using ntnu.no<mailto:sigurd.einum using ntnu.no>> wrote:
The data come from experiments that consists of a transfer of aquatic organisms from one salinity to a new one. Following the transfer, physiological traits (for these data this is NaK ATPase activity) are measured at subsequent time points. I then fit an asymptotic function to the the data for each experiment, and extract the estimates of two parameters (lambda which is the rate of change in the trait, and lFC.inf which is the asymptotic log-fold change in the trait) together with their variances and covariances. These then go into the meta-analyses.

It might be helpful to look more into the definition of these parameters, such as what the possible range (and/or typical range) of lambda values. This might inform the decision of whether or not to transform by taking logs (or using some other transformation, for that matter).


You are correct that there is skewness in the data, however the distribution of the BLUPs (which to my understanding is more crucial, https://stats.stackexchange.com/questions/451306/log-transforming-a-mean-what-to-do-with-sd) is more symmetric.

My understanding of BLUPs is that they are influenced by the model used to estimate them---you might call them "model-informed" for this reason---and so the distribution of BLUPs gets pulled towards the assumed random effects distribution. Consequently, BLUPs will necessarily tend to be closer to normally than raw estimates, even if the true distribution is not normal. In my opinion, I think the severe degree of skewness in the raw data here would be cause to question a multilevel random effects model fit to the raw estimates.

Also, I am not sure how log-transforming the estimates would help, since I cannot obtain equivalent transformed measures of variances and covariances to use in the meta-analysis? Or is there a way to do this that I am not aware of?

It might be reasonable to use delta method approximations for the variances and covariances. If the raw estimate is Y and its sampling variance is V, then
Var(ln Y) ~= V / Y^2
For raw estimates Y, Z with covariance C,
Cov(ln Y, ln Z) = C / (Y * Z)
Depending on the definition of the model parameters, it's possible that the first-stage analysis is actually estimating ln Y and ln Z and their sampling variance and covariance, then applying the reverse delta method approximation to find the variance and covariance of Y and Z. This would be the case with, for instance, an odds ratio estimated from a logistic regression model. The model is set up to estimate a log odds ratio, and then there's a delta method approximation to get exp(log odds ratio) and its standard error.

James

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