Effective degrees of freedom and its application to conditional AIC for linear mixed-effects models with correlated error structures

The effective degrees of freedom is a useful concept for describing model complexity. Recently the number of effective degrees of freedom has been shown to relate to the concept of conditional Akaike information (cAI) in the mixed effects models. This relationship was made explicit under linear mixed-effects models with i.i.d. errors, and later also extended to the generalized linear and the proportional hazards mixed models. We show that under linear mixed-effects models with correlated errors, the number of effective degrees of freedom is asymptotically equal to the trace of the usual 'hat' matrix plus the number of parameters in the error covariance matrix. Using it one can define a crude version of the conditional AIC (cAIC), which is known to be inaccurate due to the estimation of unknown variance parameters. We compare this crude version to several corrected versions of cAIC for linear mixed models with correlated errors, including one that is asymptotically unbiased counting for the unknown parameters, but one which is also difficult to compute without specific programming for each case of the error correlation structure.