Package 'blapsr' reference manual

Title:	Bayesian Inference with Laplace Approximations and P-Splines
Description:	Laplace approximations and penalized B-splines are combined for fast Bayesian inference in latent Gaussian models. The routines can be used to fit survival models, especially proportional hazards and promotion time cure models (Gressani, O. and Lambert, P. (2018) <doi:10.1016/j.csda.2018.02.007>). The Laplace-P-spline methodology can also be implemented for inference in (generalized) additive models (Gressani, O. and Lambert, P. (2021) <doi:10.1016/j.csda.2020.107088>). See the associated website for more information and examples.
Authors:	Oswaldo Gressani [aut, cre] (Author), Philippe Lambert [aut, ths] (Co-author and thesis advisor)
Maintainer:	Oswaldo Gressani <[email protected]>
License:	GPL-3
Version:	0.6.1
Built:	2024-11-19 05:56:29 UTC
Source:	https://github.com/oswaldogressani/blapsr

Bayesian additive partial linear modeling with Laplace-P-splines.

Description

Fits an additive partial linear model to data using an approximate Bayesian inference technique based on penalized regression splines and Laplace approximations. Smooth additive terms are specified as a linear combination of of a large number of cubic B-splines. To counterbalance the roughness of the fit, a discrete penalty on neighboring spline coefficients is imposed in the spirit of Eilers and Marx (1996). The error of the model is assumed to be Gaussian with zero mean and finite variance.

The optimal amount of smoothing is determined by a grid-based exploration of the posterior penalty space when the number of smooth terms is small to moderate. When the dimension of the penalty space is large, the optimal smoothing parameter is chosen to be the value that maximizes the (log-)posterior of the penalty vector.

Usage

amlps(formula, data, K = 30, penorder = 2, cred.int = 0.95)
amlps(formula, data, K = 30, penorder = 2, cred.int = 0.95)

Arguments

`formula`	A formula object where the ~ operator separates the response from the covariates of the linear part `z1,z2,..` and the smooth terms. A smooth term is specified by using the notation `sm(.)`. For instance, the formula `y ~ z1+sm(x1)+sm(x2)` specifies an additive model of the form E(y)=b0+b1z1+f1(x1)+f2(x2), where b0, b1 are the regression coefficients of the linear part and f1(.) and f2(.) are smooth functions of the continuous covariates x1 and x2 respectively.
`data`	Optional. A data frame to match the variable names provided in formula.
`K`	A positive integer specifying the number of cubic B-spline functions in the basis used to model the smooth terms. Default is `K = 30` and allowed values are `15 <= K <= 60`. The same basis dimension is used for each smooth term in the model. Also, the computational cost to fit the model increases with `K`.
`penorder`	The penalty order used on finite differences of the coefficients of contiguous B-splines. Can be either 2 for a second-order penalty (the default) or 3 for a third-order penalty.
`cred.int`	The level of the pointwise credible interval to be computed for the coefficients in the linear part of the model.

Details

The B-spline basis used to approximate a smooth additive component is computed with the function cubicbs. The lower (upper) bound of the B-spline basis is taken to be the minimum (maximum) value of the covariate associated to the smooth. For identifiability purposes, the B-spline matrices (computed over the observed covariates) are centered. The centering consists is subtracting from each column of the B-spline matrix, the corresponding column average of another B-spline matrix computed on a fine grid of equidistant values in the domain of the smooth term.

A hierarchical Gamma prior is imposed on the roughness penalty vector and Jeffreys' prior is imposed on the precision of the error. A Newton-Raphson algorithm is used to compute the posterior mode of the (log-)posterior penalty vector. The latter algorithm uses analytically derived versions of the gradient and Hessian. When the number of smooth terms in the model is smaller or equal to 4, a grid-based strategy is used for posterior exploration of the penalty space. Above that threshold, the optimal amount of smoothness is determined by the posterior maximum value of the penalty vector. This strategy allows to keep the computational burden to fit the model relatively low and to conserve good statistical performance.

Value

An object of class amlps containing several components from the fit. Details can be found in amlps.object. Details on the output printed by amlps can be found in print.amlps. Fitted smooth terms can be visualized with the plot.amlps routine.

Author(s)

Oswaldo Gressani [email protected].

References

Eilers, P.H.C. and Marx, B.D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11(2): 89-121.

Fan, Y. and Li, Q. (2003). A kernel-based method for estimating additive partially linear models. Statistica Sinica, 13(3): 739-762.

Gressani, O. and Lambert, P. (2018). Fast Bayesian inference using Laplace approximations in a flexible promotion time cure model based on P-splines. Computational Statistical & Data Analysis 124: 151-167.

Opsomer, J. D. and Ruppert, D. (1999). A root-n consistent backfitting estimator for semiparametric additive modeling. Journal of Computational and Graphical Statistics, 8(4): 715-732.

Examples

### Classic simulated data example (with simgamdata)

set.seed(17)
sim.data <- simgamdata(setting = 2, n = 200, dist = "gaussian", scale = 0.4)
data <- sim.data$data  # Simulated data frame

# Fit model
fit <- amlps(y ~ z1 + z2 + sm(x1) + sm(x2), data = data, K = 15)
fit

### Classic simulated data example (with simgamdata)

set.seed(17)
sim.data <- simgamdata(setting = 2, n = 200, dist = "gaussian", scale = 0.4)
data <- sim.data$data  # Simulated data frame

# Fit model
fit <- amlps(y ~ z1 + z2 + sm(x1) + sm(x2), data = data, K = 15)
fit

Object resulting from the fit of an additive partial linear model.

Description

An object returned by the amlps function consists in a list with various components related to the fit of an additive partial linear model with the Laplace-P-spline approach.

Value

An amlps object has the following elements:

`formula`	The formula of the additive model.
`n`	Sample size.
`q`	Total number of smooth terms.
`K`	Number of B-spline basis functions used for the fit.
`penalty.order`	Chosen penalty order.
`latfield.dim`	The dimension of the latent field. This is equal to the sum of the number of B-spline coefficients and the number of regression parameters related to the covariates in the linear part.
`linear.coeff`	Estimated linear regression coefficients. This is a matrix containing the posterior point estimate, standard deviation and lower/upper bounds of the credible interval.
`spline.estim`	The estimated B-spline coefficients. This is a list with `q` vectors of size `K-1` representing the estimated B-spline amplitudes for each smooth term.
`edf`	Estimated effective degrees of freedom for each latent field variable.
`Approx.signif`	A matrix returning the observed test statistic and p-value for the approximate significance of smooth terms.
`EDf`	The estimated effective degrees of freedom of the smooth terms.
`EDfHPD.95`	95% HPD interval for the degrees of freedom of the smooth terms.
`ED`	The estimated degrees of freedom of the additive model.
`sd.error`	The estimated standard deviation of the error.
`vmap`	The maximum a posteriori of the (log) posterior penalty vector.
`Cov.vmap`	Covariance matrix of the (log) posterior penalty vector evaluated at vmap.
`pen.family`	The family of the posterior distribution for v. It is either "skew-normal" or "gaussian".
`pendist.params`	The parameterization for the posterior distribution of v. If the posterior of v belongs to the skew-normal family, then `pendist.params` is a matrix with as many rows as the number of smooth terms `q`. Each row contains the location, scale and shape parameter of the skew-normal distribution. If the posterior of v belongs to the Gaussian family, then `pendist.params` is a vector of length `q`, corresponding to `vmap`.
`Covmaximum`	The covariance matrix of the latent field evaluated at vmap.
`latmaximum`	The latent field vector evaluated at vmap.
`fitted.values`	The fitted response values.
`residuals`	The response residuals.
`r2.adj`	The adjusted r-squared of the model indicating the proportion of the data variance explained by the model fit.
`data`	The data frame.

Author(s)

Oswaldo Gressani [email protected].

Fit a Cox proportional hazards regression model with Laplace-P-splines.

Description

Fits a Cox proportional hazards regression model for right censored data by combining Bayesian P-splines and Laplace approximations.

Usage

coxlps(formula, data, K = 30, penorder = 2, tmax = NULL)
coxlps(formula, data, K = 30, penorder = 2, tmax = NULL)

Arguments

`formula`	A formula object where the ~ operator separates the response from the covariates. In a Cox model, it takes the form response ~ covariates, where response is a survival object returned by the `Surv` function of the survival package.
`data`	Optional. A data frame to match the variable names provided in `formula`.
`K`	A positive integer specifying the number of cubic B-spline functions in the basis. Default is K = 30 and allowed values are `10 <= K <= 60`.
`penorder`	The penalty order used on finite differences of the coefficients of contiguous B-splines. Can be either 2 for a second-order penalty (the default) or 3 for a third-order penalty.
`tmax`	A user-specified value for the upper bound of the B-spline basis. The default is NULL, so that the B-spline basis is specified in the interval [0, tup], where tup is the upper bound of the follow-up times. It is required that tmax > tup.

Details

The log-baseline hazard is modeled as a linear combination of K cubic B-splines as obtained from cubicbs. The B-spline basis is specified in the interval [0, tup], where tup is the upper bound of the follow-up times, i.e. the largest observed follow-up time. Following Jullion and Lambert (2007), a robust Gamma prior is imposed on the roughness penalty parameter. A grid-based approach is used to explore the posterior penalty space and the resulting quadrature points serve to compute the approximate (joint) marginal posterior of the latent field vector. Point and set estimates of latent field elements are obtained from a finite mixture of Gaussian densities. The routine centers the columns of the covariate matrix around their mean value for numerical stability.

