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As a step towards supporting random effects in this package, the first simplest step could be to support a spatial covariance structure which provides a patient level random effects induced covariance structure.
The syntax could be sp_ref(cov1, cov2, ... | group / subject) where cov1 is the first covariate to use (potentially a factor which leads to multiple design matrix columns), etc.
This will be possible because we can marginalize over the random coefficients in closed form, such that the induced covariance matrix for the $i$th patient is a quadratic form like $Z_i \Sigma_r Z_i^\top + \sigma^2 I$ when the residuals are homoskedastic with variance $\sigma^2$ and $Z_i$ is the design matrix induced by the specified covariates and $\Sigma_r$ is the covariance matrix of the random coefficients. Here $\sigma^2$ and $\Sigma_r$ can be estimated as variance parameters from the data. For the beginning we could just assume an unstructured covariance model for $\Sigma_r$
As long as we only have subject level random effects, then the overall covariance matrix is still blockdiagonal with these entries.
The text was updated successfully, but these errors were encountered:
As a step towards supporting random effects in this package, the first simplest step could be to support a spatial covariance structure which provides a patient level random effects induced covariance structure.
The syntax could be
sp_ref(cov1, cov2, ... | group / subject)
wherecov1
is the first covariate to use (potentially a factor which leads to multiple design matrix columns), etc.This will be possible because we can marginalize over the random coefficients in closed form, such that the induced covariance matrix for the $i$th patient is a quadratic form like$Z_i \Sigma_r Z_i^\top + \sigma^2 I$ when the residuals are homoskedastic with variance $\sigma^2$ and $Z_i$ is the design matrix induced by the specified covariates and $\Sigma_r$ is the covariance matrix of the random coefficients. Here $\sigma^2$ and $\Sigma_r$ can be estimated as variance parameters from the data. For the beginning we could just assume an unstructured covariance model for $\Sigma_r$
As long as we only have subject level random effects, then the overall covariance matrix is still blockdiagonal with these entries.
The text was updated successfully, but these errors were encountered: