Laurent Bordes

Pierre Joly

Semi-and non-parametric competing risks analysis of right censored data and failure cause missing completely at random

Keywords: Additive hazards, competing risks, counting processes, missing failure cause, reliability, survival analysis

We consider a nonparametric and a semiparametric (in presence of covariates) additive hazards rate competing risks model with censoring and failure cause possibly missing completely at random. Estimators of the unknown parameters are proposed in order to satisfy some optimality criteria. Large sample results are given for all the estimators. Our nonparametric method is applied to a real data set and the behavior of the semiparametric estimators are analyzed through a Monte Carlo study.

Introduction

We consider p ≥ 2 independent competing failure causes. We assume that to each failure time T j is associated a hazard rate function (risk function) λ j with 1 ≤ j ≤ p. The failure time T is the minimum of the p failure times associated to the p failure causes (this can be seen as a 1-out-of-p system in reliability), then we have T = T 1 ∧ • • • ∧ T p . The failure time T can be censored by a censoring time C, then we observe X = T ∧ C and δ = 1(T ≤ C) where 1(•) is the indicator function. Generally, when T is uncensored, that is for δ = 1, the failure cause is known, which means that p j=1 j1(T j = T ) is observed, but from times to times, it may happen that the failure cause is unknown and no partial information about the failure cause is available. In addition, a vector of explanatory variables denoted by Z and having potentially significant effects on the p failure times may be observed. In this paper we propose a model that allows to analyze such lifetime data from a semiparametric point of view in presence of covariates, and from a nonparametric point of view otherwise. We need to emphasize that our model extends some existing models in the semi-/non-parametric direction, however, the missingness mechanism is accounted here in the simplest way.

The problem of competing risks is not new and during the last two decades many models have been proposed in order to account that a system or an individual may fail or dead from several causes (see [START_REF] Crowder | Classical Competing Risks[END_REF], for a large overview on the topic). In a number of real applications of competing risks models the authors have to face the problem of missing information (e.g. [START_REF] Miyakawa | Analysis of incomplete data in competing risks model[END_REF]; [START_REF] Usher | Maximum likelihood analysis of component reliability using masked system life-test data[END_REF][START_REF] Lin | Exact maximum likelihood estimation under masked system data[END_REF][START_REF] Schabe | Nonparametric estimation of component lifetime based on masked system life test data[END_REF][START_REF] Goetghebeur | Analysis of competing risks survival data when some failure types are missing[END_REF][START_REF] Guttman | Dependent masking and system life data analysis: Bayesian inference for two-component systems[END_REF][START_REF] Reiser | Estimating component-defect probability from masked system success/failure data[END_REF][START_REF] Basu | Bayesian analysis for masked system failure data using non-identical Weibull models[END_REF][START_REF] Flehinger | Parametric modeling for survival with competing risks and masked failure causes[END_REF][START_REF] Craiu | Inference based on the EM algorithm for the competing risks model with masked causes of failure[END_REF][START_REF] Craiu | Inference for the dependent competing risks model with masked causes of failure, Lifetime Data Analysis[END_REF].

Among the great amount of paper dealing with competing risks model some of them focus on nonparametric estimation methods (see e.g. [START_REF] Lo | Estimating a survival function with incomplete cause-of-death data[END_REF][START_REF] Schabe | Nonparametric estimation of component lifetime based on masked system life test data[END_REF]. Because in many case partial information about the failure cause can only be obtained (e.g. masked cause of failure), a large number of works developed some specific methods with accurate modeling of the missingness mechanism. Most of these models are parametric and when a latent variable represents the missingness mechanism an EM-type algorithm can be proposed to estimate the model parameters. In [START_REF] Craiu | Inference based on the EM algorithm for the competing risks model with masked causes of failure[END_REF] such estimation procedure is proposed and the missingness may depend both on the failure cause and the failure time. Recently, [START_REF] Craiu | Inference for the dependent competing risks model with masked causes of failure, Lifetime Data Analysis[END_REF] considered a very complete parametric model including dependence of failure causes.

Some authors developed estimation procedures in the semi-/non-parametric framework for two or more failure causes (see e.g. Myakawa, 1984;[START_REF] Dinse | Nonparametric prevalence and mortality estimators for animal experiments with incomplete cause-of-death data[END_REF][START_REF] Lo | Estimating a survival function with incomplete cause-of-death data[END_REF][START_REF] Schabe | Nonparametric estimation of component lifetime based on masked system life test data[END_REF]. The special case of a possibly censored single failure cause differs from the competing risks model only be the fact that in this case the censoring time is not an event of interest. However, when the censoring information is missing, we are close to the competing risks situation where failure causes are possibly missing. Some specific methods has been derived in [START_REF] Gijbels | Non-and semi-parametric analysis of failure time data with missing failure indicators[END_REF], [START_REF] Mckeague | Product-limit estimators and Cox regression with missing censoring information[END_REF], van der Laan and McKeague (1998), Sun and [START_REF] Zhou | Additive hazards regression with missing censoring information[END_REF] and [START_REF] Subramanian | Asymptotically efficient estimation of a survival function in the missing censoring indicator model[END_REF] for various models including or not covariates, and several missingness mechanisms. [START_REF] Goetghebeur | Analysis of competing risks survival data when some failure types are missing[END_REF] proposed a competing risks model with proportional hazards assumption for the different failure causes. In their model the mechanism of missingness may depend on the failure time (this is the missing at random assumption) while in the model we propose it is independent of everything (this is the missing completely at random assumption). In our model each failure type has its own semiparametric additive hazards rate model and at the contrary to Goetghebeur and Ryan (1995), these failure rates are not linked.

The paper is organized as follows. In Section 2 we describe the model and we point out that each data resulting from the model can be seen as the realization of a nonhomogeneous Markov process. In Section 3 the estimators are defined. Because data for which the failure cause is missing are informative for the whole parameters of the model, we develop a method that allows to account this information in an optimal way with respect to an efficiency criterium. In Section 4, for the Euclidean parameters, and in Section 5 for the functional parameters, the corresponding estimators are shown to be consistent and asymptotically Gaussian. For each estimator, a consistent estimator of the asymptotic variance is provided. Section 6 is devoted to numerical examples. A Monte Carlo study is performed for the case including covariates whereas our estimation method is applied to a real data set that does not include covariates. Some concluding remarks are given in the last Section.

A semiparametric additive hazards model

The complete data are as follows.

• T j is the duration associated to the jth cause (j ∈ {1, . . . , p}).

• The law of T j , conditional on the vector of covariates Z ∈ R k , is defined by the hazard rate function λ j (t|Z) = λ 0j (t) + β T j Z, t ≥ 0, where λ 0j is the baseline hazard rate function of the jth cause, and β j ∈ R k is the regression parameter associated to the jth cause. • The censoring variable C either has a hazard rate function λ C or is deterministic; it is independent of the T j 's conditionally on Z.

• The binary variable δ = 1(T ≤ C), where

T = T 1 ∧ • • • ∧ T p , is the censoring indicator.
• The binary variable M is equal to 1 when the failure cause is known (thus is is equal to 0 when δ = 0).

• The variable D = δM p j=1 j1(T j ≤ T ) reveals the failure cause when the failure time is uncensored and M = 1.

The observation coming from one indiviual is therefore (X, δ, D, Z) where X is the observed duration. Denoting

P (M = 1|X, Z, δ = 1) = P (M = 1|δ = 1) = α ∈ [0, 1] and P (M = 0|X, Z, δ = 0) = P (M = 0|δ = 0) = 1,
the observation (X, δ, D), conditional on Z, may be seen as the realization of a (p + 3)state nonhomogeneous Markov process (see Fig. 1). Writing λ0x (•|Z) the transition rate, conditional on Z, for the transition 0 → x (x ∈ {1, . . . , p, m, c}) we obtain:

   λ0j (t) = αλ j (t|Z) for j ∈ {1, . . . , p}, λ0m (t) = (1 -α) p j=1 λ j (t|Z), λ0c (t) = λ C (t).
The independence of T 1 , . . . , T p , C, conditional on Z, is a sufficient condition to obtain the above transition rates.

Remark Up to the parameter α (or 1-α), the transition rates (excepted for the transition 0 → c) are additive hazards functions. This is obvious for the transition λ0j with 1 ≤ j ≤ p but it remains true for the transition 0 → m, indeed:

λ0m (t) = (1 -α) λ m (t) + β T m Z ,
where

λ 0m = p j=1 λ 0j et β m = p j=1 β j .
3 Regression parameters estimation

Data and notations

We denote by (X i , δ i , D i , Z i ) 1≤i≤n n independent and identically distributed copies of (X, δ, D, Z). For j ∈ {1, . . . , p, m} we define the counting processes:

N ij (t) = 1(X i ≤ t, D i = j) for j = m, N im (t) = 1(X i ≤ t, δ i = 1, D i = 0).
Hereafter, in order to simplify our notations, we write m ≡ p + 1. Let Y i be the risk process defined by Y i (t) = 1(X i ≥ t). Then, by [START_REF] Andersen | Statistical Models Based on Counting Processes[END_REF] or [START_REF] Fleming | Counting Processes and Survival Analysis[END_REF], for 1 ≤ i ≤ n and j ∈ {1, . . . , p + 1} the M ij processes defined by

M ij (t) = N ij (t) - t 0 Y i (s) λ0j (s)ds, t ≥ 0,
are F-martingales with respect to the filtration F = (F t ) t≥0 defined by

F t = σ{N ij (s), Y i (s); s ≤ t; 1 ≤ i ≤ n, j ∈ {1, . . . , p + 1}}.

Estimation of the Euclidean parameters

We denote by τ < +∞ the upper bound of the interval of study. It means that individuals are observed on the time interval [0, τ ]. The particular case where the censoring variable C is deterministic corresponds to a Type-I censored sampling plan with C = τ . With the assumption made on M the parameter α is naturally estimated by

α = α(τ ) = n i=1 1(D i > 0; X i ≤ τ ) n i=1 1(δ i = 1; X i ≤ τ ) = p j=1 N •j (τ ) N •• (τ ) ,
where any point in the subscripts corresponds to summation over all the possible values of the subscript. The estimator α of α is natural since it reveals, among the individuals doing an uncensored transition, the proportion of individuals having a known cause of failure. Individuals doing the transition 0 → j (1 ≤ j ≤ p) allow to estimate β j . Indeed, it is straightforward to extend the estimating function of [START_REF] Lin | Semiparametric estimation of the additive risk model[END_REF]. Therefore β j is estimated by βj which is the solution of U j (β, α, τ ) = 0 where

U j (β, α, τ ) = n i=1 τ 0 Z i -Z(s) dN ij (s) -αβ T Z i Y i (s)ds , with Z(s) = n i=1 Y i (s)Z i n i=1 Y i (s)
.

We ever saw that the transition rate 0 → m follows an additive risk model ; using data from this transition allows to estimate β m by βm solution of U m (β, α, τ ) = 0 where

U m (β, α, τ ) = n i=1 τ 0 Z i -Z(s) dN im (s) -(1 -α)β T Z i Y i (s)ds .
At this stage, each β j has its estimator βj for j = 1, . . . , p, p + 1. Because βm is an estimator of

β m = β 1 + • • • + β p ,
it is natural to use this available information in order to improve estimates of the β j 's. We adopt the following strategy. First we consider the block matrix H defined by

H =      H 11 H 12 • • • H 1p+1 H 21 H 22 • • • H 2p+1 . . . . . . . . . H p1 H p2 • • • H pp+1      ,
where for 1 ≤ i ≤ p and 1 ≤ j ≤ p + 1 H ij is a k × k-real valued matrix. The matrix H has to satisfy

H      β 1 . . . β p β m      =    β 1 . . . β p   
for all 1 ≤ i ≤ p and β i ∈ R k . Moreover, we want H to minimize the function q(H) defined by q(H) = trace(H ΣH T ),

where Σ is an estimator of the asymptotic variance matrix of ( βT 1 , . . . , βT p , βT m ) T . In other words, we are looking for a linear transformation of ( βT 1 , . . . , βT p , βT m ) T that remains an estimator of the β j 's for j = 1, . . . , p but that has a smaller variance (in some particular sense) than the initial one. Thus, denoting by Ĥ the matrix minimizing q(H), we denote by βi = p+1 j=1 Ĥij βj (with βp+1 ≡ βm ) the final estimator of β i for i = 1, . . . , p. Later we refer this estimators as "minimal variance estimators" (in the above sense).

Remark It is worth to note that constraints imposed to H does not link its lines. Indeed, denoting by I k the identity matrix of order k, these constraints may be written

H ii + H ip+1 = I k H ij + H ip+1 = 0 for 1 ≤ i ≤ p et j ∈ {1, . . . , p}\{i}.
On the other hand we have

q(H) = p i=1 trace(H i• ΣH T i• ) = p i=1 qi (H)
where qi (H) = trace(H i• ΣH T i• ) and H i• is the ith line block of H. Because the constraints are linear and does not link the lines of H it is sufficient to solve for each i ∈ {1, . . . , p} the following problem (P i ): Looking for matrices H i1 , . . . , H ip+1 satisfying:

(P i )    H ii + H ip+1 = I k , H ij + H ip+1 = 0 for j = i, trace(H i• ΣH T i• ) is minimal.

Example for p = k = 2

We have to solve problems (P 1 ) and (P 2 ) which are identical. As a consequence we consider only (P 1 ). Let us write H (j) = H 1j for j = 1, 2, 3 and introduce the following notations:

H (j) = h (j) 11 h (j) 12 h (j) 21 h (j) 22 and Q = Σ 0 and L = h (1) 11 , h (1) 12 , h (2)
11 , h

12 , h

21 , h

= (l 1 , . . . , l 12 ). Then

q1 (H) = trace H (1) H (2) H (3) Σ H (1) H (2) H (3) T = L QL T .
The constraints are

H (1) + H (3) = I 2 H (2) + H (3) = 0 ⇔ h (1) ij + h (3) ij = 1 for 1 ≤ i, j ≤ 2, h (2) ij + h (3) ij = 0 for 1 ≤ i, j ≤ 2. ⇔ CL = d,
where

C =            
1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1

            and d =             1 0 0 0 0 1 0 0             .
Let λ be the Lagrange parameter associated to the optimization problem with linear constraints. The Lagrange function can be written

(L, λ) = 1 2 L T QL + (CL -d) T λ.
At the optimum L it satisfies

   ∂ ∂L ( L, λ) = 0 = Q L + C T λ, C L = d, so we obtain λ = -[C Q-1 C T ] -1 d, and hence L = Q-1 C T [C Q-1 C T ] -1 d.
Remark Note that Q is invertible whenever Σ is.

Asymptotic behavior of the regression parameters estimators 4.1 Asymptotics for the βj

First we need some notations. Let z be a column vector in R k , we note

x ⊗l =    1 if l = 0, z if l = 1, zz T if l = 2.
For 0 ≤ l ≤ 2 and 0 ≤ s ≤ τ we define

S l (s) = 1 n n i=1 Y i (s)Z ⊗l i ,
and for b ∈ R k we define

S 3 (s; b) = 1 n n i=1 Y i (s)Z ⊗2 i b T Z i .
Hereafter asymptotic results are given with respect to n tending to infinity. Let us introduce the following assumptions.

A1. The probability α of a known cause is strictly positive.

A2. The upper bound of the study interval satisfies 0 < τ 0 λ 0j (s)ds < +∞ for 1 ≤ j ≤ p and the vectors of covariates Z i are uniformly bounded with respect to i ≥ 1.

A3. For 0 ≤ l ≤ 2 there exist functions

s k defined on [0, τ ] such that max 0≤l≤2 sup s∈[0,τ ] S l (s) -s l (s) P -→ 0.
Moreover, s 0 is bounded by above by a strictly positive real number.

A4. Denoting by a(u) = s 2 (u) -s ⊗2 1 (u)/s 0 (u), the matrix

A(τ ) = τ 0 a(u)du
is positive definite, and A6. Functions a, aλ 0j , s 0 λ 0j , s 1 , s ⊗2 1 /s 2 0 , s 3 (•; β j ) and s 4 (•;

θ(τ ) = τ 0 s 0 (u)λ 0p+1 (u)du + β T p+1 s 1 (u) du > 0. A5. Let us consider b ∈ R k and S 4 (s; b) = n -1 n i=1 (b T Z i )Y i (s)(s 1 (s)Z T i /s 0 (s)).
β j ) (for 1 ≤ j ≤ p) are integrable on [0, τ ].
Remark Assumption A1 is clearly necessary to obtain the identifiability of the whole model parameters. Assumption A2 insures that data can be observed everywhere on [0, τ ] for each failure cause in an homogeneous way; the fact that the Z i are uniformly bounded could be relaxed, and they could also be time dependent but it would add some unimportant technicalities with respect to the main objective of this work. Assumption A3 is a simple consequence of the strong law of large numbers for i.i.d. (independent and identically distributed) processes. It is satisfied in particular when the covariates are i.i.d. with third order moment. Assumption A4 is necessary to identify β j using the transition 0 → j (1 ≤ j ≤ p + 1), whereas A5 and A6 are technical conditions. Now we introduce the vectors β = ( βT 1 , . . . , βT p+1 ) T and β = (β T 1 , . . . , β T p+1 ) T .

Theorem 4.1 Under Assumptions A1-A5, the random vector √ n( β -β) is asymptotically gaussian, centered, with positive definite covariance matrix Σ(τ ) defined hereafter.

Proof. First we denote by

U j (b, α j , τ ) = n i=1 τ 0 Z i -Z(s) dN ij (s) -α j b T Z i Y i (s)ds , for 1 ≤ j ≤ p + 1 with b ∈ R k and α j = α if 1 ≤ j ≤ p, 1 -α if j = p + 1. If we note αj = α if 1 ≤ j ≤ p, 1 -α if j = p + 1,
we have U j ( βj , αj , τ ) = 0 for 1 ≤ j ≤ p. First let us remark that for 1 ≤ j ≤ p + 1 the following equality holds

1 √ n U j (β j , α j , τ ) = Â(τ ) βj √ n( αj -α j ) + α j √ n( βj -β j ) , (1)
where

Â(τ ) = τ 0 S 2 (s) -S ⊗2 1 (s)/S 0 (s) ds.
Y i (s) λ0j (s|Z i )ds = p j=1 M •j (τ ) + α n i=1 p j=1 τ 0 Y i (s)λ j (s|Z i )ds M •• (τ ) + n i=1 p j=1 τ 0 Y i (s)λ j (s|Z i )ds = p j=1 M •j (τ ) + αn τ 0 S 0 (u)λ 0p+1 (u) + β T p+1 S 1 (u) du M •• (τ ) + n τ 0 S 0 (u)λ 0p+1 (u) + β T p+1 S 1 (u) du . It follows that √ n(α(τ ) -α) = 1 -α √ n p j=1 M •j (τ ) - α √ n M •p+1 (τ ) 1 n M •• (τ ) + τ 0 S 0 (u)λ 0p+1 (u) + β T p+1 S 1 (u) du 1(N •• (τ ) > 0) - √ nα1(N •• (τ ) = 0) = 1 -α √ n p j=1 M •j (τ ) - α √ n M •p+1 (τ ) 1 n M •• (τ ) + τ 0 S 0 (u)λ 0p+1 (u) + β T p+1 S 1 (u) du + o P (1).
Using the Lenglart inequality we show that

1 n M •• (τ ) P -→ 0.
Therefore, A3 and A4 yield

√ n(α(τ ) -α) = p+1 j=1 α * j θ(τ ) 1 √ n M •j (τ ) + o P (1), (2)
where α * j = 1 -α for 1 ≤ j ≤ p and α * p+1 = -α. By applying the Rebolledo theorem and using (2) it is easy to prove that α is √ n-consistent. Because the processes U j (β j , α j , •) are F t -martingales, and proving by using the Lenglart inequality that divided by n these processes converge to 0 uniformly in probability on [0, τ ], we obtain the convergence in probability of β to β by using additionally the consistency of α, equality (1), and Assumptions A3 and A4. Once this convergence is established, we obtain by (1) the equality

√ n( βj -β j ) = 1 α j A -1 (τ ) U j (β j , α j , τ ) √ n -β j √ n(α j -α j ) + 1 α j Â-1 (τ ) -A -1 (τ ) U j (β j , α j , τ ) √ n - √ n( αj -α j )( βj -β j ) ,
where the second term of the right hand side is an o P (1) since A -1 n (τ ) converges in probability to A -1 (τ ) from Assumptions A3 and A4, U j (β j , α j , τ )/ √ n is a O P (1) (because it is asymptotically gaussian by the Rebolledo theorem), α is √ n-consistent and β converges to β. Finally the following approximation is obtained

√ n( βj -β j ) = 1 α A -1 (τ ) U j (β j , α j , τ ) √ n -β j √ n(α -α) + o P (1), (3)
for 1 ≤ j ≤ p, and

√ n( βp+1 -β p+1 ) = 1 1 -α A -1 (τ ) U p+1 (β p+1 , α p+1 , τ ) √ n + β p+1 √ n( α -α) + o P (1). (4)
Hence, by ( 2), (3), and (4) we obtain

√ n( β -β) = Σ 1 (τ )U n (τ ) + o P (1), (5)
where

Σ 1 (τ ) =        A -1 (τ ) α 0 • • • 0 -β1 α 0 . . . . . . . . . . . . A -1 (τ ) α 0 - βp α 0 • • • 0 A -1 (τ ) 1-α βp+1 1-α        , and
U n (τ ) = 1 √ n         U 1 (β 1 , α 1 , τ ) . . . U p+1 (β p+1 , α p+1 , τ ) p+1 j=1 α * j θ(τ ) M •j (τ )        
.

As a consequence, study the asymptotic behavior of √ n( β -β) leads to study the asymptotic behavior of the process U n which is an F t -martingale. As in [START_REF] Andersen | Cox's regression model for counting processes: A large sample study[END_REF], we apply the Rebolledo theorem (see [START_REF] Rebolledo | Central limit theorems for local martingales[END_REF], and an adapted version in Andersen et al., 1993, p. 83-84). This theorem allows to show that U n converges weakly in (D[0, τ ]) k(p+1)+1 to a gaussian martingale. Thus, the random vector U n (τ ) is asymptotically gaussian and centered. Because this theorem is now of classical use we only derive the limit of the predictible variation process of U n that gives the asymptotic variance matrix of U n (τ ). On one hand we have for 1

≤ j ≤ p + 1 n -1/2 U j (β j , α, •) (t) = α j t 0 S 2 (s) -S ⊗2
1 (s)/S 0 (s) λ 0j (s)ds + α j t 0 S 3 (s; β j ) -S 4 (s; β j ) -S T 4 (s; β j ) + (β T j S 1 (s))S ⊗2 1 (s)/S 2 0 (s) ds + o P (1), which by Assumptions A2-A6 converges to

Θ j (t) = α j t 0 a(s)λ 0j (s) + s 3 (s; β j )β j -s 4 (s; β j ) -s T 4 (s; β j ) + (β T j s 1 (s))s ⊗2 1 (s)/s 2 0 (s) ds.
On the other hand, for 1 ≤ j = j ≤ p + 1, we have

U j (•), U j (•) (t) = 0,
where we note U j (t) = U j (β j , α j , t) for 1 ≤ j ≤ p + 1. Thus the asymptotic covariance of terms U j and U j is null. It remains to calculate the asymptotic covariance of √ n(α -α) with the n -1/2 U j 's, and with itself. It is easy to show that for 1 ≤ k ≤ p + 1 we have

1 n U k (•), p+1 j=1 α * j M •j (•) θ(τ ) (t) P -→ κ k α(1 -α) θ(τ ) A(t)β k ≡ ξ k (t),
where κ k = 1 if 1 ≤ k ≤ p and κ p+1 = -1, and

1 n p+1 j=1 α * j M •j (•) θ(τ ) (t) P -→ α(1 -α) θ(τ ) ≡ ξ α (t).
Then, setting

Σ 2 (τ ) =      Θ 1 (τ ) 0 0 ξ 1 (τ ) 0 . . . 0 . . . 0 0 Θ p+1 (τ ) ξ p+1 (τ ) ξ T 1 (τ ) • • • ξ T p+1 (τ ) ξ α (τ )     
we obtain by ( 5) that √ n( β -β) is asymptotically gaussian with asymptotic covariance matrix Σ(τ ) = Σ 1 (τ )Σ 2 (τ )Σ T 1 (τ ). The matrix Σ 2 (τ ) being positive definite, the same holds for Σ(τ ).

The optimal estimator

Giving the asymptotic behavior of β requires to propose an estimator Σ of the asymptotic covariance matrix Σ(τ ). In this paragraph, when it is possible, we avoid to indicate dependence in τ , for exemple Σ(τ ) will be denoted by Σ.

It is easy to check that for 1 ≤ j ≤ p + 1 we have Therefore we set

βj = 1 αj Â-1 1 n n i=1 τ 0 Z i -Z(s) dN ij (s) Then, for 1 ≤ j ≤ p + 1 we note Θj = 1 n n i=1 τ 0 Z i -Z(s) ⊗2 dN ij (s),
Σ1 =        Â-1 α 0 • • • 0 -β1 α 0 . . . . . . . . . . . . Â-1 α 0 -βp α 0 • • • 0 Â-1 1-α βp+1 1-α        and Σ2 =      Θ1 0 0 ξ1 0 . . . 0 . . . 0 0 Θp+1 ξp+1 ξT 1 • • • ξT p+1 ξα      .
Let Ĥ be the matrix defined at Section 3, β = Ĥ β and β * = (β T 1 , . . . , β T p ) T .

Theorem 4.2 Under Assumptions A1-A5, the matrix Σ = Σ1 Σ2 ΣT 1 converges in probability to the matrix Σ. If Σ is invertible, then √ n( β -β * ) has an asymptotically centered gaussian distribution, whose the covariance matrix trace is less or equal to trace(HΣH T ) for all H ∈ H. Such an estimator is said asymptotically T -optimal.

Proof. The convergence of  to A is straightforward using Assumptions A3 and A4. Otherwise, we ever proved the convergence in probability of α and βj to α and β j , respectively (for 1 ≤ j ≤ p + 1) in the proof of Theorem 4.1. As a consequence, we obtain the convergence in probability of Σ1 to Σ 1 .

The convergence in probability of θ to θ(τ ) is a consequence of the Lenglart inequality. These convergences yield the convergence in probability of the ξj 's to the ξ j (τ )'s (for 1 ≤ j ≤ p+1). Using repeatedly the Lenglart inequality allows to prove that ξα and Θj converge in probability to ξ α and Θ j , respectively (for 1 ≤ j ≤ p + 1). It follows that Σ2 converges in probability to Σ 2 , and thus, that Σ converges in probability to Σ.

Finally, as we discussed in Sections 3.2 and 3.3, the matrix Ĥ depends continuously on Σ, then when n tends to infinity, Ĥ converges in probability to the matrix H Σ ∈ H that minimizes trace(HΣH T ) whose the existency follows from the positive definiteness of Σ. Moreover, because Ĥβ = β * , by the Slutsky lemma √ n( β -β * ) = Ĥ√ n( β -β) has a centered asymptotic gaussian distribution whose the covariance matrix H Σ ΣH T Σ satisfies trace(H Σ ΣH T Σ ) ≤ trace(HΣH T ) for all H such that Hβ = β * . This finishes the proof.

5 Estimation of functional parameters

Without covariate: nonparameteric case

This is the simplest case but the model is interesting for application. Indeed the unknown parameters of the model are the failure rates λ j associated to each failure cause. The transition rates of the Markov graph (see Fig. 1) satisfy for t ≥ 0,

   λ0j (t) = αλ j (t) pour j ∈ {1, . . . , p}, λ0m (t) = (1 -α) p j=1 λ j (t) ≡ (1 -α)λ m (t), λ0c (t) = λ C (t).
At the contrary to the model by [START_REF] Goetghebeur | Analysis of competing risks survival data when some failure types are missing[END_REF], when there is no covariate in the data, the model is still of interest because it allows different failure rates for failure causes.

Principle of an optimal estimator of the cumulative hazard rate functions

Let Λ j be the cumulative hazard rate function associated to the jth failure cause (1 ≤ j ≤ p). By standard arguments it is natural to estimate Λ j by

Λj (t) = 1 α n i=1 t 0 dN ij (s) Y (s) and Λ m (t) = p j=1 Λ j (t) by Λm (t) = 1 1 - α n i=1 t 0 dN im (s) Y (s) ,
where Y (s) = n i=1 1(X i ≥ s) and α is defined in Section 3. Again we have an estimator of each Λ j (1 ≤ j ≤ p) and an estimator of their sum. Let us denote by Ĥ(t) the p × (p + 1) time dependent matrix, that will be specified later, and Λ(t) = ( Λ1 (t), . . . , Λp (t), Λm (t)) T one estimator of Λ(t) = (Λ 1 (t), . . . , Λ p (t), Λ m (t)) T . We define Λ(t) = Ĥ(t) Λ(t) as one estimator of Λ * (t) = (Λ 1 (t), . . . , Λ p (t)) T . If the family of matrices { Ĥ(t); t ∈ [0, τ ]} satisfies Ĥ(t)a = a * for all t ∈ [0, τ ], a * = (a 1 , . . . , a p ) T ∈ R p and a = (a * T , p j=1 a j ) T . Then if Ĥ(t) converges in probability to H(t) and if the sequence of processes ( √ n( Λ(t) -Λ(t)); t ∈ [0, τ ]) converges weakly in (D[0, τ ]) p+1 to a centered gaussian process with covariance function Γ(t), then the sequence of processes ( √ n( Λ(t) -Λ * (t)); t ∈ [0, τ ]) converges weakly in (D[0, τ ]) p to a gaussian process having variance function H(t)Γ(t)H T (t).

