Rethinking Individual Risk and Aggregation in Survival Analysis: A Latent Mechanism Framework

Rethinking Individual Risk and Aggregation in Surviv al Analysis: A Laten t Mec hanism F ramew ork Xijia Liu ∗ Abstract Surviv al analysis provides a well-established framew ork for mo deling time-to-even t data, with hazard and surviv al functions formally deﬁned as p opulation-lev el quan tities. In ap- plied w ork, how ev er, these quantities are often interpreted as representing individual-level risk, despite the absence of a clear generative accoun t linking individual risk mechanisms to observed surviv al data. This pap er dev elops a latent hazard framew ork that makes this relationship explicit b y mo deling even t times as arising from unobserv ed, individual- sp eciﬁc hazard mechanisms and viewing p opulation-lev el surviv al quan tities as aggregates o ver heterogeneous mechanisms. Within this framework, w e show that individual hazard tra jectories are not iden tiﬁable from surviv al data under partial information. More gen- erally , the conditional distribution of latent hazard mechanisms given cov ariates is struc- turally non-identiﬁable, ev en when p opulation-lev el surviv al functions are fully known. This non-iden tiﬁability arises from the aggregation inherent in surviv al data and p ersists inde- p enden tly of mo del ﬂexibilit y or estimation strategy . Finally , we sho w that classical surviv al mo dels can b e systematically reinterpreted according to how they handle this unresolved conditional mechanism distribution. This pap er provides a uniﬁed framew ork for under- standing heterogeneit y , identiﬁabilit y , and interpretation in surviv al analysis, and clariﬁes ho w population-level surviv al mo dels should b e interpreted when individual risk mecha- nisms are only partially observ ed, thereby establishing explicit information constraints for principled mo deling and inference. Keywor ds— Surviv al analysis; hazard function; heterogeneity; identiﬁabilit y; latent mec ha- nisms; p opulation v ersus individual risk 1 In tro duction Surviv al analysis has ac hieved remark able empirical success o ver the past ﬁve decades and has b ecome a cen tral to ol in biomedical researc h, epidemiology , and reliabilit y studies. Its funda- men tal quantities, the hazard function and the surviv al function, are deﬁned with mathematical rigor and stabilit y , and ha v e supported extensive metho dological dev elopment and applied prac- tice. How ever, alongside this success, a p ersistent in terpretational tension has remained present but has never b een systematically clariﬁed. Within the classical theoretical framework, hazard and surviv al functions are deﬁned as p opulation-lev el statistical ob jects, c haracterizing failure rates and surviv al probabilities within h yp othetical cohorts sharing the same observed co v ariates. In applied settings, how ev er, these same quan tities are often naturally endo wed with individual-level in terpretations and are used to describ e individual risk states, laten t biological pro cesses, or uncertaint y about future outcomes. This issue b ecomes particularly salien t as the fo cus of surviv al analysis has increasingly shifted from inference-oriented to prediction-orien ted ob jectives, and has b ecome diﬃcult to a v oid in contexts such as individualized prediction, risk stratiﬁcation, and decision support. More ∗ Departmen t of Statistics, Ume ˚ a Universit y , Sweden. Email : xijia.liu@umu.se 1 imp ortan tly , this tension is not conﬁned to any particular class of models or metho ds, but reﬂects a structural information constraint: surviv al data inherently aggregate o v er heterogeneous, unobserv ed hazard mechanisms, and this aggregation induces a man y-to-one mapping from individual-lev el risk pro cesses to observ able p opulation-lev el quan tities. Clarifying this issue is therefore of foundational imp ortance for understanding the in terpretational b oundaries of surviv al mo dels, the meaning of prediction, the feasibilit y of heterogeneity c haracterization, and for providing principled guidance for the design of future predictive mo dels. 1.1 Motiv ating Challenges in Surviv al Analysis The source of the in terpretational tension outlined ab o v e is not immediately apparent at the level of formal deﬁnitions. In classical surviv al theory , hazard and surviv al functions are rigorously deﬁned as p opulation-lev el quantities. In particular, the standard hazard h ( t | X ) = lim ∆ t → 0 Pr( t ≤ T < t + ∆ t | T ≥ t, X ) ∆ t , c haracterizes the instantaneous failure rate among individuals who hav e surviv ed up to time t within a hypothetical cohort sharing the same observed co v ariates. By construction, this quan tity is conditional on mem b ership in the risk set and is deﬁned only at the level of such p opulations, rather than as an intrinsic prop erty of a single s ub ject. Nev ertheless, in applied con texts, b oth classical and mo dern surviv al mo dels are often discussed, rep orted, or informally in terpreted in w ays that suggest an individual-level notion of risk, ev en though h ( t | X ) and S ( t | X ) are formally deﬁned as p opulation-lev el quan tities. This in terpretational shift is closely tied to unobserved heterogeneit y . Individuals who share the same observ ed cov ariates may diﬀer substantially in their underlying risk pro cesses. There- fore, p opulation-lev el hazards represent a v erages ov er heterogeneous individuals rather than in trinsic individual prop erties. Unless heterogeneity v anishes, a condition rarely met in biomed- ical settings, population hazards cannot b e directly equated with individual risks. Ho wev er, this distinction is often blurred in practice, complicating in terpretation, prediction, and causal re a- soning. Classical regression-based surviv al mo dels adopt sp eciﬁc functional forms for population- lev el quan tities. F or example, Cox’s prop ortional hazards model Co x [1972] and accelerated failure time (AFT) Buc kley and James [1979] form ulations describe structured relationships b et w een co v ariates and hazards or surviv al times. These mo dels are highly eﬀective descriptive to ols, but they do not explicitly sp ecify the individual-lev el pro cesses that w ould giv e rise to the assumed p opulation structures. As a result, the connection betw een mo del assumptions and underlying individual risk dynamics remains implicit. Recen t machine learning approaches to surviv al analysis intensify this issue. By learning ﬂexible mappings from co v ariates to time-dep enden t hazards or surviv al probabilities, such mo dels can further blur the distinction b et ween p opulation-lev el quantities and individual- lev el interpretations, ev en though these quantities remain formally deﬁned at the p opulation lev el. Without a generativ e framework that distinguishes individual risk mechanisms from their p opulation aggregates, the in terpretation of model outputs remains am biguous. T ak en together, these considerations highlight the absence of an explicit mechanism-based foundation that links individual-lev el risk pro cesses to observed p opulation-level surviv al patterns. 1.2 Existing P ersp ectiv es on Surviv al Mo deling The distinction b etw een individual-level and p opulation-lev el notions of hazard has long b een implicit in the surviv al analysis literature. In Cox’s original formulation [Cox, 1972], the haz- ard function is introduced as a regression ob ject describing cohort-level failure rates, with the baseline hazard treated as an arbitrary p opulation function of time. The mo del itself do es not 2 attribute ontological status to individual hazards, nor do es it attempt to sp ecify individual risk pro cesses. Ho wev er, the absence of an explicit generative interpretation has left ro om for div ergent readings in subsequent metho dological and applied w ork. A n umber of authors hav e emphasized interpretational subtleties asso ciated with hazard- based quantities. Hazard contrasts compare failure rates among individuals who hav e survived up to a given time rather than individual risks [Aalen et al., 2008, Martin ussen and V ansteelandt, 2013, Gram bsch, 2017], and unobserv ed heterogeneity can distort hazard-based interpretations ev en in randomized studies [Dumas and Stensrud, 2025]. These con tributions pro vide imp ortant clariﬁcation regarding the limitations of hazard ratios and related summaries, but they do not oﬀer a general framew ork for linking population hazards to underlying individual risk processes. P arallel developmen ts hav e sough t to address heterogeneit y through extensions suc h as frailt y mo dels V aup el et al. [1979], Hougaard [1995], accelerated failure time formulations, and laten t class surviv al analysis Lin et al. [2002], Proust-Lima et al. [2014]. Eac h of these approac hes in tro duces additional structure to account for v ariability b eyond observed cov ariates, whether through random eﬀects, discrete subp opulations, or time-scaling mechanisms. More recently , mac hine learning based surviv al mo dels hav e emphasized predictiv e p erformance by learning ﬂexible mappings from co v ariates to observ able risk or surviv al quantities [Ish waran et al., 2008, Katzman et al., 2018, Lee et al., 2018]; see W ang et al. [2021], Wiegrebe et al. [2024] for reviews. While these metho ds diﬀer substantially in form and in tent, they share a common feature: the underlying relationship b et ween individual-level risk mechanisms and p opulation-level surviv al quan tities remains implicit, particularly in predictiv e settings where individual risk estimates are diﬃcult to v alidate or calibrate [Steyerberg et al., 2010, Austin et al., 2020]. What is curren tly missing is a unifying p ersp ectiv e that makes this relationship explicit. In particular, there is a need for a framew ork that (i) deﬁnes individual risk pro cesses as latent mec hanisms, (ii) explains ho w p opulation-lev el hazards arise as aggregates of these mec hanisms, and (iii) clariﬁes the information-theoretic limitations that constrain what can b e inferred from surviv al data. The present w ork is motiv ated by this gap. 1.3 A Mec hanism-Based F ramew ork for Surviv al Analysis T o address the in terpretational and identiﬁabilit y c hallenges outlined ab o ve, this paper pro- p oses a mechanism-based analytical framew ork that explicitly characterizes individual-level risk-generating mec hanisms and clariﬁes how p opulation-level surviv al quan tities arise as ag- gregated consequences of these laten t mec hanisms, thereby la ying a foundation for a uniﬁed understanding of classical surviv al mo dels. Surviv al analysis is fundamentally concerned with the relationship b et ween individual char- acteristics and time-to-ev ent outcomes. Implicit in many mo deling approac hes, and in their common in terpretations, is the notion that each individual is asso ciated with a sp eciﬁc, though unobserv ed, hazard tra jectory o ver time. W e mak e this idea explicit by introducing the notion of an individual hazar d me chanism , denoted by Θ, which c haracterizes the sto chastic pro cess go verning an individual’s even t time. Giv en a realization of Θ, an individual hazard func- tion and surviv al function are well deﬁned, but the mechanism itself is not directly observ able. Crucially , surviv al data pro vide extremely limited information ab out individual mechanisms. Eac h individual typically con tributes at most a single ev ent-time observ ation, p ossibly sub ject to censoring. As a consequence, even in the presence of rich co v ariate information, individual hazard mec hanisms are fundamen tally not identiﬁable from surviv al data alone, ev en under idealized sampling and arbitrarily large sample sizes. This limitation is not a tec hnical artifact of sp eciﬁc mo deling choices, but a fundamen tal consequence of the information structure inher- en t in surviv al data. Observed surviv al quan tities, therefore, admit a natural p opulation-lev el in terpretation. Conditional on cov ariates X = x , the surviv al function S ( t | X = x ) arises as an aggregate o ver the distribution of individual mechanisms compatible with these cov ariates. Rather than representing the surviv al tra jectory of a representativ e individual, p opulation-lev el 3 surviv al reﬂects a mixture of heterogeneous hazard mechanisms. This distinction is essential for interpreting mo del outputs and for understanding the scop e of v alid inference in surviv al analysis. F rom this p ersp ectiv e, the cen tral ob ject linking individual-level heterogeneity and p opulation-lev el surviv al is the conditional mec hanism distribution P (Θ | X = x ). This dis- tribution enco des how muc h information observed cov ariates pro vide ab out individual hazard mec hanisms. How ev er, b ecause surviv al data do not resolve individual mechanisms, this con- ditional distribution is t ypically not identiﬁable b eyond coarse, mo del-dep enden t summaries. An y attempt to mo del surviv al data must therefore confront the fact that P (Θ | X = x ) cannot b e freely estimated from the data alone. Surviv al mo dels can thus b e understo o d as imp osing structural assumptions on the con- ditional mechanism distribution. Diﬀeren t mo deling traditions corresp ond to diﬀeren t wa ys of constraining residual heterogeneity: b y suppressing it, by restricting it to low-dimensional forms, or b y representing it through discrete components. These assumptions are often implicit, y et they fundamentally shap e what asp ects of surviv al b eha vior can b e inferred from the data and ho w mo del outputs should b e interpreted. The aim of the presen t framew ork is not to recov er individual hazard mechanisms but to pro vide a coheren t language for articulating heterogeneity , information limitations, and mo d- eling assumptions in surviv al analysis. By making the role of latent mechanisms explicit, the framew ork clariﬁes what can and cannot b e identiﬁed from surviv al data and sets the stage for a systematic examination of identiﬁabilit y , which we dev elop in the following sections. 1.4 Con tributions This pap er mak es a conceptual and structural contribution to the understanding of surviv al analysis. Rather than proposing a new estimation procedure or predictiv e model, our fo cus is on clarifying the ob jects of inference, the role of laten t heterogeneity , and the information- theoretic limitations that shap e surviv al mo deling. By reframing surviv al analysis through the lens of individual hazard mec hanisms, w e aim to provide a coheren t p ersp ectiv e that uniﬁes and con textualizes a broad range of existing approaches. Sp eciﬁcally , the contributions of this work are as follows. First, we introduce a mechanism- based form ulation that explicitly distinguishes b et w een individual-level hazard mec hanisms and p opulation-lev el surviv al quantities, thereb y resolving long-standing ambiguities in in terpreta- tion. Second, within this framework, we characterize the conditional mechanism distribution as a central but generally non-identiﬁable ob ject in surviv al analysis, and show that this non- iden tiﬁability arises from the intrinsic information structure of surviv al data rather than from metho dological or mo deling deﬁciencies. Building on these results, we reinterpret several clas- sical surviv al mo dels as imp osing distinct structural constraints on the conditional mec hanism distribution, providing a uniﬁed explanation for their assumptions, represen tational scop e, and inheren t limitations. Finally , as a unifying consequence, the prop osed framework oﬀers a prin- cipled language for comparing mo deling strategies and for articulating directions for future metho dological developmen t in surviv al analysis. By making the latent information structure explicit, the prop osed framework not only clariﬁes the limits of individual-lev el inference, but also delineates a space of mo deling choices in which metho dological assumptions can b e stated, compared, and ev aluated in a principled manner. Structure of the P ap er The remainder of the paper is organized as follows. Section 2 in tro duces the laten t hazard mec hanism framework and formalizes the relationship b et ween individual mechanisms, observ- able cov ariates, and surviv al outcomes. Section 3 examines the identiﬁabilit y implications of this form ulation, establishing general limitations imp osed by the information structure of surviv al data. In Section 4, w e revisit several classical surviv al mo dels, including prop ortional hazards 4 mo dels, frailt y mo dels, accelerated failure time mo dels, and surviv al clustering approaches, through the prop osed framework, highlighting how each handles residual heterogeneity . Sec- tion 5 concludes with a discussion of implications, limitations, and directions for future researc h. 2 Laten t Hazard F ramew ork A p ersistent source of ambiguit y in surviv al analysis concerns the relationship b et ween individual- lev el risk and p opulation-lev el observ able quantities. While hazard and surviv al functions are routinely interpreted as c haracterizing individual risk, they are, in practice, deﬁned and esti- mated at an aggregated lev el, conditional on limited information. T o analyze the iden tiﬁability and in terpretation of such quantities, it is therefore necessary to make explicit the information structure linking individual risk-generating mechanisms, observ able co v ariates, and surviv al outcomes. T o this end, w e introduce a minimal latent hazard form ulation that explicitly separates individual-lev el risk mec hanisms from observ able p opulation-lev el quantities. At the core of this form ulation is a latent hazard mechanism Θ, which represents individual-lev el risk-generating structure and is treated as a primitiv e ob ject that deterministically induces an individual hazard and surviv al tra jectory . Observ able cov ariates X are in terpreted as partial information ab out this laten t mec hanism, giving rise to a conditional distribution ov er Θ, rather than directly mo difying individual risk once the mechanism is ﬁxed. Within this p ersp ectiv e, classical p opulation-level and group-level surviv al quantities arise as observ able summaries obtained by aggregating ov er laten t heterogeneity in Θ. In particu- lar, surviv al functions corresp ond to mixture av erages of individual surviv al tra jectories, while observ able hazards emerge as survivor-w eighted av erages of individual hazards. The resulting form ulation is fully consistent with the standard formalism of surviv al analysis, but mak es ex- plicit the distinction b et w een individual-lev el mec hanisms and group-lev el observ able quan tities. This separation provides the conceptual foundation for the identiﬁabilit y analysis developed in the subsequen t sections. 2.1 Individual Hazard Mec hanism W e formalize the notion of an individual hazard mechanism, whic h serves as a conceptual represen tation of individual-level heterogeneity in surviv al outcomes. Deﬁnition 2.1 (Individual hazard mec hanism) . Let (Ω , F , P ) b e an underlying probability space. An individual hazard mechanism is a random element Θ : (Ω , F ) → ( M , B ) , taking v alues in a measurable space ( M , B ), together with a measurable mapping H : M → { h : [0 , ∞ ) → [0 , ∞ ) } , suc h that each realization θ ∈ M deterministically induces an individual hazard tra jectory h θ ( t ) = H ( θ )( t ) , t ≥ 0 . Th us, the hazard function of an individual is not itself random b ey ond the randomness of Θ; all heterogeneit y in individual time-to-even t b eha vior is represen ted, at an abstract level, through the distribution of Θ. R emark 2.1 (Mec hanism space versus function space) . Although each realization of an individual hazard mec hanism ultimately induces a nonnegativ e hazard function through the mapping H , w e do not deﬁne Θ directly as a random function. Instead, Θ is treated as an abstract random elemen t taking v alues in a general mechanism space M , whose realizations deterministically generate hazard tra jectories. This distinction separates the representation of individual risk- generating structure from the functional form of the resulting hazard and allo ws additional structure to b e incorp orated at the mec hanism level. 5 The mechanism Θ is inten tionally deﬁned at an abstract lev el, and the framew ork do es not require a unique parametrization of individual risk. Its purp ose is to formalise heterogeneity at the level of hazard-generating structure itself, rather than as sto c hastic v ariation added to an otherwise homogeneous ev ent-time distribution. Under this in terpretation, individuals diﬀer b ecause their underlying mechanisms diﬀer, and these diﬀerences induce distinct hazard tra jec- tories. As a purely illustrative example, one conv enien t w ay to instantiate suc h mec hanisms is through a hierarchical parametric representation. In this form ulation, Θ = ( C , Φ) , where C denotes a hazard class determining the qualitative shap e of the hazard function, such as monotone, unimo dal, or bathtub-shaped b ehaviour, and Φ collects class-sp eciﬁc parameters sp ecifying a particular tra jectory within that class. The induced hazard function is then given b y h Θ ( t ) = h C ( t ; Φ). This representation aligns with common taxonomic descriptions of hazard shap es and pro vides a concrete example of how a mec hanism ma y enco de b oth coarse structural patterns and within-class v ariabilit y . Imp ortantly , the general framework do es not rely on this or any other sp eciﬁc parametrisation. The admissible hazard space may b e chosen according to the scientiﬁc context and mo delling ob jectives, ranging from classical parametric families to geometric classes, mixtures, or more abstract pro cess-based representations. Accordingly , the laten t mechanism Θ should b e viewed as a general probabilistic sc hema, while concrete mo dels are obtained by sp ecifying an appropriate hazard space for the application at hand. F rom hazard mechanisms to individual surviv al. An individual hazard tra jectory h Θ ( t ) is interpreted as a rate of risk accum ulation ov er time. Accordingly , it induces a cumulativ e hazard H Θ ( t ) = Z t 0 h Θ ( u ) du, whic h summarizes the total amount of risk accrued up to time t . T o connect this risk accumu- lation with even t-time outcomes, we in tro duce an even t-time random v ariable T : Ω → [0 , ∞ ] deﬁned on the same probability space. W e assume that, conditional on the realization of the hazard mechanism, the distribution of T is fully determined by the induced hazard tra jectory . Sp eciﬁcally , if h θ is lo cally in tegrable, the asso ciated conditional surviv al function is given by S Θ ( t ) = exp( − H Θ ( t )) . F or a ﬁxed realization Θ = θ , this surviv al curve coincides with the conditional surviv al law S θ ( t ) = P ( T > t | Θ = θ ) . This is a well-deﬁned probability statement at the level of the underlying probability space. Ho wev er, it should not b e interpreted as a time-evolving probability assessmen t for a ﬁxed re- alized individual. Rather, S θ ( t ) c haracterizes the conditional distribution of even t times across individuals sharing the same hazard-generating mechanism, and serves as a deterministic func- tional summarizing how risk accumulates along the tra jectory sp eciﬁed by θ . R emark 2.2 (Mec hanism stability) . Throughout this pap er, the individual hazard mechanism Θ is assumed to b e stable ov er the observ ation window. Time enters the mo del through the ev aluation of the hazard tra jectory h Θ ( t ) and through the accumulation of risk, rather than through structural changes in the mechanism itself. Under this stability assumption, an y tem- p oral v ariation in individual risk is expressed through the form of the hazard tra jectory h Θ ( t ), rather than through the surviv al function. This assumption is implicit in classical surviv al mo d- els and is adopted here as a baseline for theoretical clariﬁcation, rather than as a claim ab out biological realit y . 6 2.2 Group-lev el surviv al and observ able hazard In this subsection, we clarify ho w classical p opulation-lev el and group-lev el surviv al quantities arise as observ able summaries of latent individual risk mechanisms. The purp ose of this section is not to in tro duce new mo deling assumptions, but to make explicit the informational structure through which laten t heterogeneity is aggregated in to observ able surviv al and hazard functions. 2.2.1 Observ able information and conditional mechanism distributions W e b egin by formalizing the role of observ able cov ariates as carriers of partial information. Deﬁnition 2.2 (Observ able information) . Let (Ω , F , P ) b e the underlying probabilit y space in tro duced in Section 2.1. An observ able information is a random element X : (Ω , F ) → ( X , A ) deﬁned on the same probabilit y space. The observ able information X is interpreted as par- tial information ab out the latent mec hanism Θ. F ormally , X induces a regular conditional distribution P Θ ( ·| X ) on ( M , B ), representing the residual heterogeneity in hazard mechanisms that remains after conditioning on the av ailable observ ations. This deﬁnition is delib erately informational rather than generative. No functional, causal, or structural relationship b et ween X and Θ is imp osed. Conditioning on X up dates the distri- bution of latent mechanisms but do es not alter the me c hanism itself. It is also worth noting that conditioning on s urviv al up to time t provides additional observ able information and induces a time-dep enden t conditional distribution P (Θ | T ≥ t, X ). Consequen tly , all risk-generating con tent is attributed to the mechanism, a principle formalized b elow as mechanism suﬃciency . Mec hanism suﬃciency . Within the prop osed formulation, the latent mec hanism Θ is in ter- preted as enco ding the complete individual-level risk-generating structure. Once a realization Θ = θ is giv en, the entire hazard tra jectory h θ ( t ) is ﬁxed, and the conditional distribution of the even t time T is fully determined. Under this in terpretation, observ able information X plays a purely informational role: it pro vides partial knowledge ab out the latent mechanism Θ but do es not mo dify the ev ent-time distribution once the mechanism is ﬁxed. This informational separation implies the conditional indep endence relation T ⊥ X | Θ . This relation is not in tro duced as an additional mo deling assumption, but follo ws directly from the deﬁnition of Θ as a hazard-generating mechanism. If Θ indeed captures all individual- lev el structure go v erning risk, then no observ able cov ariate should exert further inﬂuence on T beyond what is already enco ded in Θ. This do es not preclude cov ariates from b eing highly predictiv e of risk, but lo cates their role entirely in reﬁning information ab out the underlying mec hanism. Imp ortan tly , mec hanism suﬃciency do es not require the observ able information X to uniquely determine Θ. In most practical settings, X is only partially informative, so that substantial heterogeneit y in Θ typically remains within co v ariate-deﬁned groups. This residual uncertaint y is precisely what giv es rise to group-level surviv al and hazard functions as mixtures o v er la- ten t mechanisms, rather than as individual-level risk tra jectories. F rom a technical persp ective, mec hanism suﬃciency also plays a supp orting role in the theoretical dev elopments that follow. By treating Θ as the complete hazard-generating ob ject, this principle ensures that all sto c has- tic v ariation in the even t time T is mediated through Θ, which allows identiﬁabilit y questions 7 to b e formulated en tirely at the level of conditional mechanism distributions. In particular, this separation is used in the pro of of Theorem 2.3 to c haracterize the aggregation op erator mapping individual mec hanisms to p opulation-lev el surviv al functions. Information versus causation. Throughout this pap er, co v ariates are treated as sources of information ab out latent individual hazard mechanisms rather than as causal driv ers of those mec hanisms. This distinction is delib erate. Conditioning on observed co v ariates X reﬂects an operation of statistical conditioning rather than a statement about how interv en tions on X w ould alter the underlying hazard mechanism. Accordingly , the conditional distribution P (Θ | X = x ) enco des partial information ab out individual mechanisms, not a causal path wa y from X to Θ. This p ersp ectiv e is compatible with, but logically distinct from, causal in terpretations of surviv al mo dels. Causal questions concern h yp othetical in terven tions, such as whether m anip- ulating a treatmen t or exposure w ould change the underlying hazard mechanism. By contrast, the present framework fo cuses on the information structure induced b y observ ation: what can b e inferred ab out individual risk mec hanisms from surviv al data and cov ariates as they are observ ed. These p ersp ectiv es coincide only under strong assumptions linking co v ariates to mec hanisms through deterministic or near-deterministic relationships. Making this distinction explicit clariﬁes the in terpretation of observ able hazards and surviv al functions. Ev en when co v ariates are causally relev an t, ﬁtted hazards conditioned on X represen t survivor-w eigh ted a verages ov er heterogeneous mechanisms rather than in trinsic individual-lev el risks. The latent hazard framew ork therefore do es not deny the relev ance of causal reasoning in surviv al analysis, but emphasizes that causal questions and iden tiﬁabilit y questions op erate at diﬀeren t analytical lev els and should b e kept conceptually distinct. 2.2.2 P opulation and group-lev el surviv al W e no w show how classical surviv al and hazard functions arise as aggregated quan tities under partial information. Theorem 2.3 (Represen tation of observ able surviv al and hazard) . L et Θ b e an individual hazar d me chanism with induc e d hazar d tr aje ctory h Θ ( t ) and asso ciate d survival curve S Θ ( t ) . L et X b e an observable c ovariate interpr ete d as p artial information ab out Θ . Then the fol lowing statements hold. (i) The p opulation-level survival function S ( t ) = P ( T > t ) admits the r epr esentation S ( t ) = E [ S Θ ( t )] , that is, it arises as the aver age of individual survival curves over the latent me chanism distribution. (ii) F or any c ovariate value x with P ( X = x ) > 0 , the gr oup-level survival function satisﬁes S ( t | x ) = P ( T > t | X = x ) = E [ S Θ ( t ) | X = x ] . (iii) If S ( t | x ) is diﬀer entiable, the asso ciate d gr oup-level (observable) hazar d is given by h obs ( t | x ) = − d dt log S ( t | x ) = E [ h Θ ( t ) | T ≥ t, X = x ] . The pro of of Theorem 2.3 is pro vided in App endix A. The theorem makes explicit that surviv al and hazard functions are not primitive individual-level ob jects. P opulation-level and group-lev el surviv al functions arise as mixture av erages ov er latent mec hanisms, while observ able hazards corresp ond to surviv or-weigh ted p osterior a v erages of individual hazard tra jectories. All individual-level heterogeneity is therefore ﬁltered through the observ able surviv al function, clarifying b oth the scop e and the in trinsic limitations of hazard-based inference. 8 2.3 Mec hanism-Lev el Averages and Their Interpretation F or conceptual clarity , it is useful to distinguish the observ able hazard h obs ( t | x ) from other summaries that may b e deﬁned at the mec hanism level. In particular, one may consider the conditional mec hanism-level av erage ¯ h ( t | x ) = E [ h Θ ( t ) | X = x ] = Z M h θ ( t ) P Θ ( dθ | X = x ) , (1) whic h a verages individual hazard tra jectories with resp ect to the c onditional distribution of laten t mechanisms compatible with x . While ¯ h ( t | x ) is mathematically well deﬁned, it is not directly observ able from surviv al data. In general, h obs ( t | x )  = ¯ h ( t | x ) , b ecause the observ able hazard inv olv es additional conditioning on surviv al up to time t , as established in the previous subsection. The discrepancy betw een these tw o quantities reﬂects the com bined eﬀects of unobserved heterogeneity in individual mec hanisms and dynamic selection within the risk set. Only in the degenerate case where the conditional distribution P Θ ( ·| X = x ) collapses to a p oint mass, that is, when the cov ariates uniquely determine the underlying mec hanism, do the t wo hazards coincide. This distinction is central to the in terpretation of surviv al mo dels. Classical parametric, semiparametric, and machine-learning approaches estimate or approximate the observ able haz- ard h obs ( t | x ). They do not, and cannot without additional structural assumptions, reco ver individual hazard tra jectories or their mec hanism-level av erages. Accordingly , interpretations that treat ﬁtted hazards as individual-level risk functions implicitly imp ose strong and typically unac knowledged restrictions on the latent mec hanism distribution. Within the presen t frame- w ork, the observ able hazard should therefore b e viewed as the result of an aggregation op erator acting on latent mechanisms, { h θ ( · ) , P Θ ( ·| X = x ) } 7− → h obs ( ·| x ) , whic h integrates b oth structural heterogeneit y across mec hanisms and time-dep endent surviv or selection. This persp ective clariﬁes wh y individual-lev el hazard quan tities and their mechanism- lev el av erages are, in general, not identiﬁable from surviv al data, and motiv ates the identiﬁabilit y analysis dev elop ed in the subsequent section. 3 Iden tiﬁabilit y Theory Building on the framew ork introduced in Section 2, w e examine the identiﬁabilit y implications of mo deling surviv al data through latent individual-level mechanisms. The central ob ject of in terest is the individual hazard mechanism Θ. Identiﬁabilit y in this setting, ho wev er, is not a prop ert y of Θ alone, but of the information structure linking Θ, observ able co v ariates X , and the ev ent-time pro cess. As a consequence, what can b e learned from surviv al data is fundamentally constrained b y the wa y individual-level heterogeneity is aggregated into observ able quantities. These limitations do not hinge on censoring. Ev en in the idealized setting where all even t times are fully observed, the aggregation of heterogeneous individual mec hanisms in to population-level quan tities already entails irreducible constrain ts on identiﬁabilit y . Accordingly , this section focuses on iden tifying whic h aspects of the laten t mec hanism distri- bution are, in principle, recov erable from surviv al data, and whic h limitations are unav oidable. Rather than attempting to reconstruct individual surviv al curv es, we clarify the scop e of iden- tiﬁabilit y at the p opulation lev el under partial information. Sp eciﬁcally , we ﬁrst analyze how the information structure restricts identiﬁabilit y . W e then introduce a non-degeneracy principle under which these restrictions can b e formalized. Finally , w e sho w how the resulting non- iden tiﬁability implies that the conditional distribution of mechanisms given cov ariates should b e interpreted as an irreducible residual structure in surviv al analysis. 9 3.1 Information Structure and the Nature of Identiﬁabilit y Iden tiﬁability is fundamentally a prop ert y of the information structure rather than of any par- ticular statistical model. In statistical theory , what can b e iden tiﬁed from data is determined b y the mapping b etw een laten t quan tities and observ able v ariables, and by whether this mapping preserv es suﬃcient information to distinguish distinct underlying states (see, e.g., [Kagan et al., 1973, Lehmann and Casella, 1998]). Questions of identiﬁabilit y therefore precede mo deling c hoices, estimation pro cedures, and assumptions on functional forms. In surviv al analysis, this information structure is shap ed not only by observed cov ariates, but also b y censoring and b y the aggregation of individual risk ov er time. In applied surviv al analysis, the underlying ob ject of interest is often tak en to b e an individual-lev el surviv al or hazard function, implicitly asso ciated with an unobserved individual mec hanism, particularly in settings where the goal is individual-level risk prediction. Ho wev er, surviv al data do not pro vide direct access to such mechanisms. Instead, information ab out in- dividual surviv al b eha vior is mediated through a limited set of observ ables, t ypically consisting of the even t time, censoring indicator, and cov ariates. This mediation induces an intrinsic and irrev ersible loss of information, in the sense that multiple distinct individual mechanisms may giv e rise to the same observ able distribution. As a consequence, identiﬁabilit y issues in surviv al analysis cannot b e resolved solely by adopting more ﬂexible mo dels or ric her parameterizations. Ev en under arbitrarily ﬂexible sp eciﬁcations, the observ able data ma y b e insuﬃcien t to distinguish b etw een heterogeneous individual surviv al patterns. The resulting limitations are therefore structural in nature and arise from the aggregation of individual-level b eha vior at the p opulation level, rather than from mo del missp eciﬁcation or insuﬃcient expressiveness. This persp ective suggests that iden tiﬁabilit y questions in surviv al analysis should b e ad- dressed at the level of information structure. Rather than asking whether a particular mo del or parameterization is identiﬁable, a more fundamen tal question is which asp ects of the un- derlying surviv al mechanisms are, in principle, recov erable from the av ailable observ ables, and whic h are inherently lost. In particular, it is essential to distinguish b et w een quan tities that are iden tiﬁable only at the p opulation or group lev el and those that p ertain to individual hazard mec hanisms. The remainder of this section builds on this viewp oint to c haracterize the scop e and limits of identiﬁabilit y under partial information. 3.2 Non-degeneracy Assumptions and Iden tiﬁability Results Building on the framew ork introduced in Section 2, we view surviv al data as arising from la- ten t individual-level mec hanisms that gov ern hazard tra jectories and induce individual surviv al curv es. Within this framew ork, the observ able cov ariates X are interpreted as partial informa- tion ab out the underlying mechanism Θ, rather than as deterministic predictors of individual surviv al b eha vior. Consequently , the relationship b et ween Θ and X plays a central role in determining what asp ects of individual heterogeneity can b e learned from data. Therefore, a necessary conceptual prerequisite for interpreting surviv al quantities as reﬂecting individual- lev el risk is that the conditional distribution of mec hanisms given cov ariates, P (Θ | X ), should not collapse to a degenerate p oin t mass. This non-degeneracy principle reﬂects the fact that co v ariates t ypically provide incomplete information ab out individual surviv al mec hanisms: in- dividuals with identical or similar co v ariate v alues may still exhibit substantial heterogeneit y in their underlying hazard dynamics. In particular, we exclude settings in whic h the individual mec hanism is almost surely a deterministic function of the cov ariates, i.e., Θ = g ( X ) almost surely . This principle is esp ecially natural in contexts where the ob jective is individual-level risk prediction. While cov ariates ma y b e informativ e ab out surviv al outcomes, it is rarely plausible that they fully determine an individual hazard mechanism. T reating X as partial rather than 10 complete information ab out Θ, therefore, provides a more realistic represen tation of individual v ariability and aligns with the information structure implicit in surviv al data. Imp ortan tly , the non-degeneracy principle should b e understo o d as a structural mo deling stance rather than a technical as sumption imp osed for mathematical conv enience. It formalizes the idea that irreducible uncertain ty remains at the individual lev el even after conditioning on observed co- v ariates. T o make the ab o ve non-degeneracy principle concrete, we introduce a tec hnical formulation that is suﬃcient for deriving iden tiﬁability results. This formulation is not intended to b e ex- haustiv e or necessary; rather, it pro vides a transparent and tractable realization of the principle that co v ariates conv ey only partial information ab out individual mec hanisms Assumption 3.1 (Conditional Lo cal Ric hness) . Fix a c ovariate value x and c onsider the sp ac e of latent hazar d me chanisms that may arise under the c ondition X = x . We assume that, c onditional on X = x , the latent me chanism Θ r emains non-de gener ate in the sense that non- trivial individual-level variability p ersists at the level of survival tr aje ctories. Sp e ciﬁc al ly, we assume that ther e exist a r efer enc e me chanism θ 0 ∈ M , a b ounde d me asur able function g : [0 , ∞ ) → R that is not identic al ly zer o, a c onstant δ > 0 , and a me asur able mapping ε 7− → θ ( ε ) ∈ M , ε ∈ ( − δ, δ ) , such that for al l ε ∈ ( − δ, δ ) , the asso ciate d individual survival functions satisfy S θ ( ε ) ( t ) = S θ 0 ( t )  1 + εg ( t )  , ∀ t ≥ 0 , and deﬁne valid survival functions for e ach ε . This assumption formalizes a lo cal ric hness property of the mec hanism space: ev en after conditioning on the cov ariate v alue X = x , the space of admissible surviv al tra jectories admits non-trivial p erturbations. Consequen tly , the conditional distribution P (Θ | X = x ) do es not collapse to a degenerate p oin t mass. It is imp ortan t to emphasize that this assumption should b e understo o d as one suﬃcient technical realization of the non-degeneracy principle introduced ab o v e, rather than as a fundamental requiremen t. W e emphasize that the non-degeneracy principle itself do es not assert that all non-degenerate mec hanism spaces necessarily lead to non- iden tiﬁability . Rather, the present assumption sho ws that once non-degeneracy admits even mild lo cal v ariation in surviv al tra jectories, aggregation inevitably destroys injectivit y . Alternativ e form ulations that preserv e the conditional v ariability of mechanisms given co v ariates w ould lead to analogous iden tiﬁabilit y conclusions. The role of the present assumption is therefore to enable a precise c haracterization of identiﬁabilit y consequences under partial information, rather than to imp ose a restrictive structural mo del. In this sense, non-iden tiﬁability should b e viewed as a consequence of allo wing irreducible heterogeneity at the mechanism level, rather than as a failure of statistical mo deling. Under the conditional lo cal richness assumption, the mapping from individual surviv al tra- jectories and their conditional distribution to the p opulation-lev el surviv al function constitutes an aggregation op erator that is inherently many-to-one. The following theorem formalizes this observ ation b y establishing a fundamental non-identiﬁabilit y result at the individual lev el: the mapping from individual surviv al tra jectories and their conditional distribution to the observ- able population-level surviv al function is inheren tly non-injectiv e, and individual surviv al curv es and hazard tra jectories therefore cannot b e uniquely reco vered from p opulation-lev el surviv al data, ev en when the mec hanism space is ﬁxed and the p opulation surviv al function is fully kno wn. Theorem 3.2 (Structural non-identiﬁabilit y at the individual level) . Fix a c ovariate value x and let the individual hazar d me chanism Θ take values in a me asur able sp ac e ( M , B ) . Supp ose that the p opulation-level survival function S ( t | x ) is wel l deﬁne d on [0 , ∞ ) . Assume further 11 that, c onditional on X = x , the me chanism sp ac e M satisﬁes the c onditional lo c al richness assumption intr o duc e d ab ove. Then the observation op er ator mapping individual-level survival tr aje ctories to the p opulation-level survival function, O :  { S θ ( · ) } θ ∈M , P Θ ( ·| X = x )  7− → S ( t | x ) = Z M S θ ( t ) P Θ ( dθ | X = x ) , is non-inje ctive at the level of survival p aths. Mor e pr e cisely, even when the me chanism sp ac e M is ﬁxe d, ther e exist inﬁnitely many distinct c onditional distributions P ( k ) Θ ( ·| X = x ) , k = 1 , 2 , . . . , such that, for al l k , S ( t | x ) = Z M S θ ( t ) P ( k ) Θ ( dθ | X = x ) , ∀ t ≥ 0 , while the c orr esp onding c onditional distributions diﬀer at the p ath level. Conse quently, the induc e d c ol le ctions of individual survival curves, and the asso ciate d individual hazar d tr aje ctories h θ ( t ) , ar e not uniquely identiﬁable fr om the observable p opulation-level survival function S ( t | x ) . The pro of of Theorem 3.2 is pro vided in App endix B. This theorem shows that the lac k of iden tiﬁability at the individual level is a structural consequence of the information aggregation inheren t in surviv al data. Even when the p opulation-lev el surviv al function S ( t | x ) is fully kno wn and the mec hanism space M is ﬁxed, the mapping from individual surviv al tra jectories and their conditional distribution to the observ able p opulation-lev el surviv al curve is not inv ertible. As a result, individual surviv al curves and hazard tra jectories cannot b e uniquely recov ered from S ( t | x ). Imp ortan tly , this non-iden tiﬁabilit y do es not arise from mo del missp eciﬁcation or insuﬃ- cien t ﬂexibilit y , but from the fact that p opulation-level surviv al functions represen t av erages o ver heterogeneous individual mec hanisms. Diﬀeren t conﬁgurations of individual-lev el surviv al paths may therefore induce identical p opulation-lev el b ehavior, rendering them observ ationally indistinguishable. F rom this p ersp ectiv e, the p opulation surviv al function S ( t | x ) should b e understo od as an aggregated ob ject rather than as a direct pro xy for any individual surviv al tra jectory . The only situation in which this aggregation-induced non-identiﬁabilit y disapp ears is the degenerate case where conditioning on the observ able cov ariates eliminates all residual het- erogeneit y in the latent mec hanisms. If the conditional distribution P (Θ | X = x ) collapses to a p oin t mass at a single mec hanism, the aggregation op erator becomes trivial and the p opulation- lev el surviv al function coincides with the surviv al tra jectory induced by that mechanism. Such p erfect information scenarios represent knife-edge b oundary cases in which individual surviv al curv es are, in principle, iden tiﬁable. In realistic applications, ho wev er, observ able co v ariates rarely exhaust individual-level heterogeneity , and p opulation surviv al functions remain genuine aggregates o ver latent mechanisms. This result clariﬁes the scop e of what can and cannot b e learned from surviv al data. Al- though population-level quan tities and summaries are iden tiﬁable under appropriate conditions, individual-lev el surviv al mec hanisms remain fundamen tally underdetermined without additional structural assumptions. In the next subsection, we build on this insigh t b y interpreting the con- ditional distribution of mec hanisms given cov ariates as a residual structure in surviv al analysis. 3.3 Conditional Mec hanism Distributions as Residual Structure in Surviv al Mo dels A cen tral feature of surviv al analysis is the presence of multiple sources of randomness op erating at diﬀerent levels of the data-generating pro cess. At a minim um, randomness arises from the 12 ev ent-time mechanism itself, reﬂecting the stochastic nature of surviv al outcomes even when an individual hazard mechanism is ﬁxed. This source of randomness is explicitly represented in virtually all surviv al mo deling frameworks through hazard functions, surviv al functions, or coun ting process form ulations. In addition to ev en t-time v ariability , surviv al data also re- ﬂect heterogeneity across individuals in their underlying hazard-generating mechanisms. Ev en after conditioning on observed co v ariates, individuals may diﬀer in their latent hazard tra jec- tories in wa ys that are not directly observ able. Unlike even t-time randomness, how ever, this mec hanism-level heterogeneity is typically not represented as an explicit sto c hastic comp onent of the mo del. Instead, it is often absorb ed into p opulation-lev el quantities, constrained through lo w-dimensional parametric structures, or treated implicitly through mo deling assumptions. The latent hazard framew ork introduced in Section 2 mak es this additional lay er of randomness explicit by treating individual hazard mechanisms as latent random ob jects and by separating mec hanism-level heterogeneity from even t-time sto c hasticity . F rom an information-theoretic p ersp ectiv e, this residual heterogeneity reﬂects uncertaint y that cannot b e eliminated by conditioning on X alone. The iden tiﬁability results in the pre- vious subsection sho w that this mec hanism-level uncertain t y cannot, in general, b e resolv ed from p opulation-lev el surviv al data. Distinct conﬁgurations of individual surviv al tra jectories ma y induce iden tical observ able surviv al functions, implying that some asp ects of the underly- ing data-generating pro cess are inheren tly uniden tiﬁable. This observ ation suggests that any surviv al mo del m ust, either explicitly or implicitly , tak e a stance on how such irreducible un- certain ty is represented or absorb ed. Tw o complemen tary p ersp ectiv es on non-identiﬁabilit y . The non-iden tiﬁability of the conditional mec hanism distribution P (Θ | X = x ) can b e understoo d from tw o complemen- tary but distinct p ersp ectiv es. First, at the level of the data-generating pro cess, this non- iden tiﬁability reﬂects an inability to reco v er the structural relationship betw een latent individual mec hanisms and observ able co v ariates from surviv al data alone. In this sense, P (Θ | X = x ) pla ys a role analogous to the unknown structural form of the data-generating pro cess in regression analysis: just as the join t distribution of ( Y , X ) do es not determine whether the true relationship is linear, additiv e, or nonlinear without additional assumptions, surviv al data do not uniquely determine the conditional distribution of laten t hazard mec hanisms. Mo deling choices such as prop ortional hazards, accelerated failure time, or frailty sp eciﬁcations therefore corresp ond to structural assumptions imp osed at this level, rather than quantities identiﬁed from the data. Second, from a mo del-based p ersp ectiv e, once a particular structural form has b een adopted, the conditional mechanism distribution captures the residual heterogeneit y that remains unex- plained b y the observed cov ariates within the chosen mo del class. In this sense, P (Θ | X = x ) pla ys a role analogous to a residual comp onent: it represen ts individual-lev el v ariabilit y that cannot be attributed to X under the imp osed structural assumptions. Imp ortantly , this residual in terpretation is conditional on the mo deling framework and should not b e conﬂated with the more fundamen tal non-identiﬁabilit y that arises at the data-generating lev el. T aken together, these t wo p ersp ectiv es clarify the role of conditional mechanism distributions in surviv al analysis. Their non-iden tiﬁability is not a mo deling defect, nor a consequence of insuﬃcien t ﬂexibilit y , but a structural feature of the information av ailable in surviv al data. Diﬀeren t surviv al mo dels may parametrize, restrict, or approximate P (Θ | X = x ) in diﬀerent w ays, but none can eliminate its presence entirely unless individual surviv al mechanisms are fully determined by observed cov ariates. 4 Classical Surviv al Mo dels under a Laten t Hazard F ramew ork In the preceding sections, w e hav e argued that individual surviv al outcomes arise from la- ten t, individual-sp eciﬁc hazard mechanisms and that observ able surviv al quan tities, such as 13 p opulation-lev el or group-lev el surviv al and hazard functions, are obtained through aggregation o ver these unobserved mech anisms. A cen tral consequence of this represen tation is that, in the presence of cov ariate information X , the conditional distribution of individual mechanisms, P (Θ | X = x ) , is, in general, not iden tiﬁable from observed surviv al data. This observ ation has fundamen tal implications for surviv al analysis. It implies that any statistical mo del for surviv al data must, either explicitly or implicitly , take a stance on how this conditional mec hanism distribution is handled. Some mo dels b ypass it altogether b y operating exclusiv ely at the p opulation or group level; others imp ose restrictiv e structural assumptions that render the problem tractable; still others attempt to appro ximate or discretize the mechanism space in order to capture heterogeneit y or latent subt yp es. These choices are rarely articulated in mec hanism-lev el terms, y et they determine the class of individual heterogeneity that a mo del is capable of representing. The purp ose of this section is therefore not to review classical surviv al mo dels in terms of their regression forms or estimation procedures, but to reinterpret them within a framew ork based on mechanism conditioning. Sp eciﬁcally , we examine how diﬀerent mo del classes handle the unresolv ed v ariabilit y enco ded in P (Θ | X = x ), which we refer to as the conditional mecha- nism distribution or, b y analogy with regression analysis, the residual mec hanism comp onent. By organizing classical models according to their treatment of this ob ject, w e aim to clarify their conceptual scop e, their implicit assumptions, and the sources of their structural limitations. 4.1 Marginal Hazard Mo deling and the Co x F ramework The Cox’s prop ortional hazards mo del is a cornerstone of mo dern surviv al analysis. Within the laten t hazard framework dev elop ed in this pap er, it can b e understo o d as adopting a sp e- ciﬁc strategy for handling the conditional mechanism distribution P (Θ | X = x ). Rather than mo deling this distribution explicitly , the Cox model bypasses the mec hanism level altogether b y imp osing structure only on an aggregated, p opulation-level quan tity . In the presence of latent individual hazard mec hanisms, the ob ject of direct mo deling is the observ able or group-lev el hazard, giv en in our framew ork by the surviv or-w eighted p osterior exp ectation of individual haz- ard tra jectories conditional on remaining at risk. This observ able hazard already in tegrates o ver the conditional mechanism distribution P (Θ | X = x ) as w ell as the dynamic selection induced b y surviv al up to time t . The Cox mo del sp eciﬁes that this quantit y factorizes as h obs ( t | x ) = h 0 ( t ) exp( β ⊤ x ) , thereb y sidestepping the problem of mo deling or iden tifying P (Θ | X = x ) at the individual lev el. Under this formulation, the hazard ratio b etw een t wo cov ariate v alues x 1 and x 2 , h obs ( t | x 1 ) h obs ( t | x 2 ) = exp  β ⊤ ( x 1 − x 2 )  , should b e in terpreted as a comparison b etw een t wo survivor-weighte d p osterior mixtur es of individual hazard mechanisms. It do es not represen t a comparison of in trinsic individual-level risks, but rather reﬂects ho w cov ariates reweigh t the conditional mechanism distribution among individuals who remain in the risk set at time t . Prop ortional hazards is therefore a statement ab out the relativ e b eha vior of observ able mixtures, not ab out prop ortionality of individual hazard tra jectories. This marginal mo deling strategy has direct consequences at the mechanism level. Compati- bilit y with prop ortional hazards at the observ able level sev erely restricts the admissible forms of conditional mec hanism heterogeneity . In particular, genuine v ariation in hazard shap e, timing, or mo dalit y across individuals cannot b e represented, as all such v ariation must b e absorb ed in to risk set selection and aggregation. F rom the same p ersp ectiv e, the baseline hazard h 0 ( t ) 14 should b e view ed as a statistical device introduced to represent the observ able hazard, rather than as the baseline of an underlying individual hazard mechanism. Its form is not uniquely determined b y the data and carries no direct mec hanistic interpretation. A formal characteri- zation of the mec hanism-level restrictions implied by prop ortional hazards, including suﬃcien t conditions under whic h observ able proportionality can arise from laten t mechanisms, is pro vided in App endix C. 4.2 Lo w-Dimensional Mec hanism V ariation: F railty Mo dels F railty mo dels are commonly introduced as extensions of the Cox prop ortional hazards mo del de- signed to account for unobserved heterogeneity among individuals V aup el et al. [1979], Hougaard [1995]. F rom the p ersp ectiv e adopted in this paper, their deﬁning feature is not the introduction of random eﬀects p er se, but the particular w ay in which they restrict the conditional mecha- nism distribution P (Θ | X = x ). In a t ypical frailty form ulation, the individual hazard function is written as h ( t | X = x, Z = z ) = z h 0 ( t ) exp( β ⊤ x ) , where Z is a non-negative random v ariable, indep enden t of X , commonly referred to as the frailt y term. A t the mechanism level, this sp eciﬁcation amounts to assuming that all individual hazard mec hanisms share a common baseline shap e and diﬀer only through a multiplicativ e, time-in v arian t scaling factor. Consequently , the conditional mec hanism distribution is restricted to a one-dimensional family of prop ortional hazards indexed by Z . This explicit ackno wledgmen t of residual mechanism v ariability represents a ﬁrst step b e- y ond the Cox mo del, but it comes with sev ere structural constrain ts. Because heterogeneity is conﬁned to a single scalar dimension, frailty mo dels cannot represent diﬀerences in hazard shap e, timing, or mo dalit y; in particular, mechanisms corresp onding to early- v ersus late-onset risk, m ulti-phase hazard tra jectories, or non-prop ortional eﬀects lie outside the admissible mec h- anism space. Moreo ver, this low-dimensional represen tation do es not resolve the fundamental non-iden tiﬁability of P (Θ | X = x ), but instead renders the problem tractable b y imp osing a strong assumption: conditional on cov ariates, individual mec hanisms v ary only through a time- in v arian t random eﬀect. F rom this viewp oint, frailty mo dels should b e understo od as appro x- imations that trade expressiv e p o wer for interpretabilit y and estimability , and whose primary role is to adjust p opulation-lev el inference in the presence of unobserv ed heterogeneity rather than to recov er individual-level mechanisms or iden tify laten t subtypes. F rom this p ersp ectiv e, frailt y mo dels do not represen t a general framework for individual heterogeneit y , but a sp eciﬁc parametric closure of the conditional mec hanism distribution. The latent hazard framew ork adopted here rev erses this logic: the mechanism distribution is taken as primitive, and classical mo dels, including frailty sp eciﬁcations, are understo o d as particular restrictions imp osed for tractabilit y rather than as solutions to identiﬁabilit y . 4.3 Residual Mec hanisms and Time-Scale T ransformations: AFT Mo dels In the classical literature Kalbﬂeisch and Prentice [2002], AFT mo dels are most commonly in tro duced through a regression formulation on the logarithmic time scale, log T = µ ( X ) + ε, or, equiv alen tly , through a time–scaling representation of the surviv al function, S ( t | X ) = S 0  t e − µ ( X )  , whic h corresp onds to a systematic rescaling of the time axis t 7→ t/a ( X ), with a ( X ) = e µ ( X ) . In traditional treatments, the regression form and the distribution of the error term ε are often regarded as the primary mo deling comp onen ts. 15 Within the mec hanism-based framew ork developed in this pap er, we adopt a diﬀeren t p er- sp ectiv e. Rather than viewing AFT mo dels as regression mo dels for surviv al times, we in- terpret them as imp osing a structural constraint on the conditional mec hanism distribution P (Θ | X = x ). Speciﬁcally , AFT mo dels assume the existence of a single reference hazard mech- anism from which all individual hazard mec hanisms are generated through time–scale trans- formations. Conditional on X = x , residual heterogeneity among individual mec hanisms is therefore conﬁned to acceleration or deceleration along a common hazard shap e. Under this in terpretation, the error term ε in the classical AFT representation is not an indep endent mo d- eling choice, but an induced quan tit y reﬂecting the surviv al time generated by the reference mec hanism under a logarithmic time transformation. Consequently , the conv en tional c hoice of an error distribution implicitly corresp onds to a c hoice of the reference hazard shap e, rather than in tro ducing an additional degree of freedom in the conditional mechanism distribution. This formulation makes explicit that AFT mo dels imp ose a strong compression of P (Θ | X = x ). All conditional heterogeneity is restricted to a one-dimensional time–scaling factor, preclud- ing gen uine v ariation in hazard shap e, timing, or mo dalit y across individuals. As in the case of frailt y mo dels, this structural simplicit y confers interpretabilit y and stabilit y , but it do es not resolv e the fundamental non-iden tiﬁability of the conditional mec hanism distribution. A formal mec hanism-based generative c haracterization of AFT mo dels, together with its implications for the structure of P (Θ | X ), is pro vided in Appendix D. 4.4 Discrete Mec hanism Representations and Surviv al Clustering Surviv al clustering and latent class approaches are motiv ated b y the observ ation that hetero- geneous surviv al patterns may reﬂect the presence of distinct, unobserved subtypes within a p opulation. Within the mechanism-based framew ork developed here, these metho ds can b e understo od as adopting a sp eciﬁc and highly structured strategy for handling the conditional mec hanism distribution P (Θ | X = x ). Recen t surviv al clustering metho ds, such as the Surviv al Cluster Analysis framework Chap- fu wa et al. [2020], exemplify this strategy b y introducing a discrete latent structure in to time- to-ev ent mo dels. Rather than aiming to identify individual-level mechanisms, suc h approac hes imp ose a structured approximation to P (Θ | X = x ) in order to obtain tractable and in terpretable represen tations of heterogeneit y . At the mec hanism lev el, surviv al clustering mo dels t ypically p osit that the conditional distribution of individual mechanisms admits a ﬁnite discrete repre- sen tation, P (Θ | X = x ) = K X k =1 π k ( x ) δ Θ k , where eac h Θ k corresp onds to a distinct hazard mechanism or subtype, δ Θ k denotes a Dirac measure concentrated at Θ k , and π k ( x ) denotes cov ariate-dep enden t mixing w eights. In man y practical implementations, eac h comp onent may corresp ond to a class-sp eciﬁc family of hazard mo dels rather than a single ﬁxed hazard; here we adopt an abstracted mechanism-lev el view in whic h each comp onen t is represen ted by a representativ e mec hanism. Under this assumption, heterogeneit y is not mo deled as contin uous v ariation around a common mechanism, but rather as arising from a ﬁnite collection of qualitativ ely distinct mechanisms. F rom this p erspective, surviv al clustering replaces the unresolved v ariability in P (Θ | X = x ) with a strong discretization assumption. The conditional mechanism distribution is no longer treated as an unknown and p oten tially complex ob ject, but is approximated by a ﬁnite mixture supp orted on a small n umber of latent classes. This discretization renders the problem tractable and provides a direct interpretation in terms of subtypes, but it do es so by substantially restricting the admissible mechanism space. It is imp ortant to emphasize that this strategy do es not circumv en t the fundamen tal non- iden tiﬁability of P (Θ | X = x ). The discrete representation is not learned from the data in 16 an unconstrained sense; rather, it is imp osed as a mo deling assumption. In particular, the assignmen t of individuals to latent classes, as well as the interpretation of these classes as distinct mec hanisms, relies on the assumption that the observ ed surviv al and cov ariate information is suﬃcien t to supp ort such a discretization. In general, this assumption cannot b e justiﬁed without additional structural restrictions. View ed through the lens of mechanism conditioning, surviv al clustering metho ds are there- fore best understo o d as approximation schemes rather than iden tiﬁcation procedures. They oﬀer a conv enien t and interpretable w ay to summarize heterogeneity b y partitioning the mech- anism space into a ﬁnite num b er of representativ e comp onen ts, but this comes at the cost of in tro ducing strong and often implicit assumptions ab out the nature of the conditional mecha- nism distribution. Their primary con tribution lies in providing a low-complexit y representation of heterogeneity , rather than in recov ering individual-level mec hanisms or uncov ering uniquely iden tiﬁable subtypes. 5 Discussion and Concluding Remarks This paper w as motiv ated b y a p ersistent ambiguit y in surviv al analysis concerning the relation- ship b et w een individual-level risk and p opulation-lev el observ able quan tities. By introducing a laten t hazard framework, we made explicit the information structure linking individual hazard- generating mechanisms, observ able co v ariates, and surviv al outcomes. Within this framework, classical surviv al quan tities were shown to arise as aggregated ob jects, obtained b y av eraging o ver latent individual heterogeneit y . This p ersp ectiv e allo ws a uniﬁed treatment of iden tiﬁability and pro vides a conceptual basis for reinterpreting a broad class of surviv al mo dels. A central conclusion of the pap er is that individual-lev el hazard tra jectories are not iden tiﬁ- able from surviv al data under partial information. More strongly , ev en the conditional distribu- tion of individual mechanisms giv en cov ariates, P (Θ | X = x ), is generally not iden tiﬁable. This lac k of identiﬁabilit y is structural rather than tec hnical: it arises from the aggregation inherent in surviv al data and p ersists regardless of model ﬂexibility or estimation strategy . What is iden- tiﬁable from the data are p opulation-lev el or group-lev el surviv al and hazard functions, whic h summarize the b eha vior of heterogeneous individuals but do not corresp ond to an y in trinsic individual-lev el risk quantit y except in degenerate cases. Implications for in terpretation and prediction. These results hav e direct implications for the interpretation of ﬁtted surviv al mo dels. In particular, observ able hazards should not b e interpreted as individual risk functions. Ev en when conditioned on co v ariates, ﬁtted haz- ards represen t survivor-w eigh ted av erages ov er laten t individual mechanisms. Interpreting such quan tities as c haracterizing individual risk implicitly assumes that the conditional mec hanism distribution collapses to a p oin t, an assumption that is rarely justiﬁed in practice. The latent hazard framework clariﬁes that many common interpretations of mo del output rely on strong and often unackno wledged structural assumptions. F rom this p ersp ectiv e, a k ey implication concerns the scop e of interpretation in inferential settings. Quantities derived from ﬁtted surviv al mo dels may supp ort v alid p opulation-level in- ference within their modeling assumptions, bu t they do not automatically admit individual-lev el or mechanistic interpretations. Such in terpretations require additional structural assumptions linking the observ able representation to an underlying hazard-generating mec hanism. When these assumptions are not made explicit, inferen tial conclusions should b e understo o d as p er- taining to aggregated or pro jected quantities, rather than to in trinsic individual risk pro cesses. Clarifying this distinction do es not undermine classical inferential metho ds, but delineates the domain within which their interpretations remain conceptually coherent. These interpretational issues b ecome more severe in predictive settings. In man y con temp o- rary applications, surviv al mo dels are ev aluated primarily through predictiv e p erformance mea- 17 sures, such as discrimination or calibration. While suc h criteria are appropriate for assessing predictiv e accuracy , they do not by themselves justify mec hanistic, causal, or individual-lev el in- terpretations of the resulting risk scores. Within the laten t hazard framew ork, predictiv e success ma y arise from stable p opulation-level asso ciations, ev en when no iden tiﬁable individual-level hazard mec hanism is recov ered. Without a clear separation b etw een predictive ob jectives and inferen tial interpretation, there is a risk that predictiv e p erformance is o verstated as evidence of understanding underlying risk processes. Making these distinctions explicit is therefore essential when surviv al mo dels are deploy ed in mo dern predictiv e and mac hine-learning-based contexts. Implications for mo del c hoice and metho dological developmen t. The non-identiﬁabilit y of P (Θ | X = x ) also reframes the role of mo del c hoice in surviv al analysis. Diﬀerent mo deling strategies corresp ond to diﬀeren t wa ys of handling this unresolved structure. Some approaches, suc h as proportional hazards models, bypass the mec hanism lev el b y imposing structure directly on observ able hazards. Others, such as frailty or accelerated failure time mo dels, restrict condi- tional heterogeneit y to low-dimensional families. Latent class and surviv al clustering metho ds discretize the mechanism space. F rom this p ersp ectiv e, mo del selection is not merely a question of predictive p erformance or go o dness of ﬁt, but a choice ab out which structural assumptions one is willing to imp ose on an intrinsically unidentiﬁable ob ject. These considerations are particularly relev an t for individual-lev el risk prediction and sub- group disco very . The results of this pap er suggest that individual risk prediction is not primarily an algorithmic problem, but a structural one: meaningful individual predictions require strong assumptions ab out the form of the conditional mechanism distribution. Similarly , surviv al clus- tering metho ds should b e understo od as approximation or summarization schemes rather than as pro cedures that recov er uniquely identiﬁable subtypes. Their outputs dep end fundamentally on the imp osed discretization of an underlying distribution that is not iden tiﬁable from the data. Finally , the latent hazard framew ork p oin ts to a direction for future metho dological devel- opmen t. Rather than seeking increasingly ﬂexible regressions for observ able hazards, metho d- ological inno v ation ma y lie in dev eloping principled wa ys to constrain, appro ximate, or inter- pret the conditional mec hanism distribution. This includes explicit mo deling of mechanism heterogeneit y , incorp oration of scientiﬁc structure, or representation-based approaches that ac- kno wledge the limits of identiﬁabilit y . Suc h developmen ts would not eliminate the fundamental non-iden tiﬁability established here, but could lead to mo dels whose assumptions are clearer and whose in terpretations are b etter aligned with the information con tained in surviv al data In conclusion, the con tribution of this pap er is not the proposal of a new surviv al mo del, but a clariﬁcation of what surviv al analysis can and cannot identify under partial information. By making the latent structure explicit, the framework developed here provides a coheren t inter- pretation of classical mo dels and highlights the structural assumptions underlying individual- lev el inference. W e hop e that this p ersp ectiv e will help align metho dological developmen t and applied practice with the intrinsic limits of surviv al data, and encourage the dev elopmen t of surviv al mo dels whose assumptions and in terpretations are explicitly grounded in the av ailable information structure. 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Mac hine learning metho ds for surviv al analysis: A surv ey . ACM Computing Surveys , 54(4):1–36, 2021. doi: 10.1145/3453140. Simon Wiegrebe, Philipp Kopper, Raphael Sonabend, Bernd Bischl, and Andreas Bender. Deep learning for surviv al analysis: a review. Artiﬁcial Intel ligenc e R eview , 57(3):65, 2024. A Pro of of Theorem 2.3 Pr o of. W e prov e each statement in turn. (i) Population survival. By deﬁnition, S ( t ) = P ( T > t ) = E [ 1 { T >t } ] . Applying iterated exp ectation with resp ect to Θ yields E [ 1 { T >t } ] = E  E  1 { T >t } | Θ  . Conditional on Θ = θ , the even t time distribution is determined by the hazard h θ , and hence E [ 1 { T >t } | Θ = θ ] = P ( T > t | Θ = θ ) = S θ ( t ) . Therefore S ( t ) = E [ S Θ ( t )]. (ii) Gr oup-level survival. F or any x with P ( X = x ) > 0, S ( t | x ) = P ( T > t | X = x ) = E [ 1 { T >t } | X = x ] . Using iterated conditional exp ectation with resp ect to Θ gives E [ 1 { T >t } | X = x ] = E  E [ 1 { T >t } | Θ = θ , X = x ] | X = x  . By mechanism suﬃciency , the conditional distribution of T giv en Θ do es not dep end on X , so that E [ 1 { T >t } | Θ = θ , X = x ] = P ( T > t | Θ = θ ) = S θ ( t ) . Hence S ( t | x ) = E [ S Θ ( t ) | X = x ] . (iii) Observable hazar d. Assume that diﬀerentiation and conditional exp ectation may b e inter- c hanged. Diﬀerentiating the group-level surviv al function yields d dt S ( t | x ) = E  d dt S Θ ( t ) | X = x  = − E [ h Θ ( t ) S Θ ( t ) | X = x ] , 20 since d dt S θ ( t ) = − h θ ( t ) S θ ( t ) for each θ . Therefore, h obs ( t | x ) = − d dt log S ( t | x ) = E [ h Θ ( t ) S Θ ( t ) | X = x ] E [ S Θ ( t ) | X = x ] . Finally , by mechanism suﬃciency , noting that P (Θ ∈ dθ | T ≥ t, X = x ) ∝ S θ ( t ) P Θ ( dθ | X = x ) 1 , where the proportionality reﬂects a Bay esian update of the mechanism distribution: among individuals with cov ariate v alue x , mechanisms that are more likely to surviv e up to time t are o ver-represen ted in the risk set at time t , the ab ov e ratio is exactly the conditional exp ectation h obs ( t | x ) = E [ h Θ ( t ) | T ≥ t, X = x ] , whic h completes the pro of. B Pro of of Theorem 3.2 Pr o of. Fix a cov ariate v alue x and denote the observed p opulation-lev el surviv al function by S ( t ) := S ( t | x ) for t ≥ 0. W e construct inﬁnitely man y distinct conditional distributions P ( k ) Θ ( ·| X = x ) that induce the same surviv al function S ( t ) on a ﬁxed mechanism space ( M , B ). Since S ( ·| x ) is well deﬁned and we study the non-identiﬁabilit y of P Θ ( ·| X = x ) on a ﬁxed mec hanism space, let µ 0 b e a reference conditional distribution on ( M , B ) that induces S ( ·| x ) under the observ ation op erator, i.e., S ( t ) = Z M S θ ( t ) µ 0 ( dθ ) , ∀ t ≥ 0 . (2) In mo deling terms, µ 0 ma y b e interpreted as a candidate conditional mechanism distribution P Θ ( ·| X = x ). By the conditional lo cal richness assumption under X = x , there exist a reference mec hanism θ 0 ∈ M , a b ounded measurable function g : [0 , ∞ ) → R that is not identically zero, a constan t δ > 0, and a measurable mapping ε 7→ θ ( ε ) ∈ M for ε ∈ ( − δ, δ ) such that, for all ε ∈ ( − δ, δ ), S θ ( ε ) ( t ) = S θ 0 ( t )  1 + εg ( t )  , ∀ t ≥ 0 , (3) and S θ ( ε ) ( · ) deﬁnes a v alid surviv al function for each ε . W e now redistribute mass in the condi- tional mechanism distribution along directions that are indistinguishable under the observ ation op erator. Fix any α ∈ (0 , 1) and deﬁne δ ′ = min  δ, 1 − α α δ  . F or ε ∈ ( − δ ′ , δ ′ ), let ε ′ = − α 1 − α ε and deﬁne the tw o-p oin t mixture measure ν ε := α δ θ ( ε ) + (1 − α ) δ θ ( ε ′ ) . (4) W e then p erform a lo cal replacement of µ 0 b y deﬁning µ ε := µ 0 − η δ θ 0 + η ν ε , (5) where η ∈ (0 , 1) is chosen suc h that 0 < η ≤ µ 0 ( { θ 0 } ), ensuring that µ ε is a nonnegative measure with total mass one. (If µ 0 ( { θ 0 } ) = 0, one ma y ﬁrst p erform an equiv alen t inﬁnitesimal 1 F ormally , P (Θ ∈ dθ | T ≥ t, X = x ) denotes the regular conditional probabilit y measure on ( M , B ). The notation dθ is used in the sense of integration against measurable functions. 21 adjustmen t of µ 0 so that it assigns p ositive mass to θ 0 ; this do es not aﬀect (2) and serves only as a technical conv enience.) Com bining (2) and (5), for any t ≥ 0 we obtain Z M S θ ( t ) µ ε ( dθ ) = Z M S θ ( t ) µ 0 ( dθ ) − η S θ 0 ( t ) + η Z M S θ ( t ) ν ε ( dθ ) = S ( t ) − η S θ 0 ( t ) + η h αS θ ( ε ) ( t ) + (1 − α ) S θ ( ε ′ ) ( t ) i . Using (3), we further obtain αS θ ( ε ) ( t ) + (1 − α ) S θ ( ε ′ ) ( t ) = αS θ 0 ( t )  1 + εg ( t )  + (1 − α ) S θ 0 ( t )  1 + ε ′ g ( t )  = S θ 0 ( t ) h 1 +  αε + (1 − α ) ε ′  g ( t ) i . Since ε ′ = − α 1 − α ε , w e hav e αε + (1 − α ) ε ′ = 0 , and therefore αS θ ( ε ) ( t ) + (1 − α ) S θ ( ε ′ ) ( t ) = S θ 0 ( t ) . Substituting bac k yields Z M S θ ( t ) µ ε ( dθ ) = S ( t ) − η S θ 0 ( t ) + η S θ 0 ( t ) = S ( t ) , ∀ t ≥ 0 . Th us, for each suﬃciently small ε , the measure µ ε induces the same p opulation-lev el surviv al function S ( ·| x ). Because g ≡ 0, there exists t ∗ suc h that g ( t ∗ )  = 0. F or ε  = ˜ ε , equation (3) implies S θ ( ε ) ( t ∗ )  = S θ ( ˜ ε ) ( t ∗ ), and hence θ ( ε )  = θ ( ˜ ε ) in the sense of distinct surviv al tra jectories. Conse- quen tly , the tw o-p oin t mixtures ν ε and ν ˜ ε ha ve diﬀeren t supp orts, and therefore µ ε  = µ ˜ ε . By c ho osing a sequence ε k ↓ 0 with ε k  = ε ℓ for k  = ℓ , w e obtain inﬁnitely man y distinct condi- tional distributions { µ ε k } k ≥ 1 that all induce the same p opulation-lev el surviv al function S ( ·| x ). Hence, the observ ation operator O admits inﬁnitely man y distinct preimages for the same image S ( ·| x ) and is therefore non-injective at the path level. As a result, the conditional mechanism distribution P Θ ( ·| X = x ) cannot b e uniquely iden tiﬁed from S ( ·| x ). Finally , in settings where individual hazard tra jectories are deﬁned by h θ ( t ) = − d dt log S θ ( t ), the same argument implies that individual-level hazard paths are lik ewise not identiﬁable. C Mec hanism-Lev el Implications of the Co x Mo del In this app endix, we provide a formal characterization of the mec hanism-level restrictions im- plied by imp osing a prop ortional hazards structure on the observ able hazard. This material supp orts the in terpretation giv en in Section 4.1 and is included here to separate formal argu- men ts from the main conceptual discussion. Let Θ denote a laten t individual hazard mechanism taking v alues in a (measurable) mech- anism space M , and let H (Θ) = h Θ denote the induced individual hazard function. F or notational clarity , we consider a ﬁnite mechanism space M = { θ 1 , . . . , θ K } with corresp onding hazard tra jectories { h 1 ( t ) , . . . , h K ( t ) } . Giv en cov ariates X = x , the observ able hazard at time t can b e written as h obs ( t | x ) = E [ h Θ ( t ) | T ≥ t, X = x ] = K X k =1 w k ( t, x ) h k ( t ) , where w k ( t, x ) = P (Θ = θ k | T ≥ t, X = x ) 22 denotes the surviv or-w eighted p osterior probabilit y of mec hanism θ k . W e no w state a suﬃ- cien t condition under which prop ortional hazards at the observ able level can arise from latent mec hanism heterogeneity . Prop osition C.1 (Mec hanism-level implications of prop ortional hazards) . Assume that the observable hazar d satisﬁes h obs ( t | x ) = h 0 ( t ) exp( β ⊤ x ) for al l t ≥ 0 and al l x, and that the p osterior weights w k ( t, x ) vary nontrivial ly with x . Then ther e exists a nonne gative function h ∗ ( t ) and p ositive c onstants c 1 , . . . , c K such that h k ( t ) = c k h ∗ ( t ) , k = 1 , . . . , K . That is, pr op ortional hazar ds at the observable level ar e c omp atible with latent me chanism het- er o geneity only if al l admissible individual hazar d tr aje ctories shar e a c ommon time shap e and diﬀer solely by a multiplic ative sc ale factor. Prop osition C.1 formalizes the sense in which proportional hazards sev erely restrict the con- ditional mec hanism distribution P (Θ | X = x ). In particular, any heterogeneity in hazard shap e, timing, or mo dality across individual mechanisms is incompatible with proportionality of the ob- serv able hazard. F or exp ository clarity , the prop osition is stated for a ﬁnite mec hanism space. The restriction is not essential: analogous results hold for inﬁnite or con tin uous mec hanism spaces under mild regularit y conditions, and reﬂect a structural consequence of prop ortional hazards rather than an artifact of discretization. Pro of of Prop osition C.1 Pr o of. F or each cov ariate v alue x , the observ able hazard admits the survivor-w eigh ted mixture represen tation h obs ( t | x ) = K X k =1 w k ( t, x ) h k ( t ) , t ≥ 0 , where w k ( t, x ) = P (Θ = θ k | T ≥ t, X = x ). By assumption, the observ able hazard satisﬁes the prop ortional hazards form h obs ( t | x ) = h 0 ( t ) exp( β ⊤ x ) for all t ≥ 0 and all x. Fix co v ariate v alues x (1) , . . . , x ( K ) suc h that the corresp onding weigh t vectors w ( x ( j ) ) =  w 1 ( t, x ( j ) ) , . . . , w K ( t, x ( j ) )  , j = 1 , . . . , K , are linearly indep enden t for at least one (and hence all) t . This is p ossible by the assumption that the p osterior w eights v ary nontrivially with x . If the span of { w ( t, x ) : x } has dimension m < K , the argumen t can be restricted to an m -comp onen t eﬀectiv e submixture; the conclusion remains unc hanged. Deﬁne the K × K matrix W =  W j k  , W j k = w k ( t, x ( j ) ) , whic h is inv ertible by construction. F or each ﬁxed t , collect the individual hazard v alues into the v ector h ( t ) =  h 1 ( t ) , . . . , h K ( t )  ⊤ , and the corresp onding observ able hazards into g ( t ) =  h obs ( t | x (1) ) , . . . , h obs ( t | x ( K ) )  ⊤ . 23 By the mixture representation, g ( t ) = W h ( t ) . On the other hand, prop ortional hazards implies h obs ( t | x ( j ) ) = h 0 ( t ) exp( β ⊤ x ( j ) ) , j = 1 , . . . , K , so that g ( t ) = h 0 ( t ) c, c j = exp( β ⊤ x ( j ) ) , where the vector c do es not dep end on t . Combining the tw o expressions for g ( t ) yields W h ( t ) = h 0 ( t ) c. Since W is in vertible, we obtain h ( t ) = h 0 ( t ) W − 1 c. Th us there exists a ﬁxed v ector a = W − 1 c suc h that, for all t , h k ( t ) = a k h 0 ( t ) , k = 1 , . . . , K . Setting h ∗ ( t ) = h 0 ( t ) and c k = a k > 0 completes the pro of. On the in terpretation of the baseline hazard The prop ortional hazards representation h obs ( t | x ) = h 0 ( t ) exp( β ⊤ x ) in tro duces the baseline hazard h 0 ( t ) as a comp onen t of a factorization of the observ able hazard. F rom the mechanism-lev el p ersp ectiv e adopted in this app endix, it is imp ortan t to distinguish this statistical ob ject from an y underlying individual hazard mec hanism. Prop osition C.1 establishes that, under prop ortional hazards, all admissible individual haz- ard tra jectories m ust share a common time shap e h ∗ ( t ), diﬀering only b y a m ultiplicative scale factor. This function h ∗ ( t ) can b e interpreted as a me chanism-level b aseline shap e , in the sense that it characterizes the unique hazard geometry compatible with prop ortionality at the ob- serv able lev el. How ever, the existence of such a function do es not imply its identiﬁabilit y from surviv al data. In contrast, the Co x baseline hazard h 0 ( t ) is not an estimate of h ∗ ( t ), nor do es it corresp ond to the hazard of a represen tativ e or baseline individual. Rather, h 0 ( t ) is a mo deling device arising from marginal hazard mo deling and from the chosen factorization of the observ able hazard. Its form is not uniquely determined b y the data, as diﬀeren t choices of h 0 ( t ) may lead to the same partial likelihoo d. This distinction clariﬁes wh y baseline hazards in the Cox mo del carry no direct mec hanistic in terpretation, even though prop ortional hazards imp ose a strong and w ell-deﬁned structure on the underlying mechanism space. D Mec hanism-Based Generativ e Structure of AFT Mo dels In this app endix, we provide a formal mechanism-based characterization of accelerated failure time (AFT) mo dels. This material supp orts the interpretation given in Section 4.3 and is included here to separate formal generative arguments from the main conceptual discussion. Our goal is to make explicit the structural assumptions that AFT mo dels imp ose on the conditional mec hanism distribution P (Θ | X ). Let Θ denote a latent individual hazar d me chanism taking v alues in a measurable mechanism space M , and let H (Θ) = h Θ denote the induced individual hazard function. Let X denote 24 observ able cov ariates. W e assume that there exists a r efer enc e hazar d me chanism with hazard function h 0 and asso ciated surviv al function S 0 ( t ) = exp  − Z t 0 h 0 ( s ) ds  . The deﬁning assumption of AFT models, in the presen t framework, is that all individual hazard mec hanisms are generated from this reference mechanism through time–scale transformations. Prop osition D.1 (Mec hanism-based generative structure of AFT mo dels) . Assume that, c on- ditional on X = x , the individual hazar d me chanism Θ is gener ate d thr ough a pur e time–sc aling tr ansformation driven by a p ositive r andom variable U > 0 , namely Θ | X = x d = Φ x ( U ) , U ∼ P U , wher e the mapping Φ x satisﬁes H (Φ x ( u ))( t ) = 1 u a ( x ) h 0  t u a ( x )  , t ≥ 0 , and a ( x ) > 0 is a c ovariate–dep endent time–sc aling function. Under this me chanism structur e, the individual survival time c onditional on X = x admits the r epr esentation T | X = x = a ( x ) U · T 0 , T 0 ∼ S 0 . Deﬁning ε := log T 0 , we obtain log T = log a ( X ) + log U + ε. In this formulation, the c onditional me chanism distribution P (Θ | X ) is entir ely determine d by the r efer enc e hazar d h 0 , the time–sc aling map Φ x , and the distribution of U . Conse quently, the distribution of the err or term ε is induc e d by the r efer enc e me chanism thr ough a lo garithmic time tr ansformation, r ather than b eing an indep endent mo deling choic e. R emark D.1 (Compression of the conditional mec hanism distribution) . Prop osition D.1 clariﬁes the fundamental assumptions underlying AFT mo dels at the mechanism lev el. The existence of a single reference hazard mec hanism constitutes the primary structural assumption, while the classical error term ε arises as a deriv ed quantit y . In particular, all heterogeneity conditional on X = x is compressed in to a one-dimensional time–scaling factor U , so that the conditional mechanism distribution P (Θ | X = x ) is conﬁned to the orbit  t 7→ 1 u h 0  t u  : u > 0  . As a result, AFT models do not allow for genuine heterogeneity in hazard shap e, timing, or mo dalit y across individuals. All admissible hazard tra jectories diﬀer only by acceleration or deceleration along a common reference shap e. This strong compression of the conditional mec hanism distribution explains b oth the in- terpretabilit y and the limitations of AFT models. While the resulting structure facilitates estimation and interpretation, it do es not resolve the fundamental non-iden tiﬁability of P (Θ | X ) discussed in Section 3. These tw o asp ects are inseparable consequences of the same structural c hoice. 25 Metho dological implications The mechanism-based formulation presented ab o ve highligh ts that inno v ation in surviv al modeling need not b e conﬁned to alternative regression sp eciﬁcations or estimation techniques. Instead, new mo del classes may b e obtained by relaxing or mo difying the structural assumptions imp osed on the conditional mechanism distribution P (Θ | X ). F rom this p ersp ectiv e, generalized AFT mo dels may b e constructed by allowing multiple reference mechanisms, multi-dimensional time–scaling factors, or departures from pure time rescaling. Imp ortantly , suc h generalizations op erate at the level of the mec hanism space, rather than through ad ho c mo diﬁcations of the regression form. This viewp oin t provides a principled framew ork for balancing interpretabilit y , ﬂexibility , and iden tiﬁabilit y in the developmen t of future surviv al mo dels. Pro of of Prop osition D.1 Pr o of. Conditioning on X = x , the assumption Θ | X = x d = Φ x ( U ) together with the deﬁnition of the mapping Φ x implies that, conditional on U = u and X = x , the individual hazard function tak es the form h Θ ( t | U = u, X = x ) = 1 u a ( x ) h 0  t u a ( x )  . By the deterministic relationship b etw een the hazard and the cumulativ e hazard, w e obtain Z t 0 h Θ ( s | U = u, X = x ) ds = Z t/ ( u a ( x )) 0 h 0 ( v ) dv = H 0  t u a ( x )  , where H 0 ( t ) = R t 0 h 0 ( s ) ds . Consequently , the conditional surviv al function satisﬁes S ( t | U = u, X = x ) = exp  − H 0  t u a ( x )  = S 0  t u a ( x )  . Let T 0 b e a random v ariable with surviv al function S 0 , that is, P ( T 0 > t ) = S 0 ( t ). F or any t ≥ 0, P  a ( x ) u T 0 > t  = P  T 0 > t a ( x ) u  = S 0  t a ( x ) u  , whic h coincides with the conditional surviv al function S ( t | U = u, X = x ) deriv ed in the previous step. Therefore, in distribution, T | ( U = u, X = x ) d = a ( x ) u T 0 , and hence T | X = x = a ( x ) U · T 0 , T 0 ∼ S 0 , with T 0 indep enden t of U . T aking logarithms of the ab o ve representation and deﬁning ε := log T 0 , w e obtain log T = log a ( X ) + log U + ε. Since T 0 has surviv al function S 0 , for any z ∈ R , P ( ε ≤ z ) = P ( T 0 ≤ e z ) = 1 − S 0 ( e z ) . Th us, the distribution of ε is uniquely determined by S 0 (equiv alently , by the reference hazard h 0 ) through the logarithmic time transformation. Com bining the abov e arguments, we conclude that under the prop osed mechanism-based generativ e structure, the conditional mec hanism distribution P (Θ | X ) is fully induced by the reference mec hanism h 0 , the time–scaling map Φ x , and the distribution of U . Accordingly , the error distribution in the classical AFT represen tation is not an indep enden t mo deling choice, but a direct consequence of the underlying mec hanism structure. 26

Rethinking Individual Risk and Aggregation in Survival Analysis: A Latent Mechanism Framework

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