Deep conditional transformation models for survival analysis Gabriele Campanella12 Lucas Kook34 Ida H aggstr om56

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Deep conditional transformation models for

survival analysis

Gabriele Campanella1,2,*, Lucas Kook3,4,*, Ida H¨aggstr¨om5,6,

Torsten Hothorn3, Thomas J. Fuchs1,2

1Department of AI and Human Health, Icahn School of Medicine at Mount Sinai, New York, 10029, USA

2Hasso Plattner Institute for Digital Health at Mount Sinai, New York, 10029, USA

3Epidemiology, Biostatistics & Prevention Institute, University of Zurich, CH-8001, Switzerland

4Institute for Data Analysis and Process Design, Zurich University of Applied Sciences, CH-8400, Switzerland

5Chalmers University of Technology, Department of Electrical Engineering, 41296, Sweden

6Memorial Sloan Kettering Cancer Center, Department of Radiology, New York, 10065, USA

Abstract

An every increasing number of clinical trials features a time-to-event outcome and

records non-tabular patient data, such as magnetic resonance imaging or text data in

the form of electronic health records. Recently, several neural-network based solutions

have been proposed, some of which are binary classiﬁers. Parametric, distribution-free

approaches which make full use of survival time and censoring status have not received

much attention. We present deep conditional transformation models (DCTMs) for survival

outcomes as a unifying approach to parametric and semiparametric survival analysis.

DCTMs allow the speciﬁcation of non-linear and non-proportional hazards for both tabular

and non-tabular data and extend to all types of censoring and truncation. On real and

semi-synthetic data, we show that DCTMs compete with state-of-the-art DL approaches

to survival analysis.

1 Introduction

Arguably one of the most important aspects of health and medical research is being able to

understand and predict patient outcome in order to improve patient management and ulti-

mately extend their life span or time in remission (Hosny et al., 2018). Survival analysis is used

for these purposes to study time-to-event information relating to for example death, response

to treatment, adverse treatment eﬀects, disease relapse, and the development of new disease

∗Authors contributed equally.

Corresponding author: thomas.fuchs.ai@mssm.edu

Preprint. Version October 20, 2022. Licensed under CC-BY.

arXiv:2210.11366v2 [cs.LG] 21 Oct 2022

(Collett, 2015). Traditional approaches, such as Cox Proportional Hazards (cf. Section 3),

relied on tabular features and are not amenable to analyze high-dimensional non-tabular data

such as medical images. With recent advances in computer vision and deep leaning, there has

been increasingly more interest in performing survival analysis directly from high-dimensional

data in order to automatically learn patterns that stratify patients based on their outcome

without the need for feature engineering.

In this paper we present DCTM, a framework for parametric and semiparametric survival

analysis rooted in statistical modeling. DCTMs allow the speciﬁcation of non-linear and

non-proportional hazards for both tabular and non-tabular (image or text) data. We will

describe in detail the formalization of our proposed models and apply them to a real dataset

of medical images. Additionally, we describe how DCTMs can be used as generative models

for the generation of semi-synthetic data. To the best of our knowledge, this paper is the ﬁrst

to cover deep survival regression from the DCTM point of view.

2 Deep transformation models for survival outcomes

In the following, we introduce deep conditional transformation models (DCTMs) for survival

outcomes. We brieﬂy recap survival analysis and conditional transformation models. Then

we describe how to setup, ﬁt, evaluate and sample from our proposed models. Fig. 1 is an

overview of the proposed class of models.

Survival analysis Survival analysis characterizes the distribution of the positive real-valued

event time Tconditional on covariates X, usually on the scale of the survivor function,

ST|X=x(t) = 1−FT|X=x(t), where FT|X=xdenotes the cumulative distribution function (CDF)

of T|X=x. One of the most popular choices is the Cox proportional hazards (Cox PH)

model (Cox, 1972)

FT|X=x(t)=1−exp(−Λ(t|x)) = 1 −exp(−Λ0(t) exp(x>β)),(1)

where Λ(t|x) denotes the positive, monotone increasing cumulative hazard function. The

hazard function is assumed to be decomposable into a baseline hazard Λ0, independent of

the covariates, and a hazard ratio exp(x>β). The hazard ratio models the inﬂuence of the

covariates on the survivor function. The hazard function λ(t|x) = d

dtΛ(t|x) measures the

instantaneous risk of an event after time tconditional on the covariates xand having survived

until t(Collett, 2015).

The Cox PH model is commonly estimated using the maximum partial likelihood obtained

from proﬁling out Λ0(Cox, 1975). The connection to transformation models for distributional

regression is drawn next.

Conditional transformation models CTMs are parametric distributional regression mod-

els of the form (Hothorn et al., 2014)

FT|X=x(t) = FZ(h(t|x)),(2)

where the conditional distribution of the response T|X=xis decomposed into an a priori

chosen and parameter-free target distribution FZand a conditional transformation function

h, which depends on the input data x. In order for FTto be a valid CDF, hneeds to be

monotonically increasing in t(Hothorn et al., 2018).

CTMs can be estimated via maximum likelihood and allow various kinds of responses and

uninformative censoring. The model in (2) suggests a close connection to the Cox PH model

in (1), namely for FZ(z) = 1 −exp(−exp(z)) (the minimum extreme value distribution) and

h(t|x) = log Λ(t|x), the two models coincide.

DCTMs for survival analysis In DCTMs, the transformation function is parameterized

via (deep) neural networks. For instance, let φ:X → Rddenote a feature extractor, which

maps the input xto a feature vector of dimension d. We can then choose diﬀerent parameter-

izations for the transformation function, depending on the desired complexity of the model.

For example,

h(t|x;φ) = α+βlog(t) + φ(x)>w, β > 0,(3)

together with FZ(z) = 1 −exp(−exp(z)) is a Weibull proportional hazards model with non-

linear log hazard-ratios depending on the input x(for instance, medical images). However,

also non-proportional (time-varying) hazards can be realized in DCTMs, via

h(t|x;φ) = α+g(φ(x)>w) log(t),(4)

where g:R→R+,e.g., the soft-plus function, ensures a valid CDF. In Section 4.1, we

describe the parameterization of the DCTMs used in this paper. An overview of the proposed

method is given in Figure 1.

A parametric version of the Cox PH model can be estimated in the DCTM framework.

Here, the log cumulative baseline hazard function is estimated as a smooth basis expansion

a(t)>ϑ, instead of using the non-parametric estimate. A common choice are Bernstein poly-

nomials, which are easily constrained to be monotonically increasing (see Section 4).

Fitting DCTMs DCTMs are ﬁtted by optimizing the empirical negative log-likelihood

(NLL) via stochastic gradient descent (SGD). The likelihood contribution of a single obser-

Figure 1: Proposed DCTM model architecture. The inputs to the model consist of tabular or

non-tabular explanatory variables associated with survival data in terms of exact or right-censored

times to event. The explanatory variables can be mapped to a latent feature space via a feature

extractor. In the case of non-tabular data such as images, the feature extractor will consist of a non

linear mapping such as a convolutional neural network. The extracted features are then fed to the

transformation layer. The network is optimized by minimizing the negative log-likelihood (NLL). In

the ﬁgure we show the transformation function along with the cumulative density function (CDF),

probability density function (PDF) and NLL for an exact event and a right-censored example.

vation (t, x) can be expressed in terms of the general transformation model

L(h;t, x) = 





fZ(h(t|x))h0(t|x)texact event,

1−FZ(h(t|x)) tright censored.

(5)

However, also interval- and left-censored observations can be handled with the proposed

method (see e.g., Hothorn et al., 2018). For a general choice of the transformation func-

tion h(t|x) and error distribution F=σ, we have the following likelihood function

L(h;t, x) = 





σ(h(t|x))(1 −σ(h(t|x))h0(t|x)texact event,

1−σ(h(t|x)) tright censored.

(6)

Lastly, the NLL

NLL = −

i=1

log L(h;ti, xi),(7)

is minimized via SGD, where ntdenotes the training sample size.

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DeepconditionaltransformationmodelsforsurvivalanalysisGabrieleCampanella1,2,*,LucasKook3,4,*,IdaHaggstrom5,6,TorstenHothorn3,ThomasJ.Fuchs1,21DepartmentofAIandHumanHealth,IcahnSchoolofMedicineatMountSinai,NewYork,10029,USA2HassoPlattnerInstituteforDigitalHealthatMountSinai,NewYork,10029,USA3Epidem...

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Deep conditional transformation models for survival analysis Gabriele Campanella12 Lucas Kook34 Ida H aggstr om56

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