Value

An object of class coxlps containing various components from the fit. Details can be found in coxlps.object. Plot of estimated smooth hazard and survival curves can be obtained using plot.coxlps. If required, estimated baseline quantities on specific time values can be obtained with coxlps.baseline.

Author(s)

Oswaldo Gressani [email protected].

References

Cox, D.R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society: Series B (Methodological) 34(2): 187-202.

Jullion, A. and Lambert, P. (2007). Robust specification of the roughness penalty prior distribution in spatially adaptive Bayesian P-splines models. Computational Statistical & Data Analysis 51(5): 2542-2558.

Examples


### Example 1 (Simulated survival data)

set.seed(3)

# Simulate survival data  with simsurvdata
betas <- c(0.13, 0.52, 0.30)
simul <- simsurvdata(a = 3.8, b = 2.2, n = 250, betas = betas , censperc = 20)
simul
simdat <- simul$survdata
plot(simul) # Plot survival data

# Estimation with coxlps
fit <- coxlps(Surv(time, delta) ~ x1 + x2 + x3, data = simdat, K = 15)
# Compare coxlps and coxph
fit
summary(coxph(Surv(time, delta) ~ x1 + x2 + x3, data = simdat))

# Fitted baseline survival vs target
plot(fit, h0 = FALSE, cred.int = 0.95, overlay.km = TRUE)
domt <- seq(0, 4, length = 100)
lines(domt, simul$S0(domt), type = "l", col = "red")
legend("topright", col=c("black", "blue", "red"), lty = rep(1,3),
      c("Bayesian LPS", "Kaplan-Meier", "Target"), cex = 0.8, bty = "n")

### Example 2 (Kidney transplant data)

data(kidneytran)
Surv.obj <- Surv(kidneytran$time, kidneytran$delta)
fit <- coxlps(Surv.obj ~ age + gender + race, data = kidneytran)
coxphfit <- coxph(Surv.obj ~ age + gender + race, data = kidneytran)
## Compare coxph and coxlps results
summary(coxphfit)
fit
## Plot Kaplan-Meier curve vs Laplace-P-spline fit
plot(fit, h0 = FALSE, overlay.km = TRUE, plot.cred = FALSE)

### Example 3 (Laryngeal cancer data)

data(laryngeal)
fit <- coxlps(Surv(time, delta) ~ age + diagyr + as.factor(stage),
               data = laryngeal)
coxphfit <- coxph(Surv(time, delta) ~ age + diagyr + as.factor(stage),
                  data = laryngeal)
## Compare coxph and coxlps results
summary(coxphfit)
fit
## Plot Kaplan-Meier curve vs Laplace-P-spline fit
plot(fit, h0 = FALSE, overlay.km = TRUE, plot.cred = FALSE)


### Example 1 (Simulated survival data)

set.seed(3)

# Simulate survival data  with simsurvdata
betas <- c(0.13, 0.52, 0.30)
simul <- simsurvdata(a = 3.8, b = 2.2, n = 250, betas = betas , censperc = 20)
simul
simdat <- simul$survdata
plot(simul) # Plot survival data

# Estimation with coxlps
fit <- coxlps(Surv(time, delta) ~ x1 + x2 + x3, data = simdat, K = 15)
# Compare coxlps and coxph
fit
summary(coxph(Surv(time, delta) ~ x1 + x2 + x3, data = simdat))

# Fitted baseline survival vs target
plot(fit, h0 = FALSE, cred.int = 0.95, overlay.km = TRUE)
domt <- seq(0, 4, length = 100)
lines(domt, simul$S0(domt), type = "l", col = "red")
legend("topright", col=c("black", "blue", "red"), lty = rep(1,3),
      c("Bayesian LPS", "Kaplan-Meier", "Target"), cex = 0.8, bty = "n")

### Example 2 (Kidney transplant data)

data(kidneytran)
Surv.obj <- Surv(kidneytran$time, kidneytran$delta)
fit <- coxlps(Surv.obj ~ age + gender + race, data = kidneytran)
coxphfit <- coxph(Surv.obj ~ age + gender + race, data = kidneytran)
## Compare coxph and coxlps results
summary(coxphfit)
fit
## Plot Kaplan-Meier curve vs Laplace-P-spline fit
plot(fit, h0 = FALSE, overlay.km = TRUE, plot.cred = FALSE)

### Example 3 (Laryngeal cancer data)

data(laryngeal)
fit <- coxlps(Surv(time, delta) ~ age + diagyr + as.factor(stage),
               data = laryngeal)
coxphfit <- coxph(Surv(time, delta) ~ age + diagyr + as.factor(stage),
                  data = laryngeal)
## Compare coxph and coxlps results
summary(coxphfit)
fit
## Plot Kaplan-Meier curve vs Laplace-P-spline fit
plot(fit, h0 = FALSE, overlay.km = TRUE, plot.cred = FALSE)

Extract estimated baseline quantities from a fit with coxlps.

Description

The routine takes as input an object of class 'coxlps' and computes point estimates and credible intervals for the baseline hazard and survival on a user-specified time vector.

Usage

coxlps.baseline(object, time = NULL, compute.cred = TRUE, cred.int = 0.95,
                verbose = TRUE)
coxlps.baseline(object, time = NULL, compute.cred = TRUE, cred.int = 0.95,
                verbose = TRUE)

Arguments

`object`	An object of class 'coxlps'.
`time`	A vector of time values on which to compute the estimated baseline quantities. Each component of 'time' must be between 0 and the largest observed follow-up time. If time is 'NULL' (the default), then only the baseline median lifetime (if available) is computed.
`compute.cred`	Should the credible intervals be computed? Default is TRUE.
`cred.int`	The level for an approximate pointwise credible interval to be computed for the baseline hazard and survival curves. Default is 0.95.
`verbose`	Should the table of estimated values be printed to console? Default is TRUE.

Value

A list with the following components:

`fit.time`	A matrix with point and set estimates of the baseline hazard and survival curves for values provided in 'time'. Only available if 'time' is not 'NULL'. Column Time summarizes the provided values in 'time'. Columns named h0, S0, are the point estimates of the baseline hazard and baseline survival respectively. low and up give the lower and upper bound respectively of the approximate pointwise credible interval.
`median.lifetime`	The estimated baseline median lifetime.
`cred.int`	The chosen level to construct credible intervals.

Author(s)

Oswaldo Gressani [email protected].

Examples


## Simulate survival data
set.seed(2)
betas <- c(0.15, 0.82, 0.41) # Regression coefficients
data <- simsurvdata(a = 1.8, b = 2, n = 300, betas = betas, censperc = 15)
simdat <- data$survdata

# Fit model
fit <- coxlps(Surv(time, delta) ~ x1 + x2 + x3, data = simdat, K = 20)
coxlps.baseline(fit, time = seq(0, 2, by = 0.5), cred.int = 0.90)

## Simulate survival data
set.seed(2)
betas <- c(0.15, 0.82, 0.41) # Regression coefficients
data <- simsurvdata(a = 1.8, b = 2, n = 300, betas = betas, censperc = 15)
simdat <- data$survdata

# Fit model
fit <- coxlps(Surv(time, delta) ~ x1 + x2 + x3, data = simdat, K = 20)
coxlps.baseline(fit, time = seq(0, 2, by = 0.5), cred.int = 0.90)

Object from a Cox proportional hazards fit with Laplace-P-splines.

Description

An object returned by the coxlps function consists in a list with various components related to the fit of a Cox model using the Laplace-P-spline methodology.

Value

A coxlps object has the following elements:

`formula`	The formula of the Cox model.
`K`	Number of B-spline basis functions used for the fit.
`penalty.order`	Chosen penalty order.
`latfield.dim`	The dimension of the latent field. This is equal to the sum of the number of B-spline coefficients and the number of regression parameters related to the covariates.
`n`	Sample size.
`num.events`	The number of events that occurred.
`event.times`	The standardized event times, i.e. if t denotes the original time scale, then `event.times = t / sd(t)`, where `sd` is the standard deviation.
`tup`	The upper bound of the follow-up, i.e. `max(event.times)`.
`sd.time`	The standard deviation of the event times in original scale.
`event.indicators`	The event indicators.
`regcoeff`	Posterior estimates of the regression coefficients. coef gives the point estimate, sd.post gives the posterior standard deviation, z is the Wald test statistic, lower .95 and upper .95 the posterior approximate 95% quantile-based credible interval.
`penalty.vector`	The selected grid of penalty values.
`vmap`	The maximum a posteriori of the (log) penalty parameter.
`spline.estim`	The estimated B-spline coefficients.
`edf`	Estimated effective degrees of freedom for each latent field variable.
`ED`	The effective model dimension.
`Covthetamix`	The posterior covariance matrix of the B-spline coefficients.
`X`	The matrix of covariate values.
`loglik`	The log-likelihood evaluated at the posterior latent field estimate.
`p`	Number of parametric coefficients in the model.
`AIC.p`	The AIC computed with the formula -2loglik+2p, where p is the number of parametric coefficients.
`AIC.ED`	The AIC computed with the formula -2loglik+2ED, where ED is the effective model dimension.
`BIC.p`	The BIC computed with the formula -2loglik+plog(ne), where p is the number of parametric coefficients and ne the number of events.
`BIC.ED`	The BIC computed with the formula -2loglik+EDlog(ne), where ED is the effective model dimension and ne the number of events.

Author(s)

Oswaldo Gressani [email protected].

Construct a cubic B-spline basis.

Description

Computation of a cubic B-spline basis matrix.

Usage

cubicbs(x, lower, upper, K)
cubicbs(x, lower, upper, K)

Arguments

`x`	A numeric vector containing the values on which to evaluate the B-spline basis.
`lower`, `upper`	The lower and upper bounds of the B-spline basis domain. Must be finite with lower < upper.
`K`	A positive integer specifying the number of B-spline functions in the basis.

Value

An object of class cubicbs for which print and plot methods are available. The cubicbs class consists of a list with the following components:

`x`	A numeric vector on which the basis is evaluated.
`lower`, `upper`	The lower and upper bounds of the basis domain.
`K`	The number of cubic B-spline functions in the basis.
`knots`	The knot sequence to build the basis.
`nknots`	Total number of knots.
`dimbasis`	The dimension of the B-spline basis matrix.
`Bmatrix`	The B-spline basis matrix.

The print method summarizes the B-spline basis and the plot method gives a graphical representation of the basis with dashed vertical lines indicating knot placement and blue ticks the coordinates of x.

Author(s)

Oswaldo Gressani [email protected].

The core algorithm of the cubicbs function owes much to a code written by Phlilippe Lambert.

References

Eilers, P.H.C. and Marx, B.D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11(2): 89-121.

Examples

lb <- 0  # Lower bound
ub <- 1  # Upper bound
xdom <- runif(100, lb, ub) # Draw uniform values between lb and ub
Bsmat <- cubicbs(xdom, lb, ub, 25) # 100 x 25 B-spline matrix
Bsmat
plot(Bsmat) # Plot the basis
lb <- 0  # Lower bound
ub <- 1  # Upper bound
xdom <- runif(100, lb, ub) # Draw uniform values between lb and ub
Bsmat <- cubicbs(xdom, lb, ub, 25) # 100 x 25 B-spline matrix
Bsmat
plot(Bsmat) # Plot the basis

Promotion time cure model with Laplace P-splines.

Description

Fits a promotion time cure model with the Laplace-P-spline methodology. The routine can be applied to survival data for which a plateau is observed in the Kaplan-Meier curve. In this case, the follow-up period is considered to be sufficiently long to intrinsically account for long-term survivors and hence a cured fraction. The user can separately specify the model covariates influencing the cure probability (long-term survival) and the population hazard dynamics (short-term survival).

Usage

curelps(formula, data, K = 30, penorder = 2, tmax = NULL, constr = 6)
curelps(formula, data, K = 30, penorder = 2, tmax = NULL, constr = 6)

Arguments

`formula`	A formula object where the ~ operator separates the response from the covariates. In a promotion time cure model, it takes the form response ~ covariates, where response is a survival object returned by the `Surv` function of the survival package. The model covariates influencing the long-term survival can be specified in the function `lt(.)` separated by '+', while the covariates affecting the short-term survival can be specified in `st(.)`. For instance, a promotion time cure model with covariates specified as `lt(x1+x2)+st(x1)`, means that `x1` will jointly influence the long- and short-term survival, while `x2` will only influence the long-term survival.
`data`	Optional. A data frame to match the variable names provided in `formula`.
`K`	A positive integer specifying the number of cubic B-spline functions in the basis. Default is `K = 30` and allowed values are `10 <= K <= 60`.
`penorder`	The penalty order used on finite differences of the coefficients of contiguous B-splines. Can be either 2 for a second-order penalty (the default) or 3 for a third-order penalty.
`tmax`	A user-specified value for the upper bound of the B-spline basis. The default is NULL, so that the B-spline basis is specified in the interval [0, tup], where tup is the upper bound of the follow-up times. It is required that tmax > tup.
`constr`	Constraint imposed on last B-spline coefficient (default is 6).

Details

The log-baseline hazard is modeled as a linear combination of K cubic B-splines as obtained from cubicbs. A robust Gamma prior is imposed on the roughness penalty parameter. A grid-based approach is used to explore the posterior penalty space and the resulting quadrature points serve to compute the approximate (joint) posterior of the latent field vector. Point and set estimates of latent field elements are obtained from a finite mixture of Gaussian densities. The routine centers the columns of the covariate matrix around their mean value for numerical stability. See print.curelps for a detailed explanation on the output printed by the curelps function.

Value

An object of class curelps containing various components from the promotion time cure model fit. Details can be found in curelps.object. Estimates on the baseline survival, population survival (for a chosen covariate profile) and cure probability can be obtained with the plot.curelps and curelps.extract routines.

Author(s)

Oswaldo Gressani [email protected].

References

Cox, D.R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society: Series B (Methodological) 34(2): 187-202.

Bremhorst, V. and Lambert, P. (2016). Flexible estimation in cure survival models using Bayesian P-splines. Computational Statistical & Data Analysis 93: 270-284.

Lambert, P. and Bremhorst, V. (2019). Estimation and identification issues in the promotion time cure model when the same covariates influence long- and short-term survival. Biometrical Journal 61(2): 275-289.

Examples


## Fit a promotion time cure model on malignant melanoma data

data(melanoma)
medthick <- median(melanoma$thickness)

# Kaplan-Meier estimate to check the existence of a plateau
KapMeier <- survfit(Surv(time,status) ~ 1, data = melanoma)
plot(KapMeier, mark.time = TRUE, mark = 4, xlab = "Time (in years)")

# Fit with curelps
fit <- curelps(Surv(time , status) ~ lt(thickness + ulcer) +
                   st(thickness + ulcer), data = melanoma, K = 40)
fit

# Cure prediction for median thickness and absence of ulceration
curelps.extract(fit, time = c(2, 4 ,6, 8), curvetype = "probacure",
                cred.int = 0.90, covar.profile = c(medthick, 0, medthick, 0))

# Plot of baseline and population survival functions
opar <- par(no.readonly = TRUE)
par(mfrow = c(1, 2))
# Baseline survival
plot(fit, curvetype = "baseline", plot.cred = FALSE, ylim = c(0,1))
# Population survival
plot(fit, curvetype = "population", covar.profile = c(medthick, 0, medthick, 0),
plot.cred = FALSE, ylim = c(0,1))
par(opar)

## Fit a promotion time cure model on malignant melanoma data

data(melanoma)
medthick <- median(melanoma$thickness)

# Kaplan-Meier estimate to check the existence of a plateau
KapMeier <- survfit(Surv(time,status) ~ 1, data = melanoma)
plot(KapMeier, mark.time = TRUE, mark = 4, xlab = "Time (in years)")

# Fit with curelps
fit <- curelps(Surv(time , status) ~ lt(thickness + ulcer) +
                   st(thickness + ulcer), data = melanoma, K = 40)
fit

# Cure prediction for median thickness and absence of ulceration
curelps.extract(fit, time = c(2, 4 ,6, 8), curvetype = "probacure",
                cred.int = 0.90, covar.profile = c(medthick, 0, medthick, 0))

# Plot of baseline and population survival functions
opar <- par(no.readonly = TRUE)
par(mfrow = c(1, 2))
# Baseline survival
plot(fit, curvetype = "baseline", plot.cred = FALSE, ylim = c(0,1))
# Population survival
plot(fit, curvetype = "population", covar.profile = c(medthick, 0, medthick, 0),
plot.cred = FALSE, ylim = c(0,1))
par(opar)

Extract estimates of survival functions and cure probability for the promotion time cure model.

Description

The routine takes as input an object of class curelps and computes estimates of the baseline survival curve, the population survival curve and the cure probability on a specified time vector. Approximate pointwise credible intervals are available.

Usage

curelps.extract(object, time = NULL, curvetype = c("baseline", "population", "probacure"),
                covar.profile, compute.cred = TRUE, cred.int = 0.95, verbose = TRUE)
curelps.extract(object, time = NULL, curvetype = c("baseline", "population", "probacure"),
                covar.profile, compute.cred = TRUE, cred.int = 0.95, verbose = TRUE)

Arguments

`object`	An object of class `curelps`.
`time`	A vector of time values on which to compute the estimates. Each component of `time` must be between 0 and the largest observed follow-up time.
`curvetype`	The curve on which estimates are computed ; `baseline` (the default) is for the baseline survival, `population` is for the population survival function for a profile of covariates given in `covar.profile`, and `probacure` is for the probability to be cured (for a profile of covariates given in `covar.profile`) given that the subject has survived until time t.
`covar.profile`	A numeric vector of the same length as the number of covariates in the model. This corresponds to the profile of covariates for which to compute the population survival function and cure probability estimates. The order of the covariates in `covar.profile` is the same as the order specified in `formula` of the `curelps` routine. Each component of `covar.profile` should be in the range of the observed values for the corresponding covariate. If `covar.profile` is left unspecified by the user, the default will be to take the median covariate values.
`compute.cred`	Should credible intervals be computed? Default is TRUE.
`cred.int`	The level for an approximate pointwise credible interval. Default is 0.95.
`verbose`	Should estimates be printed to console?

Value

A list with the following components:

`fit.time`	Estimates on the time values provided in `time`.
`cred.int`	The chosen level to construct approximate pointwise credible intervals.
`covar.profile`	The chosen profile of covariates.

Author(s)

Oswaldo Gressani [email protected].

Examples


# Example on phase III clinical trial e1684 on melanoma data

data(ecog1684)

# Kaplan-Meier curve
plot(survfit(Surv(time, status) ~ 1, data = ecog1684), mark.time = TRUE)
fit <- curelps(Surv(time, status) ~ lt(age + trt+ sex) +
             st(age + trt + sex), data = ecog1684, K = 20, penorder = 2)
fit
profile1 <- c(0, 1, 1, 0, 1, 1) # Mean age, trt = IFN, sex = Female.
profile2 <- c(0, 0, 1, 0, 0, 1) # Mean age, trt = control, sex = Female.

# Extract cure probabilities
curelps.extract(fit, time = c(0, 1, 2, 3), curvetype = "probacure",
                covar.profile = profile1, cred.int = 0.90)
curelps.extract(fit, time = c(0, 1, 2, 3), curvetype = "probacure",
                covar.profile = profile2, cred.int = 0.90)
# Example on phase III clinical trial e1684 on melanoma data

data(ecog1684)

# Kaplan-Meier curve
plot(survfit(Surv(time, status) ~ 1, data = ecog1684), mark.time = TRUE)
fit <- curelps(Surv(time, status) ~ lt(age + trt+ sex) +
             st(age + trt + sex), data = ecog1684, K = 20, penorder = 2)
fit
profile1 <- c(0, 1, 1, 0, 1, 1) # Mean age, trt = IFN, sex = Female.
profile2 <- c(0, 0, 1, 0, 0, 1) # Mean age, trt = control, sex = Female.

# Extract cure probabilities
curelps.extract(fit, time = c(0, 1, 2, 3), curvetype = "probacure",
                covar.profile = profile1, cred.int = 0.90)
curelps.extract(fit, time = c(0, 1, 2, 3), curvetype = "probacure",
                covar.profile = profile2, cred.int = 0.90)

Object from a promotion time model fit with Laplace-P-splines.

Description

An object returned by the curelps function consists in a list with various components related to the fit of a promotion time cure model using the Laplace-P-spline methodology.

Value

A curelps object has the following elements:

`formula`	The formula of the promotion time cure model.
`K`	Number of B-spline basis functions used for the fit.
`penalty.order`	Chosen penalty order.
`latfield.dim`	The dimension of the latent field. This is equal to the sum of the number of B-spline coefficients and the number of regression parameters related to the covariates.
`event.times`	The observed event times.
`n`	Sample size.
`num.events`	The number of events that occurred.
`tup`	The upper bound of the follow up, i.e. `max(event.times)`.
`event.indicators`	The event indicators.
`coeff.probacure`	Posterior estimates of the regression coefficients related to the cure probability (or long-term survival).
`coeff.cox`	Posterior estimates of the regression coefficients related to the population hazard dynamics (or short-term survival).
`vmap`	The maximum a posteriori of the (log-)posterior penalty parameter.
`vquad`	The quadrature points of (log-) posterior penalty parameters used to compute the Gaussian mixture posterior of the latent field vector.
`spline.estim`	The estimated B-spline coefficients.
`edf`	Estimated effective degrees of freedom for each latent field variable.
`ED`	The effective model dimension.
`Covtheta.map`	The posterior covariance matrix of the B-spline coefficients for a penalty fixed at its maximum posterior value.
`Covlatc.map`	The posterior covariance matrix of the latent field for a penalty fixed at its maximum posterior value.
`X`	The covariate matrix for the long-term survival part.
`Z`	The covariate matrix for the short-term survival part.
`loglik`	The log-likelihood evaluated at the posterior latent field estimate.
`p`	Number of parametric coefficients in the model.
`AIC.p`	The AIC computed with the formula -2loglik+2p, where p is the number of parametric coefficients.
`AIC.ED`	The AIC computed with the formula -2loglik+2ED, where ED is the effective model dimension.
`BIC.p`	The BIC computed with the formula -2loglik+plog(ne), where p is the number of parametric coefficients and ne the number of events.
`BIC.ED`	The BIC computed with the formula -2loglik+EDlog(ne), where ED is the effective model dimension and ne the number of events.

Author(s)

Oswaldo Gressani [email protected].

Phase III Melanoma clinical trial.

Description

Melanoma data from the phase III Eastern Cooperative Oncology Group (ECOG) two-arm clinical trial studied in Kirkwood et al. (1996).

Usage

data(ecog1684)
data(ecog1684)

Format

A data frame with 284 rows and 5 columns.

trt: Treatment: 0=control, 1=Interferon alpha-2b (IFN).
time: Relapse-free survival (in years).
status: 1=death or relapse, 0=censored.
age: Age centered to the mean.
sex: 0=Male, 1=Female.

Source

https://CRAN.R-project.org/package=smcure

References

Kirkwood, J. M., Strawderman, M. H., Ernstoff, M. S., Smith, T. J., Borden, E. C. and Blum, R. H. (1996). Interferon alfa-2b adjuvant therapy of high-risk resected cutaneous melanoma: the Eastern Cooperative Oncology Group Trial EST 1684. Journal of clinical oncology 14(1): 7-17.

Corbiere, F. and Joly, P. (2007). A SAS macro for parametric and semiparametric mixture cure models. Computer methods and programs in Biomedicine 85(2): 173-180.

Bayesian generalized additive modeling with Laplace-P-splines.

Description

Fits a generalized additive model (GAM) to data using an approximate Bayesian inference technique based on penalized regression splines and Laplace approximations. Smooth additive terms are specified as a linear combination of a large number of cubic B-splines. To counterbalance the roughness of the fit, a discrete penalty on neighboring spline coefficients is imposed in the spirit of Eilers and Marx (1996). The effective degrees of freedom of the smooth terms are also estimated.

Usage

gamlps(formula, data, K = 30, family = c("gaussian", "poisson", "bernoulli", "binomial"),
       gauss.scale, nbinom, penorder = 2, cred.int = 0.95)
gamlps(formula, data, K = 30, family = c("gaussian", "poisson", "bernoulli", "binomial"),
       gauss.scale, nbinom, penorder = 2, cred.int = 0.95)

Arguments

`formula`	A formula object where the ~ operator separates the response from the covariates of the linear part `z1,z2,..` and the smooth terms. A smooth term is specified by using the notation `sm(.)`. For instance, the formula `y ~ z1+sm(x1)+sm(x2)` specifies a generalized additive model of the form g(mu) = b0+b1z1+f1(x1)+f2(x2), where b0, b1 are the regression coefficients of the linear part and f1(.) and f2(.) are smooth functions of the continuous covariates x1 and x2 respectively. The function g(.) is the canonical link function.
`data`	Optional. A data frame to match the variables names provided in formula.
`K`	A positive integer specifying the number of cubic B-spline functions in the basis used to model the smooth terms. Default is `K = 30` and allowed values are `15 <= K <= 60`. The same basis dimension is used for each smooth term in the model. Also, the computational cost to fit the model increases with `K`.
`family`	The error distribution of the model. It is a character string that must partially match either `"gaussian"` for Normal data with an identity link, `"poisson"` for Poisson data with a log link, `"bernoulli"` or `"binomial"` for Bernoulli or Binomial data with a logit link.
`gauss.scale`	The scale parameter to be specified when `family = "gaussian"`. It corresponds to the variance of the response.
`nbinom`	The number of experiments in the Binomial family.
`penorder`	The penalty order used on finite differences of the coefficients of contiguous B-splines. Can be either 2 for a second-order penalty (the default) or 3 for a third-order penalty.
`cred.int`	The level of the pointwise credible interval to be computed for the coefficients in the linear part of the model.

Details

A hierarchical Gamma prior is imposed on the roughness penalty vector. A Newton-Raphson algorithm is used to compute the posterior mode of the (log-)posterior penalty vector. The latter algorithm uses analytically derived versions of the gradient and Hessian. When the number of smooth terms in the model is smaller or equal to 4, a grid-based strategy is used for posterior exploration of the penalty space. Above that threshold, the optimal amount of smoothness is determined by the posterior maximum value of the penalty vector. This strategy allows to keep the computational burden to fit the model relatively low and conserve good statistical performance.

Value

An object of class gamlps containing several components from the fit. Details can be found in gamlps.object. Details on the output printed by gamlps can be found in print.gamlps. Fitted smooth terms can be visualized with the plot.gamlps routine.

Author(s)

Oswaldo Gressani [email protected].

References

Hastie, T. J. and Tibshirani., RJ (1990). Generalized additive models. Monographs on statistics and applied probability, 43, 335.

Eilers, P.H.C. and Marx, B.D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11(2): 89-121.

Examples


set.seed(14)
sim <- simgamdata(n = 300, setting = 2, dist = "binomial", scale = 0.25)
dat <- sim$data
fit <- gamlps(y ~ z1 + z2 + sm(x1) + sm(x2), K = 15, data = dat,
              penorder = 2, family = "binomial", nbinom = 15)
fit

# Check fit and compare with target (in red)
opar <- par(no.readonly = TRUE)
par(mfrow = c(1, 2))
domx <- seq(-1, 1 ,length = 200)
plot(fit, smoo.index = 1, cred.int = 0.95, ylim = c(-2, 2))
lines(domx, sim$f[[1]](domx), type= "l", lty = 2, lwd = 2, col = "red")
plot(fit, smoo.index = 2, cred.int = 0.95, ylim = c(-3, 3))
lines(domx, sim$f[[2]](domx), type= "l", lty = 2, lwd = 2, col = "red")
par(opar)

set.seed(14)
sim <- simgamdata(n = 300, setting = 2, dist = "binomial", scale = 0.25)
dat <- sim$data
fit <- gamlps(y ~ z1 + z2 + sm(x1) + sm(x2), K = 15, data = dat,
              penorder = 2, family = "binomial", nbinom = 15)
fit

# Check fit and compare with target (in red)
opar <- par(no.readonly = TRUE)
par(mfrow = c(1, 2))
domx <- seq(-1, 1 ,length = 200)
plot(fit, smoo.index = 1, cred.int = 0.95, ylim = c(-2, 2))
lines(domx, sim$f[[1]](domx), type= "l", lty = 2, lwd = 2, col = "red")
plot(fit, smoo.index = 2, cred.int = 0.95, ylim = c(-3, 3))
lines(domx, sim$f[[2]](domx), type= "l", lty = 2, lwd = 2, col = "red")
par(opar)

Object resulting from the fit of a generalized additive model.

