A fitted GAM object returned by function `gam`

and of class
`"gam"`

inheriting from classes `"glm"`

and `"lm"`

. Method
functions `anova`

, `logLik`

, `influence`

, `plot`

,
`predict`

, `print`

, `residuals`

and `summary`

exist for
this class.

All compulsory elements of `"glm"`

and `"lm"`

objects are present,
but the fitting method for a GAM is different to a linear model or GLM, so
that the elements relating to the QR decomposition of the model matrix are
absent.

A `gam`

object has the following elements:

- aic
AIC of the fitted model: bear in mind that the degrees of freedom used to calculate this are the effective degrees of freedom of the model, and the likelihood is evaluated at the maximum of the penalized likelihood in most cases, not at the MLE.

- assign
Array whose elements indicate which model term (listed in

`pterms`

) each parameter relates to: applies only to non-smooth terms.- boundary
did parameters end up at boundary of parameter space?

- call
the matched call (allows

`update`

to be used with`gam`

objects, for example).- cmX
column means of the model matrix (with elements corresponding to smooths set to zero ) --- useful for componentwise CI calculation.

- coefficients
the coefficients of the fitted model. Parametric coefficients are first, followed by coefficients for each spline term in turn.

- control
the

`gam`

control list used in the fit.- converged
indicates whether or not the iterative fitting method converged.

- data
the original supplied data argument (for class

`"glm"`

compatibility). Only included if`gam`

`control`

argument element`keepData`

is set to`TRUE`

(default is`FALSE`

).- db.drho
matrix of first derivatives of model coefficients w.r.t. log smoothing parameters.

- deviance
model deviance (not penalized deviance).

- df.null
null degrees of freedom.

- df.residual
effective residual degrees of freedom of the model.

- edf
estimated degrees of freedom for each model parameter. Penalization means that many of these are less than 1.

- edf1
similar, but using alternative estimate of EDF. Useful for testing.

- edf2
if estimation is by ML or REML then an edf that accounts for smoothing parameter uncertainty can be computed, this is it.

`edf1`

is a heuristic upper bound for`edf2`

.- family
family object specifying distribution and link used.

- fitted.values
fitted model predictions of expected value for each datum.

- formula
the model formula.

- full.sp
full array of smoothing parameters multiplying penalties (excluding any contribution from

`min.sp`

argument to`gam`

). May be larger than`sp`

if some terms share smoothing parameters, and/or some smoothing parameter values were supplied in the`sp`

argument of`gam`

.- F
Degrees of freedom matrix. This may be removed at some point, and should probably not be used.

- gcv.ubre
The minimized smoothing parameter selection score: GCV, UBRE(AIC), GACV, negative log marginal likelihood or negative log restricted likelihood.

- hat
array of elements from the leading diagonal of the `hat' (or `influence') matrix. Same length as response data vector.

- iter
number of iterations of P-IRLS taken to get convergence.

- linear.predictors
fitted model prediction of link function of expected value for each datum.

- method
One of

`"GCV"`

or`"UBRE"`

,`"REML"`

,`"P-REML"`

,`"ML"`

,`"P-ML"`

,`"PQL"`

,`"lme.ML"`

or`"lme.REML"`

, depending on the fitting criterion used.- mgcv.conv
A list of convergence diagnostics relating to the

`"magic"`

parts of smoothing parameter estimation - this will not be very meaningful for pure`"outer"`

estimation of smoothing parameters. The items are:`full.rank`

, The apparent rank of the problem given the model matrix and constraints;`rank`

, The numerical rank of the problem;`fully.converged`

,`TRUE`

is multiple GCV/UBRE converged by meeting convergence criteria and`FALSE`

if method stopped with a steepest descent step failure;`hess.pos.def`

Was the hessian of the GCV/UBRE score positive definite at smoothing parameter estimation convergence?;`iter`

How many iterations were required to find the smoothing parameters?`score.calls`

, and how many times did the GCV/UBRE score have to be evaluated?;`rms.grad`

, root mean square of the gradient of the GCV/UBRE score at convergence.

- min.edf
Minimum possible degrees of freedom for whole model.

- model
model frame containing all variables needed in original model fit.

- na.action
The

`na.action`

used in fitting.- nsdf
number of parametric, non-smooth, model terms including the intercept.

- null.deviance
deviance for single parameter model.

- offset
model offset.

- optimizer
`optimizer`

argument to`gam`

, or`"magic"`

if it's a pure additive model.- outer.info
If `outer' iteration has been used to fit the model (see

`gam`

argument`optimizer`

) then this is present and contains whatever was returned by the optimization routine used (currently`nlm`

or`optim`

).- paraPen
If the

`paraPen`

argument to`gam`

was used then this provides information on the parametric penalties.`NULL`

otherwise.- pred.formula
one sided formula containing variables needed for prediction, used by

`predict.gam`

- prior.weights
prior weights on observations.

- pterms
`terms`

object for strictly parametric part of model.- R
Factor R from QR decomposition of weighted model matrix, unpivoted to be in same column order as model matrix (so need not be upper triangular).

- rank
apparent rank of fitted model.

- reml.scale
The scale (RE)ML scale parameter estimate, if (P-)(RE)ML used for smoothness estimation.

- residuals
the working residuals for the fitted model.

- rV
If present,

`rV%*%t(rV)*sig2`

gives the estimated Bayesian covariance matrix.- scale
when present, the scale (as

`sig2`

)- scale.estimated
`TRUE`

if the scale parameter was estimated,`FALSE`

otherwise.- sig2
estimated or supplied variance/scale parameter.

- smooth
list of smooth objects, containing the basis information for each term in the model formula in the order in which they appear. These smooth objects are what gets returned by the

`smooth.construct`

objects.- sp
estimated smoothing parameters for the model. These are the underlying smoothing parameters, subject to optimization. For the full set of smoothing parameters multiplying the penalties see

`full.sp`

. Divide the scale parameter by the smoothing parameters to get, variance components, but note that this is not valid for smooths that have used rescaling to improve conditioning.- terms
`terms`

object of`model`

model frame.- var.summary
A named list of summary information on the predictor variables. If a parametric variable is a matrix, then the summary is a one row matrix, containing the observed data value closest to the column median, for each matrix column. If the variable is a factor the then summary is the modal factor level, returned as a factor, with levels corresponding to those of the data. For numerics and matrix arguments of smooths, the summary is the mean, nearest observed value to median and maximum, as a numeric vector. Used by

`vis.gam`

, in particular.- Ve
frequentist estimated covariance matrix for the parameter estimators. Particularly useful for testing whether terms are zero. Not so useful for CI's as smooths are usually biased.

- Vp
estimated covariance matrix for the parameters. This is a Bayesian posterior covariance matrix that results from adopting a particular Bayesian model of the smoothing process. Paricularly useful for creating credible/confidence intervals.

- Vc
Under ML or REML smoothing parameter estimation it is possible to correct the covariance matrix

`Vp`

for smoothing parameter uncertainty. This is the corrected version.- weights
final weights used in IRLS iteration.

- y
response data.

Simon N. Wood simon.wood@r-project.org

This model object is different to that described in Chambers and Hastie (1993) in order to allow smoothing parameter estimation etc.

A Key Reference on this implementation:

Wood, S.N. (2017) Generalized Additive Models: An Introduction with R (2nd edition). Chapman & Hall/ CRC, Boca Raton, Florida

Key Reference on GAMs generally:

Hastie (1993) in Chambers and Hastie (1993) Statistical Models in S. Chapman and Hall.

Hastie and Tibshirani (1990) Generalized Additive Models. Chapman and Hall.

`gam`