Let Γ(t) be a consistent estimator of Γ(t), and Ĥ(t) be defined by

Ĥ(t) = arg min H∈H trace(H Γ(t)H T ), ( 6
)
where H is the set of p × (p + 1) matrices satisfying Ha = a * for all a * = (a 1 , . . . , a p ) T ∈ R p and a = (a * T , p j=1 a j ) T . We denote by L the column vector in R p(p+1) defined by L = (H 1 , . . . , H p ) where H i is the ith line of H ∈ H. The link between L = (l i ) 1≤i≤p(p+1) and H = (h ij ) 1≤i≤p;1≤j≤p+1 is therefore h ij = l (i-1)(p+1)+j . Because linear constraints on H are transmitted to L we have

h ii + h i,p+1 = 1, for 1 ≤ i ≤ p, h ij + h i,p+1 = 0, for 1 ≤ i ≤ p, 1 ≤ j ≤ p + 1 and i = j, ⇔ l (i-1)(p+1)+i + l i(p+1) = 1, for 1 ≤ i ≤ p, l (i-1)(p+1)+j + l (i+1)p = 0, for 1 ≤ i ≤ p, 1 ≤ j ≤ p + 1 and i = j, ⇔ CL = d.
Denoting by Q(t) the block diagonal matrix defined by

Q(t) =    Γ(t) . . . Γ(t)         p-times,
we show that q(L) ≡ trace(H Γ(t)H T ) = L T Q(t)L. Following the method of Section 3.3, with the linear constraint CL = d, q(L) reaches it minimum at

L(t) = Q-1 (t)C T (C Q-1 (t)C T ) -1 d.
The Ĥ(t) matrix is therefore defined by ĥij (t) = l(i-1)(p+1)+j (t) for 1 ≤ i ≤ p and 1 ≤ j ≤ p + 1.

Asymptotics for Λ and Λ

Let us introduce two assumptions that allow to obtain the asymptotic behavior of estimators Λ and Λ.

B1. τ satisfies 0 < Λ j (τ ) < +∞ for 1 ≤ j ≤ p.

B2. There exists a function y, defined on [0, τ ], and bounded away from 0, such that sup

s∈[0,τ ] |Y (s)/n -y(s)| P -→ 0.
Note that functions yλ j and λ j /y are integrable on [0, τ ] and we define Theorem 5.1 Under Assumption A1, B1, and B2, the sequence ( √ n( Λ(t)-Λ(t)); t ∈ [0, τ ]) converges weakly on (D[0, τ ]) p+1 to a centered gaussian process with variance function Γ(t) defined subsequently.

Proof. The arguments we need here are close to those we used in the proof of Theorem 4.1, as a consequence we only give the main lines of this proof. We begin with α. Following the proof of Theorem 4.1, the equality (2) becomes

√ n(α -α) = p+1 j=1 α * j θ 0 (τ ) 1 √ n M •j (τ ) + o P (1), (7)
where θ 0 (τ ) = τ 0 y(s)λ m (s)ds > 0 by B1 and B2. The Rebolledo theorem insures that α(τ ) converges to α at √ n-rate. With this result and the Lenglart inequality we show that the Λj 's converge in probability to the Λ j 's, uniformly on [0, τ ] for 1 ≤ j ≤ p + 1 (p + 1 ≡ m). In consequence we obtain the following uniform (in t ∈ [0, τ ]) approximation results

√ n( Λj (t) -Λ j (t)) = - Λ j (t) α j √ n(α j -α j ) + 1 α j √ n t 0 dM •j (s) Y (s) + o P (1) (8)
for 1 ≤ j ≤ p + 1. By the approximations ( 7) and ( 8) we obtain

√ n( Λ(t) -Λ(t)) = Σ 3 (t)V n (t) + o P (1), (9)
where

Σ 3 (t) =        1 α 0 • • • 0 -Λ1(t) α 0 . . . . . . . . . . . . 1 α 0 - Λp(t) α 0 • • • 0 1 1-α Λm(t) 1-α        and V n (t) = √ n       t 0 dM•1(s) Y (s)
. . .

t 0 dM•p+1(s) Y (s) 1 n p+1 j=1 1-αj θ0(τ ) M •j (τ )       .
The Rebolledo theorem allows to prove that V n converges weakly in (D[0, τ ]) p+2 to a centered gaussian process. It remains to specify the variance function Σ 4 of this process. First we can write

V n (t) = V 1,n (t) + V 2,n (t) where V 1,n (t) = √ n       t 0 dM•1(s) Y (s)
. . .

t 0 dM•p+1(s) Y (s) 1 n p+1 j=1 1-αj θ0(τ ) M •j (t)       and V 2,n (t) = 1 √ n      0 . . . 0 p+1 j=1 1-αj θ0(τ ) (M •j (τ ) -M •j (t))     
.

Because the M •j 's are martingales, we have

E V ⊗2 n (t) = E V ⊗2 1,n (t) + E V ⊗2 2,n (t) .
We are thus looking for

Σ 4 (t) = lim n→∞ E V ⊗2 1,n (t) + lim n→∞ E V ⊗2 2,n (t) . Setting ρ j (t) = κ j α(1 -α)Λ j (t)/θ 0 (τ ) for 1 ≤ j ≤ p + 1 = m and ρ α (τ ) = α(1 -α)/θ 0 (τ ), it is straightforwardly established that Σ 4 (t) =      η 1 (t) 0 0 ρ 1 (t) 0 . . . 0 . . . 0 0 η p+1 (t) ρ p+1 (t) ρ 1 (t) • • • ρ p+1 (t) ρ α (τ )      .
Then by ( 9) we obtain Γ(t) = Σ 3 (t)Σ 4 (t)Σ T 3 (t), which finishes the proof.

For t ∈]0, τ ] we define

Σ3 (t) =        1 α 0 • • • 0 -Λ1(t) α 0 . . . . . . . . . . . . 1 α 0 -Λp(t) α 0 • • • 0 1 1-α Λm(t) 1-α        and Σ4 (t) =      η1 (t) 0 0 ρ1 (t) 0 . . . 0 . . . 0 0 ηp+1 (t) ρp+1 (t) ρ1 (t) • • • ρp+1 (t) ρα     
where for 1 ≤ j ≤ p + 1 we note

ηj (t) = n t 0 dN •j (s) Y 2 (s) ,
and for 1 ≤ j ≤ p we note

ρj (t) = κ j α(1 -α) Λj (t) θ0 with θ0 = N •• (τ )/n and ρα = α(1 -α)/ θ0 . Thus we define Γ(t) = Σ3 (t) Σ4 (t) ΣT 3 (t).
Theorem 5.2 Under Assumptions A1, B1 and B2, for all t ∈ [0, τ ], the matrix Γ(t) converges in probability to the matrix Γ(t). Let us assume moreover that for all t ∈]0, τ ] the matrix Γ(t) is invertible. If Ĥ(t) is the unique solution of (6), then √ n( Λ -Λ) converges weakly in (D[0, τ ]) p to a centered gaussian process, and is asymptotically T -optimal with variance function equal to H(t)Γ(t)H T (t) (see ( 6)).

Proof. Since the proof follows the lines of Theorem 4.2 proof, it is omitted. textitRemark Quantities Λ(t) and Γ(t) are piecewise constant, therefore it is sufficient to calculate the matrix Ĥ(t) at points t ∈ [0, τ ] where N •• has jumps, that is at points X i ∈ [0, τ ] such that δ i = 1.

Survival functions estimation

One purpose of this paper is to improve the estimators of p parameters when in addition we are able to estimate their sum. Our model is especially well adapted to this problem for estimation of both regression parameters and cumulative hazard rate functions. Because the survival function coming from transition 0 → m has not the additive property of the above mentioned parameters it is desirable to estimate the survival functions using the T -optimal estimates of the cumulative hazard functions. As a consequence, the Kaplan-Meier type estimator of S j is defined by Sj (t) = s≤t (1 -∆ Λj (s)), t ≥ 0, for 1 ≤ j ≤ p, with ∆F (t) = F (t) -F (t-) for any F : R → R k . Because estimates Λj are piecewise constant with jumps at points X i for which δ i = 1, we have

Sj (t) = {1≤i≤n;Xi≤t,δi=1}
(1 -∆ Λj (X i )), t ≥ 0, which are easily computed. Now let us define 1 p = (1, . . . , 1) T ∈ R p and for t ∈ [0, τ ], S(t) = ( S1 (t), . . . , Sp (t)) T and S(t) = (S 1 (t), . . . , S p (t)) T . Theorem 5.3 Under Assumptions A1, B1 and B2, the process ( √ n( S(t) -S(t)); t ∈ [0, τ ]) converges weakly in (D[0, τ ]) p to a centered Gaussian process with covariance function Ω(t) = (ω ij (t)) 1≤i,j≤p defined by

Then we have S(t) =

ω ij (t) = S i (t)S j (t)(H(t)Γ(t)H T (t)) ij .
Then Ω(t) = (ω ij (t)) 1≤i,j≤p defined by

ωij (t) = Si (t) Sj (t)( Ĥ(t) Γ(t) ĤT (t)) ij ,
converges in probability to Ω(t) for t ∈]0, τ ].

With explanatory variables

Let us consider j ∈ {1, . . . , p}. For simplicity we only base our estimator of the conditional cumulative hazard function Λ j (•|Z) for cause j, on transition 0 → j (see Figure 1). Let us recall that Λ j (•|Z) is defined by

Λ j (t|Z) = Λ 0j (t) + β T j Zt, t ≥ 0.
The conditional survival function S j (•|Z) associated to the jth cause is therefore defined by .

S j (t|Z) = exp(-β T j Zt)φ(Λ 0j )(t
Let L j (t) be the (p + 2) × 1-matrix and Ξ j (t) be the (p + 1) × (p + 2)-matrix respectively defined by

L j (t) = 1 α   1 Λ j (t|Z) -2(k(t) + tZ) T β j A -1 (τ )(k(t) + tZ)   , and
Ξ j (t) =   µ 1j (t) µ 2j (t) µ T 3j (t) µ 2j (t) ξ α (τ ) ξ T j (τ ) µ 3j (t) ξ j (τ ) Θ j (τ )   ,
where ξ α (τ ), ξ j (τ ) and Θ j (τ ) are defined in Section 4 and for 1

≤ j ≤ p k(t) = t 0 s 1 (s) s 0 (s) ds, µ 1j (t) = α t 0 s 0 (s)λ 0j (s) + β T j s 1 (s) s 2 0 (s) ds, µ 2j (t) = α θ(τ ) t 0 s 0 (s)λ 0j (s) + β T j s 1 (s) s 0 (s) ds, and
µ 3j (t) = α t 0 a(s) s 0 (s) dsβ j .
Let us define Lj and Ξj the empirical versions of L j and Ξ j , by The proof of the next theorem is basically the same as the preceding ones then it is omitted.

Lj (t) = 1 α   1 Λj (t|Z) -2( K(t) + tZ) T βj Â-1 (τ )( K(t) + tZ)   , with K(t) = t 0 S 1 (s) S 0 (s) ds, and
Ξj (t) =   μ1j (
Theorem 5.4 Under Assumptions A1-A5 and if for all t ∈ [0, τ ] the matrices L(t) and Ξ(t) are well defined, then for 1 ≤ j ≤ p, the sequence of processes ( √ n( Λj (t|Z) -Λ j (t|Z)); t ∈ [0, τ ]) converges weakly in D[0, τ ] to a centered Gaussian process with variance function γ j defined by γ j (t) = L T j (t)Ξ j (t)L j (t). Moreover γj (t) = LT j (t) Ξj (t) Lj (t) converges in probability to γ j (t) for all t ∈ [0, τ ]. The sequence of processes ( √ n( Ŝj (t|Z) -S j (t|Z)); t ∈ [0, τ ]) converges weakly in D[0, τ ] to a centered Gaussian process with variance function π j defined by π j (t) = S 2 j (t|Z)γ j (t) and πj (t) = Ŝ2 j (t|Z)γ j (t) converges in probability to π j (t) for all t ∈ [0, τ ].

Numerical study 6.1 A simulated example with covariates

In this section we give an example where we have two competing risks with a single covariate Z. We assume that Z is 1 or 2 with equal probabilities. Conditional on Z, we suppose that the hazard rate functions of T 1 and T 2 are respectively given by λ 1 (t|Z) = t/2 + 2Z and λ 2 (t|Z) = 2t/9 + 3Z, therefore we have β 1 = 2 and β 2 = 3. The censoring time C is exponentially distributed with mean 1 whereas the rate of missingness α is equal to 0.5. As a consequence, on the whole simulated data the rate of censored data is about 12%, the rates of observed failures from cause 1 and 2 are respectively about 15% and 28%, and the rate of missingness is about 45%.

In Table 1 we compare the performances of estimators ( βi ) i=1,2 based on transitions 0 → i with T -optimal estimators ( βi ) i=1,2 that use observations coming from the three informative transitions. We computed the mean and standard errors of 1000 estimates of β 1 and β 2 . We can see in Table 1 that from the bias point of view the T -optimal estimators are generally better (except for β 1 when n = 100) whereas the standard errors of T -optimal estimates are always smaller than the standard errors of estimates based on transitions 1 and 2.

A reliability example without covariates

This example deals with the hard drives data sample that may be found in [START_REF] Flehinger | Parametric modeling for survival with competing risks and masked failure causes[END_REF]. These authors consider a scenario in which a company manufacturing hard drives for computers tries to analyze causes of failures of a certain sub-assembly. Some of these causes, such as "defective head", are related to components, but others (e.g., "particle contamination") are not; in this example, there are three major causes of failures which, without going into details, are denoted as causes 1, 2 and 3. We assume that these causes act independently and in series. 10,000 drives were manufactured and then information about failures was collected in a database during 4 years. The number of failures observed in this period was 172. Some of the failures were masked and a selected number of those were analyzed to complete resolution in the defect isolation laboratory. The only observed masked groups were {1, 2, 3} and {1, 3}. Considering causes 1 and 3 as a single failure cause we obtain data from a competing risks model with two failure modes. Mode 1 (corresponding to causes 1 , 3 or masked group {1, 3} of the original data set) and mode 2 (that corresponds to cause 2 in the original data set). The failure cause is missing when none of the three original cause of failure is known (corresponding to the masking group {1, 2, 3} of the original data set). Finally, we obtain a data set with 119 failures of type 1, 19 failures of type 2 and 34 failures for which the failure cause is unavailable. The lifetimes of the 9828 drives still functioning at the end of the study are censored by the 4 years of the study duration. Because no information is available after 4 years we fix τ to 4. The probability α that the two failure causes are missing is estimated by α = 0.802. Figure 2 shows, for each failure cause, estimates of cumulative hazard rate functions with and without using the transition 0 → 3 and the corresponding 95% pointwize confidence intervals for each estimate. We can see that there is a little gain to use transition 0 → 3. We can see also that the modified Nelson-Aalen estimators that we propose are slightly more regular (smaller size jumps) than the corresponding Nelson-Aalen estimators, in exchange of which our estimators can be non legitimate (they can be locally decreasing). However this drawback disapears as the sample size increases because of uniform convergence of our estimators. This is also true for for the two causes reliability estimates that are given on Figure 3 with pointwize 95% confidence intervals.

Conclusions

In this paper we consider a semiparametric competing risks model that accounts covariates and the fact that information on failure cause can be missing completely at random. In this paper the missingness mechanism is very simple because it is independent of everything (time, failure causes, covariates, etc.). However, because of nonparametric assumptions for baseline hazard rate function of every type of failure cause, this model is quite adaptable. It is certainly possible to extend this model in several directions: dependent risks, missingness mechanism dependent on failure cause, etc. This model can also be extended to the case of masked cause of failure for which many sophisticated parametric models and inference methods have been developed over the two past decades. The estimation method that consists in seeing data as realizations of a nonhomogeneous markov process is inspired by McKeague and Subramanian (1998) while the estimators of the regression parameters of each transition is inspired from the Lin and Ying (1994) method. In addition we propose a linear transformation of these estimators which is shown to be asymptotically optimal in the sense of variance reduction.

Figure 1 :

 1 Figure 1: Markov graph associated to (X, M, D, Z)

  For all b ∈ R k , there exist functions s 3 (s; b) and s 4 (s; b) such that max 3≤l≤4 sup s∈[0,τ ] S l (s; b) -s l (s; b) P -→ 0.

η j (t) = α j t 0 λ

 0 j (s)/y(s)ds for 1 ≤ j ≤ m and η m = m j=1 η j . Assumption B2 is fulfilled whenever τ 0 λ C (s)ds < +∞.

s≤t 1 p

 1 -∆ Λ(s) , or equivalently, S = Φ( Λ) where the operator Φ from (D[0, τ ]) p to (D[0, τ ]) p is defined from its p identical components product-integral φ from D[0, τ ] to D[0, τ ]. As φ is Hadamard differentiable on E K ⊂ D[0, τ ] the set of functions with variations bounded by K < +∞ (see Andersen et al., 1993, p. 114) the same result holds for Φ on E p K with Hadamard derivative (dΦ(X).H)(t) = Φ(X)(t) H(t), where for a, b ∈ R p , a b ∈ R p is the component-wise product of a and b. From Theorem II.8.2 in Andersen et al. (1993, p. 112), √ n( S -S) is asymptotically equivalent to dΦ(Λ) √ n( Λ -Λ * ) = S √ n( Λ -Λ * ). The next theorem follows from the preceding result and Theorem 5.2.

  ), t ≥ 0, where φ is still the product-integral operator.First let us propose an estimator of Λ 0j . Using

	dN •j (s) = dM •j (s) + αS 0 (s)λ 0j (s)ds + αS T 1 (s)β j ds,
	we obtain the following estimator of Λ 0j (t) defined by
	Λ0j (t) =	1 α	0	t	dN •j (s) S 0 (s)	-βT j	0	t	S 1 (s) S 0 (s)	ds, t ≥ 0,
	where we recall that βj satisfies U j ( βj , α j , τ ) = 0 (see Section 3.2). Then we propose to
	estimate Λ j (•|Z) by Λj (•|Z) defined for t ≥ 0 by
	Λj (t|Z) =	1 α	0	t	dN •j (s) S 0 (s)		+ βT j	tZ -	0	t	S 1 (s) S 0 (s)	ds ,
	=	{1≤i≤n;Xi≤t,Di=j}	1 αS 0 (X i )	+ βT j	tZ -	0	t	S 1 (s) S 0 (s)	ds .
	The survival function S j (•|Z) is therefore estimated by Ŝj (•|Z) defined by
	Ŝj (t|Z) = exp -t βT j Z + βT j	0	t	S 1 (s) S 0 (s)	ds	1≤i≤n;Xi≤t,Di=j	1 -	1 αS 0 (X i )

Table 1 :

 1 Comparison of means and standard errors (within brackets) of N = 1000 estimates of βi and βi for i = 1, 2.

	n	β1	β1	β2	β2
	100 2.116 [1.583] 2.117 [1.438] 3.024 [1.923] 3.004 [1.610]
	200 2.055 [1.111] 2.023 [1.000] 3.083 [1.367] 3.039 [1.146]
	400 2.021 [0.768] 2.001 [0.687] 3.043 [0.919] 3.011 [0.784]
	1000 1.998 [0.500] 2.001 [0.454] 2.989 [0.584] 2.993 [0.502]


Gabriel Girard

San Patten

Marc-André Leblanc

Barry D Adam

Edward Jackson

Is HIV prevention creating new biosocialities among gay men? Treatment as prevention and pre-exposure prophylaxis in Canada

The advancements of "treatment as prevention" (TasP), "undetectable viral load" (UVL) and "pre-exposure prophylaxis" (PrEP) are redefining HIV prevention standards. Relying on the concept of biosociality, this article explores how gay men rally around, debate, and sometimes disagree about these emerging HIV prevention technologies. This article is based on data from the Resonance Project, a Canadian community-based research project. Twelve focus groups (totalling 86 gay and bisexual men) were held in three Canadian cities (Montreal, Toronto, Vancouver) in 2013-2014. Respondents view UVL and PrEP through the prism of their generational experience of HIV prevention. In this respect, biosocialities highlight an experiential dimension that is tied to the context of the HIV epidemic. The biosocialities of HIV prevention are also built around serological identities. However, our study shows the diversity of these positions. Analysis grounded in biosocialities is useful for better understanding how scientific information circulates, is made sense of, and generates debate among gay men.

1 Treatment as prevention refers to the use of antiretroviral therapy to suppress viral load among HIVpositive people, thereby preventing onward transmission of HIV.

2 Undetectable viral load refers to an amount of HIV below the level that can be detected by current biomedical tests, typically 40 copies per ml. There is now medical consensus that people with undetectable viral are untransmittable, although this statement was not as widely accepted at the time of this study. 3 Pre-Exposure Prophylaxis describes the use of antiretroviral medication by HIV-negative people, before sexual intercourse, in order to minimize the risk of being infected.

Introduction

The use of antiretroviral therapy to prevent transmission is one of the major developments in the past decade in the field of HIV/AIDS. This approach has not only demonstrated its potential in terms of reducing risks, it also promises to transform the understanding of the danger and fear of passing on or acquiring the virus and, therefore, the implications of condomless sex. The concepts of "treatment as prevention" (TasP) 1 , "undetectable viral load" (UVL) 2 and "preexposure prophylaxis" (PrEP) 3 have the potential to redefine HIV prevention standards across public health responses, as well as at the community level and within relationships. In Canada, as in most industrialised countries, gay and bisexual men are particularly affected by these changes. Each year, this population accounts for over half of new HIV infections, particularly in large urban centres such as Montreal, Toronto and Vancouver. The social and political issues associated with the diffusion of the concepts of TasP and PrEP in gay communities have been discussed in a number of critical publications [START_REF] Patton | The cost of science, Knowledge and ethics in HIV pre-exposure prophylaxy trials[END_REF]Kim 2012, Rosengarten and[START_REF] Rosengarten | The performative function of expectations in translating treatment to prevention: the case of HIV pre-exposure prophylaxis, or PrEP[END_REF]. Biomedical developments in HIV prevention have also been approached from the perspective of the effects of these technologies on behaviours [START_REF] Holt | HIV-negative and HIV-positive gay Men's attitudes to medicines, HIV Treatments and Antiretroviral-Based Prevention[END_REF][START_REF] Grace | HIV-negative gay men's accounts of using context-dependent sero-adaptative strategies[END_REF][START_REF] Holt | Configuring the users of new HIV-prevention technologies: the case of HIV preexposure prophylaxis[END_REF], social relationships and/or cultural representations of risk [START_REF] Grace | Becoming "undetectable": longitudinal narratives of gay men's sex lives after a recent HIV diagnosis[END_REF][START_REF] Young | Can a pill prevent HIV?, Negotiating the biomedicalisation of HIV prevention[END_REF]. This article focuses on how socialities are formed around new prevention technologies. Using the concept of "biosociality" developed by anthropologist Paul Rabinow, this article analyses the complex and changing relationships among biomedicine, biotechnologies and social relationships [START_REF] Rabinow | Essays on the Anthropology of Reason[END_REF][START_REF] Rabinow | Biopower today[END_REF]. This term refers to the formation of social relationships among individuals impacted or affected by the same health condition or disease [START_REF] Rabinow | Essays on the Anthropology of Reason[END_REF]. Early work on biosociality centred on genetic issues around the Human Genome Project and sought to analyse how specific biological features could be the starting point for new kinds of relationships and mobilization among patients and their loved ones. The emergence of biosocialities is integral to analysing the impact of the mechanisms of power on life in contemporary societies [START_REF] Rabinow | Biopower today[END_REF]. Biopower is no longer systematically based on constraint, but mobilises selfefficacy and individual accountability. In the field of public health, such mechanisms are apparent in obesity prevention campaigns or breast cancer screening [START_REF] Lupton | Risk[END_REF]. Indeed, according to Rabinow and Rose, biopower involves population-based intervention strategies composed of, or based on, biosocial communities that may be specified in terms of race, gender, sexuality or genes. This theoretical perspective has been applied to various fields [START_REF] Meloni | The biosocial: sociological themes and issues[END_REF]. It has also led to an analysis of the political dimensions of biosocialities expressed in terms of "biological citizenship" [START_REF] Petryna | Life Exposed: Biological Citizens After Chernobyl[END_REF], "biolegitimacy" [START_REF] Fassin | Another politics of life is possible[END_REF] or "therapeutic citizenship" [START_REF] Nguyen | Adherence as therapeutic citizenship: impact of the history of access to antiretroviral drugs on adherence to treatment[END_REF]). These various approaches have shed light on the complex subjectivation processes, defined as "the forming of oneself as an ethical subject" (Foucault 1985, 28). Therefore, biosocialities and biosocial groups refer to "collectivities formed around a biological conception of a shared identity" [START_REF] Rose | Biological citizenship[END_REF]. If epidemiology and biomedicine define "at risk" or "vulnerable" situations and identities, the individuals and groups involved take on and interpret this objectivation work and contribute to shaping it [START_REF] Hacking | The Social Construction of What? Cambridge[END_REF]. This article draws on the Resonance Project, a community-based research project that was conducted in three Canadian cities (Montreal, Toronto, Vancouver) in 2013 and 2014. The purpose of the project was to gain a better understanding of how gay and bisexual men in Canada perceive, discuss and integrate the developments in HIV prevention in a context where TasP and PrEP continue to spark debate in gay communities.

TasP, PrEP and debate about HIV prevention

Since the beginning of epidemic, gay men's sexual behaviours and their adaptation to HIV risks have been the subject of multiple analyses [START_REF] Kippax | Sustaining safe practice: twenty years on[END_REF]. Over time, gay communities developed various prevention strategies beyond condom use: reduction of the number of partners, frequent HIV testing, strategic positioning and negotiated safety. These techniques, which often emerged from gay cultures themselves, have been progressively used and rationalized by public health discourses. Since the late 1990s, in response to reduced condom use in gay communities, a broad field of scientific literature has been dedicated to the study of the development and implementation of biomedical prevention alternatives to condoms [START_REF] Hart | Sexual risk behaviour of men who have sex with Men: emerging patterns and new challenges[END_REF]. Relationships among sexual practices, community knowledge and scientific categorisations of risk are part of a long history [START_REF] Race | The Gay Science. Intimate Experiments with the Problem of HIV[END_REF].

Without question, the Swiss Statement, announced by Professor Hirschel and colleagues in late 2007, marked an important turning point in HIV prevention [START_REF] Vernazza | Les personnes séropositives ne souffrant d'aucune autre MST et suivant un traitment antirétroviral efficace ne transmettent pas le VIH par voie sexuelle[END_REF]. Having observed zero infections in a cohort of heterosexual serodiscordant couples, the doctors stated that HIV-positive individuals with an undetectable viral load could not transmit HIV, provided they had been treated for at least 6 months, adhered to treatment, and did not have any STIs. In 2011, the HPTN052 study demonstrated the effectiveness of ART-related protection for serodiscordant heterosexual couples [START_REF] Cohen | Prevention of HIV-1 infection with early antiretroviral therapy[END_REF]; in 2014 and 2016, the PARTNER study provided data on the effectiveness of undetectable viral load for serodiscordant heterosexual and same-sex male couples. In both studies, no infections occurred when the HIV-positive partner had an undetectable viral load.

Around the same time, another approach was being developed and tested: pre-exposure prophylaxis. The principle was not new: taking ART to prevent infection had been the basis for preventing perinatal transmission since the early 1990s. For "at-risk" men and trans women who have sex with men, three main studies took place to test daily -then intermittent -ART administration. In 2010, data from the iPrEx study demonstrated the strong potential for risk reduction with this strategy that was strongly correlated with adherence [START_REF] Grant | Preexposure chemoprophylaxis for HIV prevention in men who have sex with men[END_REF][START_REF] Grant | Uptake of pre-exposure prophylaxis, sexual practices, and HIV incidence in men and transgender women who have sex with men: a cohort study[END_REF]. In 2014, the IPERGAY and PROUD studies confirmed the effectiveness of PrEP [START_REF] Mccormack | Pre-exposure prophylaxis to prevent the acquisition of HIV-1 infection (PROUD): effectiveness results from the pilot phase of a pragmatic open-label randomised trial[END_REF][START_REF] Molina | On-demand preexposure prophylaxis in men at high risk for HIV-1 infection[END_REF] for prevention.

How these biomedical developments are affecting sexual interactions and risk management is not always straightforward. TasP has been studied as an area of new identity development among gay men [START_REF] Race | The undetectable crisis: changing technologies of risk[END_REF] and undetectability is changing notions of the "normal" or "responsible" especially for HIV-positive men [START_REF] Grace | Becoming "undetectable": longitudinal narratives of gay men's sex lives after a recent HIV diagnosis[END_REF].

Redefining gay socialities in the context of antiretroviral therapy has been the subject of a significant amount of literature that highlights the transformation of community ties and growing individualism, but also the creation of new real or virtual peer networks [START_REF] Race | The undetectable crisis: changing technologies of risk[END_REF][START_REF] Davis | The "loss of community" and other problems for sexual citizenship in recent HIV prevention[END_REF][START_REF] Rowe | Sex, love, friendship, belonging and place: is there a role for "gay community" in HIV prevention today?[END_REF][START_REF] Holt | Gay men and ambivalence about 'gay community': from gay community attachment to personal communities[END_REF][START_REF] Girard | HIV risk and sense of community, French gay male discourses on barebacking[END_REF]). This article focuses on the understandings and perceptions about new HIV prevention technologies in focus groups of gay, bisexual and other men who have sex with men living in Canada. It deals with how gay men rally around, debate, and sometimes disagree with each other concerning these HIV prevention technologies.

The study of biosocialities examines how the subjects of biomedicine integrate, adhere to, or challenge the biotechnologies intended for them and this study shows the extent to which relationships of trust and scepticism vis-_a-vis expert knowledge translate into different modalities of belonging to an experiential community. Therefore, in this article, we aim to analyse biosocialities among gay men, in a context of a long-term epidemic. This involves focusing on the relationships created through HIV prevention but it also requires considering how individuals and their relationships are situated and enacted through biotechnologies. Our data show this double dimension.

Methods

The Resonance Project employed a community-based research (CBR) approach through all phases. In order to ensure that the Resonance Project reflected and represented gay men's perceptions, the team privileged their perspectives and interpretations throughout the research process. The research team included gay men, gay men's health service providers, educators from community-based organisations, biomedical prevention technology experts, public health experts and representatives of the collaborating organisations. The research team provided advice related to the research design, including development of research tools, participant recruitment, data collection, data analysis, interpretation and dissemination through various knowledge exchange methods.

Participants

The Resonance Project gathered perspectives of gay men through four focus groups with gay men in each city (Vancouver, Toronto, Montreal), for a total of 12 focus groups. The choice of focus group method aimed to create the conditions for discussion and debate among gay and bisexual men in order to better understand how their arguments are developed.

Above is a table summarising the inclusion criteria for the focus groups. All participants had to be at least 18 years old, to identify as cisgender or transgender men, and to have been sexually active with men in the last 6 months. Focus groups were recruited purposively to engage men with potentially different relationships to new prevention technologies: men who had contact with AIDS service organisations (ASOs) and those who did not, men in serodiscordant relationships who were managing everyday HIV prevention, sexually active HIV-positive men, and men at elevated risk of HIV but who had no formal contact with ASOs (Table 1).

Recruitment

The local partners -Health Initiative for Men (HIM) in Vancouver, REZO in Montreal and Gay Men's Sexual Health Alliance (GMSH) in Toronto -recruited participants for the focus groups in ways that were adapted to the local gay scene. Each city developed its own recruitment materials, based on a common template (see poster/card graphics below). Recruitment posters and ads were posted in various physical and virtual spaces, including clinics, bars, bathhouses, coffee shops, websites, social media and hook-up sites. Outreach workers and volunteers they enlisted found participants in community venues, such as social clubs, gay men's health clinics, at community meetings and events, bars, bathhouses and outside subway stations.

There were also more targeted efforts for identifying potential participants for specific focus groups. The table below summarises the number of gay men who participated in focus groups across the three cities (Montreal, Toronto and Vancouver) (Table 2).

86

The figures below show the demographic characteristics of the men who participated in the study (Fig. 1 and2).

Focus group methods

The focus group guide was developed collaboratively by the research team to cover the following topics: what biomedical information gay men are hearing and from whom; level of interest in biomedical technologies such as home testing, PrEP and Post-Exposure Prophylaxis (PEP)4 ; types of biomedical logic and reasoning that they are using; barriers to using biomedical approaches to prevention; and how biomedical knowledge of HIV informs their risk calculations. Rather than directly ask the focus group participants about their knowledge levels and how they know what they know in relation to the biomedical concepts, mock online profiles Table 2 Total number of participants in focus groups from common hook-up sites/apps, dating scenarios, and real media headlines were used to stimulate conversations to see how study participants would react, for example, to prospective partners who indicate they bareback, are undetectable, or are taking PrEP. Focus groups in Vancouver and Toronto were conducted in English; in Montreal in French.

Analysis

Once cleaned of identifiers, the transcripts were loaded into NVivo10. Data analysis was conducted by a subgroup of the research team, namely the co-authors of this article. Using a process of interpretive description [START_REF] Thorne | Interpretive description[END_REF], the authors repeatedly read the data to confirm, test, explore, and expand on the key concepts and patterns, emerging themes, indicative examples from the data, and connections to relevant theoretical perspectives. We analysed collectively a sample of focus groups (from different cities and focus group types), in order to establish a coding scheme. Two members of the subgroup then coded all the transcripts, allowing all co-authors to proceed to a collective reading and analysis of the entire coded data, leading to article writing. As part of a commitment to critical discourse analysis, we also analysed the use of language and metaphors to uncover interpretations of biomedical knowledge of HIV. The inductive process for thematic analysis, drawing on [START_REF] Boyatzis | Transforming Qualitative Information. Thematic Analysis and Code Development[END_REF], included developing a coding scheme for organising segments of similar or related texts; identifying initial themes; applying the codes from the coding scheme to the text with the intent of identifying meaningful units of text; and finally, connecting codes and developing additional themes.

Results

The focus groups discussed in depth two of the most recent prevention technologies: preexposure prophylaxis (PrEP) and undetectable viral load (UVL). The level of knowledge and understanding of these two approaches was heterogeneous according to type of focus group, but also according to city, HIV status and age. There seemed to be a better understanding of PrEP among Montreal participants, possibly because of the IPERGAY study, which began in 2013 in Quebec, but also due to ongoing advertisements about PrEP published by private clinics in the community press. In Montreal, two participants were taking PrEP and mentioned this during the focus groups. There seemed to be a higher level of knowledge of UVL in Vancouver, possibly because British Columbia health authorities actively promote TasP as a strategy. Otherwise, participants had highly variable levels of knowledge: some were discovering the strategies for the first time within the focus group; others had heard of them but were not able to define them with certainty; while still others were able to explain how they worked by referring to the scientific data.

Mapping of social and moral risk

PrEP and UVL contribute to redefining the concept of sexual risk by introducing the possibility that condomless sex does not necessarily mean exposure to HIV. In focus group discussions, participants expressed their views about norms of responsibility for HIV prevention; PrEP and UVL led some participants to reposition or reaffirm their values regarding risk.

Risk reduction and responsibility

In the focus groups, discussions about PrEP inevitably triggered debate about responsibility with respect to HIV risk. Some participants expressed skepticism concerning the motives for taking PrEP while others viewed PrEP as both rational and responsible. Claim about the lack of responsibility of "others" was often a way to define a moral boundary between "good" and "bad" subjects with respect to prevention.

Participant A (Neg, <35, M4): I find that people see this anti-HIV pill as a pretext for not using protection. A miracle pill not to protect yourself, exposing yourself to risk of infection. . . This makes no sense to me. (All excerpts from Montreal (M) groups are translated from the French.) Participant B (Neg, <35, M1): I'm sorry but I am on PrEP myself and I am not a case of "I bareback every week." I saw a doctor and had a discussion with her and it was at my request that she prescribed it to me.

Participant C (Pos, 50+, M1):

They are not all like you.

In these last two excerpts, two participants express, each in his own forceful way, a suspicion about PrEP users. Even when another participant (B) taking PrEP speaks up to explain his own risk relationship, the speaker singles him out as being unique among a group of men who are perceived as reckless and irresponsible. A number of participants feel that PrEP is still a minority strategy and is "not very well regarded in the community", as was expressed by another man who is also taking PrEP (Participant D,Neg,(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(49)M2). In Montreal, PrEP use was starting to increase when the focus groups were conducted (January 2014), but it remained limited.

In the three cities, discussions about PrEP led the participants to position users of this strategy in terms of responsibility criteria but assigning this responsibility did not necessarily mean moral disapproval. For a number of men, PrEP was considered an affirmation of a fulfilling sex life: In this excerpt, the technology creates an affinity around the idea of sexuality that is free from the requirements of conventional prevention. In another vein, some participants position PrEP as a potential guarantee of a responsible approach to managing one's health.

Participant F (Pos, 50+, V3): It would just prove to me that they're being responsible if they're sexually active. If they're on PrEP they're probably getting tested on a regular basis too. The people that take PrEP aren't the ones that never get tested I would imagine, I would think. I would hope. The people on PrEP are the ones that probably would be getting tested regularly because they're taking responsibility for their health… The range of reactions about PrEP points to the fact that this prevention technology is emerging in an existing moral context. For some participants, PrEP translates into a concern for self and others, providing reliable protection while allowing sexual liberation. Other participants feel that this strategy promotes sexual freedom at the expense of sexual health. The latter placed condom use on a higher moral plane than PrEP use.

Undetectable viral load also presents moral dilemmas. For HIV-positive men, the risk of exposing an HIV-negative partner triggers intense reflection, given the context of criminalization in Canada [START_REF] Adam | Impacts of criminalization on the everyday lives of people living in with HIV in Canada Sexuality Research and Social Policy[END_REF]. Within the focus groups, this concern was sometimes expressed as self-justification in the face of the assumed judgement of other participants: Participant I (Pos, <35, M3): I'm going to go a bit opposite to the group, but personally, what this information tells me, it's simply that indirectly, I tell myself that if I have sex with someone who does not use protection. . . I did not force him. So we did not necessarily talk about our status or anything, but we had exciting sex, and as we get into sex. . . I tell myself. . . I am going to do it without feeling persecuted, like I was a criminal or a killer because I know that I have an undetectable load, I'm really good about my meds, it's been 3 years that I've been undetectable every time I do a blood test every 3 months. I know the risk of transmission is low, so that means I will go ahead with unprotected sex. But I will cum outside his ass though. . . that's my gauge. So after, in terms of my ethics, is this good or bad? I don't know, we could talk about this if you want, put me on trial. This man expresses his interpretation of responsibility with respect to TasP: because he has an undetectable viral load, he feels he is not putting his partners at risk. Therefore, it is up to them to ask about using a condom. The normative tension is explicit here in his discourse and his use of the expressions "feeling persecuted, like I was a criminal" and "put me on trial".