Description

An object returned by the gamlps function consists in a list with various components related to the fit of a generalized additive model with the Laplace-P-spline approach.

Value

A gamlps object has the following elements:

`formula`	The formula of the generalized additive model.
`family`	The chosen exponential family.
`link`	The link function used for the fit.
`n`	Sample size.
`q`	Total number of smooth terms.
`K`	Number of B-spline basis functions used for the fit.
`penalty.order`	Chosen penalty order.
`latfield.dim`	The dimension of the latent field. This is equal to the sum of the number of B-spline coefficients and the number of regression parameters related to the covariates in the linear part.
`linear.coeff`	Estimated linear regression coefficients. This is a matrix containing the posterior point estimate, standard deviation, z-score and lower/upper bounds of the credible interval.
`spline.estim`	The estimated B-spline coefficients. This is a list with `q` vectors of size `K-1` representing the estimated B-spline amplitudes for each smooth term.
`edf`	Estimated effective degrees of freedom for each latent field variable.
`Approx.signif`	A matrix returning the observed test statistic and p-value for the approximate significance of smooth terms.
`EDf`	The estimated effective degrees of freedom of the smooth terms.
`EDfHPD.95`	95% HPD interval for the degrees of freedom of the smooth terms.
`ED`	The estimated degrees of freedom of the GAM model.
`vmap`	The maximum a posteriori of the (log) posterior penalty vector.
`Cov.vmap`	Covariance matrix of the (log) posterior penalty vector evaluated at vmap.
`pen.family`	The family of the posterior distribution for v. It is either "skew-normal" or "gaussian".
`pendist.params`	The parameterization for the posterior distribution of v. If the posterior of v belongs to the skew-normal family, then `pendist.params` is a matrix with as many rows as the number of smooth terms `q`. Each row contains the location, scale and shape parameter of the skew-normal distribution. If the posterior of v belongs to the Gaussian family, then `pendist.params` is a vector of length `q`, corresponding to `vmap`.
`Covmaximum`	The covariance matrix of the latent field evaluated at the posterior maximum value of the penalty vector.
`latmaximum`	The latent field value evaluated at the posterior maximum value of the penalty vector.
`fitted.values`	The fitted response values.
`residuals`	The response residuals.
`r2.adj`	The adjusted r-squared of the model indicating the proportion of the data variance explained by the model fit.
`data`	The data frame of the model.

Author(s)

Oswaldo Gressani [email protected].

Survival data of kidney transplant patients.

Description

Survival data of kidney transplant patients from Section 1.7 of Klein and Moeschberger (2003).

Usage

data(kidneytran)
data(kidneytran)

Format

A data frame with 863 rows and 6 columns.

obs: Observation number.
time: Time to death.
delta: Event indicator, 1=Dead, 0=Alive.
gender: Gender, 1=Male, 2=Female.
race: Race, 1=White, 2=Black.
age: Age in years.

Source

https://cran.r-project.org/package=KMsurv

References

Klein, J.P. and Moeschberger, M. L. (2003). Survival analysis: Techniques for Censored and Truncated Data (Second edition), Springer. ISBN 978-1-4419-2985-3

Survival data of male laryngeal cancer patients.

Description

Survival data of male patients with larynx cancer from Section 1.8 of Klein and Moeschberger (2003).

Usage

data(laryngeal)
data(laryngeal)

Format

A data frame with 90 rows and 5 columns.

stage: Stage of disease.
time: Time to death in months.
age: Age at diagnosis of larynx cancer.
diagyr: Year of diagnosis of larynx cancer.
delta: Event indicator, 1=Dead, 0=Alive.

Source

https://cran.r-project.org/package=KMsurv

References

Klein, J.P. and Moeschberger, M. L. (2003). Survival analysis: Techniques for Censored and Truncated Data (Second edition), Springer. ISBN 978-1-4419-2985-3

Data from the 1986 Medicaid Consumer Survey.

Description

Data from the 1986 Medicaid survey sponsored by the Health Care Financing Administration (USA). It can be used to illustrate generalized additive models with a log link for the number of doctor visits as a response variable. The dataset is studied in Gurmu (1997).

Usage

data(medicaid)
data(medicaid)

Format

A data frame with 485 rows and 10 columns.

numvisits: Count of doctor office/clinic and health centre visits.
exposure: Length of observation period for ambulatory care in days.
children: Number of children in the household.
age: Age of the respondent.
income1000: Annual household income in US dollars.
access: Access to health services, 0=Low access, 100=High access.
pc1times1000: First principal component of three health status variables: functional limitations, acute conditions and chronic conditions.
maritalstat: Marital status, 0=Other, 1=Married.
sex: Gender, 1=Female, 0=Male.
race: Race, 0=Other, 1=White.

Source

http://qed.econ.queensu.ca/jae/1997-v12.3/gurmu/

References

Gurmu, S.(1997). Semi-parametric estimation of hurdle regression models with an application to medicaid utilization. Journal of Applied Econometrics 12(3):225-242.

Melanoma survival data.

Description

Melanoma survival dataset with 205 patients suffering from skin cancer and operated for malignant melanoma at Odense University Hospital in Denmark.

Usage

data(melanoma)
data(melanoma)

Format

A data frame with 205 rows and 7 columns.

time: Survival time in years.
status: 1 Died from melanoma, 0 still alive or died from another event.
sex: 1=Male, 0=Female.
age: Age in years.
year: Year of operation.
thickness: Tumour thickness measured in mm.
ulcer: 1=Presence of ulceration, 0=Absence of ulceration.

Source

http://www.stats.ox.ac.uk/pub/MASS4/

References

Venables W.N., and Ripley, B.D. (2002). Modern Applied Statistics with S, Fourth edition. Springer, New York. ISBN 0-387-95457-0.

Andersen, P.K., Borgan, O., Gill, R.D., and Keiding, N. (1993) Statistical Models based on Counting Processes. Springer.

Plot the approximate posterior distribution of the penalty vector.

Description

The routine gives a graphical representation of the univariate approximate posterior distribution of the (log-)penalty parameters for objects of class coxlps, curelps, amlps and gamlps.

Usage

penaltyplot(object, dimension, ...)
penaltyplot(object, dimension, ...)

Arguments

`object`	An object of class `coxlps`, `curelps`, `amlps` or `gamlps`.
`dimension`	For objects of class `amlps` and `gamlps`, the penalty vector can have a dimension larger than one, i.e. more than a single smooth term is present in the considered additive model. In that case, `dimension` is the penalty dimension to be plotted corresponding either to a scalar indicating the desired dimension or to a vector indicating more than one dimension. For instance, dimension = c(1,3) displays two separate plots of the (approximate) posterior distribution of the (log-)penalty parameter associated to the first and the third smooth function respectively.
`...`	Further arguments to be passed to the routine.

Details

When q, the number of smooth term in a (generalized) additive model is smaller than five, the exploration of the posterior penalty space is based on a grid strategy. In particular, the multivariate grid of dimension q is constructed by taking the Cartesian product of univariate grids in each dimension j = 1,...q. These univariate grids are obtained from a skew-normal fit to the conditional posterior p(vj|vmap[-j]),D), where vj is the (log-)penalty value associated to the jth smooth function and vmap[-j] is the posterior maximum of the (log-)penalty vector omitting the jth dimension. The routine displays the latter skew-normal distributions. When q>=5, inference is based on vmap and the grid is omitted to avoid computational overflow. In that case, the posterior distribution of the (log-)posterior penalty vector v is approximated by a multivariate Gaussian and the routine shows the marginal distributions.

Author(s)

Oswaldo Gressani [email protected].

Examples

### Classic simulated data example (with simgamdata)

set.seed(123)
sim.data <- simgamdata(setting = 2, n = 250, dist = "gaussian", scale = 0.25)
plot(sim.data)         # Scatter plot of response
data <- sim.data$data  # Simulated data frame
# Fit model
fit <- amlps(y ~ z1 + z2 + sm(x1) + sm(x2), data = data, K = 15)
fit

# Penalty plot
opar <- par(no.readonly = TRUE)
par(mfrow = c(1, 2))
penaltyplot(fit, dimension = c(1, 2))
par(opar)

### Classic simulated data example (with simgamdata)

set.seed(123)
sim.data <- simgamdata(setting = 2, n = 250, dist = "gaussian", scale = 0.25)
plot(sim.data)         # Scatter plot of response
data <- sim.data$data  # Simulated data frame
# Fit model
fit <- amlps(y ~ z1 + z2 + sm(x1) + sm(x2), data = data, K = 15)
fit

# Penalty plot
opar <- par(no.readonly = TRUE)
par(mfrow = c(1, 2))
penaltyplot(fit, dimension = c(1, 2))
par(opar)

Plot smooth functions of an additive model object.

Description

Displays a plot of the fitted additive smooth components of an amlps object. The routine can also be used to print a table of point and set estimates of an additive smooth term for a user-specified grid of values.

Usage

## S3 method for class 'amlps'
plot(x, xp, smoo.index, cred.int = 0.95, plot.cred = TRUE,
     np = 100, fit.col = "blue", shade.col = "gray75", show.plot = TRUE,
     show.info = TRUE, ...)
## S3 method for class 'amlps'
plot(x, xp, smoo.index, cred.int = 0.95, plot.cred = TRUE,
     np = 100, fit.col = "blue", shade.col = "gray75", show.plot = TRUE,
     show.info = TRUE, ...)

Arguments

`x`	An object of class `amlps`.
`xp`	A numeric vector of grid values on which to compute a point estimate and pointwise credible interval for the smooth function specified in `smoo.index`. The components of `xp` must be within the range of the observed covariate values for the corresponding smooth function. Results will be displayed in a table.
`smoo.index`	The index of the smooth function. For instance `smoo.index = 2` refers to the second smooth function specified in the `formula` of the `amlps` routine.
`cred.int`	The level of the pointwise credible interval to be computed for the smooth additive term. Default is `0.95`.
`plot.cred`	Logical. Should the credible intervals be plotted? Default is `TRUE`.
`np`	The number of points used to construct the plot of the smooth additive function. Default is 100 and allowed values are between 20 and 200.
`fit.col`	The color of the fitted curve.
`shade.col`	The shading color for the credible intervals.
`show.plot`	Logical. Should the plot be displayed? Default is `TRUE`.
`show.info`	Logical. Should the table of point and set estimates of the smooth function on the specified `xp` values be displayed? Default is `TRUE`.
`...`	Further arguments to be passed to `plot`.

Details

Produces a plot of a smooth additive term fitted with the amlps function. On the y-axis, the estimated effective dimension of the smooth term is also displayed. At the bottom of each plot, vertical ticks indicate the location of the covariate values. The labels on the x-axis correspond to the covariate name associated to the smooth term.

Value

If xp is unspecified (the default), the routine will only return a plot of the estimated smooth curve. Otherwise, it provides a list with the following components:

`xp`	The chosen points on which to compute the smooth fit.
`sm.xp`	The estimated smooth fit at points specified in `xp`.
`sm.low`	The lower bound of the pointwise credible interval for the smooth additive function at points specified in `xp`.
`sm.up`	The upper bound of the pointwise credible interval for the smooth additive function at points specified in `xp`.
`cred.int`	The chosen level to compute credible intervals.
`smoo.index`	The index of the smooth function.

Author(s)

Oswaldo Gressani [email protected].

Examples


### Classic simulated data example

set.seed(3)

sim.data <- simgamdata(setting = 2, n = 200, dist = "gaussian", scale = 0.3)
plot(sim.data)         # Scatter plot of response
data <- sim.data$data  # Simulated data frame

# Fit model
fit <- amlps(y ~ z1 + z2 + sm(x1) + sm(x2), data = data, K = 20)
fit

# Plot fit of second function and results for a specific grid x
plot(fit, xp = c(-0.8, -0.4, 0, 0.4, 0.8), smoo.index = 2, ylim=c(-3, 3))

### Classic simulated data example

set.seed(3)

sim.data <- simgamdata(setting = 2, n = 200, dist = "gaussian", scale = 0.3)
plot(sim.data)         # Scatter plot of response
data <- sim.data$data  # Simulated data frame

# Fit model
fit <- amlps(y ~ z1 + z2 + sm(x1) + sm(x2), data = data, K = 20)
fit

# Plot fit of second function and results for a specific grid x
plot(fit, xp = c(-0.8, -0.4, 0, 0.4, 0.8), smoo.index = 2, ylim=c(-3, 3))

Plot baseline hazard and survival curves from a coxlps object.

Description

Produces a plot of the baseline hazard and/or survival based on a coxlps object.

Usage

## S3 method for class 'coxlps'
plot(x, S0 = TRUE, h0 = TRUE, cred.int = 0.95, overlay.km = FALSE,
     plot.cred = FALSE, np = 50, show.legend = TRUE, ...)
## S3 method for class 'coxlps'
plot(x, S0 = TRUE, h0 = TRUE, cred.int = 0.95, overlay.km = FALSE,
     plot.cred = FALSE, np = 50, show.legend = TRUE, ...)

Arguments

`x`	An object of class `coxlps`.
`S0`	Logical. Should the estimated baseline survival be plotted?
`h0`	Logical. Should the estimated baseline hazard be plotted?
`cred.int`	The level for an approximate pointwise credible interval to be computed for the baseline curves. Default is 0.95.
`overlay.km`	A logical value indicating whether the Kaplan-Meier estimate should be plotted together with the smooth baseline survival curve. The default is `FALSE`.
`plot.cred`	Logical. Should the credible intervals be plotted ? Default is `FALSE`.
`np`	The number of points used to plot the smooth baseline functions. Default is 50 and allowed values are between 20 and 200.
`show.legend`	Logical. Should a legend be displayed?
`...`	Further arguments to be passed to plot.

Details

Plots for the baseline hazard and survival curves are computed on a grid (of length np) between 0 and the 99th percentile of follow-up times. When plot.cred is FALSE, the fit omits to compute the approximate pointwise credible intervals for plotting and hence is less computationally intensive. Vertical ticks on the x-axis correspond to the observed follow-up times.

Author(s)

Oswaldo Gressani [email protected].

Examples



## Simulate survival data
set.seed(6)
betas <- c(0.35, -0.20, 0.05, 0.80) # Regression coefficients
data <- simsurvdata(a = 1.8, b = 2, n = 200, betas = betas, censperc = 25)
simdat <- data$survdata

# Fit model
fit <- coxlps(Surv(time, delta) ~ x1 + x2 + x3 + x4, data = simdat)
plot(fit, h0 = FALSE, S0 = TRUE, overlay.km = FALSE, show.legend = FALSE)
domt <- seq(0, 5.5, length = 500)
lines(domt, data$S0(domt), type = "l", col = "red")
legend("topright", c("Bayesian LPS", "Target"), col = c("black", "red"),
       lty = c(1, 1), bty = "n", cex = 0.8)

## Simulate survival data
set.seed(6)
betas <- c(0.35, -0.20, 0.05, 0.80) # Regression coefficients
data <- simsurvdata(a = 1.8, b = 2, n = 200, betas = betas, censperc = 25)
simdat <- data$survdata

# Fit model
fit <- coxlps(Surv(time, delta) ~ x1 + x2 + x3 + x4, data = simdat)
plot(fit, h0 = FALSE, S0 = TRUE, overlay.km = FALSE, show.legend = FALSE)
domt <- seq(0, 5.5, length = 500)
lines(domt, data$S0(domt), type = "l", col = "red")
legend("topright", c("Bayesian LPS", "Target"), col = c("black", "red"),
       lty = c(1, 1), bty = "n", cex = 0.8)

Plot estimated survival functions and cure probability for the promotion time cure model.

Description

The routine takes as input an object of class curelps and plots the estimated baseline survival curve, the population survival curve for a specific covariate profile and a a smooth curve for the cure probability. Approximate pointwise credible intervals are available.

Usage

## S3 method for class 'curelps'
plot(x, cred.int = 0.95, curvetype = c("baseline", "population",
     "probacure"), covar.profile, fit.col = "black", shade.col = "gray75",
      plot.cred = FALSE, np = 50, show.legend = TRUE, ...)
## S3 method for class 'curelps'
plot(x, cred.int = 0.95, curvetype = c("baseline", "population",
     "probacure"), covar.profile, fit.col = "black", shade.col = "gray75",
      plot.cred = FALSE, np = 50, show.legend = TRUE, ...)

Arguments

`x`	An object of class `curelps`.
`cred.int`	The level for an approximate pointwise credible interval to be computed for the smooth curves. Default is 0.95.
`curvetype`	The curve to be plotted; `baseline` (the default) will plot the estimated baseline survival, `population` the population survival function for a profile of covariates given in `covar.profile`, and `probacure` the probability to be cured (for a profile of covariates given in `covar.profile`) given that the subject has survived until time t.
`covar.profile`	A numeric vector of the same length as the number of covariates in the model. This corresponds to the profile of covariates for which to compute the population survival function and cure probability estimates. The order of the covariates in `covar.profile` is the same as the order specified in `formula` of the `curelps` routine. Each component of `covar.profile` should be in the range of the observed values for the corresponding covariate. If `covar.profile` is left unspecified by the user, the default will be to take the median covariate values.
`fit.col`	The color used for the estimated survival curve.
`shade.col`	The color used for the shading of the credible intervals.
`plot.cred`	Logical. Should the credible intervals be plotted? Default is `FALSE`.
`np`	The number of points used to plot the smooth curves. Default is 50 and allowed values are between 20 and 200.
`show.legend`	Show the legend? Default is TRUE.
`...`	Further arguments to be passed to `plot`.

Details

When plot.cred is FALSE, the routine omits to compute the approximate pointwise credible intervals for plotting and hence is less computationally intensive.

Author(s)

Oswaldo Gressani [email protected].

Examples


# Example on phase III clinical trial e1684 on melanoma data

data(ecog1684)

# Kaplan-Meier curve
plot(survfit(Surv(time, status) ~ 1, data = ecog1684), mark.time = TRUE)
fit <- curelps(Surv(time, status) ~ lt(age + trt + sex) +
             st(age + trt + sex), data = ecog1684, K = 20, penorder = 2)
fit
profile1 <- c(0, 1, 1, 0, 1, 1) # Mean age, trt = IFN, sex = Female.
profile2 <- c(0, 0, 1, 0, 0, 1) # Mean age, trt = control, sex = Female.
opar <- par(no.readonly = TRUE)
par(mfrow = c(1, 2))
plot(fit, curvetype = "probacure", plot.cred = TRUE, ylim = c(0,1),
     covar.profile = profile1, cred.int = 0.90,
     main = "Mean age, trt = IFN, sex = Female", cex.main = 0.8,
     show.legend = FALSE)
plot(fit, curvetype = "probacure", plot.cred = TRUE, ylim = c(0,1),
     covar.profile = profile2, cred.int = 0.90,
     main = "Mean age, trt = control, sex = Female", cex.main = 0.8,
     show.legend = FALSE)
par(opar)
# Example on phase III clinical trial e1684 on melanoma data

data(ecog1684)

# Kaplan-Meier curve
plot(survfit(Surv(time, status) ~ 1, data = ecog1684), mark.time = TRUE)
fit <- curelps(Surv(time, status) ~ lt(age + trt + sex) +
             st(age + trt + sex), data = ecog1684, K = 20, penorder = 2)
fit
profile1 <- c(0, 1, 1, 0, 1, 1) # Mean age, trt = IFN, sex = Female.
profile2 <- c(0, 0, 1, 0, 0, 1) # Mean age, trt = control, sex = Female.
opar <- par(no.readonly = TRUE)
par(mfrow = c(1, 2))
plot(fit, curvetype = "probacure", plot.cred = TRUE, ylim = c(0,1),
     covar.profile = profile1, cred.int = 0.90,
     main = "Mean age, trt = IFN, sex = Female", cex.main = 0.8,
     show.legend = FALSE)
plot(fit, curvetype = "probacure", plot.cred = TRUE, ylim = c(0,1),
     covar.profile = profile2, cred.int = 0.90,
     main = "Mean age, trt = control, sex = Female", cex.main = 0.8,
     show.legend = FALSE)
par(opar)

Plot smooth functions of a generalized additive model object.

Description

Displays a plot of the fitted additive smooth components of an gamlps object. The routine can also be used to print a table of point and set estimates of an additive smooth term for a user-specified grid of values.

Usage

## S3 method for class 'gamlps'
plot(x, xp, smoo.index, cred.int = 0.95, plot.cred = TRUE,
     np = 100, fit.col = "blue", shade.col = "gray75", show.plot = TRUE,
     show.info = TRUE, ...)
## S3 method for class 'gamlps'
plot(x, xp, smoo.index, cred.int = 0.95, plot.cred = TRUE,
     np = 100, fit.col = "blue", shade.col = "gray75", show.plot = TRUE,
     show.info = TRUE, ...)

Arguments

`x`	An object of class `gamlps`.
`xp`	A numeric vector of grid values on which to compute a point estimate and pointwise credible interval for the smooth function specified in `smoo.index`. The components of `xp` must be within the range of the observed covariate values for the corresponding smooth function. Results will be displayed in a table.
`smoo.index`	The index of the smooth function. For instance `smoo.index = 2` refers to the second smooth function specified in the `formula` of the `amlps` routine.
`cred.int`	The level of the pointwise credible interval to be computed for the smooth additive term. Default is `0.95`.
`plot.cred`	Logical. Should the credible intervals be plotted? Default is `TRUE`.
`np`	The number of points used to construct the plot of the smooth additive function. Default is 100 and allowed values are between 20 and 200.
`fit.col`	The color of the fitted curve.
`shade.col`	The shading color for the credible intervals.
`show.plot`	Logical. Should the plot be displayed? Default is `TRUE`.
`show.info`	Logical. Should the table of point and set estimates of the smooth function on the specified `xp` values be displayed? Default is `TRUE`.
`...`	Further arguments to be passed to `plot`.

Details

Produces a plot of a smooth additive term fitted with the gamlps function. On the y-axis, the estimated effective dimension of the smooth term is also displayed. At the bottom of each plot, vertical ticks indicate the location of the covariate values. The labels on the x-axis correspond to the covariate name associated to the smooth term.

Value

If xp is unspecified (the default), the routine will only return a plot of the estimated smooth curve. Otherwise, it provides a list with the following components:

`xp`	The chosen points on which to compute the smooth fit.
`sm.xp`	The estimated smooth fit at points specified in `xp`.
`sm.low`	The lower bound of the pointwise credible interval for the smooth additive function at points specified in `xp`.
`sm.up`	The upper bound of the pointwise credible interval for the smooth additive function at points specified in `xp`.
`cred.int`	The chosen level to compute credible intervals.
`smoo.index`	The index of the smooth function.

Author(s)

Oswaldo Gressani [email protected].

Print an additive partial linear model object.

Description

Print an additive partial linear model object.

Usage

## S3 method for class 'amlps'
print(x, ...)
## S3 method for class 'amlps'
print(x, ...)

Arguments

`x`	An object of class `amlps`.
`...`	Additional arguments to be passed to print.

Details

Prints informative output of a fitted additive partial linear model. In particular, the model formula, sample size, number of B-splines in basis, number of smooth terms, the chosen penalty order, the latent field dimension, the estimated coefficients of the linear part, the estimated standard deviation of the error and the effective dimension of the smooth terms are displayed.

Author(s)

Oswaldo Gressani [email protected].

Print a coxlps object.

Description

Print method for a coxlps object.

Usage

## S3 method for class 'coxlps'
print(x, ...)
## S3 method for class 'coxlps'
print(x, ...)

Arguments

`x`	An object of class `coxlps`.
`...`	Further arguments passed to print.

Details

Prints informative output of a fitted Cox proportional hazards model with Laplace-P-splines.

Author(s)

Oswaldo Gressani [email protected].

Print the fit of a promotion time cure model.

Description

Print method for a curelps object.

Usage

## S3 method for class 'curelps'
print(x, ...)
## S3 method for class 'curelps'
print(x, ...)

Arguments

`x`	An object of class `curelps`.
`...`	Further arguments to be passed to print.

Details

Prints informative output of a fitted promotion time cure model with the Laplace-P-spline approach. In particular, the model formula, number of B-splines in basis, chosen penalty order, latent field dimension, sample size, number of events and effective model dimension are provided. The estimated model coefficients related to the cure probability (long-term survival) and the population hazard dynamics (short-term survival) are also provided, where coef is the point estimate, sd.post the posterior standard deviation, z is the Wald test statistic and lower.95 and upper.95 the lower, respectively upper bounds of the approximate 95% pointwise credible interval.

Author(s)

Oswaldo Gressani [email protected].

Print a generalized additive model object.

Description

Print a generalized additive model object.

Usage

## S3 method for class 'gamlps'
print(x, ...)
## S3 method for class 'gamlps'
print(x, ...)

Arguments

`x`	An object of class `gamlps`.
`...`	Further arguments to be passed to print.

Details

Prints informative output of a fitted generalized additive model. In particular, the model formula, sample size, number of B-splines in basis, number of smooth terms, the chosen penalty order, the latent field dimension, model degrees of freedom, the estimated coefficients of the linear part and the estimated effective degrees of freedom of the smooth terms are displayed.

Author(s)

Oswaldo Gressani [email protected].

Simulation of survival times for the promotion time cure model.

Description

Generates right censored time-to-event data with a plateau in the Kaplan-Meier estimate.

Usage

simcuredata(n, censor = c("Uniform", "Weibull"), cure.setting = 1,
            info = TRUE, KapMeier = FALSE)
simcuredata(n, censor = c("Uniform", "Weibull"), cure.setting = 1,
            info = TRUE, KapMeier = FALSE)

Arguments

`n`	Sample size.
`censor`	The censoring scheme. Either Uniform (the default) or Weibull.
`cure.setting`	A number indicating the desired cure percentage. If `cure.setting = 1` (default) the cure percentage is around 20%. With `cure.setting = 2` the cure percentage is around 30%.
`info`	Should information regarding the simulation setting be printed to the console? Default is `TRUE`.
`KapMeier`	Logical. Should the Kaplan-Meier curve of the generated data be plotted? Default is `FALSE`.

Details

Latent event times are generated following Bender et al. (2005), with a baseline distribution chosen to be a Weibull with mean 8 and variance 17.47. When cure.setting = 1 the regression coefficients of the long-term survival part are chosen to yield a cure percentage around 20%, while cure.setting = 2 yields a cure percentage around 30%. Censoring is either governed by a Uniform distribution on the support [20, 25] or by a Weibull distribution with shape parameter 3 and scale parameter 25.

Value

A list with the following components:

`n`	Sample size.
`survdata`	A data frame containing the simulated data.
`beta.coeff`	The regression coefficients pertaining to long-term survival.
`gamma.coeff`	The regression coefficients pertaining to short-term survival.
`cure.perc`	The cure percentage.
`censor.perc`	The percentage of censoring.
`censor`	The censoring scheme.
`S0`	The baseline survival function under the chosen Weibull parameterization.

Author(s)

Oswaldo Gressani [email protected].

This function is based on a routine used to describe a simulation setting in Bremhorst and Lambert (2016). Special thanks go to Vincent Bremhorst who shared this routine during his PhD thesis.

References

Bender, R., Augustin, T. and Blettner, M. (2005). Generating survival times to simulate Cox proportional hazards models, Statistics in Medicine 24(11): 1713-1723.

Bremhorst, V. and Lambert, P. (2016). Flexible estimation in cure survival models using Bayesian P-splines. Computational Statistics & Data Analysis 93: 270-284.

Examples

set.seed(10)
sim <- simcuredata(n = 300, censor = "Weibull", KapMeier = TRUE)

set.seed(10)
sim <- simcuredata(n = 300, censor = "Weibull", KapMeier = TRUE)

Simulation of data for (Generalized) additive models.

Description

Simulation of a data set that can be used to illustrate the amlps or gamlps routines to fit (generalized) additive models with the Laplace-P-spline methodology.

