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

[James Pustejovsky]

Hi Sigurd,

Some comments inline below.

James

On Thu, Nov 7, 2024 at 1:52 AM Sigurd Einum <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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