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For an ordinary unpenalized fit from lrm or ols and for a vector or list of penalties,
fits a series of logistic or linear models using penalized maximum likelihood
estimation, and saves the effective degrees of freedom, Akaike Information
Criterion (pentrace can
use the nlminb function to solve for the optimum penalty factor or
combination of factors penalizing different kinds of terms in the model.
The effective.df function prints the original and effective
degrees of freedom for a penalized fit or for an unpenalized fit and
the best penalization determined from a previous invocation of
pentrace if method="grid" (the default).
The effective d.f. is computed separately for each class of terms in
the model (e.g., interaction, nonlinear).
A plot method exists to plot the results, and a print method exists
to print the most pertinent components. Both penalty. Otherwise,
the first two types of penalty factors are plotted, showing only the
pentrace(fit, penalty, penalty.matrix,
method=c('grid','optimize'),
which=c('aic.c','aic','bic'), target.df=NULL,
fitter, pr=FALSE, tol=1e-7,
keep.coef=FALSE, complex.more=TRUE, verbose=FALSE, maxit=12,
subset, noaddzero=FALSE)effective.df(fit, object)
# S3 method for pentrace
print(x, …)
# S3 method for pentrace
plot(x, method=c('points','image'),
which=c('effective.df','aic','aic.c','bic'), pch=2, add=FALSE,
ylim, …)
a result from lrm or ols with x=TRUE, y=TRUE and without using penalty or
penalty.matrix
(or optionally using penalization in the case of effective.df)
can be a vector or a list. If it is a vector, all types of terms in
the model will be penalized by the same amount, specified by elements in
penalty, with a penalty of zero automatically added. penalty can
also be a list in the format documented in the lrm function, except that
elements of the list can be vectors. The expand.grid function is
invoked by pentrace to generate all possible combinations of
penalties. For example, specifying
penalty=list(simple=1:2, nonlinear=1:3) will generate 6 combinations
to try, so that the analyst can attempt to determine whether penalizing
more complex terms in the model more than the linear or categorical
variable terms will be beneficial. If complex.more=TRUE, it is assumed
that the variables given in penalty are listed in order from less
complex to more complex. With method="optimize" penalty specifies
an initial guess for the penalty or penalties. If all term types are
to be equally penalized, penalty should be a single number,
otherwise it should be a list containing single numbers as elements,
e.g., penalty=list(simple=1, nonlinear=2). Experience has shown that the optimization algorithm is more likely to find a reasonable solution when the starting value specified in penalty is too large rather than too small.
an object returned by pentrace. For effective.df, object can be
omitted if the fit was penalized.
see lrm
The default is method="grid" to print various indexes for all
combinations of penalty parameters given by the user. Specify
method="optimize" to have pentrace use nlminb to solve for the
combination of penalty parameters that gives the maximum value of the
objective named in which, or, if target.df is given, to find the
combination that yields target.df effective total degrees of freedom
for the model. When target.df is specified, method is set to
"optimize" automatically.
For plot.pentrace this parameter applies only if more than one
penalty term-type was used. The default is to use open triangles
whose sizes are proportional to the ranks of the AICs, plotting the
first two penalty factors respectively on the x and y axes. Use
method="image" to plot an image plot.
the objective to maximize for either method. Default is "aic.c" (corrected
AIC).
For plot.pentrace, which is a vector of names of criteria to show;
default is to plot all 4 types, with effective d.f. in its own separate plot
applies only to method="optimize". See method. target.df makes
sense mainly when a single type of penalty factor is specified.
a fitting function. Default is lrm.fit (lm.pfit is always used for ols).
set to TRUE to print intermediate results
tolerance for declaring a matrix singular (see lrm.fit, solvet)
set to TRUE to store matrix of regression coefficients for all the fits (corresponding
to increasing values of penalty) in object Coefficients in the
returned list. Rows correspond to penalties, columns to regression
parameters.
By default if penalty is a list, combinations of penalties for which
complex terms are penalized less than less complex terms will be
dropped after expand.grid is invoked. Set complex.more=FALSE to
allow more complex terms to be penalized less. Currently this option
is ignored for method="optimize".
set to TRUE to print number of intercepts and sum
of effective degrees of freedom
maximum number of iterations to allow in a model fit (default=12).
This is passed to the appropriate fitter function with the correct
argument name. Increase maxit if you had to when fitting the
original unpenalized model.
a logical or integer vector specifying rows of the design and response
matrices to subset in fitting models. This is most useful for
bootstrapping pentrace to see if the best penalty can be estimated
with little error so that variation due to selecting the optimal
penalty can be safely ignored when bootstrapping standard errors of regression
coefficients and measures of predictive accuracy. See an example below.
set to TRUE to not add an unpenalized model to
the list of models to fit
a result from pentrace
used for method="points"
set to TRUE to add to an existing plot. In that case, the effective
d.f. plot is not re-drawn, but the AIC/BIC plot is added to.
2-vector of y-axis limits for plots other than effective d.f.
other arguments passed to plot, lines, or image
a list of class "pentrace"
with elements penalty, df, objective, fit, var.adj, diag, results.all, and
optionally Coefficients.
The first 6 elements correspond to the fit that had the best objective
as named in the which argument, from the sequence of fits tried.
Here fit is the fit object from fitter which was a penalized fit,
diag is the diagonal of the matrix used to compute the effective
d.f., and var.adj is Gray (1992) Equation 2.9, which is an improved
covariance matrix for the penalized beta. results.all is a data
frame whose first few variables are the components of penalty and
whose other columns are df, aic, bic, aic.c. results.all thus
contains a summary of results for all fits attempted. When
method="optimize", only two components are returned: penalty and
objective, and the object does not have a class.
Gray RJ: Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. JASA 87:942--951, 1992.
Hurvich CM, Tsai, CL: Regression and time series model selection in small samples. Biometrika 76:297--307, 1989.
# NOT RUN {
n <- 1000 # define sample size
set.seed(17) # so can reproduce the results
age <- rnorm(n, 50, 10)
blood.pressure <- rnorm(n, 120, 15)
cholesterol <- rnorm(n, 200, 25)
sex <- factor(sample(c('female','male'), n,TRUE))
# Specify population model for log odds that Y=1
L <- .4*(sex=='male') + .045*(age-50) +
(log(cholesterol - 10)-5.2)*(-2*(sex=='female') + 2*(sex=='male'))
# Simulate binary y to have Prob(y=1) = 1/[1+exp(-L)]
y <- ifelse(runif(n) < plogis(L), 1, 0)
f <- lrm(y ~ blood.pressure + sex * (age + rcs(cholesterol,4)),
x=TRUE, y=TRUE)
p <- pentrace(f, seq(.2,1,by=.05))
plot(p)
p$diag # may learn something about fractional effective d.f.
# for each original parameter
pentrace(f, list(simple=c(0,.2,.4), nonlinear=c(0,.2,.4,.8,1)))
# Bootstrap pentrace 5 times, making a plot of corrected AIC plot with 5 reps
n <- nrow(f$x)
plot(pentrace(f, seq(.2,1,by=.05)), which='aic.c',
col=1, ylim=c(30,120)) #original in black
for(j in 1:5)
plot(pentrace(f, seq(.2,1,by=.05), subset=sample(n,n,TRUE)),
which='aic.c', col=j+1, add=TRUE)
# Find penalty giving optimum corrected AIC. Initial guess is 1.0
# Not implemented yet
# pentrace(f, 1, method='optimize')
# Find penalty reducing total regression d.f. effectively to 5
# pentrace(f, 1, target.df=5)
# Re-fit with penalty giving best aic.c without differential penalization
f <- update(f, penalty=p$penalty)
effective.df(f)
# }
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