It is a challenging task to model the emerging highdimensional clinical data with survival outcomes. For its simplicity and efficiency, penalized Cox models are significantly useful for accomplishing such tasks.
hdnom
streamlines the workflow of highdimensional Cox model building, nomogram plotting, model validation, calibration, and comparison. To load the package in R, simply type:
library("hdnom")
To build a penalized Cox model with good predictive performance, some parameter tuning is usually needed. For example, the elasticnet model requires to tune the \(\ell_1\)\(\ell_2\) penalty tradeoff parameter \(\alpha\), and the regularization parameter \(\lambda\).
To free the users from the tedious and errorprone parameter tuning process, hdnom
provides several functions for automatic parameter tuning and model selection, including the following model types:
Function Name  Model Type 


Lasso 

Adaptive lasso 

Fused lasso 

Elasticnet 

Adaptive elasticnet 

MCP 

Mnet (MCP + elasticnet) 

SCAD 

Snet (SCAD + elasticnet) 
In the next, we will use the imputed SMART study data (Steyerberg 2008) to demonstrate a complete process of model building, nomogram plotting, model validation, calibration, and comparison with hdnom
.
Load the smart
dataset:
data("smart")
x = as.matrix(smart[, c(1, 2)])
time = smart$TEVENT
event = smart$EVENT
library("survival")
y = Surv(time, event)
The dataset contains 3873 observations with corresponding survival outcome (time
, event
). 27 clinical variables (x
) are available as the predictors. See ?smart
for a detailed explanation of the variables.
Fit a penalized Cox model by adaptive elasticnet regularization, with hdcox.aenet
:
# enable parallel parameter tuning
suppressMessages(library("doParallel"))
registerDoParallel(detectCores())
fit = hdcox.aenet(x, y, nfolds = 10, rule = "lambda.1se",
seed = c(5, 7), parallel = TRUE)
names(fit)
## [1] "seed" "enet_best_alpha" "enet_best_lambda"
## [4] "enet_model" "aenet_best_alpha" "aenet_best_lambda"
## [7] "aenet_model" "pen_factor"
Adaptive elasticnet includes two estimation steps. The random seed used for parameter tuning, the selected best \(\alpha\), the selected best \(\lambda\), the model fitted for each estimation step, and the penalty factor for the model coefficients in the second estimation step are all stored in the model object fit
.
Before plotting the nomogram, we need to extract some necessary information about the model, namely, the model object and parameters, from the result of the last step:
model = fit$aenet_model
alpha = fit$aenet_best_alpha
lambda = fit$aenet_best_lambda
adapen = fit$pen_factor
To plot the nomogram, first we make x
available as a datadist
object for the rms
package (Harrell 2013), then generate a hdnom.nomogram
object with hdnom.nomogram()
, and plot the nomogram:
suppressMessages(library("rms"))
x.df = as.data.frame(x)
dd = datadist(x.df)
options(datadist = "dd")
nom = hdnom.nomogram(model, model.type = "aenet",
x, time, event, x.df, pred.at = 365 * 2,
funlabel = "2Year Overall Survival Probability")
plot(nom)
According to the nomogram, the adaptive elasticnet model selected 6 variables from the original set of 27 variables, effectively reduced the model complexity.
Information about the nomogram itself, such as the pointlinear predictor unit mapping and total pointssurvival probability mapping, can be viewed by printing the nom
object directly.
It is a common practice to utilize resamplingbased methods to validate the predictive performance of a penalized Cox model. Bootstrap, \(k\)fold crossvalidation, and repeated \(k\)fold crossvalidation are the most employed methods for such purpose.
hdnom
supports both internal model validation and external model validation. Internal validation takes the dataset used to build the model and evaluates the predictive performance on the data internally with the above resamplingbased methods, while external validation evaluates the model’s predictive performance on a dataset which is independent to the dataset used in model building.
hdnom.validate()
allows us to assess the model performance internally by timedependent AUC (Area Under the ROC Curve) with the above three resampling methods.
