smoothSurvReg.object {smoothSurv}

Smoothed Survival Regression Object

Description

This class of objects is returned by the smoothSurvReg class of functions to represent a fitted smoothed survival regression model.

Objects of this class have methods for the functions print, summary, plot, residuals, survfit.

COMPONENTS I

The following components must be included in a legitimate smoothSurvReg object.

fail

Indicator of the failure of the fitting procedure. Possible values are 0 for no problems, 3 if the iteration process was stopped because of non-positive definite minus Hessian, 4 if the eiteration process was stopped because too many halving steps were performed, 5 if it was not possible to find the three reference knots (it was not then possible to perform optimization with respect to the full parameter vector), 6 if the maximal number of iterations was performed without reaching a convergence. The fail component is increased by 10 if the final minus Hessian of the penalized log-likelihood was not positive definite. The fail component is further increased by 20 if the computed effective degrees of freedom were non-positive. The fail component is further increased by 40 if there are negative estimates of standard errors for some regression parameters. The fail component is 99 or higher if the fitting procedure failed at all and there is no fit produced.

COMPONENTS II

The following components must be included in a legitimate smoothSurvReg object if fail is lower than 99.

regres

Estimates of the regression parameters \alpha, \beta, \sigma if these have been estimated with their standard errors stored in a data frame with colnames “Value”, “Std.Error”, “Std.Error2” and rownames derived from the names of the design matrix with “(Intercept)” for the intercept, “Scale” for the scale and “Log(scale)” for the log-scale. If the log-scale depends on covariates then rows named “LScale.(Intercept)”, “LScale.cov1” etc. give estimates of regression parameters for log-scale. The two standard errors are computed using either var or var2 described below.

spline

Description of the fitted error density. A data frame with colnames “Knot”, “SD basis”, “c coef.”, “Std.Error.c”, “Std.Error2.c”, “a coef.”, “Std.Error.a” and “Std.Error2.a” and rownames knot[1], ..., knot[g] where g stands for the number of basis G-splines. The column “Knot” contains the knots in ascending order, “SD basis” the standard deviation of an appropriate basis G-spline, “c coef.” estimates of the G-spline coefficients and “Std.Error.c” and “Std.Error2.c” the estimates of their standard errors based either on var or var2. The column “a coef.” contains the estimates of transformed c coefficients where

c_j = \frac{\exp(a_j)}{\sum_{l=1}^{g}\exp(a_l)}, j = 1,\dots, g.

If the error distribution is estimated, one of the a coefficients is set to zero and two other a's are expressed as a function of the remaining a coefficients (to avoid equality constraints concerning the mean and the variance of the error distribution). The standard error for these three a coefficients is then not available (it is equal to NA). Standard error is set to NaN is a diagonal element of the appropriate covariance matrix was negative.

loglik

Maximized penalized log-likelihood, log-likelihood and the penalty term. A data frame with one row and three columns named “Log Likelihood”, “Penalty” and “Penalized Log Likelihood”.

aic

Akaike's information criterion of the fitted model computed as a maximized value of the penalized log-likelihood minus the effective degrees of freedom.

degree.smooth

Effective degrees of freedom, number of parameters and related information. A data frame with one row and columns named “Lambda”, “Log(Lambda)”, “df”, “Number of parameters”, “Mean param.”, “Scale param.”, “Spline param.” where “Lambda” gives the value of the tunning parameter used in the final (optimal) fit, “df” the effective degrees of freedom, “Number of parameters” the real number of parameters and “Mean param.”, “Scale param.” and “Spline param.” its decomposition. Note that if G-spline coefficients are estimated “Spline param.” is equal to the number of basis G-spline with non-zero coefficients minus three.

var

The estimate of the covariance matrix of the estimates based on the Bayesian approximation. It is equal to the inverse of the converged minus Hessian of the penalized log-likelihood. Note that there are no columns and rows corresponding to the three transformed G-spline coefficients since these are functions of the remaining transformed G-spline coefficients (to avoid equality constraints).

var2

The estimate of the covariance matrix of the estimates based on the asymptotic theory for penalized models. It is equal to H^{-1}\,I\,H^{-1} where H is converged minus Hessian of the penalized log-likelihood and I is converged minus Hessian of the log-likelihood component of the penalized log-likelihood.

dCdD

A matrix with derivatives of c spline coefficients with respect to d spline coefficients (these are a coefficients with three of them omitted). This matrix can be used later to compute estimates and standard errors of functions of original parameters using a Delta method. For closer definition of d coefficients see an enclosed document.

iter

Used number of iterations to fit the model with the optimal \lambda.

estimated

Indicator of what has really been estimated and not fixed. A four-component vector with component names “(Intercept)”, “Scale”, “ccoef”, “common.logscale”. The first component is TRUE if the intercept was included in the regression model. The second component is TRUE if the scale parameter was not fixed, the third component is TRUE is the G-spline coefficients were not fixed. The fourth component is TRUE if the log-scale does not depend on covariates.

warning

A data frame with one column called “warnings” and three rows called “Convergence”, “Final minus Hessian” and “df” containing a string information corresponding to the value of the fail component of the object. It contains a string “OK” if there are no problems with the appropriate part of the fitting process.

H

Converged minus Hessian of the penalized log-likelihood.

I

Converged minus Hessian of the log-likelihood component of the penalized log-likelihood. I = H - G.

G

Converged minus Hessian of the penalty term of the penalty term of the penalized log-likelihood. G = H - I.

U

Converged score vector based on the penalized log-likelihood.

na.action

The na.action attribute, if any, that was returned by the na.action routine.

terms

The terms object used.

formula

A symbolic description of the model to be fit.

call

The matched call.

init.dist

A string indicating the error distribution of the untransformed response to find the initial values. Possible values are “lognormal”, “loglogistic”, “weibull”.

model

If requested, the model frame used.

x

The model matrix used.

y

The response matrix used (two columns if there were no interval censored observations, three columns if there were some interval censored observations). The last column indicates the death status.

z

The model matrix used for the expression of log-scale.

init.spline

A data frame describing the initial error density. It has columns named “Knot”, “SD basis”, “c coef.” and rows named “knot[1]”, ..., “knot[g]”.

init.regres

Initial estimates of the regression parameters. A data frame with one column named “Value” and rows named as in the regres component of the smoothSurvReg object.

adjust

Adjusted intercept and scale. A data frame with a column named “Value” and rows named “(Intercept)” and “Scale”. “(Intercept)” gives the overall intercept taking into account the mean of the fitted error distribution, “Scale” gives the overall scale taking into account the variance of the fitted error distribution. If the error distribution is standardized (always when G-spline coefficients are estimated) then the “(Intercept)” is equal to the “(Intercept)” from the regres component and “Scale” is equal to the “Scale” of either regres or init.regres component. NA's appeare in this data.frame in the case that log-scale depends on covariates.

error.dist

A data frame with columns named “Mean”, “Var” and “SD” and a row named “Error distribution: ” giving the mean, variance and the standard deviation of the fitted error distribution. These are equal to 0, 1 and 1 if the G-spline coefficients were estimated.

searched

Information concerning the searched values of the tunning paramater \lambda when looking for the best AIC. A data frame with columns named “Lambda”, “Log(Lambda)”, “AIC”, “df”, “PenalLogLik”, “LogLik”, “nOfParm”, “fail”.

Author(s)

Arnošt Komárek [e-mail]