PERSONALIZED TREATMENT SELECTION VIA PRODUCT PARTITION MODELS WITH COVARIATES Matteo Pedone

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PERSONALIZED TREATMENT SELECTION VIA PRODUCT

PARTITION MODELS WITH COVARIATES

Matteo Pedone

Department of Statistics, Computer Science and Applications

University of Florence

matteo.pedone@unifi.it

Raffaele Argiento

Department of Economics

University of Bergamo

raffaele.argiento@unibg.it

Francesco C. Stingo

Department of Statistics, Computer Science and Applications

University of Florence

francescoclaudio.stingo@unifi.com

ABSTRACT

Precision medicine is an approach for disease treatment that deﬁnes treatment strategies based on

the individual characteristics of the patients. Motivated by an open problem in cancer genomics,

we develop a novel model that ﬂexibly clusters patients with similar predictive characteristics and

similar treatment responses; this approach identiﬁes, via predictive inference, which one among a

set of treatments is better suited for a new patient. The proposed method is fully model-based,

avoiding uncertainty underestimation attained when treatment assignment is performed by adopting

heuristic clustering procedures, and belongs to the class of product partition models with covariates,

here extended to include the cohesion induced by the Normalized Generalized Gamma process. The

method performs particularly well in scenarios characterized by considerable heterogeneity of the

predictive covariates in simulation studies. A cancer genomics case study illustrates the potential

beneﬁts in terms of treatment response yielded by the proposed approach. Finally, being model-

based, the approach allows estimating clusters’ speciﬁc response probabilities and then identifying

patients more likely to beneﬁt from personalized treatment.

1 Introduction

Cancer comprises a collection of complex diseases characterized by heterogeneous cellular alterations across patients

and cancer cells within the same neoplasm (Bedard et al.,2013). Patients with similar clinical diagnoses may show

diverse responses to the same treatment due to tumor heterogeneity. A treatment for a particular diagnosis may

be effective on average, but its effectiveness may vary across subpopulations. In recent years many attempts have

been made to devise personalized treatment strategies that leverage patients’ characteristics, including the tumor’s

genome, to identify the treatment with the highest likelihood of success (Simon,2010). Within this precision medicine

paradigm, there is an increasing interest in discovering individualized treatment rules (ITRs) for patients that show

heterogeneous responses to treatment, e.g., when the treatment effect varies across groups of patients. An ITR is

a decision rule that assigns the patient to the treatment given patient/disease characteristics (Ma et al.,2015). The

optimal ITR is the one that maximizes the population mean outcome. Statistical methodology research in precision

medicine is devoted to developing personalized treatment rules to inform decision-making. The distinctive mark

of statistical inference under the precision medicine paradigm is to leverage heterogeneity to improve therapeutic

strategies (Kosorok and Laber,2019).

Our interest speciﬁcally lies in developing frontline treatment selection rules rather than estimating treatment’s causal

effects, as commonly done within the ITR framework. Conventional methods for treatment selection rules are based

on semi- and non-parametric procedures to identify subgroups of patients more likely to beneﬁt from a treatment

leveraging few baseline markers (Bonetti and Gelber,2000;Song and Pepe,2004). The subgroup approach can provide

arXiv:2210.06030v2 [stat.ME] 1 Sep 2023

Personalized Treatment Selection via Product Partition Models with Covariates

valuable information when performed according to a prespeciﬁed analysis plan. Nonetheless, stratiﬁed subsets of

patients deﬁned by one or few biomarkers are often inadequate to account for patient heterogeneity and ultimately fail

to establish effective treatment selection rules (Pocock et al.,2002). Other approaches account for patient heterogeneity

by including covariates (Zhang et al.,2012;Zhao et al.,2012). However, for these methods, the correct deﬁnition of

treatment-by-markers interactions is crucial and relies on sensitive assumptions, which are difﬁcult to specify in the

clinical practice and may be limited to generalized linear models (Ma et al.,2016).

