In fauvernierma/survPen: Multidimensional Penalized Splines for (Excess) Hazard Models, Relative Mortality Ratio Models and Marginal Intensity Models

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survPen

hazard and excess hazard modelling with multidimensional penalized splines

In survival and net survival analysis, in addition to modelling the effect of time (via the baseline hazard), one has often to deal with several continuous covariates and model their functional forms, their time-dependent effects, and their interactions. Model specification becomes therefore a complex problem and penalized regression splines represent an appealing solution to that problem as splines offer the required flexibility while penalization limits overfitting issues.

Current implementations of penalized survival models can be slow or unstable and sometimes lack some key features like taking into account expected mortality to provide net survival and excess hazard estimates. In contrast, survPen provides an automated, fast, and stable implementation (thanks to explicit calculation of the derivatives of the likelihood) and offers a unified framework for multidimensional penalized hazard and excess hazard models.

Later versions (>2.0.0) include penalized models for relative mortality ratio, and marginal intensity in recurrent event setting.

survPen may be of interest to those who 1) analyse any kind of time-to-event data: mortality, disease relapse, machinery breakdown, unemployment, etc 2) wish to describe the associated hazard and to understand which predictors impact its dynamics, 3) wish to model the relative mortality ratio between a cohort and a reference population, 4) wish to describe the marginal intensity for recurrent event data.

You can install this R package from GitHub:

install.packages("devtools")
library(devtools)
install_github("fauvernierma/survPen")

or directly from the CRAN repository:

install.packages("survPen")

The functionalities of the survPen package are extensively detailed in the following vignette

Available from GitHub or from CRAN

Quick start

In the following section, we will use a simulated dataset that contains artificial data from 2,000 women diagnosed with cervical cancer between 1990 and 2010. End of follow-up is June 30th 2013.

Let's fit a penalized hazard model and a penalized excess hazard model. Without covariates for now.

library(survPen)

f1 <- ~ smf(fu) # penalized natural cubic spline of time with 10 knots placed at the quantiles

mod.total <- survPen(f1,data=datCancer,t1=fu,event=dead,method="LAML")

mod.excess <- survPen(f1,data=datCancer,t1=fu,event=dead,expected=rate,method="LAML")

Now let's look at the predicted dynamics of the hazard (and excess hazard) as well as the predicted survival curve (and net survival curve)

lwd1 = 2

# compare the predictions of the models
new.time <- seq(0,5,length=100)
pred.total <- predict(mod.total,data.frame(fu=new.time))
pred.excess <- predict(mod.excess,data.frame(fu=new.time))

# hazard vs excess hazard
par(mfrow=c(1,2))
plot(new.time,pred.total$haz,type="l",ylim=c(0,0.2),main="hazard vs excess hazard",
xlab="time since diagnosis (years)",ylab="hazard",lwd=lwd1)
lines(new.time,pred.excess$haz,col="red",lwd=lwd1,lty=2)
legend("topright",legend=c("total","excess"),col=c("black","red"),lty=c(1,2), lwd=rep(lwd1,2))

plot(new.time,pred.total$surv,type="l",ylim=c(0,1),main="survival vs net survival",
xlab="time since diagnosis (years)",ylab="survival",lwd=lwd1)
lines(new.time,pred.excess$surv,col="red",lwd=lwd1,lty=2)
legend("bottomleft",legend=c("overall survival","net survival"), col=c("black","red"), lty=c(1,2), lwd=rep(lwd1,2)) 

Now we want to include the effect of age at diagnosis on the excess hazard. The dynamics of the excess hazard may be different from one age to another. Therefore we need to take into account an interaction between time and age. Thus, let's fit a penalized tensor product spline of time and age

f.tensor <- ~ tensor(fu,age) # penalized tensor product spline of time and age with 5*5 = 25 knots placed 
# at the quantiles

mod.tensor <- survPen(f.tensor,data=datCancer,t1=fu,event=dead,expected=rate)

summary(mod.tensor)

Now let's predict the excess hazard surface

new.age <- seq(50,90,length=50)
new.time <- seq(0,5,length=50)

Z.tensor <- outer(new.time,new.age,function(t,a) predict(mod.tensor,data.frame(fu=t,age=a))$haz)

# color settings
col.pal <- colorRampPalette(c("white", "red"))
colors <- col.pal(100)

facet <- function(z){

    facet.center <- (z[-1, -1] + z[-1, -ncol(z)] + z[-nrow(z), -1] + z[-nrow(z), -ncol(z)])/4
    cut(facet.center, 100)

}

theta1 = 30
zmax=1.1

# plot the excess hazard surface
par(mfrow=c(1,1))
persp(new.time,new.age,Z.tensor,col=colors[facet(Z.tensor)],main="tensor",theta=theta1,
xlab="\n time since diagnosis",ylab="\n age",zlab="\n excess hazard",
ticktype="detailed",zlim=c(0,zmax))

As you can see, tensor product splines allow capturing complex effects and interactions while the penalization limits overfitting issues. The example above shows a bi-dimensional function but you can use tensor product splines to model the interaction structure of more than two covariates.

Comparison with existing approaches and simulation study

survPen is based on the following method article (with associated supplementary material) https://rss.onlinelibrary.wiley.com/doi/full/10.1111/rssc.12368

The supplementary provides a comparison between survPen and existing approaches. The code associated with this comparison and with the simulation study from the article is available here https://github.com/fauvernierma/code_Fauvernier_JRSSC_2019

Contributing

File an issue here if there is a feature, or a dataset, that you think is missing from the package, or better yet submit a pull request!

Please note that the survPen project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.

Citing

If using survPen please consider citing the package in the relevant work. Citation information can be generated in R using the following (after installing the package),

citation("survPen")

You may also consider citing the method article: Fauvernier, M., Roche, L., Uhry, Z., Tron, L., Bossard, N. and Remontet, L. (2019). Multi-dimensional penalized hazard model with continuous covariates: applications for studying trends and social inequalities in cancer survival, Journal of the Royal Statistical Society, series C. doi: 10.1111/rssc.12368

fauvernierma/survPen documentation built on Dec. 3, 2024, 8:06 p.m.