R-squared

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)

Deviance

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).

AIC

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.

BIC

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 DF

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.

Value/DF

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