# Can I use partial eta and/or partial omega squared for effect size estimation of each independent variable in linear mixed model regression?

I need to estimate effect size estimation of each independent variable in linear mixed model regression.

Here is a code:

library(lme4)
library(MASS)
library(sjstats)
data(oats)
names(oats) = c('block', 'variety', 'nitrogen', 'yield')
oats$$mainplot = oats$$variety
oats$$subplot = oats$$nitrogen

m1.lme4 = lmer(yield ~ variety + nitrogen + (1|block/mainplot),
data = oats)
summary(m1.lme4)

eta_sq(m1.lme4, partial = TRUE, n = 10000)
omega_sq(m1.lme4, partial = TRUE, n = 10000)


output:

Linear mixed model fit by REML ['lmerMod']
Formula: yield ~ variety + nitrogen + (1 | block/mainplot)
Data: oats

REML criterion at convergence: 568.1

Scaled residuals:
Min       1Q   Median       3Q      Max
-1.84135 -0.66280 -0.06694  0.63822  1.66067

Random effects:
Groups         Name        Variance Std.Dev.
mainplot:block (Intercept) 109.7    10.47
block          (Intercept) 214.5    14.65
Residual                   162.6    12.75
Number of obs: 72, groups:  mainplot:block, 18; block, 6

Fixed effects:
Estimate Std. Error t value
(Intercept)         79.917      8.220   9.722
varietyMarvellous    5.292      7.079   0.748
varietyVictory      -6.875      7.079  -0.971
nitrogen0.2cwt      19.500      4.250   4.588
nitrogen0.4cwt      34.833      4.250   8.196
nitrogen0.6cwt      44.000      4.250  10.353

Correlation of Fixed Effects:
(Intr) vrtyMr vrtyVc ntr0.2 ntr0.4
vartyMrvlls -0.431
varityVctry -0.431  0.500
ntrgn0.2cwt -0.259  0.000  0.000
ntrgn0.4cwt -0.259  0.000  0.000  0.500
ntrgn0.6cwt -0.259  0.000  0.000  0.500  0.500
>
> eta_sq(m1.lme4, partial = TRUE, n = 10000)
term partial.etasq
1  variety         0.052
2 nitrogen         0.695
> omega_sq(m1.lme4, partial = TRUE, n = 10000)
term partial.omegasq
1  variety           0.026
2 nitrogen           0.688


It gives me partial eta squired and partial omega squared, but:

1. This web page says that you can use eta squared in regression. Is it valid to use partial eta and/or omega squared for effect size estimation of each independent variable in linear mixed model regression?
2. If answer to question 1 is "yes": The documentation from sjstats package talks only about use of partial eta and/or partial omega squared for ANOVA but not about other models. But if I give it mixed model it gives me result. Is that result valid?

Unless the sjstats eta_sq / omega_sq take into account the random effects (or more particularly, the variance of the random effects), I would say that is method may not be valid.