To overcome these limitations, Ma et al. (2016,2018,2019) have established a hybrid two-step predictive model for

personalized treatment selection. In the ﬁrst step, a clustering algorithm based on a pre-deﬁned genomic signature

(predictive markers) is used to obtain a heuristic measure of the patients’ molecular similarity. In the second step,

given this measure of patients’ similarity and a set of prognostic markers, a Bayesian model is used for treatment

selection; speciﬁcally, for a new, untreated patient, the model predicts the treatment response probabilities for each

competing treatment. This framework establishes two signiﬁcant improvements over existing methods. Firstly, the

common assumption of statistical exchangeability among patients is relaxed. Since each tumor is unique, patients are

considered partially exchangeable only to the extent to which their tumors are molecularly similar. Moreover, this

approach utilizes complementary sources of information for treatment selection, integrating predictive and prognostic

characteristics of a patient.

This paper proposes a Bayesian predictive model for personalized treatment selection that builds upon Ma et al. (2019)

and overcomes some of its main limitations. As in Ma et al. (2019), we leverage prognostic determinants and predictive

biomarkers for treatment selection. We propose a fully Bayesian integrative framework for clustering and prediction

that performs all inferential tasks in a single model avoiding multi-step procedures; the proposed approach results in

a treatment selection rule that fully accounts for patients’ heterogeneity. Note that in Ma et al. (2019), the patients’

similarities were estimated in the ﬁrst step and included as known quantities in the second step; moreover, in the ﬁrst

step, two arbitrary choices had to be made, namely the clustering algorithm and the number of clusters. The proposed

method accounts for the uncertainty in all modeling steps, resulting in improved prediction performances. In particular,

we use a product partition model with covariates (PPMx, M¨

uller et al.,2011) to cluster observations that are similar in

terms of the values of the predictive covariates; speciﬁcally, the predictive covariates enter the model through the prior

for the random partition. The resulting partitions are only partially exchangeable, and patients with similar covariates

are a priori more likely to be clustered together. In this paper, we use the cohesion function induced by the Normalized

Generalized Gamma process (NGGP) as a building block of our PPMx model to mitigate the rich-get-richer property

of the Bayesian nonparametric (BNP) priors. Namely, the rich-get-richer is the tendency for a small number of clusters

to become overrepresented as more data points are added to the process, resulting in few large clusters and potentially

many singletons. Despite being well studied in the Bayesian nonparametric literature as a prior inducing a Gibbs-type

random partition (Lijoi et al.,2007), NGGP still has no common use. To the best of our knowledge, this is one of the

ﬁrst attempts the NGGP is employed as cohesion function in a PPMx model (see Argiento et al.,2022).

We devise a method that, given the patients’ prognostic and predictive markers, assigns them to the treatment with

the highest likelihood of positive response. Prognostic covariates inﬂuence disease progression regardless of the

treatments given to the patient, whereas predictive covariates change the likelihood of a positive response to a particular

treatment. Conceptually, our strategy for selecting the optimal treatment for the new, untreated patient can be broken

down into three steps. First, we consider historical patients and cluster them separately for each treatment according to

their predictive markers. In this way, patients that underwent the same treatment are divided into homogeneous clusters

with respect to predictive biomarkers. Then, we compute the utility provided by each competing treatment to the new

untreated patient by assigning the new patient to the subgroup of historical patients with whom he shows the largest

similarity in terms of predictive markers. The utility function relies on the model’s posterior predictive distribution,

which depends on both prognostic and predictive biomarkers. Finally, we select the treatment that ensures the largest

predicted beneﬁt.

We apply the proposed method to a brain cancer dataset (Ma et al.,2019), comprising 158 patients equally assigned to

either standard or targeted treatment. For each patient, prognostic and predictive biomarkers, both consisting of pre-

selected genomics markers, are available in addition to their categorical response to treatment. To facilitate optimal

treatment selection, we assign numerical utilities to each treatment response level. This leads to a median utility score,

which serves as a one-dimensional criterion for treatment selection. Our model shows good predictive performances

and provides a sound framework for the identiﬁcation and interpretation of clusters of patients.

