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 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.
Array whose elements indicate which model term (listed in
pterms) each parameter relates to: applies only to non-smooth terms.
did parameters end up at boundary of parameter space?
the matched call (allows update to be used with gam objects, for example).
column means of the model matrix (with elements corresponding to smooths set to zero ) --- useful for componentwise CI calculation.
the coefficients of the fitted model. Parametric coefficients are first, followed by coefficients for each spline term in turn.
the gam control list used in the fit.
indicates whether or not the iterative fitting method converged.
the original supplied data argument (for class "glm" compatibility).
Only included if gam control argument element
keepData is set to TRUE (default is FALSE).
matrix of first derivatives of model coefficients w.r.t. log smoothing parameters.
model deviance (not penalized deviance).
null degrees of freedom.
effective residual degrees of freedom of the model.
estimated degrees of freedom for each model parameter. Penalization means that many of these are less than 1.
similar, but using alternative estimate of EDF. Useful for testing.
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 object specifying distribution and link used.
fitted model predictions of expected value for each datum.
the model formula.
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.
Degrees of freedom matrix. This may be removed at some point, and should probably not be used.
The minimized smoothing parameter selection score: GCV, UBRE(AIC), GACV, negative log marginal likelihood or negative log restricted likelihood.
array of elements from the leading diagonal of the `hat' (or `influence') matrix. Same length as response data vector.
number of iterations of P-IRLS taken to get convergence.
fitted model prediction of link function of expected value for each datum.
One of "GCV" or "UBRE", "REML", "P-REML", "ML",
"P-ML", "PQL", "lme.ML" or "lme.REML", depending on the fitting
criterion used.
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.defWas 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.
Minimum possible degrees of freedom for whole model.
model frame containing all variables needed in original model fit.
The na.action used in fitting.
number of parametric, non-smooth, model terms including the intercept.
deviance for single parameter model.
model offset.
optimizer argument to gam, or "magic" if it's a pure
additive model.
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).
If the paraPen argument to gam was used then this provides
information on the parametric penalties. NULL otherwise.
one sided formula containing variables needed for prediction, used by predict.gam
prior weights on observations.
terms object for strictly parametric part of model.
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).
apparent rank of fitted model.
The scale (RE)ML scale parameter estimate, if (P-)(RE)ML used for smoothness estimation.
the working residuals for the fitted model.
If present, rV%*%t(rV)*sig2 gives the estimated Bayesian covariance matrix.
when present, the scale (as sig2)
TRUE if the scale parameter was estimated, FALSE otherwise.
estimated or supplied variance/scale parameter.
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.
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 object of model model frame.
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.
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.
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.
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.
final weights used in IRLS iteration.
response data.
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.