[R-sig-ME] Number of random effects estimated with different lmer specifications

[Thierry Onkelinx]

Dear Sointu,

You only need to count the number of parameters contributing to the linear
predictor. Hence not the parameters of the variance-covariance matrix.
1 random intercept + 2 random slopes = 3 parameters per ID
Times 10 ID yields 30 parameters
Your mod1 actually requires 2 times (1 random intercept + 1 random slope)
times 10 ID = 40 random effect parameters. IMHO lme4 should warn for this
too.
ranef(mod1)$id

Best regards,

ir. Thierry Onkelinx
Statisticus / Statistician

Vlaamse Overheid / Government of Flanders
INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR NATURE AND
FOREST
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Op wo 8 mei 2024 om 10:51 schreef Leikas, Sointu S <
sointu.leikas using helsinki.fi>:

> Hello!
>
> I'm confused about the error the lmer function sometimes gives, "Error:
> number of observations (=n) <= number of random effects (=n) for term (x|
> id); the random-effects parameters and the residual variance (or scale
> parameter) are probably unidentifiable"
> Regarding this error, I'm confused about how the "number of random
> effects" is defined. My naïve understanding was that if you have, say, 10
> clusters, then a random intercept model estimates 10 random effects (i.e.,
> a random intercept for all clusters). If you have one random intercept and
> one random slope, the model would estimate 10+10+1=21 random effects
> (random intercept and slope for all participants plus the correlation
> between them). With 2 random intercepts and 2 slopes it would be
> 10+10+10+10+2 (or 3, depending on the exact random effects syntax)=42 (or
> 43).
>
> However, experimenting a little showed me that the correlations between
> random effects do not seem to be included into the number of random
> effects, so I'll forget them for now.
>
> My main puzzlement comes from this: I generated a simple dataset of 10
> clusters ("id") and 3 observations per cluster ("time"), as well as a level
> 1 predictor ("x") and ran the following models:
>
>
> mod1<-lmer(y ~ x + time + (time|id) + (x|id), data=d)
>
> This model converges (though the correlations between random effects are 1
> and -1, which is probably just due to my sloppy data generation process)
> with no errors or warnings.
>
> Then I ran this model:
>
>
> mod2<-lmer(y ~ x + time + (time + x |id), data=d)
>
>
>
> This model won't converge and I get the error:
>
> Error: number of observations (=30) <= number of random effects (=30) for
> term (time + x | id); the random-effects parameters and the residual
> variance (or scale parameter) are probably unidentifiable
>
> ********************
> I thought that the first model would estimate 40 random effects (two
> intercepts and two slopes for each of the 10 clusters), and the second
> model would estimate 30 (1 intercept and 2 slopes for each cluster). This
> seems to be correct regarding the second model, but why is the first model
> seemingly estimating less random effects (less than 40, and apparently also
> less than 30)?
>
> I do apologize if this is very basic; I don't have math or proper stats
> background, just applied stats. I did read the manual, and several online
> discussions regarding this error before posting. I didn't find the answer
> in the manual (this may well be due to my own incompetence), and the online
> discussions (e.g.
> https://stats.stackexchange.com/questions/193678/number-of-random-effects-is-not-correct-in-lmer-model)
> seem to support my initial intuition, which however is clearly wrong. What
> am I missing?
>
> Thank you in advance if someone can help!
>
> -Sointu
>
> *********************
> My code for the above:
>
> set.seed(12345)
> id1<-c(1:10)
> id<-rep(id1, each=3)
> t<-c(1:3)
> time<-rep(t, times=10)
> x<-rnorm(30, 3,1)
> err<-rnorm(10,0,1)
> err2<-rep(err, each=3)
> y<-3+0.2*x+0.3*time+rnorm(30)+err2
> d<-data.frame(id, time, x, y)
> mod1<-lmer(y ~ x + time + (time|id) + (x|id),data=d)
> summary(mod1)
> mod2<-lmer(y ~ x + time + (time + x|id),data=d)
>
>
>
>
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