Plot Function For Additive Cox Proportional Hazard Regression

Note: This initial blog post is discontinued. If you want to have the latest development of the function have a look at the static page PlotHR

Usually a Cox-regression is achieved in R by
library(survival)
model <- coxph ( SuvivalObject ~ Covariate1 + Covariate2 + Factor1 + Factor2 , data = Dataset )

The covariates can be enclosed in other funtions:

  • factors should be enclosed by factor()
  • strata, which allow to adjust for a factor without getting an estimate, should be enclosed by strata()
  • non-log linear continuous terms can be enclosed by
    pspline()

In the latter case the model might look like
model <- coxph (SurvivalObject ~ pspline(Covariate1) + Covariate2 + factor(Factor1) + strata(Factor2) , data = Dataset )

The functional form of the covariates (including the factors) can now be plotted with
termplot(model)

Though the termplot() function fails with plotting just one covariate and leaves no cusomization.

The function plotHR() plots the functional form of the desired term: plotHR(model)
plots the first term in the model by default but other terms can be accessed by calling their number (e.g. the second one):
plotHR(model , terms = 2)

In order to use the function you have to “source” it into R. It is the same procedure as calling a package, but using “source” instead of “library”.

Paste the function syntax into a textfile and safe it (as plot.HR.R) on your harddisk, remember the path and include
source("C:Path/to/plotHR_0.6.R")
before using the function.

Download plotHR()

Note: I have rewritten the function several times since I wrote the initial post … using version numbers now …

  • V0.6 – removed the y.log option, since the scale should be logaritmic anyway. Later I will also rewrite the x.log option, since the feature is already incorporated in the plot.default() function. I also removed the dottet line at HR=1 level, since some complained about it overstating the importance of the log(HR) intercept. I included it, since it gives a hint about the significance of the smooth term, in case the confidence intervalls cross over the line… Those who miss it can add manually lines( h = 0 , type = 2 )

    I rewrote the “rugs” option. Try rugs = "density" It is still “beta”ish, but some like it.

  • V0.5 – bug fix for the y-scale and slight adjustment of the default plotting colors (paler CI shade and stronger term-line)
  • V0.4 – the y-scale should be logarithmic; a HR of 0.5 (50% reduced Hazard) should show the same distance from HR = 1 as a doubled Hazard (HR = 2); this is now default. The linear scale I used initially is biased in this concern (Hat-tip: Arve Ulvik, Eva Pedersen and Roy Nilsen). The option y.log allows both ways (linear and log-scale); the axis labels denote Hazard Ratio instead of log(HR).

Usage:
plotHR( model , terms = 1 , se = TRUE , rug = "ticks" , x.log = FALSE , xlab = "" , ylab = "Hazard Ratio" , main = NULL , xlim = NULL , ylim = NULL, col.term = "#08519C", lwd.term = 3, col.se = "#DEEBF7", cex = 1 , bty = "n" , axes = TRUE )

  • model – a coxph model
  • terms – integer; the number of the term to plot
  • se – logical TRUE/FALSE; plotting the CI
  • rug – “ticks” or “density”; rug plot or density plot at x-axis. Any other value for “rug” will omit the rugplot.
  • x.log – logical TRUE/FALSE; log-transformed exposure variable
  • xlab – character; x-axis label
  • ylab – character; y-axis label
  • main – character; main plot title
  • xlim – 2×1 column vector; x-range of plot
  • ylim – 2×1 column vector; y-range of plot
  • col.term – color of HR-curve
  • lwd.term – line width of HR-curve
  • col.se – color of CI (if plotted)
  • cex – numeric; size factor of labels
  • bty – specifies the boxtype around the plot. See ?plot.default
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GAM Plot with 95% Confidence Shade

GAM plot with conficence "shade" and customized rugs
The lightblue shade denoting the 95% pointwise confidence limits of the GAM estimate is a polygon() object in R.

Usually one would plot the GAM model with the default termplot() function and specifiy se=T to get the confidence limits as lines. I showed in a recent post how to plot the fitted GAM smooth and the confidence limits manually.