2 Bayesian Integrative Model

We consider nhistorical patients treated with Talternative treatments, whose predictive and prognostic biomarkers

are measured along with a discrete set of response levels of the clinical outcome. Let a= 1, . . . , T index treatments

Personalized Treatment Selection via Product Partition Models with Covariates

and n=PT

a=1 nabe the total number of treated patients, of which naassigned to therapy a. Note that, in our

notation the superscript ais solely a treatment index. The treatment response ya

iof patient iis a categorical variable

with Klevels that encodes the residual disease extent after a clinically relevant post-therapy follow-up period. In

particular, ya

ifollows a multinomial distribution ya

i|πa

ind

∼Multinomial(1,πa

i), for i= 1, . . . , na, with associated

probability vector πa

i= (πa

i1, . . . , πa

iK )⊤;πa

ik is the probability of observing outcome kfor the i−th patient under

treatment a, for k= 1,...K. These probabilities will depend on za

iand xa

i, the P−and Q−dimensional vector of

prognostic and predictive features measured on the i−th patient that received treatment a, respectively. We assume

that patients with similar predictive biomarkers and the same prognostic covariates will respond similarly to a given

treatment. To quantify the effectiveness of each competing treatment for patients with similar values of the predictive

biomarkers, we adopt a covariate-dependent random partition model (RPM). For each treatment a= 1, . . . , T , patients

receiving treatment aare partitioned into clusters based on their predictive biomarkers xa. Namely, we make the

random partitions depend on predictive biomarkers. Section 4will describe the covariate-dependent RPM we use to

achieve this goal. In this section, we assume that Pa

na={Sa

1, . . . , Sa

na}is a given treatment-speciﬁc partition of the

indices {1, . . . , na}, where Ca

nais the number of clusters among patients treated with therapy aand na

j=|Sa

j|is the

cardinality of cluster j, for j= 1, . . . , Ca

na. Since we will later treat the partition of the units as a random quantity,

the partition itself and the number of clusters depend on the number of observations, na. Following a common

convention, we identify cluster-speciﬁc quantities using the superscript “⋆”. For example, when considering cluster

j, the response vector is ya⋆

j={ya

i:i∈Sa

j}, while xa⋆

j={xa

i:i∈Sa

j}is the partitioned covariate matrix. We

deﬁne the following hierarchical model for the response variables:

i|πa

ind

∼Multinomial(1,πa

πa

1,...,πa

na|ηa⋆

1,...,ηa⋆

na,Pa

na,β∼

j=1 Y

i∈Sa

Dirichlet(γa

i(ηa⋆

j,β)),(2.1)

where γa

i(ηa⋆

j,β)=(γa

i1(ηa⋆

j1,β1), . . . , γa

iK (ηa⋆

jK ,βK))⊤is a vector of log-linear functions of the prognostic markers

and cluster-speciﬁc parameters deﬁned as follows:

log(γa

ik(ηa⋆

jk ,βk)) = ηa⋆

jk +β1kza

i1+· · · +βP kza

iP .(2.2)

Model (2.1) is robust with respect to overdispersion (Corsini and Viroli,2022), which is usually observed in multi-

variate categorical data (Chen and Li,2013). Predictive biomarkers, xa, enter equation (2.2) through the parameter

vectors ηa

1,...,ηa

naand the partition Pa

nathat depends on the predictive covariates xa, as we will elaborate in Sec-

tion 3. The K-dimensional vectors ηa⋆

1,...,ηa⋆

naare cluster-speciﬁc parameters; high values of ηa⋆

jk correspond to

an high probability of observing response kfor an individual treated with treatment ain cluster j. We enforce ηa⋆

to be treatment-speciﬁc, and, as a consequence, the partitions {Pa

na}a=1,...,T are independent across treatments. This

construction provides a comparison among competing treatments. In fact, it allows patients with close genetic proﬁles

that received different treatments to have distinct response probabilities. Finally, β= (β1,...,βK)is a P×Kmatrix

of regression parameters shared across treatments. Prognostic biomarkers enter equation (2.2) as linear terms. Since

prognostic determinants impact the likelihood of achieving a given therapeutic response regardless of the treatment,

the associated coefﬁcients are deﬁned across therapies. Thus, prognostic covariates set a baseline response probability

measure. Since patients should not be regarded as statistically exchangeable with respect to predictive biomarkers (Ma

et al.,2016), we leverage predictive biomarkers to drive the clustering process within each treatment. The resulting

