PC-Hazard Example

In this notebook, we will present a simple example of the PCHazard method described in this paper.

For a more verbose introduction to pycox see this notebook.

In [1]:
import numpy as np
import matplotlib.pyplot as plt

# For preprocessing
from sklearn.preprocessing import StandardScaler
from sklearn_pandas import DataFrameMapper 

import torch # For building the networks 
import torchtuples as tt # Some useful functions

from pycox.datasets import metabric
from pycox.models import PCHazard
from pycox.evaluation import EvalSurv
In [2]:
## Uncomment to install `sklearn-pandas`
# ! pip install sklearn-pandas
In [3]:
np.random.seed(1234)
_ = torch.manual_seed(123)

Dataset

We load the METABRIC data set as a pandas DataFrame and split the data in in train, test and validation.

The duration column gives the observed times and the event column contains indicators of whether the observation is an event (1) or a censored observation (0).

In [4]:
df_train = metabric.read_df()
df_test = df_train.sample(frac=0.2)
df_train = df_train.drop(df_test.index)
df_val = df_train.sample(frac=0.2)
df_train = df_train.drop(df_val.index)
In [5]:
df_train.head()
Out[5]:
x0 x1 x2 x3 x4 x5 x6 x7 x8 duration event
0 5.603834 7.811392 10.797988 5.967607 1.0 1.0 0.0 1.0 56.840000 99.333336 0
1 5.284882 9.581043 10.204620 5.664970 1.0 0.0 0.0 1.0 85.940002 95.733330 1
3 6.654017 5.341846 8.646379 5.655888 0.0 0.0 0.0 0.0 66.910004 239.300003 0
4 5.456747 5.339741 10.555724 6.008429 1.0 0.0 0.0 1.0 67.849998 56.933334 1
5 5.425826 6.331182 10.455145 5.749053 1.0 1.0 0.0 1.0 70.519997 123.533333 0

Feature transforms

The METABRIC dataset has 9 covariates: x0, ..., x8. We will standardize the 5 numerical covariates, and leave the binary covariates as is. Note that PyTorch require variables of type 'float32'.

We like using the sklearn_pandas.DataFrameMapper to make feature mappers.

In [6]:
cols_standardize = ['x0', 'x1', 'x2', 'x3', 'x8']
cols_leave = ['x4', 'x5', 'x6', 'x7']

standardize = [([col], StandardScaler()) for col in cols_standardize]
leave = [(col, None) for col in cols_leave]

x_mapper = DataFrameMapper(standardize + leave)
In [7]:
x_train = x_mapper.fit_transform(df_train).astype('float32')
x_val = x_mapper.transform(df_val).astype('float32')
x_test = x_mapper.transform(df_test).astype('float32')

Label transforms

The survival methods require individual label transforms, so we have included a proposed label_transform for each method. In this case label_transform is just a shorthand for the class pycox.preprocessing.label_transforms.LabTransDiscreteTime.

The PCHazard is a continuous-time method, but requires defined intervals in which the hazard is constant. Hence, we need to perform an operation similar to a discretization of the time scale. We let num_durations define the size of this (equidistant) interval grid.

In [8]:
num_durations = 10
labtrans = PCHazard.label_transform(num_durations)
get_target = lambda df: (df['duration'].values, df['event'].values)
y_train = labtrans.fit_transform(*get_target(df_train))
y_val = labtrans.transform(*get_target(df_val))

train = (x_train, y_train)
val = (x_val, y_val)

# We don't need to transform the test labels
durations_test, events_test = get_target(df_test)
In [9]:
type(labtrans)
Out[9]:
pycox.preprocessing.label_transforms.LabTransPCHazard

Note that y_train now consist of three labels: the interval index, the event indicator, and the proportion of the interval before the event/censoring occur (i.e, $\rho(t_i)$ in the paper).

In [10]:
y_train
Out[10]:
(array([2, 2, 6, ..., 1, 5, 3]),
 array([0., 1., 0., ..., 1., 0., 0.], dtype=float32),
 array([0.7965465 , 0.69519496, 0.7370492 , ..., 0.06606603, 0.5865238 ,
        0.96302533], dtype=float32))

Neural net

We make a neural net with torch. For simple network structures, we can use the MLPVanilla provided by torchtuples. For building more advanced network architectures, see for example the tutorials by PyTroch.

The following net is an MLP with two hidden layers (with 32 nodes each), ReLU activations, and num_nodes output nodes. We also have batch normalization and dropout between the layers.

In [11]:
in_features = x_train.shape[1]
num_nodes = [32, 32]
out_features = labtrans.out_features
batch_norm = True
dropout = 0.1

net = tt.practical.MLPVanilla(in_features, num_nodes, out_features, batch_norm, dropout)

If you instead want to build this network with torch you can uncomment the following code. It is essentially equivalent to the MLPVanilla, but without the torch.nn.init.kaiming_normal_ weight initialization.

In [12]:
# net = torch.nn.Sequential(
#     torch.nn.Linear(in_features, 32),
#     torch.nn.ReLU(),
#     torch.nn.BatchNorm1d(32),
#     torch.nn.Dropout(0.1),
    
#     torch.nn.Linear(32, 32),
#     torch.nn.ReLU(),
#     torch.nn.BatchNorm1d(32),
#     torch.nn.Dropout(0.1),
    
#     torch.nn.Linear(32, out_features)
# )

Training the model

To train the model we need to define an optimizer. You can choose any torch.optim optimizer, but here we instead use one from tt.optim as it has some added functionality. We use the Adam optimizer, but instead of choosing a learning rate, we will use the scheme proposed by Smith 2017 to find a suitable learning rate with model.lr_finder. See this post for an explanation.