The same approach can be used to have the confidence limits as a shade. In order to achieve this:

  1. Fit the model
  2. library(mgcv)
    model <- gam( MyResponse ~ MyCovariate , data = MyDataset)

  3. get the estimated values of the fitted smoothing spline
  4. fit <- predict( model , se = TRUE )$fit

  5. and pointwise standard errors
  6. se <- predict( model , se = TRUE)$se.fit

  7. calculate the values of the upper and lower 95%-confidence limits
  8. lcl <- fit - 1.96 * se
    ucl <- fit + 1.96 * se

  9. plot an empty coordinate system with right scaling, labels , titles and so on. I prefer to include axes = FALSE and add the axes manually. See here for an example.
  10. plot( 0 , type = "n" , bty = "n" , xlab = "The x-axis lable" , ylab = "The y-axis lable" , main = "The Title" , xlim = c(0,5) , ylim = c(0,3) )

  11. No it gets a bit tricky with sorting the coordinates in the right order. R provides a very effective way to achieve this, though it is not easily understandable at once. We create two indices on the dataset i.for and i.back which number the dataset according to the independend variable on the x-axis – forward and backward respectively:
  12. i.for <- order( MyDataset$MyCovariate )
    i.back <- order( MyDataset$MyCovariate , decreasing = TRUE )

  13. The points of the shade polygon follow here first the upper confidence line (ucl) forward and then the lower confidence line backward. The coordinates of the shade-polygon are
  14. x.polygon <- c( MyDataset$MyCovariate[i.for] , MyDataset$MyCovariate[i.back] )
    y.polygon <- c( ucl[i.for] , lcl[i.back] )

  15. First plot the polygon with the confidence shade – otherwise the GAM plot vanishes behind it.
  16. polygon( x.polygon , y.polygon , col = "#A6CEE3" , border = NA )

  17. now the mainplot afterwards
  18. lines( MyDataset$MyCovariate[i.for] , fit[i.for], col = "#1F78B4" , lwd = 3 )

  19. add a horizontal line marking the intercept ( the mean of the covariate) – note: univariate only. I will post a remark on this later…
  20. abline( h = mean(MyDataset$MyCovariate) , lty = 2 )

      Hope no typos. Comment if you find some.

Fancy Rugs in Regression Plots

Additive Regression Model with Customized Rug
Additive Regression Model with Customized Rug

Rugplots along the axes show the distribution of the underlying data in regression model plots. This is particulary useful in connection with additive (nonparametric) models where the plotted smooth function is the exclusive representation of the model in order to assess how much data contributed to the model fit at the different values of the exlanatory variable.

The custom plot.gam() function includes the possibility of such rugs and pointwise conficence intervalls by default.

Adding quartiles to the rugs requires some customization, though. I included the complete code to produce the above plot underneath.

The example for this GAM is borrowed from the excellent book of Alain Zuur et. al. Mixed Effects Models and Extensions in Ecology with R p.55ff. The ISIT data to run the code above is included in the R package AED which can be downloaded from the books website.

Note: the package AED is needed for the example dataset only. It is NOT necessary to use the example code on ones own dataset.

The main points concerning the rugs and quantile lables on the x-axis are:

  1. Plot the coordinate system without lables
  2. plot( ... , axes = FALSE )

  3. Plot the x-axis with x-lables
  4. axis(side = 1 , line = 0.3 , at = 0:5*1000 , tick = TRUE)

  5. Plot rugs – the jitter() is necessary since a lot of datapoints sit on the same values of SampleDepth – so shake them a bit.
  6. axis(side = 1 , line = -0.9 , at = jitter(ISIT$SampleDepth) , labels = F , tick = T , tcl = 0.8 , lwd.ticks = 0.1 , lwd = 0)

  7. Print the lables “1Q”, “Median” and “3Q” or whatever you like to call them on the right position. The line and padj parameter set the position of the text and cex.axis the textsize.
  8. axis(side = 1 , line = -0.8 , at = fivenum(ISIT$SampleDepth)[2:4], lwd = 0 , tick = F , labels = c("1Q","median","3Q"), cex.axis = 0.7, col.axis = "black" , padj = -2.8)

  9. Plot thick tickmarks crossing through the rug cloud at 1Q, median and 3Q
  10. axis(side = 1 , line = -0.8 , at = fivenum(ISIT$SampleDepth)[2:4], lwd = 0 , tick = T, tcl = 1.1 , lwd.ticks = 1 , col.ticks = "black", labels = FALSE)

  11. and finally short thick tickmarks under the text touching the x-axis
  12. axis(side = 1 , line = 0.3, at = fivenum(ISIT$SampleDepth)[2:4], lwd = 0 , tick = T, tcl = 0.2 , lwd.ticks = 1 , col.ticks = "black", labels = FALSE)

Here goes the complete code:
library(mgcv)
library(AED)
data(ISIT)
#
# Fit a univariate GAM model
model <- gam(Sources ~ s(SampleDepth) , data = ISIT)
fit <- predict(model, se = T)$fit
se <- predict(model, se = T)$se.fit
lcl <- fit - 1.96 * se
ucl <- fit + 1.96 * se
#
# open a jpeg
jpeg("FancyRugs.jpg" , width=400, height=400)
#
# set plotting options: 1 plot per page, horizontal labels and textsize
par(mfrow = c(1,1) , las = 1 , cex = 1)
#
# plot coordinatesystem and labels
plot(0 , bty = "n" , type = "n" , xlim = c(0,5000) , ylim = c(-10,50) , xlab = "Depth (m)" , ylab = expression(paste("Number of sources (" , m^-3 , ")")) , axes = FALSE)
#
title(main="Association between number of sources of\nbioluminescent organisms and ocean depth" , cex.main = 0.8)
#
## _____ X-AXIS ______
# x-axis values
axis(side = 1 , line = 0.3 , at = 0:5*1000 , tick = TRUE)
#
# rugs at datapoints
axis(side = 1 , line = -0.9 , at = jitter(ISIT$SampleDepth) , labels = F , tick = T , tcl = 0.8 , lwd.ticks = 0.1 , lwd = 0)
#
# labels at 1Q, median and 3Q
axis(side = 1 , line = -0.8 , at = fivenum(ISIT$SampleDepth)[2:4], lwd = 0 , tick = F , labels = c("1Q","median","3Q"), cex.axis = 0.7, col.axis = "black" , padj = -2.8)
#
# tick marks at 1Q, median and 3Q
axis(side = 1 , line = 0.3, at = fivenum(ISIT$SampleDepth)[2:4], lwd = 0 , tick = T, tcl = 0.2 , lwd.ticks = 1 , col.ticks = "black", labels = FALSE)
#
axis(side = 1 , line = -0.8 , at = fivenum(ISIT$SampleDepth)[2:4], lwd = 0 , tick = T, tcl = 1.1 , lwd.ticks = 1 , col.ticks = "black", labels = FALSE)
#
## _____ Y-AXIS ______
# y-axis values
axis(side = 2 , at = 0:5*10)
#
# rugs at datapoints
axis(side = 2 , line = -0.9 , at = jitter(ISIT$Sources) , labels = F , tick = T , tcl = 0.8 , lwd.ticks = 0.1 , lwd = 0)
#
# labels at 1Q, median and 3Q
axis(side = 2 , line = -0.7 , at = fivenum(ISIT$Sources)[2:4], lwd = 0 , tick = F , labels = c("1Q","median","3Q"), cex.axis = 0.7, col.axis = "black")
#
# thicker tick marks at 1Q, median and 3Q
axis(side = 2 , line = 0.3, at = fivenum(ISIT$Sources)[2:4], lwd = 0 , tick = T, tcl = 0.3 , lwd.ticks = 1 , col.ticks = "black", labels = FALSE , padj = -2)
axis(side = 2 , line = -0.7 , at = fivenum(ISIT$Sources)[2:4], lwd = 0 , tick = T, tcl = 1.1 , lwd.ticks = 1 , col.ticks = "black" , labels = FALSE)
#
# horizontal line marking the intercept = mean(Sources) (for univariate model only)
abline(h=mean(ISIT$Sources), lty=3)
#
# Scatterplot
lines(ISIT$SampleDepth , ISIT$Source , type = "p" , cex = 0.4 , lwd = 0.2 , col = "grey")
#
# plot main figure
lines(ISIT$SampleDepth[order(ISIT$SampleDepth)] , fit[order(ISIT$SampleDepth)] , col = "black" , lwd = 2)
#
# plot lower confidence limit (lcl)
lines(ISIT$SampleDepth[order(ISIT$SampleDepth)] , lcl[order(ISIT$SampleDepth)] , col = "grey" , lwd = 1)
#
# plot upper confidence limit (ucl)
lines(ISIT$SampleDepth[order(ISIT$SampleDepth)] , ucl[order(ISIT$SampleDepth)] , col = "grey" , lwd = 1)
#
# closing the jpg file
dev.off()

How2plot nicer GAM curves

Generalized additive models visualize potential non-linear associations between a predictor and a response variable.

The default plotting method produces clean plots for all covariates in the model (or a selection) but: They do not have presentation quality by any means in terms of colors, understandable axes-labels, scaling, etc.

This example shows a customization variant for a additive regression model with one covariate. The goal was to display the absolute value of the response variable on the y-axis and not the “difference from intercept” which is default.

This is only meaningful for a single covariate in the model.

1 The default plot.gam() method

MyGAM1<- with(MyData[MyData$Strata==1,], gam(Y ~ s(Covariate)))
MyGAM0<- with(MyData[MyData$Strata==0,], gam(Y ~ s(Covariate)))

par( mfcol=c(1,2))
plot(MyGAM0)
plot(MyGAM1)

This is the resulting plot:

Stratified Additive Regression Model
Stratified Additive Regression Model

2 The fancy way

1. Extract the values of the model response from the GAM object:

response1 <- predict(MyGAM1, type="response", se.fit=T)
response0 <- predict(MyGAM0, type="response", se.fit=T)

2. Print the response values against the covariate (note: this works just with one covariate)

par(mfcol=c(1,1))

plot(0, type="n", bty="n", main="Fancy GAM plot", xlab="MyCovariate", ylab="MyResponse", lwd=3,ylim=c(0,60), xlim=c(0,200))
legend("bottomright", bty="n", lwd=5, col=c("green","red"), legend=c("Strata = 0", "Strata = 1"))

lines(sm.spline(MyGAM1$model$Covariate , response1$fit) , lwd = 3 , col = "red")
lines(sm.spline(MyGAM1$model$Covariate , response1$fit+1.96*response1$se) , lty = 3 , lwd = 2 , col = "red")
lines(sm.spline(MyGAM1$model$Covariate , response1$fit-1.96*response1$se) , lty = 3 , lwd = 2 , col = "red")

lines(sm.spline(MyGAM0$model$Covariate , response0$fit) , lwd = 3 , col = "green")
lines(sm.spline(MyGAM0$model$Covariate, response0$fit + 1.96 * response0$se) , lty = 3 , lwd = 2, col = "green")
lines(sm.spline(MyGAM0$model$Covariate, response0$fit - 1.96 * response0$se) , lty = 3 , lwd = 2 , col = "green")

abline(h=gam.dm1$coefficients[1], lty=2, lwd=1, col="red")
abline(h=gam.dm0$coefficients[1], lty=2, lwd=1, col="green")

Stratified Additive Regression Model on Response Scale
Stratified Additive Regression Model on Response Scale

 

Additive COX-regression

Update:I have written a much more detailed static page about the additive COX model: http://rforge.org/plothr/
The page has a download link to the function plotHR() which does all the fuzz. It is extensively commented. It should be easy to understand the syntax and modify it for individual purposes.

Therneau et al. refer to the proportional hazards model or COX-regression model as “the workhorse of regression analysis for censored data”. They show how to implement the additive form of this model in SAS and S-pluss; already mentioned by Hastie and Tibshirany in 1986 when introducing Generalized Additive Models (GAM).

I found modelling the functional form of the covariates in a regression model for rightcensored survival times with smoothing splines extremely useful. And the implementation is absolutely straightforward in R.

The only thing needed is the installation of the R-libraries “survival” and “pspline”:

install.packages("pspline")
and
install.packages("survival")

In the following code I will refer to a dataset “MyData” with a binary status variable “death” and a time-to-event variable “days2death”.
The status variable “death” should be (not necessarily) 1 if the event of interesst occured to the subject and “days2death” gives then the time to this event.

Viualizing the functional form of a covariate takes the following steps:

  1. create the survival object of interesst
  2. fit a proportional hazards model with smoothing splines,
  3. predict the functional form of the covariate of interesst and
  4. plot it!

Note that there is the termplot() function in R which gives you the GAM plots after the modelfit, so step 3 would not be necessary – BUT: it has a bug and fails plotting a single covariate; and it does not allow all to much customizing.

This is the R code to achieve the analysis:

1 Create survival object:

surv.death <- Surv(MyData$days2death, MyData$death)

2 Fit proportional hazards model with smoothing splines for continuous covariates:

library(survival)
library(pspline)
pham.fit <- coxph( surv.death ~ pspline(EF, df=4) + pspline(Age, df=4) + strata (Sex, df=4) , data = MyData)

The model above includes the continuous covariates “EF” (ejection fraction) and “Age” and stratifies for “Sex”.

3 Produce the fitted smoothing spline for the first covariate in the above model formula with standard errors

predicted <- predict(pham.fit , type = "terms" , se.fit = TRUE , terms = 1)
“terms=1” refers to “pspline(EF,df=4)”

4 Plot it

First plotting axes and labels
plot(0 , xlab="Ejection Fraction" , ylab = "Hazard Ratio" , main = "All-cause Death" , type = "n" , xlim=c(0,100) , ylim=c(0,3))
the range of values on the x-axis (“xlim=c(0,100)”) is chosen manually for this specific covariate; of course it is possible to use something like ylim = c( 0 , max(MyData$EF) ).

Now plot the fitted smoothing spline using the lines() function:
lines( sm.spline(MyData$EF , exp(predicted$fit)) , col = "red" , lwd = 0.8)
Note that the term prediction gives log-hazard-ratios; therefore exp(predicted$fit) is plotted against the values of the covariate. The sm.spline() function is necessary since the points of the plot appear in random order and density, according to the underlying dataset; a plain lines() function would produce just a chaotic pattern. Alternative:
plot(MyData$EF , exp(predicted$fit) , col = "red" , cex = 0.2)
produces a scattered plot that reflects the distribution of the underlying data – I do prefer adding a rug-plot on the bottom of the graph to illustrate this (see under).

… upper and lower confidence limits with dashed thinner lines

lines(sm.spline(MyData$EF , exp(predicted$fit + 1.96 * predicted$se)) , col = "orange" , lty = 2 , lwd = 0.4)
and
lines(sm.spline(MyData$EF , exp(predicted$fit - 1.96 * predicted$se)) , col = "orange" , lty = 2 , lwd = 0.4)

… a tiny horizontal line at hazard level 1, do see where the confidence limits cross:
abline( h = 1 , col = "lightgrey" , lty = 2 , lwd = 0.4)

… tiny tickmarks on the x-axes to reflect the distribution of the underlying data:
axis( side = 1 , at = MyData$EF, labels = F , tick = T , tcl = 0.4 , lwd.ticks = 0.1)

… and some fancy red tickmarks to mark minimum, lower hinge, median, upper hinge and maximum of the covariate in the dataset:
axis( side = 1 , at = fivenum(MyData$EF), labels = F , tick = T , tcl = -0.2 , lwd.ticks = 1 , col.ticks = "red")

Fancy customized smoothing spline fitted to the functional form of a covariate in a additive proportional hazard model
Fancy customized smoothing spline fitted to the functional form of a covariate in a additive proportional hazard model

Thats it!

4b) The easy way (works ONLY with MORE then 1 continous covariate) – predicting the terms can be omitted:

termplot(pham.fit, se=T, rug=T)

Resulting in …

The default termplot method for fitted smoothing splines
The default termplot method for fitted smoothing splines