cluster-speciﬁc parameters ηa⋆

jassess the beneﬁt offered by a speciﬁc treatment on groups of similar patients. Note

that the linear predictor is a function of the prognostic biomarkers only: the predictive covariates enter non-linearly

equation (2.2) only through the cluster- and treatment-speciﬁc parameters ηa⋆

j. This construction results in a random

intercept that estimates the adjustment provided by predictive biomarkers to the baseline prognostic response prob-

ability on account of groups of patients with close predictive determinants. Note that, while the Multinomial logit

model (Agresti,2019) could have provided similar predictive performance, its interpretation would have been less

straightforward since the parameters represent log odds ratios with respect to a speciﬁc baseline response level.

3 Prior distributions

We assume independent shrinkage priors for the parameters βk. In particular, we adopt horseshoe priors (Carvalho

et al.,2010), which belong to the class of global-local scale mixtures of normals. More in details, for p= 1, . . . , P

and k= 1, . . . , K

βpk

iid

∼N(0, λ2

pkτ2

k), λpk, τk

iid

∼HC(0,1),

Personalized Treatment Selection via Product Partition Models with Covariates

where HC denotes a half-Cauchy distribution, {λpk}are local shrinkage parameters, and {τk}are global shrinkage

parameters. All coefﬁcients will be nonzero, but only those supported by the data will have large values due to

the heavy tails of the prior. The joint distribution of the clustering and the cluster-speciﬁc parameters (Pa

na,ηa⋆

j), is

assumed to be independent across treatments. Therefore we will omit the superscript athroughout Sections 3and 4. In

particular, we assume a product partition model with covariates (PPMx, M¨

uller et al.,2011), that induces independence

across clusters and conditional independence within clusters. We detail our proposal for the PPMx on Pnin Section

4. Here, given Pn, we details the prior for η⋆

j,j= 1, . . . , Cn. We assume conditional independence between clusters,

that is η⋆

j∼G0, for j= 1, . . . , Cn, where G0is a prior for cluster-speciﬁc parameters. Then, the joint law of (Pn,η⋆

is assigned hierarchically as:

η⋆

iid

∼G0,for j= 1, . . . , Cn,Pn|x∼P P Mx(x).

Speciﬁcally, we take G0to be a K−dimensional multivariate normal distribution and assume that η⋆

j|θ,Λiid

∼

NK(θ,Λ−1). To achieve more ﬂexibility, we add an extra layer of hierarchy by assuming θ|µ0,Λ, ν0∼

NK(µ0,(ν0Λ)−1)and Λ|s0,Λ0

iid

∼W(Λ0, s0), where Wis a Wishart distribution, with mean s0Λ0. As custom-

ary hyperparameter choice, we set µ0to be the K−dimensional vector of 0,s0=K+ 2,Λ0to be a K×Kdiagonal

matrix with elements on the diagonal being equal to 10, and ν0= 10. Elicitation for the latter two parameters is

discussed in Supplementary Material A.

4 Bayesian Nonparametric Covariate Driven Clustering

In this section, we introduce the Product Partition Model (PPM) and describe its extension to incorporate the Normal-

ized Generalized Gamma process (NGGP). We follow M¨

uller et al. (2011)’s approach to integrate predictive biomark-

ers into the model, making the random partition dependent on predictive markers. We devise a covariate-dependent

prior on the random partition that enables predictive markers to drive the clustering process. Thereby, we induce clus-

ters of homogeneous observations in terms of predictive biomarkers. The resulting model deﬁnes independence across

clusters and exchangeability only within clusters. The joint evaluation of prognostic and predictive covariates guides

the optimal treatment selection, our main inferential goal. Still, only the predictive markers identify patients likely to

beneﬁt from a particular therapy. In this way, we may quantify the extent of beneﬁt offered by a speciﬁc treatment on

groups of patients characterized by similar values of the predictive markers. We denote with Pn:= {S1, . . . , SCn}

the partition of the data label set {1, . . . , n}into Cnsubsets Sj, for j= 1, . . . , Cnand with nj=|Sj|being the

cardinality of cluster j. In the seminal paper by Hartigan (1990) the prior on Pnis assigned by letting

