=Paper=
{{Paper
|id=Vol-3068/short3
|storemode=property
|title=PseudoNAM: A Pseudo Value Based Interpretable Neural Additive Model for Survival Analysis
|pdfUrl=https://ceur-ws.org/Vol-3068/short3.pdf
|volume=Vol-3068
|authors=Md Mahmudur Rahman,Sanjay Purushotham
|dblpUrl=https://dblp.org/rec/conf/aaaifs/RahmanP21
}}
==PseudoNAM: A Pseudo Value Based Interpretable Neural Additive Model for Survival Analysis==
https://ceur-ws.org/Vol-3068/short3.pdf
PseudoNAM: A Pseudo Value Based Interpretable Neural Additive Model for
Survival Analysis
Md Mahmudur Rahman, Sanjay Purushotham
Department of Information Systems, University of Maryland Baltimore County, Baltimore, Maryland, USA
mrahman6@umbc.edu, psanjay@umbc.edu
Abstract els (Rahman et al. 2021; Zhao and Feng 2020; Ishwaran
et al. 2008; Katzman et al. 2018) achieve more accurate
Deep learning models have achieved the start-of-the-art per-
formance in survival analysis as they can handle censor- predictions but may require specialized objective functions
ing while learning complex nonlinear hidden representa- to handle censoring (Lee et al. 2018). Moreover, many of
tions directly from the raw data. However, the covariate ef- these approaches are not interpretable or explainable, which
fects on survival probabilities are difficult to explain using makes them opaque and unsuitable for medical applications.
deep learning models. To address this challenge, we propose To address the limitations of the existing methods, we
PseudoNAM - an interpretable model which uses pseudo val- propose a pseudo value based neural additive model, called
ues to efficiently handle censoring and uses neural additive PseudoNAM, which directly models the complex non-
networks to capture the nonlinearity in the covariates of the linear time-varying effect of the covariate on the survival
survival data. In particular, PseudoNAM uses neural addi- function. Our PseudoNAM uses neural additive models
tive models to jointly learn a linear combination of neural
(NAM) (Agarwal et al. 2020) to jointly learn a linear com-
networks corresponding to each covariate and identifies the
effect of the individual covariate on the output, and thus, is bination of neural networks corresponding to each covariate
inherently interpretable. We show that our PseudoNAM out- and determines the magnitude of covariates’ effect on the
puts can be used in other survival models such as random survival outcome, and thus, is inherently interpretable. The
survival forests to obtain improved survival prediction per- neural networks for each feature in the PseudoNAM are in-
formance. Our experiments on three real-world survival anal- dependent, and thus, can provide the individual feature con-
ysis datasets demonstrate that our proposed models achieve tribution towards output survival prediction. Like in NAM,
similar or better performance (in terms of C-index and Brier we sum up the individual feature contributions (neural net-
scores) than the state-of-the-art survival methods. We show- work outputs), followed by a logit transformation using
case that PseudoNAM provides overall feature importance sigmoid activation function to predict the survival proba-
scores and feature-level interpretations (covariate effect on
bility at different time points. We show different types of in-
survival risk) for survival predictions at different time points.
terpretations from PseudoNAM, including 1) the mean fea-
ture contributions to the survival probability predictions at
Introduction different time points (overall feature importance scores) and
Survival analysis (Kleinbaum and Klein 2010), a well- 2) feature-level interpretations which show the time-varying
studied problem, aims to estimate the risk of a subject’s fail- covariate effect on the survival predictions.
ure from an event, such as death due to breast cancer at a Our experiments on three real-world datasets demonstrate
particular time point. One key challenge in survival analy- that PseudoNAM performs similar or better than the state-
sis is the presence of censored subjects for whom the actual of-the-art survival analysis models while providing inter-
survival times remain unknown. A good survival analysis pretable results. We further improve the performance of
model should handle censoring, accurately discriminate the PseudoNAM by proposing PseudoNRSF, a random sur-
predicted risks, and should be interpretable. Traditional sta- vival forest approach that takes as input the learned outputs
tistical survival analysis models such as Cox Proportional of individual neural networks from PseudoNAM and pre-
Hazard models (Cox 1972), regression models based on dicts the survival probabilities. We show that PseudoNRSF
pseudo-observations (Andersen, Klein, and Rosthøj 2003; achieves state-of-the-art results in terms of c-index and Brier
Andersen and Pohar Perme 2010) are interpretable but are scores.
less accurate and limited by strong assumptions on the un-
derlying stochastic process, such as linearity, parametric, Related Works
and proportional hazards assumptions. Recent survival ap- Cox-based statistical and deep learning survival models
proaches based on machine learning and deep learning mod- (Cox 1972; Faraggi and Simon 1995; Katzman et al. 2018;
Copyright © 2021 for this paper by its authors. Use permitted under Kvamme and Borgan 2019) are widely studied for analyzing
Creative Commons License Attribution 4.0 International (CC BY time-to-event data. However, these models make strong pro-
4.0) portional hazard and linearity assumptions that may not hold
for real data, thus leading to less accurate survival results. function (e.g., logit link function), β is the bias and each
Machine learning models such as Random survival fi (.) is parametrized by a neural network. y(t|X) is the
forests (Ishwaran et al. 2008) and multi-task logistic regres- pseudo values for survival probability at time t in the pres-
sion (MTLR) (Yu et al. 2011; Fotso 2018) relax some of ence of covariates. Each of the networks learn the complex
these assumptions and outperform statistical-based methods. shape function of a specific covariate, and all the networks
Recently proposed deep learning models (Lee et al. 2018; are trained jointly. A n × p matrix of p baseline covariates
Nagpal, Li, and Dubrawski 2021) and conditional gener- with n individuals are used as input in the input layer. Out-
ative adversarial networks (Chapfuwa et al. 2018) achieve put layer returns the survival probabilities at M evaluation
state-of-the-art results for time-to-event analysis. However, time points. PseudoNAM model provides interpretability
these methods require either making assumptions on the un- because it jointly trains a set of neural networks correspond-
derlying stochastic process or design a specialized objective ing to each individual covariate and returns the covariates’
function to handle censoring. Moreover, these methods lack contribution scores to the output (i.e., the output of the neu-
interpretability which is required in the medical domain. To ral networks) for all each covariate and for the M evalua-
address the censoring challenge, (Zhao and Feng 2020; Rah- tion time points. Then we sum up the contribution scores of
man et al. 2021) have respectively proposed pseudo value all covariates followed by applying a sigmoid activation
based deep learning models for survival and competing risk function to get the final output, i.e., the survival probabilities
analysis. However, even these methods are not directly in- at the M time points. The non-overlapping neural networks
terpretable and rely on off-the-shelf explainable AI methods for individual covariates allow to identify the individual co-
such as LRP (Montavon et al. 2019) for providing explana- variate effect on the survival probabilities.
tions.
𝐟𝟏 (𝐗 𝟏 )
Our Proposed Models
FC Layer Output Layer 1
To address the censoring and interpretability challenges
X1
...
FC Layer
of existing survival analysis approaches, in this work, we
propose two interpretable pseudo value based deep learning FC Layer {y11 , y12 , … , y1M }
models, PseudoNAM and PseudoNRSF. Before describing
our models in detail, we will briefly introduce pseudo values. 𝐟𝟐 (𝐗 𝟐 )
FC Layer Output Layer 2
What are Pseudo values? Pseudo values for the survival X2 𝜎 𝐲
...
FC Layer
probability are derived from the non-parametric population- FC Layer {y21 , y22 , … , y2M }
based Kaplan-Meier (KM) estimator, an approximately un-
biased estimator of the survival probability under indepen- ... Add
dent censoring (Andersen et al. 2012). For the ith subject, 𝐟𝐩 (𝐗 𝐩 ) 𝜎 Sigmoid
a Jackknife pseudo value, based on the KM estimate of the FC Layer Output Layer p
survival probability (Klein et al. 2008), is computed at time Xp β Bias
...
FC Layer
horizon t∗ as FC Layer {yp1 , yp2 , … , ypM }
∗ ∗ −i ∗
Ŝi (t ) = nŜ(t ) − (n − 1)Ŝ (t ) β
∗
where, Ŝ(t ) is the Kaplan-Meier estimate of the survival
probability at time t∗ based on a sample with n subjects
and Ŝ −i (t∗ ) is the Kaplan-Meier estimate of the survival Figure 1: Architecture of PseudoNAM. X =
probability at time t∗ based on a leave-one-out sample {X1 , X2 , .., Xp } is a p dimensional vector of covariates.
with (n − 1) subjects, obtained by omitting the ith subject. fi (Xi ) is the neural network corresponding to covariate Xi
Pseudo values are calculated for both uncensored subjects and β is the bias. Sigmoid (σ) is worked as inverse logit link
and censored subjects (incompletely observed) at a specified function. y is the output, i.e., survival probability at M time
time point. points. FC Layer means Fully Connected Layer.
PseudoNAM: Inspired by the success of pseudo value PseudoNRSF: While PseudoNAM provides inter-
based deep models, DNNSurv (Zhao and Feng 2020) and pretable predictions; its performance is limited by the
DeepPseudo (Rahman et al. 2021) to handle censoring, we NAM model architecture. To improve the performance of
propose PseudoNAM - a multi-output neural additive model PseudoNAM model and to obtain global and local inter-
which predicts pseudo values for survival risk analysis. pretations like Random survival forests (RSF) (Ishwaran
PseudoNAM, shown in Figure 1, learns non-linear repre- et al. 2008), we propose PseudoNRSF - a two-stage deep
sentations in the data and uses pseudo values to handle cen- learning model. In the first stage, PseudoNAM model is
soring efficiently. PseudoNAM has the following form: used to learn the individual feature contribution scores
for predicting pseudo values. In the second stage, these
g(E[y(t|X)]) = β + f1 (X1 ) + f2 (X2 ) + + fp (Xp ) (1)
learned feature contribution scores are input to an RSF with
Here, Xi = (Xi1 , Xi2 , ..., Xip ) is a p-dimensional covari- the goal of directly predicting survival probabilities. Thus,
ate vector for ith individual; i = 1, 2, .., n. g(.) is the link PseudoNRSF returns the subject-specific survival proba-
Feature Importance Plot for METABRIC Dataset
Mean Feature Contribution
MKI67 EGFR PGR ERBB2 Hormone Radio- Chemo- ER- Age at
treatment therapy therapy positive diagnosis
Figure 2: Mean individual feature contributions on survival probabilities at different time points on METABRIC dataset. Here,
10th Perc., 20th Perc. are representing 10th percentile and 20th percentile of the survival time distribution at which we get the
PseudoNAM model predictions.
METABRIC: This data (Katzman et al. 2018)1 contains
patients’ gene expressions and clinical variables for breast
cancer survival prediction.
SUPPORT: This dataset (Knaus et al. 1995) is from the
Vanderbilt University study to estimate survival of 9,105
seriously ill hospitalized patients.
WHAS: This dataset was collected to examine the effects
of a patient’s factors on acute myocardial infraction (MI)
survival (Hosmer and Lemeshow 2002).
Figure 3: Importance of feature (mean weight ± sd) on the
survival probability predictions measured by PseudoNRSF Implementation Details: The (ground-truth) pseudo
on METABRIC dataset. values for survival probabilities are obtained using the
jackknife function of R package prodlim at each
evaluation time point (separately for training and validation
bilities at different time points as output, and the predictions sets). We performed stratified 5-fold cross-validation so
are interpretable due to the use of RSF. PseudoNRSF has that the ratio of censored and uncensored subjects remained
the interpretation property of the RSF. We can easily get the the same in each fold. We jointly train our PseudoNAM’s
effect of each covariate (importance score) on the overall feature networks based on an early stopping criterion and
survival probability. However, using the PseudoNAM, we choose the best model based on the model’s performance on
can see the change of covariate effect on survival probabili- validation data. Each feature network consists of 3 hidden
ties at different time points. layers with a number of units [128, 64, 32]. We used relu
activation function in the hidden layer of each covariate’s
neural network and tanh activation function in the output
Experiments layer of the neural networks. In the final output layer, we
We conducted experiments on three real-world datasets to sum up the output of individual feature neural networks and
answer the following questions: a) how well our proposed use the sigmoid activation function to get the survival
models perform compared to the state-of-the-art survival probability at 10th , 20th , 30th , 40th , 50th , 60th percentile
models? b) how well can our PseudoNAM explain their of the maximum survival time of the training data. We did
predictions? not perform hyperparameter tuning. We set the learning
rate 0.0001, output penalty coefficient 0.001, weight decay
Datasets: Table 1 shows statistics of the following coefficient 0.000001, dropout rate 0.0, and feature dropout
datasets.
1
https://github.com/jaredleekatzman/DeepSurv
Feature Contribution
EGFR ER-positive ERBB2 MKI67 PGR
Feature Contribution
Age at Chemotherapy Hormone
Radiotherapy
diagnosis treatment
Figure 4: Feature-level contributions to survival probability at different time points (10th , 20th , 30th , 40th , 50th , 60th percentile
of survival time distribution) for individual features in METABRIC dataset. The darker the brown bars indicates higher density
of the data. Age at diagnosis, EGFR, ERBB2, MKI67, PGR are continuous valued features, while others are discrete valued.
Table 1: Descriptive Statistics of the three Real-World Survival Datasets
No. of No. of No. of No. of Event Time Censoring Time
Dataset
Observation uncensored (%) censored (%) features Min Max Mean Median Min Max Mean Median
METABRIC 1904 1103(57.9) 801(42.1) 9 0.1 355.2 100.0 85.9 0.1 337.0 159.6 158.0
SUPPORT 8873 6036 (68.0) 2837 (32.0) 14 3.0 1944.0 205.5 57.0 344.0 2029.0 1059.9 918.0
WHAS 1638 690 (42.1) 948 (57.9) 5 0.1 1965.9 696.7 515.5 371.0 1999.0 1298.9 1347.5
rate 0. We used Adam optimizer with batch size 128 during sion [MTLR] (Yu et al. 2011)
training and used Penalized Mean Squared Error loss func-
tion to train our models. For performance metrics, we used • Deep Learning models: DNNSurv (Zhao and Feng
(a) time-dependent concordance index (Antolini, Boracchi, 2020), DeepHit (Lee et al. 2018), DeepSurv (Katz-
and Biganzoli 2005) adjusted with an inverse propensity of man et al. 2018), CoxTime (Kvamme, Borgan, and
censoring estimate for evaluating the discriminative-ability, Scheel 2019), Deep Survival Machine [DSM] (Nagpal,
and (b) Integrated IPCW Brier Score (denoted as Brier Li, and Dubrawski 2021), Piecewise Constant Hazard
Score) (Gerds and Schumacher 2006) metric for evaluating [PCHazard] (Kvamme and Borgan 2019), and our pro-
the predictive-ability. To encourage reproducibility, the posed models: PseudoNAM and PseudoNRSF.
source codes for our proposed models are available at this
link: https://github.com/umbc-sanjaylab/PseudoNAM SA Results and Discussion
Model Comparisons: We compared the following Table 2 shows the performance comparison of the survival
survival analysis models: models based on time dependent concordance index and
Brier scores. From this table, we see that our PseudoNAM
• Statistical models: Cox Proportional Hazard Model obtains similar or comparable performance to other survival
[CoxPH] (Cox 1972) analysis models, while PseudoNRSF outperforms all the
survival models on the WHAS dataset, and obtains similar
• Machine learning models: Random Survival Forest performance as the state-of-the-art models on the other two
[RSF] (Ishwaran et al. 2008), Multi-task Logistic Regres- datasets. We notice that the independent neural networks for
Table 2: Model comparisons of the performance metrics (mean and 95% confidence interval) evaluated on survival datasets
Time-dependent Concordance Index
PseudoNRSF PseudoNAM DNNSurv CoxPH CoxTime DeepHit DeepSurv DSM MTLR PCHazard RSF
METABRIC 0.645±0.038 0.616±0.025 0.617±0.014 0.622±0.013 0.660±0.055 0.655±0.045 0.641±0.017 0.616±0.040 0.550±0.043 0.614±0.041 0.616±0.058
SUPPORT 0.619±0.019 0.613±0.017 0.581±0.009 0.568±0.016 0.616±0.012 0.593±0.012 0.589±0.009 0.595±0.005 0.550±0.024 0.589±0.022 0.638±0.010
WHAS 0.865±0.038 0.740±0.022 0.721±0.018 0.739±0.013 0.783±0.027 0.851±0.038 0.787±0.030 0.739±0.013 0.618±0.104 0.685±0.038 0.768±0.041
Brier Score
PseudoNRSF PseudoNAM DNNSurv CoxPH CoxTime DeepHit DeepSurv DSM MTLR PCHazard RSF
METABRIC 0.171±0.005 0.245±0.013 0.243±0.010 0.313±0.020 0.168±0.011 0.178±0.022 0.165±0.013 0.249±0.020 0.225±0.024 0.201±0.014 0.296±0.016
SUPPORT 0.196±0.01 0.207±0.007 0.221±0.002 0.206±0.005 0.192±0.008 0.211±0.008 0.198±0.006 0.212±0.003 0.263±0.019 0.225±0.018 0.190±0.007
WHAS 0.099±0.010 0.267±0.022 0.290±0.029 0.234±0.029 0.136±0.018 0.140±0.054 0.132±0.018 0.201±0.005 0.162±0.028 0.141±0.009 0.206±0.033
individual covariates limits PseudoNAM to learn the shared (i.e., the outputs of the individual neural networks of
effect on the survival probability at different times, thus re- PseudoNAM) on survival probability at different time
sulting in comparable but not the best results. points (i.e., time-varying covariate effect on survival pre-
dictions) for the METABRIC dataset. Here x-axis shows
Model Intepretations the feature values, and the y-axis shows their contribu-
tions. In other words, this plot provides feature-level in-
The main advantage of our PseudoNAM models is that they
terpretations. For example, the survival probability for
can provide interpretations. Here, we discuss the two ways
the feature age at diagnosis at all the time points
of interpreting the PseudoNAM model predictions: overall
starts decreasing after 65 years; and we see that the feature
feature importance scores and feature-level interpretations.
chemotherapy is biased to the patients who did not re-
PseudoNAM first learns the individual feature contri-
ceive chemotherapy since the density is much higher for this
butions for pre-specified time points (here, we choose
group (darker brown bar). The plot also shows that the model
10th , 20th , ..., 60th percentile of the event horizon). Then
predicted a decrease in survival probability for a few patients
we sum up these feature contributions followed by the
who received chemotherapy, especially at later time points.
sigmoid transformation to get the final survival proba-
bilities at the pre-specified time points. Figure 2 shows the Why PseudoNAM is suited for healthcare domain?
overall feature importance scores measured as mean in-
dividual feature contributions on the survival probability at As shown in Table 2, our PseudoNAM models obtain good
the pre-specified time points for the METABRIC dataset. predictive and discriminative performance on all the survival
We see that the features can have a positive or negative datasets. Moreover, using our models, one can visualize each
impact (overall effect) on survival probability predictions covariates’ contribution to the survival probability. There-
at different time points for breast cancer patients. For ex- fore, PseudoNAM helps to identify the potential risk factors
ample, the covariates such as MKI67, radiotherapy, for an event, such as death due to breast cancer. The visual-
and chemotherapy have positive feature contribu- ization of the feature-level interpretations can be a step to-
tions at the initial time points (10th percentile), which means wards transparency in the deep learning models, which can
that they influence better survival outcomes (higher survival inform clinical decision-making and perhaps lead to trust in
probabilities). However, at later time points (such as 60th the model. Thus, PseudoNAM models with high predictive-
percentile), these features have negative feature contri- ability and inherent interpretability could be well-suited for
butions - meaning they result in mortality. This is expected survival analysis in the healthcare domain.
because the survival probability remains higher at initial
time points, and it decreases over time. Therefore, the treat- Conclusion
ment like chemotherapy fails to reduce the risk of death In this paper, we proposed interpretable pseudo value-based
at later time points, and the older people (age at diagnosis) deep learning approaches PseudoNAM and PseudoNRSF
are at greater mortality risk. to model the nonlinear time-varying covariate effect on sur-
Figure 3 shows the permutation feature importance, vival predictions. Our proposed models use 1) pseudo val-
which is measured by observing how random re-shuffling ues to handle censoring and 2) neural additive networks to
of each covariate influences model performance. We capture the complex nonlinear relationships and to obtain in-
use eli52 library to compute the feature importance terpretable predictions. Empirical results show that our pro-
for PseudoNRSF model. We observe that age at posed models achieve similar or better performance than the
diagnosis has the highest importance on the survival state-of-the-art survival methods. Our PseudoNAM model
probability predictions and ER-positive has the lowest provides both overall feature importance scores and feature-
feature importance. level interpretations of predicted survival probabilities at dif-
Figure 4 shows the individual feature contribution ferent time points. For future work, we study and compare
the interpretability of our proposed models with other para-
2
https://github.com/eli5-org/eli5 metric survival approaches.
Acknowledgements Knaus, W. A.; Harrell, F. E.; Lynn, J.; Goldman, L.; Phillips,
R. S.; Connors, A. F.; Dawson, N. V.; Fulkerson, W. J.;
This work is partially supported by grant IIS–1948399
Califf, R. M.; Desbiens, N.; et al. 1995. The SUPPORT
from the US National Science Foundation and grant
prognostic model: Objective estimates of survival for seri-
80NSSC21M0027 from the National Aeronautics and Space
ously ill hospitalized adults. Annals of internal medicine .
Administration.
Kvamme, H.; and Borgan, Ø. 2019. Continuous and
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