Here, we validate the performance of the adaptive elasticnet model with bootstrap resampling, at every half year from the first year to the fifth year:
val.int = hdnom.validate(x, time, event, model.type = "aenet",
alpha = alpha, lambda = lambda, pen.factor = adapen,
method = "bootstrap", boot.times = 10,
tauc.type = "UNO", tauc.time = seq(1, 5, 0.5) * 365,
seed = 42, trace = FALSE)
val.int
## HighDimensional Cox Model Validation Object
## Random seed: 42
## Validation method: bootstrap
## Bootstrap samples: 10
## Model type: aenet
## glmnet model alpha: 0.15
## glmnet model lambda: 2.559313e+12
## glmnet model penalty factor: specified
## Timedependent AUC type: UNO
## Evaluation time points for tAUC: 365 547.5 730 912.5 1095 1277.5 1460 1642.5 1825
summary(val.int)
## TimeDependent AUC Summary at Evaluation Time Points
## 365 547.5 730 912.5 1095 1277.5
## Mean 0.6633218 0.6908359 0.6835247 0.6775386 0.7095764 0.7301827
## Min 0.6568324 0.6864658 0.6783314 0.6747490 0.7025513 0.7224083
## 0.25 Qt. 0.6589450 0.6878048 0.6820924 0.6759995 0.7091919 0.7298811
## Median 0.6656771 0.6917197 0.6840430 0.6777887 0.7099562 0.7310424
## 0.75 Qt. 0.6664824 0.6932608 0.6850584 0.6788960 0.7120672 0.7327024
## Max 0.6679094 0.6953842 0.6868593 0.6802138 0.7149459 0.7349118
## 1460 1642.5 1825
## Mean 0.6716061 0.6731475 0.6883837
## Min 0.6647284 0.6693436 0.6783940
## 0.25 Qt. 0.6693304 0.6703827 0.6875514
## Median 0.6721205 0.6735065 0.6889860
## 0.75 Qt. 0.6746232 0.6753697 0.6906602
## Max 0.6775177 0.6770954 0.6952586
The mean, median, 25%, and 75% quantiles of timedependent AUC at each time point across all bootstrap predictions are listed above. The median and the mean can be considered as the biascorrected estimation of the model performance.
It is also possible to plot the model validation result:
plot(val.int)
The solid line represents the mean of the AUC, the dashed line represents the median of the AUC. The darker interval in the plot shows the 25% and 75% quantiles of AUC, the lighter interval shows the minimum and maximum of AUC.
It seems that the bootstrapbased validation result is stable: the median and the mean value at each evaluation time point are close; the 25% and 75% quantiles are also close to the median at each time point.
Bootstrapbased validation often gives relatively stable results. Many of the established nomograms in clinical oncology research are validated by bootstrap methods. \(K\)fold crossvalidation provides a more strict evaluation scheme than bootstrap. Repeated crossvalidation gives similar results as \(k\)fold crossvalidation, and usually more robust. These two methods are more applied by the machine learning community. Check ?hdnom.validate
for more examples about internal model validation.
Now we have the internally validated model. To perform external validation, we usually need an independent dataset (preferably, collected in other studies), which has the same variables as the dataset used to build the model. For penalized Cox models, the external dataset should have at least the same variables that have been selected in the model.
For demonstration purposes, here we draw 1000 samples from the smart
data and assume that they form an external validation dataset, then use hdnom.external.validate()
to perform external validation:
x_new = as.matrix(smart[, c(1, 2)])[1001:2000, ]
time_new = smart$TEVENT[1001:2000]
event_new = smart$EVENT[1001:2000]
# External validation with timedependent AUC
val.ext =
hdnom.external.validate(fit, x, time, event,
x_new, time_new, event_new,
tauc.type = "UNO",
tauc.time = seq(0.25, 2, 0.25) * 365)
val.ext
## HighDimensional Cox Model External Validation Object
## Model type: aenet
## Timedependent AUC type: UNO
## Evaluation time points for tAUC: 91.25 182.5 273.75 365 456.25 547.5 638.75 730
summary(val.ext)
## TimeDependent AUC Summary at Evaluation Time Points
## 91.25 182.5 273.75 365 456.25 547.5 638.75
## AUC 0.4328241 0.5608442 0.6299909 0.6289888 0.6530596 0.6728417 0.6795879
## 730
## AUC 0.6922393
plot(val.ext)
The timedependent AUC on the external dataset is shown above.
Measuring how far the model predictions are from actual survival outcomes is known as calibration. Calibration can be assessed by plotting the predicted probabilities from the model versus actual survival probabilities. Similar to model validation, both internal model calibration and external model calibration are supported in hdnom
.
hdnom.calibrate()
provides nonresampling and resampling methods for internal model calibration, including direct fitting, bootstrap resampling, \(k\)fold crossvalidation, and repeated crossvalidation.
For example, to calibrate the model internally with the bootstrap method:
cal.int = hdnom.calibrate(x, time, event, model.type = "aenet",
alpha = alpha, lambda = lambda, pen.factor = adapen,
method = "bootstrap", boot.times = 10,
pred.at = 365 * 5, ngroup = 3,
seed = 42, trace = FALSE)
cal.int
## HighDimensional Cox Model Calibration Object
## Random seed: 42
## Calibration method: bootstrap
## Bootstrap samples: 10
## Model type: aenet
## glmnet model alpha: 0.15
## glmnet model lambda: 2.559313e+12
## glmnet model penalty factor: specified
## Calibration time point: 1825
## Number of groups formed for calibration: 3
summary(cal.int)
## Calibration Summary Table
## Predicted Observed Lower 95% Upper 95%
## 1 0.8002909 0.7614210 0.7329893 0.7909556
## 2 0.8940495 0.8903038 0.8687012 0.9124435
## 3 0.9245110 0.9343037 0.9168044 0.9521371
We split the samples into three risk groups. In practice, the number of risk groups is decided by the users according to their needs.
The model calibration results (the median of the predicted survival probability; the median of the observed survival probability estimated by KaplanMeier method with 95% CI) are summarized as above.
Plot the calibration result:
plot(cal.int, xlim = c(0.5, 1), ylim = c(0.5, 1))
In practice, you may want to perform calibration for multiple time points separately, and put the plots together in one figure. See ?hdnom.calibrate
for more examples about internal model calibration.
To perform external calibration with an external dataset, use hdnom.external.calibrate()
:
cal.ext =
hdnom.external.calibrate(fit, x, time, event,
x_new, time_new, event_new,
pred.at = 365 * 5, ngroup = 3)
cal.ext
## HighDimensional Cox Model External Calibration Object
## Model type: aenet
## Calibration time point: 1825
## Number of groups formed for calibration: 3
summary(cal.ext)
## External Calibration Summary Table
## Predicted Observed Lower 95% Upper 95%
## 1 0.7940258 0.7533312 0.7057400 0.8041316
## 2 0.8916822 0.8667762 0.8296887 0.9055215
## 3 0.9214927 0.9387588 0.9122184 0.9660715
plot(cal.ext, xlim = c(0.5, 1), ylim = c(0.5, 1))
The external calibration results have the similar interpretations as the internal calibration results, except the fact that external calibration is performed on the external dataset.
Internal calibration and external calibration both classify the testing set into different risk groups. For internal calibration, the testing set means all the samples in the dataset that was used to build the model, for external calibration, the testing set means the samples from the external dataset.
We can further analyze the differences in survival time for different risk groups with KaplanMeier survival curves and a number at risk table. For example, here we plot the KaplanMeier survival curves and evaluate the number at risk from one year to six years for the three risk groups, with the function hdnom.kmplot()
:
hdnom.kmplot(cal.int, group.name = c('High risk', 'Medium risk', 'Low risk'),
time.at = 1:6 * 365)
hdnom.kmplot(cal.ext, group.name = c('High risk', 'Medium risk', 'Low risk'),
time.at = 1:6 * 365)
The \(p\)value of the logrank test is also shown in the plot.
To compare the differences between the survival curves, logrank test is often applied. hdnom.logrank()
performs such tests on the internal calibration and external calibration results:
cal.int.logrank = hdnom.logrank(cal.int)
cal.int.logrank
## Call:
## survdiff(formula = formula("Surv(time, event) ~ grp"))
##
## n=3872, 1 observation deleted due to missingness.
##
## N Observed Expected (OE)^2/E (OE)^2/V
## grp=1 1290 274 152 97.67 146.1
## grp=2 1291 121 157 8.07 12.2
## grp=3 1291 65 151 49.26 73.5
##
## Chisq= 155 on 2 degrees of freedom, p= 0
cal.int.logrank$pval
## [1] 1.962792e34
cal.ext.logrank = hdnom.logrank(cal.ext)
cal.ext.logrank
## Call:
## survdiff(formula = formula("Surv(time, event) ~ grp"))
##
## n=999, 1 observation deleted due to missingness.
##
## N Observed Expected (OE)^2/E (OE)^2/V
## grp=1 333 81 45.2 28.37 41.11
## grp=2 333 42 49.5 1.15 1.74
## grp=3 333 24 52.3 15.28 23.75
##
## Chisq= 45 on 2 degrees of freedom, p= 1.72e10
cal.ext.logrank$pval
## [1] 1.720597e10
The exact \(p\)values for logrank tests are stored as cal.int.logrank$pval
and cal.ext.logrank$pval
. Here \(p < 0.001\) indicates significant differences between the survival curves for different risk groups.
Given all the available model types, it is a natural question to ask: which type of model performs the best for my data? Such questions about model type selection can be answered by builtin model comparison functions in hdnom
.
We can compare the model performance using timedependent AUC by the same (internal) model validation approach as before. For example, here we compare lasso and adaptive lasso by 5fold crossvalidation:
cmp.val =
hdnom.compare.validate(x, time, event,
model.type = c("lasso", "alasso"),
method = "cv", nfolds = 5, tauc.type = "UNO",
tauc.time = seq(0.25, 2, 0.25) * 365,
seed = 42, trace = FALSE)
cmp.val
## HighDimensional Cox Model Validation Object
## Random seed: 42
## Validation method: kfold crossvalidation
## Crossvalidation folds: 5
## Model type: lasso
## glmnet model alpha: 1
## glmnet model lambda: 0.01952497
## glmnet model penalty factor: not specified
## Timedependent AUC type: UNO
## Evaluation time points for tAUC: 91.25 182.5 273.75 365 456.25 547.5 638.75 730
##
## HighDimensional Cox Model Validation Object
## Random seed: 42
## Validation method: kfold crossvalidation
## Crossvalidation folds: 5
## Model type: alasso
## glmnet model alpha: 1
## glmnet model lambda: 1.786852
## glmnet model penalty factor: specified
## Timedependent AUC type: UNO
## Evaluation time points for tAUC: 91.25 182.5 273.75 365 456.25 547.5 638.75 730
summary(cmp.val)
## Model type: lasso
## 91.25 182.5 273.75 365 456.25 547.5
## Mean 0.5213013 0.6134751 0.6475741 0.6482503 0.6557042 0.6793388
## Min 0.4144015 0.5453072 0.5849391 0.6145979 0.6175435 0.6621891
## 0.25 Qt. 0.4775529 0.5846219 0.6309895 0.6180075 0.6199005 0.6641325
## Median 0.5112358 0.6210905 0.6542970 0.6565121 0.6645453 0.6643884
## 0.75 Qt. 0.5504564 0.6499560 0.6750810 0.6753578 0.6842037 0.6917158
## Max 0.6528601 0.6664000 0.6925638 0.6767762 0.6923281 0.7142681
## 638.75 730
## Mean 0.6770071 0.6701842
## Min 0.6576044 0.6232936
## 0.25 Qt. 0.6597050 0.6241340
## Median 0.6614052 0.6685402
## 0.75 Qt. 0.6996538 0.7139283
## Max 0.7066672 0.7210249
##
## Model type: alasso
## 91.25 182.5 273.75 365 456.25 547.5
## Mean 0.5002315 0.6052897 0.6402225 0.6450094 0.6577948 0.6709570
## Min 0.3796394 0.5066813 0.5064970 0.5272463 0.5529251 0.6061399
## 0.25 Qt. 0.4399829 0.5692907 0.6293616 0.6558905 0.6652881 0.6614951
## Median 0.4535396 0.5992165 0.6329806 0.6564025 0.6755343 0.6713710
## 0.75 Qt. 0.6120902 0.6322008 0.6499734 0.6724999 0.6903455 0.6862749
## Max 0.6159053 0.7190593 0.7822996 0.7130080 0.7048811 0.7295041
## 638.75 730
## Mean 0.6720233 0.6747241
## Min 0.6101307 0.6376704
## 0.25 Qt. 0.6656156 0.6463165
## Median 0.6694466 0.6611572
## 0.75 Qt. 0.6823622 0.6825825
## Max 0.7325616 0.7458939
plot(cmp.val)
plot(cmp.val, interval = TRUE)
The solid line, dashed line and intervals have the same interpretation as above. For this comparison, there seems to be no substantial difference (AUC difference \(< 5\%\)) between lasso and adaptive lasso in predictive performance, although lasso performs slightly better than adaptive lasso for the first three time points, adaptive lasso performs slightly better than lasso for the last few time points.
The model comparison functions in hdnom
have a minimal input design so you do not have to set the parameters for each model type manually. The functions will try to determine the best parameter settings automatically for each model type to achieve the best performance.
We can compare the models by comparing their (internal) model calibration performance. To continue the example, we split the samples into five risk groups, and compare lasso to adaptive lasso via calibration:
cmp.cal =
hdnom.compare.calibrate(x, time, event,
model.type = c("lasso", "alasso"),
method = "cv", nfolds = 5,
pred.at = 365 * 9, ngroup = 5,
seed = 42, trace = FALSE)
cmp.cal
## HighDimensional Cox Model Calibration Object
## Random seed: 42
## Calibration method: kfold crossvalidation
## Crossvalidation folds: 5
## Model type: lasso
## glmnet model alpha: 1
## glmnet model lambda: 0.01952497
## glmnet model penalty factor: not specified
## Calibration time point: 3285
## Number of groups formed for calibration: 5
##
## HighDimensional Cox Model Calibration Object
## Random seed: 42
## Calibration method: kfold crossvalidation
## Crossvalidation folds: 5
## Model type: alasso
## glmnet model alpha: 1
## glmnet model lambda: 1.786852
## glmnet model penalty factor: specified
## Calibration time point: 3285
## Number of groups formed for calibration: 5
summary(cmp.cal)
## Model type: lasso
## Calibration Summary Table
## Predicted Observed Lower 95% Upper 95%
## 1 0.5943124 0.4891464 0.4167302 0.5741465
## 2 0.7015301 0.6923917 0.5810899 0.8250122
## 3 0.7535589 0.8143767 0.7611762 0.8712957
## 4 0.7889556 0.7877060 0.6853574 0.9053388
## 5 0.8303876 0.8965615 0.8617767 0.9327503
##
## Model type: alasso
## Calibration Summary Table
## Predicted Observed Lower 95% Upper 95%
## 1 0.5617432 0.5138413 0.4381910 0.6025521
## 2 0.7233005 0.7026239 0.6386473 0.7730095
## 3 0.7919996 0.8038972 0.7316597 0.8832667
## 4 0.8259658 0.8365244 0.7891229 0.8867733
## 5 0.8576949 0.9053904 0.8448153 0.9703090
plot(cmp.cal, xlim = c(0.3, 1), ylim = c(0.3, 1))
The summary output and the plot show the calibration results for each model type we want to compare. Lasso and adaptive lasso have comparable performance in this case, since their predicted overall survival probabilities are both close to the observed survival probabilities in a similar degree. Adaptive lasso seems to be slightly more stable than lasso in calibration.
To predict the overall survival probability on certain time points for new samples with the established models, simply use predict()
on the model objects and the new data.
As an example, we will use the samples numbered from 101 to 105 in the smart
dataset as the new samples, and predict their overall survival probability from one year to ten years:
predict(fit, x, y, newx = x[101:105, ], pred.at = 1:10 * 365)
## 365 730 1095 1460 1825 2190 2555
## [1,] 0.9476831 0.9203352 0.8883491 0.8572830 0.8171430 0.7841366 0.7430376
## [2,] 0.9714625 0.9562557 0.9382028 0.9203799 0.8969040 0.8771987 0.8521190
## [3,] 0.9786224 0.9671660 0.9535052 0.9399529 0.9220000 0.9068385 0.8874163
## [4,] 0.8971762 0.8456666 0.7873705 0.7327622 0.6651361 0.6120032 0.5489623
## [5,] 0.9736111 0.9595251 0.9427806 0.9262253 0.9043816 0.8860127 0.8625882
## 2920 3285 3650
## [1,] 0.7047801 0.6547589 0.6547589
## [2,] 0.8281917 0.7959836 0.7959836
## [3,] 0.8687509 0.8434080 0.8434080
## [4,] 0.4933861 0.4252340 0.4252340
## [5,] 0.8401909 0.8099641 0.8099641
The hdnom
package has 4 unique builtin color palettes available for all above plots, inspired by the colors commonly used by scientific journals. Users can use the col.pal
argument to select the color palette. Possible values for this argument are listed in the table below:
Value  Color Palette Related Journals 


Journal of Clinical Oncology 

Lancet journals, such as Lancet Oncology 

NPG journals, such as Nature Reviews Cancer 

AAAS Journals, such as Science 
By default, hdnom
will use the JCO color palette (col.pal = "JCO"
).
To cite our paper (preprint), please use (Xiao, Xu, and Li 2016). For further information about the hdnom project, please visit:
Harrell, Frank E. 2013. Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis. Springer Science & Business Media.
Steyerberg, Ewout W. 2008. Clinical Prediction Models: A Practical Approach to Development, Validation, and Updating. Springer Science & Business Media.
Xiao, Nan, QingSong Xu, and MiaoZhu Li. 2016. “Hdnom: Building Nomograms for Penalized Cox Models with HighDimensional Survival Data.” bioRxiv. Cold Spring Harbor Labs Journals. doi:10.1101/065524.