# [R-meta] Variance-covariance is not positive definite.

Viechtbauer, Wolfgang (SP) wo||g@ng@v|echtb@uer @end|ng |rom m@@@tr|chtun|ver@|ty@n|
Thu Sep 26 14:38:48 CEST 2019

```Hi Ana,

No, this code is not from me, but I know the nearPD() function. But that's just slapping a bandaid on, not really fixing the issue. In principle, for the scenario you are describing, the V matrix should automatically be positive definite. The fact that it isn't suggest that there might be a more fundamental problem. If so, simply adjust the V matrix to the 'nearest' PD matrix isn't a proper fix for that.

Best,
Wolfgang

-----Original Message-----
From: Ana Benítez [mailto:abenitez81 using gmail.com]
Sent: Thursday, 26 September, 2019 13:33
To: Viechtbauer, Wolfgang (SP)
Cc: r-sig-meta-analysis using r-project.org
Subject: Re: [R-meta] Variance-covariance is not positive definite.

Dear Wolfgang,

Apologies for sending the text in rich-text format. Here I copy-paste the original message and I additionally attach a piece of code to solve the issue. I got the code from Alfredo Sánchez-Tojar but he claims he got it from someone else, maybe even you Wolfgang. The code is this:

# this function is to make sure that a is matrix positive-definitive
# it uses the function nearPD in the Matrix package to compute the
# nearest positive definite matrix to an approximate one
# Function reused from: https://osf.io/ayxrt/

PDfunc <- function(matrix){
require(Matrix)
if(corpcor::is.positive.definite(matrix) == T){
x <- corDiag
}else{
x <- Matrix::nearPD(matrix)
}
mat <- x\$mat
return(mat)
}

# is the matrix positive-definitive? No, run PDfunc to make it
# so. If a matrix is not positive-definite, it won't work

is.positive.definite(varcovar) # FALSE
varcovar  <-PDfunc(varcovar)
is.positive.definite(varcovar) # TRUE

And this is my original email in case someone is interested:

Dear Wolfgang and meta-analysis list subscribers,

I am running a muti-level meta-regression in which I have several comparisons using a common control for some of the studies. My effect size is the log response ratio (RR) and I have as random effects Study, Effect size ID and Species nested in Family and Order. I am also rerunning the same analysis using a phylogenetic correlation matrix instead of nesting species in higher taxonomic taxa (see end of the message).

To calculate the var-covar matrix I have used Wolfgang’s code:

calc.v <- function(x) {
v <- matrix(x\$sd_m[1]^2 / (x\$N_m[1] * x\$Mean_m[1]^2), nrow=nrow(x), ncol=nrow(x))
diag(v) <- x\$var
v
}
#make sure we order the database by the variable we are splitting on (CommonControl)
mamdata <- mamdata[order(mamdata\$CommonControl),]

Vmam<- bldiag(lapply(split(mamdata, mamdata\$CommonControl), calc.v))

Yet, when I run the meta-analysis, and although I get reasonable results, I get the warning: 'V' appears to be not positive definite. See results here:

Multivariate Meta-Analysis Model (k = 705; method: REML)

logLik   Deviance        AIC        BIC       AICc
116.6641  -233.3282  -219.3282  -187.4407  -219.1671

Variance Components:

estim    sqrt  nlvls  fixed                        factor
sigma^2.1  0.0149  0.1223    131     no                     Reference
sigma^2.2  0.0275  0.1657    705     no                            ID
sigma^2.3  0.0013  0.0363     13     no                         Order
sigma^2.4  0.0069  0.0831     37     no                  Order/Family
sigma^2.5  0.0242  0.1554    167     no  Order/Family/Species.nominal

Test of Moderators (coefficient(s) 2):
QM(df = 1) = 18.0854, p-val < .0001

Model Results:

estimate      se     zval    pval    ci.lb    ci.ub
intrcpt       0.1768  0.0636   2.7780  0.0055   0.0521   0.3015   **
logmass   -0.0880  0.0207  -4.2527  <.0001  -0.1286  -0.0475  ***

I have also tried computing the var-covar matrix using the script provided by Moatt et al. 2016, The effect of dietary restriction on reproduction: a meta-analytic perspective. BMC evolutionary biology, 16(1), 199, available at https://datadryad.org/stash/dataset/doi:10.5061/dryad.3fc02. I slightly modified the script to calculate the var-covar matrix for RR as:

# create square matrix matching N of ES, filled with zeros
V <- matrix(0,nrow = dim(mamdata)[1],ncol = dim(mamdata)[1])
rownames(V) <- mamdata\$ID
colnames(V) <- mamdata\$ID

# find start and end coordinates for the subsets
shared_coord <- which(mamdata\$CommonControl%in%mamdata\$CommonControl[duplicated(mamdata\$CommonControl)]==TRUE)
shared_coord

# matrix of combinations of coordinates for each experiment with shared control
combinations <- do.call("rbind", tapply(shared_coord, mamdata[shared_coord,"CommonControl"], function(x) t(combn(x,2))))
combinations

# calculate covariance values between ES values at the positions in shared_list and place them on the matrix
for (i in 1:dim(combinations)[1]){
p1 <- combinations[i,1]
p2 <- combinations[i,2]
p1_p2_cov <- mamdata\$sd_m[1]^2 / (mamdata\$N_m[1] * mamdata\$Mean_m[1]^2)
V[p1,p2] <- p1_p2_cov
V[p2,p1] <- p1_p2_cov
}

# add the diagonal - use df\$var as matrix diagonal
diag(V) <-  mamdata\$var

Using this approach the results are pretty similar but I still get the warning: 'V' appears to be not positive definite.

Multivariate Meta-Analysis Model (k = 705; method: REML)

logLik   Deviance        AIC        BIC       AICc
115.2436  -230.4872  -216.4872  -184.5997  -216.3261

Variance Components:

estim    sqrt  nlvls  fixed                        factor
sigma^2.1  0.0153  0.1238    131     no                     Reference
sigma^2.2  0.0274  0.1656    705     no                            ID
sigma^2.3  0.0014  0.0370     13     no                         Order
sigma^2.4  0.0070  0.0836     37     no                  Order/Family
sigma^2.5  0.0242  0.1554    167     no  Order/Family/Species.nominal

Test of Moderators (coefficient(s) 2):
QM(df = 1) = 18.3750, p-val < .0001

Model Results:

estimate      se     zval    pval    ci.lb    ci.ub
intrcpt       0.1799  0.0639   2.8159  0.0049   0.0547   0.3051   **
logmass   -0.0890  0.0208  -4.2866  <.0001  -0.1298  -0.0483  ***

Needless to say that when I run the phylogenetic meta-analysis I also get the warning. Results below:

Multivariate Meta-Analysis Model (k = 702; method: REML)

logLik   Deviance        AIC        BIC       AICc
115.2222  -230.4445  -218.4445  -191.1380  -218.3233

Variance Components:

estim    sqrt  nlvls  fixed           factor    R
sigma^2.1  0.0135  0.1161    131     no        Reference   no
sigma^2.2  0.0276  0.1662    702     no               ID   no
sigma^2.3  0.0274  0.1654    164     no             SPID   no
sigma^2.4  0.0090  0.0951    161     no  Species.nominal  yes

Test of Moderators (coefficient(s) 2):
QM(df = 1) = 17.6159, p-val < .0001

Model Results:

estimate      se     zval    pval    ci.lb    ci.ub
intrcpt   0.1486  0.0744   1.9979  0.0457   0.0028   0.2943    *
logmass   -0.0792  0.0189  -4.1971  <.0001  -0.1162  -0.0422  ***

My question is whether I should be worried by the warning or not and if so, how I can fix it.

Best,

Ana

El mar., 24 sept. 2019 a las 13:12, Viechtbauer, Wolfgang (SP) (<wolfgang.viechtbauer using maastrichtuniversity.nl>) escribió:
>
> Dear Ana,
>
> Please send messages to this list in plain text (no HTML / rich-text emails). As you can see below, your message includes lots of superfluous empty lines, which makes it hard to read.
>
> Based on what you posted, it is not possible to determine why the construction of the V matrix leads to a non-positive definite matrix. We would have to see the actual dataset (mamdata). The only thing I can suggest to check is that the common control group is always given by variables Mean_m, sd_m, and N_m and not the other three variables that are used for the comparison groups. Also, how was variable 'var' computed? Was it computed using escalc() or was this computed based on other software? In the latter case, the computation used may not match up with what the calc.v() function below uses for computing the covariances.
>
> Best,
> Wolfgang
```