p(Pn) = Vn,Cn

j=1

ρ(Sj),(4.1)

where ρ(·)is referred to as cohesion function, and quantiﬁes the unnormalized probability of each cluster (M¨

uller

et al.,2011). Moreover, Vn,Cnis a normalizing constant assuring that the prior sum up to one over the space of all

partitions of the integers {1, . . . , n}. If ρ(Sj)is only a function of nj=|Sj|, then the resulting model for Pnis

invariant under permutations of the labels of the set of integers {1, . . . , n}. Under this assumption, the resulting model

for Pnfalls in the class of Gibbs-type priors (Gnedin and Pitman,2006). In this framework, the cohesion assume

the analytical expression ρ(Sj) = (1 −σ)nj−1with σ < 1and (1 −σ)nj−1being the rising factorials, deﬁned as

(a)n=a(a+1) . . . (a+n−1), with (a)0= 1;p(Pn)is denoted as exchangeable partition probability function (eppf)

and the normalizing constant Vn,Cnmust satisfy the triangular recursion Vn,Cn=Vn+1,Cn(n−σCn) + Vn+1,Cn+1

for each n > 1and 1≤k≤nwith the proviso that V1,1= 1. Note that, since ρ(Sj)is an increasing function

of the cluster size nj, heavily populated clusters are more likely. This leads to the rich-get-richer behaviour in the

clustering induced by the BNP prior. The connection between product partition models and Gibbs-type prior has

been deeply investigated since the seminal paper by Quintana and Iglesias (2003), see also De Blasi et al. (2013).

In this paper we choose σ≥0and introduce a new parameter κ > 0such that: Vn,Cn=1

Γ(n)R∞

0un−1exp−

(1/σ)[(κ+u)σ−κσ](κ+u)−n+σCndu. In this way, the law of Pncoincides with the one induced by the Normalized

generalized gamma process (NGGP, Lijoi et al.,2007). The NGGP encompasses the well known Dirichlet process

(DP) when σ= 0. In particular, Lijoi et al. (2007) highlighted the role of σin the predictive mechanism of an

NGGP: p(˜

η∈ · | Pn,η⋆

1,...,η⋆

Cn) = Vn+1,Cn+1

Vn,Cn

G0(·) + Vn+1,Cn

Vn,CnPCn

j=1(nj−σ)δη⋆

j.The above formula describes

the rule used to assign a new observation to a cluster, where the summand on the left represents the probability of

forming a new group, and the one on the right represents the probability of being assigned to an already observed

Personalized Treatment Selection via Product Partition Models with Covariates

group. It is apparent that larger values of σincrease the probability of generating new groups. From our simulation

study (Supplementary Material B.1; see also Lijoi et al.,2007, Section 3.2) large values of σalso reduce the number

of estimated singletons. These behaviours result in mitigating the rich-get-richer property of BNP priors. We also

assume a discrete prior distribution for (κ, σ). In this way, we let the data choose the appropriate reinforcement rate

(Lijoi et al.,2007), and we overcome a critical “trade-off” occurring when κand σare set to a ﬁxed value. Indeed,

both the parameters σand κhave an effect on the number of clusters Cnand on the reinforcing mechanism (see Lijoi

et al.,2007;Favaro et al.,2013;Argiento et al.,2016, for a deep discussion). We mention that both have an increasing

effect on the probability of observing a new cluster and the prior (and posterior) number of clusters. Interestingly,

σalso enters the expression of the weights of existing clusters and, as observed before, reduces the probability of

clusters with few elements. We refer to this double effect of σas the “trade-off” between the number of clusters and

reinforcement. In particular, for (κ, σ)we adopted a discrete prior on a 10×10 grid in (0,15)×(0.0,0.6), such that the

marginal distribution are discrete approximation of κ∼Gamma(2,1) and σ∼Beta(5,23), respectively. Extending

the work by M¨

uller et al. (2011), we aim at obtaining a prior for the random partition that encourages two subjects to

co-cluster when they have similar covariates, i.e., predictive biomarkers. In particular, the prior on the random partition

is deﬁned perturbing the cohesion function of a product partition model in equation (4.1) via a similarity function g

inducing the desired dependence on covariates. More in detail, the similarity function gis a non-negative function that

depends on the covariates associated with subjects in each cluster. Let xidenote the covariates for the i−th unit, while

x⋆

j= (xi, i ∈Sj)represents the covariates arranged by cluster. The product partition distribution with covariates is

p(Pn)∝Vn,Cn

j=1

ρ(nj)g(x⋆

j).(4.2)

The choice of the similarity function is of paramount importance for our modeling. It measures the homogeneity of

covariates arranged by clusters, and thus, the more the covariates take similar values, the larger the value of gmust be.