We also set duration_index which connects the output nodes of the network the the discretization times. This is only useful for prediction and does not affect the training procedure.

In [13]:
model = PCHazard(net, tt.optim.Adam, duration_index=labtrans.cuts)
In [14]:
batch_size = 256
lr_finder = model.lr_finder(x_train, y_train, batch_size, tolerance=8)
_ = lr_finder.plot()
In [15]:
lr_finder.get_best_lr()
Out[15]:
0.10722672220103299

Often, this learning rate is a little high, so we instead set it manually to 0.01

In [16]:
model.optimizer.set_lr(0.01)

We include the EarlyStopping callback to stop training when the validation loss stops improving. After training, this callback will also load the best performing model in terms of validation loss.

In [17]:
epochs = 100
callbacks = [tt.callbacks.EarlyStopping()]
log = model.fit(x_train, y_train, batch_size, epochs, callbacks, val_data=val)
0:	[0s / 0s],		train_loss: 2.6413,	val_loss: 2.5174
1:	[0s / 0s],		train_loss: 2.3406,	val_loss: 2.3192
2:	[0s / 0s],		train_loss: 2.1494,	val_loss: 2.0976
3:	[0s / 0s],		train_loss: 1.9278,	val_loss: 1.8521
4:	[0s / 0s],		train_loss: 1.7077,	val_loss: 1.6468
5:	[0s / 0s],		train_loss: 1.5645,	val_loss: 1.5185
6:	[0s / 0s],		train_loss: 1.4957,	val_loss: 1.4789
7:	[0s / 0s],		train_loss: 1.4887,	val_loss: 1.4919
8:	[0s / 0s],		train_loss: 1.4821,	val_loss: 1.4874
9:	[0s / 0s],		train_loss: 1.4499,	val_loss: 1.4807
10:	[0s / 0s],		train_loss: 1.4514,	val_loss: 1.4747
11:	[0s / 0s],		train_loss: 1.4425,	val_loss: 1.4793
12:	[0s / 0s],		train_loss: 1.4182,	val_loss: 1.4848
13:	[0s / 0s],		train_loss: 1.4268,	val_loss: 1.4878
14:	[0s / 0s],		train_loss: 1.4193,	val_loss: 1.4861
15:	[0s / 0s],		train_loss: 1.4099,	val_loss: 1.4978
16:	[0s / 0s],		train_loss: 1.3999,	val_loss: 1.5006
17:	[0s / 0s],		train_loss: 1.4148,	val_loss: 1.4996
18:	[0s / 0s],		train_loss: 1.4060,	val_loss: 1.5080
19:	[0s / 0s],		train_loss: 1.3819,	val_loss: 1.5106
20:	[0s / 0s],		train_loss: 1.3740,	val_loss: 1.5111
In [18]:
_ = log.plot()

Prediction

For evaluation we first need to obtain survival estimates for the test set. This can be done with model.predict_surv which returns an array of survival estimates, or with model.predict_surv_df which returns the survival estimates as a dataframe.

However, we need to define at how many points we want to get the predictions. The default (model.sub = 1) is just to use the interval knots, but by increasing the model.sub argument, we replace the knots with and equidistant number of points in each interval. This is very similar to the interpolation of the discrete methods such as LogisticHazard and PMF.

In [19]:
surv = model.predict_surv_df(x_test)
In [20]:
surv.iloc[:, :5].plot(drawstyle='steps-post')
plt.ylabel('S(t | x)')
_ = plt.xlabel('Time')
In [21]:
model.sub = 10
In [22]:
surv = model.predict_surv_df(x_test)
surv.iloc[:, :5].plot(drawstyle='steps-post')
plt.ylabel('S(t | x)')
_ = plt.xlabel('Time')

Evaluation

The EvalSurv class contains some useful evaluation criteria for time-to-event prediction. We set censor_surv = 'km' to state that we want to use Kaplan-Meier for estimating the censoring distribution.

In [23]:
ev = EvalSurv(surv, durations_test, events_test, censor_surv='km')

Concordance

We start with the event-time concordance by Antolini et al. 2005.

In [24]:
ev.concordance_td('antolini')
Out[24]:
0.6573402857263979

Brier Score

We can plot the the IPCW Brier score for a given set of times. Here we just use 100 time-points between the min and max duration in the test set. Note that the score becomes unstable for the highest times. It is therefore common to disregard the rightmost part of the graph.

In [25]:
time_grid = np.linspace(durations_test.min(), durations_test.max(), 100)
ev.brier_score(time_grid).plot()
plt.ylabel('Brier score')
_ = plt.xlabel('Time')

Negative binomial log-likelihood

In a similar manner, we can plot the the IPCW negative binomial log-likelihood.

In [26]:
ev.nbll(time_grid).plot()
plt.ylabel('NBLL')
_ = plt.xlabel('Time')

Integrated scores

The two time-dependent scores above can be integrated over time to produce a single score Graf et al. 1999. In practice this is done by numerical integration over a defined time_grid.

In [27]:
ev.integrated_brier_score(time_grid) 
Out[27]:
0.16404163283537904
In [28]:
ev.integrated_nbll(time_grid) 
Out[28]:
0.4844648690929724
In [ ]: