[R-meta] extracting variances

[Gram, Gil (IITA)]

Dear all,

I have a question regarding extracting the variances from my model.

Say I want to analyse the yields (tonnes per hectare) of 4 treatments (control, OR, MR, ORMR) running across different sites and times. A simplified version of my model would then be:

dat = escalc(measure="MN", mi=yield, sdi=sdYield, ni=nRep, data=temp)
dat$yi = sqrt(dat$yi) # sqrt transformation
dat$vi = dat$vi/(4*dat$yi) # variance adjustment to sqrt transformation

mod = rma.mv(yi, as.matrix(vi), method = 'REML', struct="HCS", sparse=TRUE, data=dat,
                               mods = ~ rateORone + kgMN + I(rateORone^2) + I(kgMN^2)
                               + rateORone:kgMN + I(rateORone^2):I(kgMN^2) + […],
                               random = list(~1|ref, ~1|idRow, ~ treatment | idSite, ~ treatment | idSite.time))


I’m interested in the yield variance responses over time, of OR and ORMR versus control. So I extract the variance-covariance matrix H = mod$H:

          Control        MR        OR      ORMR
Control 0.1098190 0.1179042 0.1055471 0.1216751
MR      0.1179042 0.1360579 0.1174815 0.1354332
OR      0.1055471 0.1174815 0.1090329 0.1212389
ORMR    0.1216751 0.1354332 0.1212389 0.1449001

The variance responses I then calculate with e.g. responseOR = varianceOR + varianceControl - 2*covar(OR, Control):

resOR
= (H['OR','OR'] + H['Control','Control'] - 2*H['Control','OR'])
= 0.1090329 + 0.1098190 - 2* 0.1055471
~ 0.00775

resORMR
~ 0.0114


I understand therefore that the variance responses over time for treatments OR and ORMR are about 0.77% and 1.1%. These values are extremely small, hence my questions:

- Am I correct that this was the correct way to estimate the yield variability (responses) over time?

If this is all correct, then this means that there is hardly any variability associated with these components. And one could start wondering what the point is of even looking at this. I tried looking at the values of the other components, and see whether these are larger.

- Keeping in mind the original data was sqrt transformed, can these values still be considered as variances? or as standard deviations instead?
- If this makes up so little variance, then where is the variance coming from? How much variability is associated with the error term? Or the other components. Are these then magnitudes larger? How do I check if the sum of all variance components equals 100% with the model output below?


I hope my questions are clear…

Thanks a lot in advance for your help,

Gil


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Multivariate Meta-Analysis Model (k = 1161; method: REML)

Variance Components:

            estim    sqrt  nlvls  fixed  factor
sigma^2.1  0.0604  0.2458     40     no     ref
sigma^2.2  0.0285  0.1688   1161     no   idRow

outer factor: idSite    (nlvls = 71)
inner factor: treatment (nlvls = 4)

            estim    sqrt  k.lvl  fixed    level
tau^2.1    0.1285  0.3584    275     no  Control
tau^2.2    0.0952  0.3086    374     no       MR
tau^2.3    0.1217  0.3488    234     no       OR
tau^2.4    0.0711  0.2666    278     no     ORMR
rho        0.7172                    no

outer factor: idSite.time (nlvls = 271)
inner factor: treatment   (nlvls = 4)

              estim    sqrt  k.lvl  fixed    level
gamma^2.1    0.1098  0.3314    275     no  Control
gamma^2.2    0.1361  0.3689    374     no       MR
gamma^2.3    0.1090  0.3302    234     no       OR
gamma^2.4    0.1449  0.3807    278     no     ORMR
phi          0.9646                    no

Test for Residual Heterogeneity:
QE(df = 1151) = 501266.0717, p-val < .0001

Test of Moderators (coefficients 2:10):
QM(df = 9) = 441.0373, p-val < .0001

Model Results:

                          estimate      se     zval    pval    ci.lb    ci.ub
intrcpt                     1.2855  0.0691  18.6010  <.0001   1.1501   1.4210  ***
rateORone                   0.0059  0.0007   8.5224  <.0001   0.0045   0.0072  ***
kgMN                        0.0096  0.0009  10.5108  <.0001   0.0078   0.0114  ***
I(rateORone^2)             -0.0000  0.0000  -5.2103  <.0001  -0.0000  -0.0000  ***
I(kgMN^2)                  -0.0000  0.0000  -6.6753  <.0001  -0.0000  -0.0000  ***
[…]
rateORone:kgMN             -0.0000  0.0000  -3.7035  0.0002  -0.0000  -0.0000  ***
I(rateORone^2):I(kgMN^2)    0.0000  0.0000   2.5775  0.0100   0.0000   0.0000   **

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