The default choice, proposed by M¨

uller et al. (2011), deﬁnes gas the marginal probability of an auxiliary Bayesian

model. Several alternatives can be taken (see for example Page and Quintana,2018;Argiento et al.,2022), since the

only requirement for gis to be a symmetric non-negative function. We implement the “Double Dipper” similarity

function because it has been shown to work well both in settings with a large number of covariates and in settings

where prediction is the main inferential goal (Page and Quintana,2016,2018):

g(x⋆

j) =

q=1 ZY

i∈Sj

p(xiq|ξ⋆

j)p(ξ⋆

j|x⋆

jq)dξ⋆

j,(4.3)

with p(ξ⋆

j|x⋆

jq)∝Qi∈Sjp(xiq|ξ⋆

j)p(ξ⋆

j). This structure is not due to any probabilistic properties since the covariates

are not considered random, but it measures the similarity of the covariates in cluster Sj. The name comes from the

fact that the covariates are used twice and correspond to the x⋆

j’s posterior predictive. The model in equation (4.3) is

completed by assuming p(·|ξ⋆

j) = N(·|m⋆

j, v⋆

j), where N(·|m, v)is a Gaussian density with mean mand variance v,

and p(ξ⋆

j) = p(m⋆

j, v⋆

j) = NIG(m⋆

j, v⋆

j|m0, k0, v0, n0)is the Normal-Inverse-Gamma density function. The resulting

similarity function can model scenarios with heterogeneous within-cluster variability. We follow Page and Quintana

(2018) and set the parameters of the Normal-Inverse-Gamma density to the default values m0= 0, k0= 1.0, v0=

1.0, n0= 2; since there is no notion of the xibeing random, parameters ξ⋆

jare not updated. Approaches based

on covariate-dependent random partition perform well if the clustering is not completely driven by covariates. As

the number of covariates increases, similarity functions tend to overwhelm the information provided by the response,

completely driving the clustering process. To counteract this behavior, we calibrate the inﬂuence of covariates on

clustering. To this end, with an abuse of notation, gin equation (4.2) is taken to be g(x⋆

j) := g(x⋆

j)1/√Q, namely

a small variation of the coarsened similarity function by Page and Quintana (2018). The impact of the cohesion and

similarity functions on the number of clusters is evaluated in a simulation study reported in Supplementary Material B;

in summary, this simulation study demonstrates the effectiveness of the NGGP in controlling the prior mass allocated

to different partitions through the reinforcement mechanism induced by σ. Additionally, we observe that the covariates

included in the prior effectively drive the clustering process, as desired.

5 Posterior Inference and Treatment Selection

We implement an MCMC algorithm to simulate from the posterior distribution of the parameters of interest. The

core part of the MCMC algorithm is the update of cluster membership; the computation associated with the joint law

of (Pa

na,ηa⋆

j)is based on Neal (2000)’s Algorithm 8 with a reuse strategy (Favaro et al.,2013). Conditional on the

updated cluster labels, all the remaining parameters are easily updated with Gibbs sampler or Metropolis-Hastings

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PERSONALIZEDTREATMENTSELECTIONVIAPRODUCTPARTITIONMODELSWITHCOVARIATESMatteoPedoneDepartmentofStatistics,ComputerScienceandApplicationsUniversityofFlorencematteo.pedone@unifi.itRaffaeleArgientoDepartmentofEconomicsUniversityofBergamoraffaele.argiento@unibg.itFrancescoC.StingoDepartmentofStatistics,Co...

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PERSONALIZED TREATMENT SELECTION VIA PRODUCT PARTITION MODELS WITH COVARIATES Matteo Pedone

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