GAMLj version ≥ 2.4.4
In this page some details about the GAMLj GMixed (Generalized mixed model) implementation are given. When the code is showed, it is meant to be R code underlying the GAMLj module.
R-squared corresponds pseudo R-squared implemented here and described in (Lefcheck 2016) and in (Johnson 2014)
Two R-squares values are reported:
Conditional: It is the estimated proportion of reduced error of the model as compared with a null-model without fixed and random effects. It corresponds to the goodness of fit of the model due to fixed and random effects. In other words, conditional \(R^2\) indicates the variance explained by the fixed and random effects as a proportion of the sum of all the variance components (Johnson 2014)
Marginal: It is the estimated proportion of reduced error of the model as compared with a null-model without fixed effects. It corresponds to the goodness of fit of the model due to the fixed effects. In other words, marginal \(R^2\) indicates the variance explained by the fixed effects as a proportion of the sum of all the variance components (Johnson 2014)
The implementation follows exactly the indication of R package
lme4 (Douglas Bates 2020). In generalized
mixed models, deviance can be defined in different ways. One dichotomy
is the reference to a saturated model. The absolute deviance of
the model is simply the model deviance, the relative deviance
is the deviance resulting from subtracting the deviance of a saturated
model from the model deviance. A crossing factor is whether the deviance
is conditional to the random effects (all effects affect the deviance)
or it is not conditional to them (only the fixed effects affect the
deviance). GAMLj computes two cells of
the potential 2x2 table of possible deviance definitions:
-2*LogLikel.,
Unconditioal absolute deviance: Computed by
-2*logLik(model)Deviance, Conditional relative deviance :
Computed by stats::deviance(model)Furthermore, GAMLj outputs also the
logLikelihood (logLik), which is simply the absolute log
likelihood computed as stats::logLik(model).
Aikake Information Criterion: it can be used for model comparison. A
model has a better fit than another when its AIC is smaller than the
other’s. It is implemented by simply estracting it from the R
glmer estimated model:
stats::extractAIC(model). Details in (Bates et al.
2015; Douglas Bates
2020). The AIC is computed based on the uncoditional absolute
deviance.
Bayesian Information Criterion: it can be used for model comparisons.
A model has a better fit than another when its BIC is smaller. It is
implemented by simply estracting it from the R glm
estimated model: stats::BIC(model). Details in (Bates et al.
2015; Douglas Bates
2020). The BIC is computed based on the uncoditional absolute
deviance.
Residual variance degrees of freedom: \(DF_{M_s} -DF_{M_e}\), where \(M_s\) is the saturated model and \(M_e\) is the estimated model.
a measure of dispersion for Poisson-like model and binomial models.
It is given by the Pearson \(\chi^2\)
statistics divided by the residual degrees of freedom. It is expected to
be 1, thus larger number (usually > 3) indicate overdispersion.
Values smaller than 1 (usually < .333) indicate underdispersion. It
is useful to decide whether the Poisson model is presenting
overdispersion, in which case Quasipoisson or
negative binomial models may be preferred.
It is implemented as follows:
value <- sum(residuals(model, type = "pearson")^2)
result <- value/model$df.residual
Post-hoc tests are model-based: Each comparison comparares two groups means using the standard error derived from the model error. This means that the comparisons are consisistent to the model they belong to and that different models may produce different results for the same set of comparisons.
Post-hocs tests are performed as implemented in the emmeans
package. For all GZLM models estimated with glm
function (all but the multinomial model) post hoc are implemented as
follows (for any given model and term selected
by the user) :
referenceGrid <- emmeans::emmeans(model, formula)
none <- summary(pairs(referenceGrid, adjust='none'))
tukey <- summary(pairs(referenceGrid, adjust='tukey'))
scheffe <- summary(pairs(referenceGrid, adjust='scheffe'))
bonferroni <- summary(pairs(referenceGrid, adjust='bonferroni'))
holm <- summary(pairs(referenceGrid, adjust='holm'))
Return to main help page: Generalized Mixed Models module
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