Exactly how the random effects are implemented is best seen by example. Consider the model
term `s(x,z,bs="re")`

. This will result in the model matrix component corresponding to `~x:z-1`

being added to the model matrix for the whole model. The coefficients associated with the model matrix
component are assumed i.i.d. normal, with unknown variance (to be estimated). This assumption is
equivalent to an identity penalty matrix (i.e. a ridge penalty) on the coefficients. Because such a
penalty is full rank, random effects terms do not require centering constraints.

If the nature of the random effect specification is not clear, consider a couple more examples:
`s(x,bs="re")`

results in `model.matrix(~x-1)`

being appended to the overall model matrix,
while `s(x,v,w,bs="re")`

would result in `model.matrix(~x:v:w-1)`

being appended to the model
matrix. In both cases the corresponding model coefficients are assumed i.i.d. normal, and are hence
subject to ridge penalties.

If the random effect precision matrix is of the form \(\sum_j \lambda_j S_j\) for known matrices \(S_j\) and unknown parameters \(\lambda_j\), then a list containing the \(S_j\) can be supplied in the `xt`

argument of
`s`

. In this case an array `rank`

should also be supplied in `xt`

giving the ranks of the \(S_j\) matrices. See simple example below.

Note that smooth `id`

s are not supported for random effect terms. Unlike most smooth terms, side
conditions are never applied to random effect terms in the event of nesting (since they are identifiable
without side conditions).

Random effects implemented in this way do not exploit the sparse structure of many random effects, and
may therefore be relatively inefficient for models with large numbers of random effects, when `gamm4`

or `gamm`

may be better alternatives. Note also that `gam`

will not support
models with more coefficients than data.

The situation in which factor variable random effects intentionally have unobserved levels requires special handling.
You should set `drop.unused.levels=FALSE`

in the model fitting function, `gam`

, `bam`

or `gamm`

, having first ensured that any fixed effect factors do not contain unobserved levels.

The implementation is designed so that supplying random effect factor levels to `predict.gam`

that were not levels of
the factor when fitting, will result in the corresponding random effect (or interactions involving it) being set to zero (with zero standard error) for prediction. See `random.effects`

for an example. This is achieved by the `Predict.matrix`

method zeroing any rows of the prediction matrix involving factors that are `NA`

. `predict.gam`

will set any factor observation to `NA`

if it is a level not present in the fit data.