[R-meta] Meta-analysis questions

Tue Sep 14 10:17:32 CEST 2021

Dear James,

Thank you for your detailed and very helpful reply!

I have checked whether the degree of heterogeneity is constant across temperature differentials, and it is, so I don’t think having LnRR/LnSR as the response would be needed for this particular model, but thanks for letting me know about that, as it might come in useful when I look at the effect of other moderators later!

I have a further question about checking for normality of random effects. I have only checked for normality of the model as a whole and the response variable so far (I am not sure whether these things also are assumed to be normally distributed?) My random effects are categorical, so I am not sure how these can end up as a distribution. Is it still expected that categorical random effects should be normally distributed? If so, are you able to let me know the code to use to check this?

Best wishes,

Bethan

________________________________________
Bethan Lang
PhD Candidate
ARC Centre of Excellence for Coral Reef Studies
James Cook University
Townsville, QLD, Australia
4811
[signature_1454588885]

From: James Pustejovsky <jepusto using gmail.com>
Date: Tuesday, 14 September 2021 at 5:33 am
To: Bethan Lang <bethan.lang using my.jcu.edu.au>
Cc: "r-sig-meta-analysis using r-project.org" <r-sig-meta-analysis using r-project.org>, Jennifer Donelson <jennifer.donelson using my.jcu.edu.au>
Subject: Re: [R-meta] Meta-analysis questions

Hi Bethan,

I'll respond to your first questions and add a query of my own.

First, non-normality and high kurtosis are indeed problems for the modeling approach you've taken. One consequence is that you will likely have very high levels of estimated heterogeneity. Another consequence, and potential concern, is that the standard errors and confidence intervals that come from rma.mv<https://aus01.safelinks.protection.outlook.com/?url=http%3A%2F%2Frma.mv%2F&data=04%7C01%7Cbethan.lang%40my.jcu.edu.au%7C71cbd862a4f045a7780708d976ed3d34%7C2eba4cf8af764db3bcaf81b5592535ef%7C0%7C0%7C637671584024790169%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C2000&sdata=zfI7RIxL6czOcAeym2SfkfmOk%2FH9cPZtXTySiUQMkdo%3D&reserved=0>() should probably not be trusted because they are predicated on assumptions about normality of the random effects in the specified model. There are (at least) two ways that you could deal with this issue. First, you could ignore the standard errors from rma.mv<https://aus01.safelinks.protection.outlook.com/?url=http%3A%2F%2Frma.mv%2F&data=04%7C01%7Cbethan.lang%40my.jcu.edu.au%7C71cbd862a4f045a7780708d976ed3d34%7C2eba4cf8af764db3bcaf81b5592535ef%7C0%7C0%7C637671584024790169%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C2000&sdata=zfI7RIxL6czOcAeym2SfkfmOk%2FH9cPZtXTySiUQMkdo%3D&reserved=0>() and instead use robust variance estimation methods, which are asymptotically robust to non-normality as well as mis-specification of the model covariance structure. For example, the following will give you robust confidence intervals for your model:

conf_int(Traitcat_model, vcov = "CR2")

An alternative to RVE is to use bootstrap confidence intervals, as you've attempted. However, the usual implementation of bootstrapping will not work here. Because you've got dependent effect sizes with a multi-level structure, you'll need to bootstrap re-sample *at the study level* instead of re-sampling individual observations. Here's a rough sketch of how to cluster bootstrap:

library(boot)
study_ids <- unique(Traitcatdata$Study_number)

boot.func <- function(study_ids, indices) {

  row_indices <- Traitcatdata$Study_number %in% study_ids[indices]
    Traitcat_model2 <- try(suppressWarnings(rma.mv<https://aus01.safelinks.protection.outlook.com/?url=http%3A%2F%2Frma.mv%2F&data=04%7C01%7Cbethan.lang%40my.jcu.edu.au%7C71cbd862a4f045a7780708d976ed3d34%7C2eba4cf8af764db3bcaf81b5592535ef%7C0%7C0%7C637671584024800164%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C2000&sdata=NgG7PipW0ZRmnmPNGHCLkZfkjyS%2FZEChmp3K44DMYyY%3D&reserved=0>(yi=Ln_response_corrected,V=Variance, data=Traitcatdata, mods=~LnSR:Trait_cat - 1, test="t", random = list(~1|Study_number/Response_id/Effect_size_id), method = "REML", subset=row_indices)))
    if (inherits(Traitcat_model2, "try-error")) {
    NA
  } else {
    c(coef(Traitcat_model2), Traitcat_model2$sigma2)
  }
}

res.boot2 <- boot(study_ids, boot.func, R=5000)

Cluster-bootstrapped confidence intervals will usually give you results that are quite similar to the RVE approach.

To your question about reporting the bootstrap CIs with the original model estimates versus with "bootstrapped estimates", I assume that "bootstrapped estimates" means the average (arithmetic mean) of the bootstrap distribution. Usually these should be quite close to the point estimates from the original model, particularly with a linear model such as yours. If they're discrepant, then something weird is going on that probably warrants seeking help from a statistician.

And an additional question for you: I see in your model specification that you've dropped the intercepts, so for each trait category, you're modeling the slope of the relationship between the log response ratio and the log of the temperature differential. By dropping the intercept, you are assuming that the magnitude of the effect size is multiplicatively related to the magnitude of the temperature differential (i.e., if you double the log-temperature difference, you should expect to get twice as large a log response ratio). By interacting the slope with trait category, you're allowing this multiplicative relationship to differ for each trait category. But then you're also including random effects in the model, which are assumed to have a constant variance on the scale of the log response ratio and across trait categories. Let's ignore the hierarchical structure for the moment and just think about one effect per study. For a given trait category, the model would be

LRR_i = beta * log( temp diff )_i + v_i + e_i

where e_i is the sampling error with known variance Var(e_i) = V_i and v_i is a random effect with variance Var(v_i) = tau^2. Note that this model assumes that the degree of heterogeneity is constant across temperature differentials, so the degree of heterogeneity in a set of studies that all looked at very small temperature differentials is the same as the degree of heterogeneity in a set of studies that all examined very large temperature differentials. Does that make theoretical sense in your scientific context? (This is an honest question--I don't know anything about your research area!)

Alternatively, I wonder whether it might be plausible to assume that the degree of heterogeneity is *also* multiplicatively related to the magnitude of the temperature differential. Under that assumption, you would divide the effect sizes and their standard errors by the log of the temperature differential, so the new model for a given category would become

[ LRR_i / log( temp diff )_i ] = beta + v_i + e_i,

where now Var(e_i) = V_i / [log( temp diff )_i]^2 and Var(v_i) = tau^2. Here, the variance parameter tau^2 represents heterogeneity in the response *per unit change* in the log temperature differential. This implies that studies with larger log temperature differentials would have proportionately larger degrees of heterogeneity in their responses. To take into account multiple trait categories, you could fit a model that has indicators for each category (dropping the intercept) but you would no longer necessarily need to include th interaction with lnSR.

James

On Sun, Sep 12, 2021 at 8:42 PM Bethan Lang <bethan.lang using my.jcu.edu.au<mailto:bethan.lang using my.jcu.edu.au>> wrote:
Hi there,

I am working on a meta-analyses looking at the effect of ocean warming on various traits of marine animals. My model is as follows:

Traitcat_model <- rma.mv<https://aus01.safelinks.protection.outlook.com/?url=http%3A%2F%2Frma.mv%2F&data=04%7C01%7Cbethan.lang%40my.jcu.edu.au%7C71cbd862a4f045a7780708d976ed3d34%7C2eba4cf8af764db3bcaf81b5592535ef%7C0%7C0%7C637671584024800164%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C2000&sdata=NgG7PipW0ZRmnmPNGHCLkZfkjyS%2FZEChmp3K44DMYyY%3D&reserved=0>(yi = Ln_response_corrected, V = Variance, mods = ~LnSR:Trait_cat - 1, test="t", random = list(~1|Study_number/Response_id/Effect_size_id), method = "REML", data = Traitcatdata)

Ln_response_corrected is the lnRR, i.e. the natural logarithm of the difference between the experimental (high temperature) and control mean.
Trait_cat is the measure e.g. survival, metabolic rate, growth. I have added in the an interaction with LnSR (the natural logarithm of the difference between the control and experimental temperature). I included this to account for the fact that some studies might test say a control and +10C, while others may only test at +2C. The effect size id is the number of the observation, which is nested within response_id. All observations of one measure e.g. metabolic rate at the different experimental temperatures tested in a study will have the same response Id. This is nested within the study number, which is the number assigned to each paper included.

Firstly, when I test the normality and heteroskedasticity of the model as a whole as well as the response variable, there is a high level on non-normality and heteroskedasticity (see attached screenshots for the diagnostic plots of the model). I am not sure if this means that the model output cannot be trusted? Or if this level of non-normality and heteroskedasticity is ok?

If not, I was thinking about bootstrapping the confidence intervals to account for the non-normality. However, I cannot seem to figure out the code to do this. This is the code that I tried, at the moment I am getting the error: <text>:1:6: unexpected symbol 1: .arg show_col_types ^
But I was getting a different error when I tried to run the code yesterday, I can’t remember what this one was.

library(boot)
boot.func <- function(dat, indices) {
    Traitcat_model2 <- try(suppressWarnings(rma.mv<https://aus01.safelinks.protection.outlook.com/?url=http%3A%2F%2Frma.mv%2F&data=04%7C01%7Cbethan.lang%40my.jcu.edu.au%7C71cbd862a4f045a7780708d976ed3d34%7C2eba4cf8af764db3bcaf81b5592535ef%7C0%7C0%7C637671584024810157%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C2000&sdata=COVetfkKSkWV%2F4S6RSNO0PIRQ%2BsWhrqSL0Y2XQKl22w%3D&reserved=0>(yi=Ln_response_corrected,V=Variance, data=Traitcatdata, mods=~LnSR:Trait_cat - 1, test="t", random = list(~1|Study_number/Response_id/Effect_size_id), method = "REML", subset=indices)))
    if (inherits(Traitcat_model2, "try-error")) {
    NA
  } else {
    c(coef(Traitcat_model2), vcov(Traitcat_model2), Traitcat_model2$sigma2)
  }
}
res.boot2 <- boot(Traitcatdata, boot.func, R=5000)
res.boot2

Please let me know how I need to adjust this code to get the bootstrapped CI’s for each Trait_cat group.

Also, would it be ok to use bootstrapped CI’s along with the estimates from the original model, or should I use bootstrapped estimates too?

Secondly, I am trying to figure out how to do an Orchard plot, which I know you cannot create from a model with two multiple moderators. I was wondering whether there is any way to back calculate from the model to get data points that have been adjusted for LnSR, and then use them for the Orchard plots?

Please let me know if you need any further details in order to help me with these queries.

Best wishes,

Bethan

________________________________________
Bethan Lang
PhD Candidate
ARC Centre of Excellence for Coral Reef Studies
James Cook University
Townsville, QLD, Australia
4811
[signature_1249144060]

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