I have read many times that random effects (BLUPs/conditional modes for, say, subjects) are not parameters of a linear mixed effects model but instead can be derived from the estimated variance/covariance parameters. E.g. Reinhold Kliegl et al. (2011) state:

Random effects are subjects’ deviations from the grand mean RT and subjects’ deviations from the fixed-effect parameters. They are assumed to be independently and normally distributed with a mean of 0. It is important to recognize that these random effects are not parameters of the LMM – only their variances and covariances are. [...] LMM parameters in combination with subjects’ data can be used to generate “predictions” (conditional modes) of random effects for each subject.

Can someone give an intuitive explanation how the (co)variance parameters of the random effects can be estimated without actually using/estimating the random effects?

I have read many times that random effects (BLUPs/conditional modes for, say, subjects) are not parameters of a linear mixed effects model but instead can be derived from the estimated variance/covariance parameters. E.g. Reinhold Kliegl et al. (2011) state:

Random effects are subjects’ deviations from the grand mean RT and subjects’ deviations from the fixed-effect parameters. They are assumed to be independently and normally distributed with a mean of 0. It is important to recognize that these random effects are not parameters of the LMM – only their variances and covariances are. [...] LMM parameters in combination with subjects’ data can be used to generate “predictions” (conditional modes) of random effects for each subject.

Can someone give an intuitive explanation how the (co)variance parameters of the random effects can be estimated without actually using/estimating the random effects?

I have read many times that random effects (BLUPs/conditional modes for, say, subjects) are not parameters of a linear mixed effects model but instead can be derived from the estimated variance/covariance parameters. E.g. Reinhold Kliegl et al. (2011) state

Random effects are subjects’ deviations from the grand mean RT and subjects’ deviations from the fixed-effect parameters. They are assumed to be independently and normally distributed with a mean of 0. It is important to recognize that these random effects are not parameters of the LMM – only their variances and covariances are. [...] LMM parameters in combination with subjects’ data can be used to generate “predictions” (conditional modes) of random effects for each subject.

Can someone give an intuitive explanation how the (co)variance parameters of the random effects can be estimated without actually using/estimating the random effects?

I have read many times that random effects (BLUPs/conditional modes for, say, subjects) are not parameters of a linear mixed effects model but instead can be derived from the estimated variance/covariance parameters. E.g. Reinhold Kliegl et al. (2011) state:

Random effects are subjects’ deviations from the grand mean RT and subjects’ deviations from the fixed-effect parameters. They are assumed to be independently and normally distributed with a mean of 0. It is important to recognize that these random effects are not parameters of the LMM – only their variances and covariances are. [...] LMM parameters in combination with subjects’ data can be used to generate “predictions” (conditional modes) of random effects for each subject.

Can someone give an intuitive explanation how the (co)variance parameters of the random effects can be estimated without actually using/estimating the random effects?

I have read many times that random effects (BLUPs/conditional modes for, say, subjects) are not parameters of a linear mixed effects model but instead can be derived from the estimated variance/covariance parameters.

Can someone give an intuitive explanation how the variance/covariance parameters of the random effects can be estimated without actually using the random effects?

I have read many times that random effects (BLUPs/conditional modes for, say, subjects) are not parameters of a linear mixed effects model but instead can be derived from the estimated variance/covariance parameters. E.g. Reinhold Kliegl et al. (2011) state

Random effects are subjects’ deviations from the grand mean RT and subjects’ deviations from the fixed-effect parameters. They are assumed to be independently and normally distributed with a mean of 0. It is important to recognize that these random effects are not parameters of the LMM – only their variances and covariances are. [...] LMM parameters in combination with subjects’ data can be used to generate “predictions” (conditional modes) of random effects for each subject.

Can someone give an intuitive explanation how the (co)variance parameters of the random effects can be estimated without actually using/estimating the random effects?