Usage

simgamdata(setting = 1, n, dist = "gaussian", scale = 0.5, info = TRUE)
simgamdata(setting = 1, n, dist = "gaussian", scale = 0.5, info = TRUE)

Arguments

`setting`	The simulation setting. The default is `setting = 1` for a setting with three smooth terms, while `setting = 2` is another setting with only two smooth terms. The coefficients of the linear part of the predictor are also different in the two settings.
`n`	The sample size to simulate.
`dist`	A character string to specify the response distribution. The default is `"gaussian"`. Other distributions can be `"poisson"`, `"bernoulli"` and `"binomial"`.
`scale`	Used to tune the noise level for Gaussian and Poisson distributions.
`info`	Should information regarding the simulation be printed? Default is true.

Details

The simulation settings contain two covariates in the linear part of the predictor, namely z1 ~ Bern(0.5) and z2 ~ N(0,1). The smooth additive terms are inspired from Antoniadis et al. (2012). For Binomial data, the number of trials is fixed to 15.

Value

An object of class simgam. Plot of a simgam object yields a scatter plot of the generated response values.

`data`	A data frame.
`f`	The true smooth functions.
`betas`	The regression coefficients of the linear part. The first term is the intercept.
`dist`	The distribution of the response.

Author(s)

Oswaldo Gressani [email protected].

References

Antoniadis, A., Gijbels, I., and Verhasselt, A. (2012). Variable selection in additive models using P-splines. Technometrics 54(4): 425-438.

Examples

set.seed(10)
sim <- simgamdata(n = 150, dist = "poisson", scale = 0.3)
plot(sim)

set.seed(10)
sim <- simgamdata(n = 150, dist = "poisson", scale = 0.3)
plot(sim)

Simulation of right censored survival times for the Cox model.

Description

Generates right censored time-to-event data. Latent event times are drawn from a Weibull distribution, while censoring times are generated from an exponential distribution.

Usage

simsurvdata(a, b, n, betas, censperc, tmax = NULL)
simsurvdata(a, b, n, betas, censperc, tmax = NULL)

Arguments

`a`, `b`	The shape parameter 'a>0' and scale parameter 'b>0' of the Weibull.
`n`	Sample size.
`betas`	A numeric vector of regression coefficients. Allowed components of 'betas' are in the interval [-1 ,1] and the total number of components cannot exceed 5.
`censperc`	A numeric value in [0,100] corresponding to the targeted percentage of censoring.
`tmax`	A maximum upper bound for the generated latent event times. Especially useful for a simulation study in which the observed event times are constrained to be generated in a fixed range.

Details

The Weibull baseline hazard is parameterized as follows (see Hamada et al. 2008 pp. 408-409) :

$h_0(t) = (a/(b^a)) t^(a-1), t > 0.$

The ith latent event time is denoted by T_i and is generated following Bender et al. (2005) as follows:

$T_i = b (-log(U_i) exp(-\beta^T x_i))^(1/a),$

where U_i is a uniform random variable obtained with 'runif(1)' , x_i is the ith row of a covariate matrix X of dimension 'c(n, length(betas))' where each component is generated from a standard Gaussian distribution and $\beta$ is the vector of regression coefficients given by 'betas'.

Value

An object of class 'simsurvdata' which is a list with the following components:

`sample.size`	Sample size.
`censoring`	Censoring scheme. Either No censoring or Exponential.
`num.events`	Number of events.
`censoring.percentage`	The effective censoring percentage.
`survdata`	A data frame containing the simulated data.
`regcoeffs`	The true regression coefficients used to simulate the data.
`S0`	The baseline survival function under the chosen Weibull parameterization.
`h0`	The baseline hazard function under the chosen Weibull parameterization.
`Weibull.mean`	The mean of the Weibull used to generate latent event times.
`Weibull.variance`	The variance of the Weibull used to generate latent event times.

The 'print' method summarizes the generated right censored data and the 'plot' method produces a graph with time on the x axis and horizontal bars on the y axis corresponding either to an event or a right censored observation. If 'n > 25', only the 25 first observations are plotted.

Author(s)

Oswaldo Gressani [email protected].

References

Bender, R., Augustin, T. and Blettner, M. (2005). Generating survival times to simulate Cox proportional hazards models, Statistics in Medicine 24(11): 1713-1723.

Hamada, M. S., Wilson, A., Reese, C. S. and Martz, H. (2008). Bayesian Reliability. Springer Science and Business Media.

Examples

set.seed(10)
sim <- simsurvdata(a = 2, b = 1, n = 300, betas = c(0.8, -0.6), censperc = 25)
sim
plot(sim)

set.seed(10)
sim <- simsurvdata(a = 2, b = 1, n = 300, betas = c(0.8, -0.6), censperc = 25)
sim
plot(sim)

Fit a skew-normal distribution to a target density.

Description

The routine fits a skew-normal univariate distribution to a target density. Parameters of the resulting skew-normal fit are estimated by the method of moments.

Usage

snmatch(x, y, p = c(0.025, 0.5, 0.975))
snmatch(x, y, p = c(0.025, 0.5, 0.975))

Arguments

`x`	A numeric vector on the domain of the target density.
`y`	The y-coordinates of the target density on grid x.
`p`	Vector of probabilities at which to compute quantiles of the skew-normal fit.

Details

The skew-normal density is parameterized by a location parameter $\mu$ , a scale parameter $\omega$ and a shape parameter $\rho$ that regulates skewness. The probability density function at any x on the real line is:

$p(x) = (2/\omega) \phi((x-\mu)/\omega) \psi(\rho (x-\mu)/\omega),$

where $\phi()$ and $\psi()$ denote the standard Gaussian density and cumulative distribution function respectively (see Azzalini 2018). The first moment and second and third central moments of the target density are computed based on the x, y coordinates using the trapezoidal rule and matched against the theoretical moments of a skew-normal distribution. The solution to this system of equations is the method of moment estimate of the location, scale and shape parameters of a skew-normal density.

Value

A list with the following components:

`location`	Estimated location parameter.
`scale`	Estimated scale parameter.
`shape`	Estimated shape parameter.
`snfit`	Fitted values of the skew-normal density computed on an equally spaced grid between min(x) and max(x).
`quant`	Vector of quantiles of the skew-normal fit computed on the input vector of probabilities p.
`xgrid`	Equidistant grid on which the skew-normal fitted density is computed.

Author(s)

Oswaldo Gressani [email protected].

References

Azzalini, A. (2018). The Skew-Normal and Related families. Cambridge University Press.

Examples

# Pdf of skew-normal density
sn.target <- function(x, location, scale, shape){
               val <- 2 * stats::dnorm(x, mean = location, sd = scale) *
               pnorm(shape * (x - location) / scale)
               return(val)
              }

# Extract x and y coordinates from target
x.grid <- seq(-2, 6, length = 200)
y.grid <- sapply(x.grid, sn.target, location = 0, scale = 2, shape = 3)

# Computation of the fit and graphical illustration
fit <- snmatch(x.grid, y.grid)
domx <- seq(-2, 6, length = 1000)
plot(domx, sapply(domx, sn.target, location = 0, scale = 2, shape = 3),
     type = "l", ylab = "f(x)", xlab = "x", lwd= 2)
lines(fit$xgrid, fit$snfit, type="l", col = "red", lwd = 2, lty = 2)
legend("topright", lty = c(1,2), col = c("black", "red"), lwd = c(2, 2),
       c("Target","SN fit"), bty="n")

# Extract estimated parameters
fit$location # Estimated location parameter
fit$scale    # Estimated scale parameter
fit$shape    # Estimated shape parameter

# Pdf of skew-normal density
sn.target <- function(x, location, scale, shape){
               val <- 2 * stats::dnorm(x, mean = location, sd = scale) *
               pnorm(shape * (x - location) / scale)
               return(val)
              }

# Extract x and y coordinates from target
x.grid <- seq(-2, 6, length = 200)
y.grid <- sapply(x.grid, sn.target, location = 0, scale = 2, shape = 3)

# Computation of the fit and graphical illustration
fit <- snmatch(x.grid, y.grid)
domx <- seq(-2, 6, length = 1000)
plot(domx, sapply(domx, sn.target, location = 0, scale = 2, shape = 3),
     type = "l", ylab = "f(x)", xlab = "x", lwd= 2)
lines(fit$xgrid, fit$snfit, type="l", col = "red", lwd = 2, lty = 2)
legend("topright", lty = c(1,2), col = c("black", "red"), lwd = c(2, 2),
       c("Target","SN fit"), bty="n")

# Extract estimated parameters
fit$location # Estimated location parameter
fit$scale    # Estimated scale parameter
fit$shape    # Estimated shape parameter

Package 'blapsr'

Help Index

Bayesian additive partial linear modeling with Laplace-P-splines.

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Object resulting from the fit of an additive partial linear model.

Description

Value

Author(s)

See Also

Fit a Cox proportional hazards regression model with Laplace-P-splines.

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Extract estimated baseline quantities from a fit with coxlps.

Description

Usage

Arguments

Value

Author(s)

Examples

Object from a Cox proportional hazards fit with Laplace-P-splines.

Description

Value

Author(s)

See Also

Construct a cubic B-spline basis.

Description

Usage

Arguments

Value

Author(s)

References

Examples

Promotion time cure model with Laplace P-splines.

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Extract estimates of survival functions and cure probability for the promotion time cure model.

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Object from a promotion time model fit with Laplace-P-splines.

Description

Value

Author(s)

See Also

Phase III Melanoma clinical trial.

Description

Usage

Format

Source

References

Bayesian generalized additive modeling with Laplace-P-splines.

Description

Usage

Arguments