Ich denke, ich kann Ihre Fragen zumindest hinsichtlich der nicht angepassten Wiederholbarkeitsschätzungen, dh der klassischen klasseninternen Korrelationen (ICCs), beantworten. Was die "angepassten" Wiederholbarkeitsschätzungen anbelangt, habe ich das von Ihnen verknüpfte Papier überflogen und nicht wirklich gesehen, wo sich die Formel, die Sie anwenden, im Papier befindet? Basierend auf dem mathematischen Ausdruck scheint es sich um die Wiederholbarkeit von Mittelwerten zu handeln (und nicht um Einzelwerte). Aber es ist nicht klar, dass dies ein kritischer Teil Ihrer Frage ist, deshalb werde ich es ignorieren.
(1.) Sind die obigen Berechnungen zum Erhalten der Punktschätzung der Wiederholbarkeit eines Effekts sinnvoll?
Ja, der von Ihnen vorgeschlagene Ausdruck ist sinnvoll, aber eine geringfügige Änderung Ihrer vorgeschlagenen Formel ist erforderlich. Im Folgenden zeige ich, wie Sie Ihren vorgeschlagenen Wiederholungskoeffizienten ableiten können. Ich hoffe, dass dies sowohl die begriffliche Bedeutung des Koeffizienten verdeutlicht als auch zeigt, warum es wünschenswert wäre, ihn geringfügig zu ändern.
Zunächst nehmen wir den Wiederholungskoeffizienten in Ihrem ersten Fall und klären, was er bedeutet und woher er kommt. Das Verständnis dieses Sachverhalts hilft uns, den komplizierteren zweiten Fall zu verstehen.
Nur zufällige Abschnitte
ij
yij=β0+u0j+eij,
u0jσ2u0eijσ2e
xy
corr=cov(x,y)var(x)var(y)−−−−−−−−−−√.
The expression for ICC / repeatability coefficient then comes from letting the two random variables x and y be two observations drawn from the same j group,
ICC=cov(β0+u0j+ei1j,β0+u0j+ei2j)var(β0+u0j+ei1j)var(β0+u0j+ei2j)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√,
and if you simplify this using the definitions given above and the properties of variances/covariances (a process which I will not show here, unless you or others would prefer that I did), you end up with
ICC=σ2u0σ2u0+σ2e.
What this means is that the ICC or "unadjusted repeatability coefficient" in this case has a simple interpretation as the expected correlation between a pair observations from the same cluster (net of the fixed effects, which in this case is just the grand mean). The fact that the ICC is also interpretable as a
proportion of variance in this case is coincidental; that interpretation is not true in general for more complicated ICCs. The interpretation as some sort of correlation is what is primary.
Random intercepts and random slopes
Now for the second case, we have to first clarify what precisely is meant by "the reliability of effects (i.e. sum contrast effect of a variable with 2 levels)" -- your words.
First we lay out the model. The mixed model for the ith response in the jth group under the kth level of a contrast-coded predictor x is
yijk=β0+β1xk+u0j+u1jxk+eijk,
where the random intercepts have variance
σ2u0, the random slopes have variance
σ2u1, the random intercepts and slopes have covariance
σu01, and the residuals
eij have variance
σ2e.
So what is the "repeatability of an effect" under this model? I think a good candidate definition is that it is the expected correlation between two pairs of difference scores computed within the same j cluster, but across different pairs of observations i.
So the pair of difference scores in question would be (remember that we assumed x is contrast coded so that |x1|=|x2|=x):
yi1jk2−yi1jk1=(β0−β0)+β1(xk2−xk1)+(u0j−u0j)+u1j(xk2−xk1)+(ei1jk2−ei1jk1)=2xβ1+2xu1j+ei1jk2−ei1jk1
and
yi2jk2−yi2jk1=2xβ1+2xu1j+ei2jk2−ei2jk1.
Plugging these into the correlation formula gives us
ICC=cov(2xβ1+2xu1j+ei1jk2−ei1jk1,2xβ1+2xu1j+ei2jk2−ei2jk1)var(2xβ1+2xu1j+ei1jk2−ei1jk1)var(2xβ1+2xu1j+ei2jk2−ei2jk1)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√,
which simplifies down to
ICC=2x2σ2u12x2σ2u1+σ2e.
Notice that the ICC is technically a function of
x! However, in this case
x can only take 2 possible values, and the ICC is identical at both of these values.
As you can see, this is very similar to the repeatability coefficient that you proposed in your question, the only difference is that the random slope variance must be appropriately scaled if the expression is to be interpreted as an ICC or "unadjusted repeatability coefficient." The expression that you wrote works in the special case where the x predictor is coded ±12√, but not in general.
(2.) When I have multiple variables whose repeatability I want to estimate, adding them all to the same fit (e.g. lmer(dv~(iv1+iv2|unit)+iv1+iv2
) seems to yield higher repeatability estimates than creating a separate model for each effect. This makes sense computationally to me, as inclusion of multiple effects will tend to decrease the residual variance, but I'm not positive that the resulting repeatability estimates are valid. Are they?
I believe that working through a similar derivation as presented above for a model with multiple predictors with their own random slopes would show that the repeatability coefficient above would still be valid, except for the added complication that the difference scores we are conceptually interested in would now have a slightly different definition: namely, we are interested in the expected correlation of the differences between adjusted means after controlling for the other predictors in the model.
If the other predictors are orthogonal to the predictor of interest (as in, e.g., a balanced experiment), I would think the ICC / repeatability coefficient elaborated above should work without any modification. If they are not orthogonal then you would need to modify the formula to take account of this, which could get complicated, but hopefully my answer has given some hints about what that might look like.