On the other hand, another HIV-positive participant reaffirms an interpretation of prevention based on avoiding situations with risk.

Participant E (Pos, 50+, V2): Then sometimes you start to get through that little bit of the conversation which is sex possible. . . if you're HIV negative, I'm going to think two or three times before I touch you. I watched a lot of people die. When I have made an error that way and it's happened once or twice, it has cost me weeks of depression. I feel like a gatekeeper in a way. I feel like every time I do it with another positive guy, bareback or not, in Darwinian terms I'm selecting for a weaker strain of the virus, just like if you use a condom or whatever else. That's the harm reduction. Now they're talking about undetectable viral loads. I wouldn't still. I wouldn't want to put a negative guy at risk. By refusing to have condomless sex with his HIV-negative partners, this man is responding to his own moral imperative. Following a very strict prevention approach for him is consistent with the experience of the dark years of the epidemic.

In the focus groups, the issue of assigning responsibility was the topic of much discussion. Participant views were based on pre-existing moral conceptions that guide their understanding of prevention choices (Davis 2008, Rangel andAdam 2014). These conceptions cannot be disassociated from the trajectory and perception of risk or the definition of a fulfilling sex life. Conflicting views expressed in the focus groups reflected different decisional processes related to these aspects. With TasP and PrEP, these differences are not necessarily based on HIV status. The experience of risk -condomless sexual practices or a partner with a different HIV statuswas then part of redefining preventive biosocialities [START_REF] Grace | HIV-negative gay men's accounts of using context-dependent sero-adaptative strategies[END_REF]. Biosocial groups appear to be anchored in different conceptions of sexual authenticity in times of HIV epidemics. Here, we can clearly see the twofold process: the appropriation of biotechnologies to redefine preventive interactions, and the very notion of responsibility, and how incorporating these technologies produces "healthy sexuality" as a new moral imperative.

Undetectability: Areas of uncertainty

In comparison to PrEP, participants were more familiar with the protective effects of undetectable viral load (UVL). This did not however preclude disparities within the groups or geographic differences. The Vancouver groups were on the whole better informed, which can be explained by the active policy of the province's health authorities promoting TasP as a strategy.

However, in each of the three cities, some participants became aware of the concept of UVL within the focus group: These are minority situations but they show the heterogeneous nature of the gay community with respect to information about prevention. More generally, knowledge and lack of knowledge about UVL were components of discussions in the groups. These exchanges often involved the level of information among people other than themselves: other members of the gay community, sexual partners or users of hook-up sites.

Participant L (Pos, <35, T1): On the Internet especially, if you say the word HIVpositive, people are more. . . hmm. . .. The minute you say healthy and undetectable, it's becoming more of 'okay what is undetectable? Can you explain that to me?' Some people don't really know. The problem is the information out there is not, in my opinion, not clear enough to define what undetectable is. Participant N: Not everyone. They'll ask, what's that? They'll say, is that undetectable lubricant that you can't feel? What's that, what does that mean? They won't find you with their GPS? HIV-negative partners know about it because their boyfriend is HIV positive.

In these two dialogues, the lack of knowledge about the strategy is attributed to lack of information about the subject. However, one of the Montreal participants also points out how knowledge of this approach cannot be separated from the biological and relational experience of HIV, such as being part of a serodiscordant couple. Other participants suggest that knowledge of UVL is indicative of an unequal distribution of knowledge about prevention even within the gay community: Participant B (Pos, <35, M1): In Montreal, I would say that in the Village, everyone knows what undetectable is. When you go beyond that area, when I say I'm a poz undetectable guy, then you get a bunch of question marks. I don't even take the time anymore to explain because it's too long and ultimately, the person will still be extremely afraid and then not understand what it was. There are truly two realities I find, in the Village and then outside of the Village in other areas.

Here, the lack of knowledge about UVL defines a geographic divide between "the Village" and the urban periphery. These spatial issues can also be part of the biographical trajectory, as with this Vancouver participant who explains his own journey in terms of UVL:

Participant O (Neg, 35-49, V2): I actually didn't even hear about undetectable viral load until I moved here. Then I learned it from him [his partner]. I did a lot of research. I went to the web and looked up everything I could and spoke to some health professionals at the HIM [clinic]5 . They all said the same thing; having sex with somebody that is known to have low viral load is safer than having sex with a stranger with a condom. I couldn't believe that they were all saying the same thing. Wow. So I started having unprotected sex with him [HIVpositive partner]. I'm always the top. That's the stipulation.

In this case, the integration of information is directly related to a change in environment and access to community outreach. These two examples show the spatial dimension of biosocialities in HIV prevention. However, knowledge of new technologies such as UVL aligns with participation in a community experience [START_REF] Friedner | Biopower, biosociality, and community formation: how biopower is constitutive of the deaf community[END_REF].

Undetectability and self-identification

For a number of participants, achieving undetectable viral load means a new relationship to the self and to others; namely being defined as "undetectable" instead of simply as "positive" or "HIVpositive". In the first focus group of the study, one of the men made this very clear statement:

Participant F (Pos, 50 + , V3): I always say undetectable is the new negative.

F: In what ways and why?

Participant F: Just because it's just low risk, next to no risk.

Here, undetectability is placed on equal footing with absence of risk; in broader terms, it removes the need to be identified with HIV. In the subsequent focus groups, we introduced the expression "Undetectable is the new negative" to other participants in order to generate discussion about these self-identification implications, especially during online cruising. The statement led to much debate during the focus groups: F: Participant D, What do you think about 'undetectable is the new negative'? Participant D (Neg, 35-49, M2): Well, as far as I know, there is no such thing. Unless this is a question of . . . it's me who is not aware, that I missed something there. But positive undetectable, that's one thing. Positive, that's another thing. And then negative, that's another thing. You can't stick your head on the sand. Although yes, the risk is minimised, even nearly impossible, but if you forget your medication one day then everything is wiped out and you start back at zero.

Participant P (Pos, 50 + , T3):

Undetectable, a lot of people think, oh I'm never going to infect, I'm never going to get infected, or I'm never going to infect anybody else so therefore, get rid of the condoms now because you're undetectable and I'm undetectable. But there's still that less than .05% that you are. People seem to have this belief that you're safe again or that they can play unsafe and not worry about it.

Far from building a broad consensus, the statement, "Undetectable is the new negative," revealed a series of conflicting views within the groups, both among HIV-negative and HIVpositive participants with regard to assessing risk statistics or the credibility of scientific sources. As shown in other research, undetectability reinforces the feeling of responsibility on people living with HIV [START_REF] Grace | Becoming "undetectable": longitudinal narratives of gay men's sex lives after a recent HIV diagnosis[END_REF]. Undetectability allows some respondents to renegotiate their relationships to others. However, for some, it seems that UVL does not tie into an identification process, because unstable biological data are involved.

Refusal and criticism of medicalisation: other biosocialities

For some participants, new prevention technologies generate skepticism, fear, or rejection. This criticism also illustrates other forms of biosocialities whose legitimacy is rooted in values, individual and collective histories, or adherence to other, older prevention tools (condoms, for example). Analysing these discourses of defiance-which sometimes extend to conspiracy theories-as forms of biosociality allows for another way to consider the tensions around biomedical innovations (Callon andRabeharisoa 1999, Raman and[START_REF] Raman | Life, science, and biopower[END_REF].

PrEP, a risk technology?

Throughout the groups, there was a great deal of criticism and reluctance with respect to PrEP. Fears about the potential toxicity of antiretroviral treatments (ART) for uninfected people came up consistently in the concerns voiced, particularly by HIV-positive men and by HIV-negative men who have (or had) someone close to them living with HIV: Participant Q (Neg, 50 + , V1): Part of what would not make PrEP a choice for me at this point in time is that it's fairly new and we don't know what it does to the body in the long term to a healthy immune system because of the toxicity of the drugs. We all know that. Whenever somebody goes on a new regimen of drugs, you have to go through that adjustment period. The toxicity, in spite of people's longevity with HIV these days, is still there. That to me is not a desirable thing if I don't need it. We don't know what the long-term impact of that is yet, at least I don't.

Participant R (Pos, 35-49, T3): But I think it definitely leads to a false sense of security because how long will it be before you know in 50 years a study finds out that this stuff is actually causing liver damage or something else that we weren't aware of? Then it's 50 years down the road too late. Some fear that PrEP raises a false sense of security, a concern rooted in generations of gay men that experienced the long-term effects of the earlier sets of anti-HIV drug treatments on the body. This echoes several critical discourses about "over-medicalisation" in modern societies. On a related note, criticism of the pharmaceutical industry came up regularly in the groups with respect to PrEP: Participant I (Pos, <35, M3): So for me, this raises ethical issues with respect to the pharmaceutical companies that make huge profits in selling the drugs and ultimately now we don't just have HIVpositive guys taking medications but now HIV-negative guys as well. Similar treatments. Ok, I don't know if there are as many taking them as those who are HIV positive but… it's become a way for not transmitting HIV. That's ok fine, I don't know. . . I don't know what kind of world we're heading toward.

For Participant I, the pharmaceuticalisation of prevention for people who are HIV-negative is part of uncontrollable changes in a world led by profit motives. In the same vein, users of PrEP were sometimes described as laboratory animals in terms of the potential risks.

Participant O (Neg, 35-49, V2): And that's one of the reasons I chose against it. I'd be taking on a lot of risk by taking the drugs when I didn't feel like I was in an unsafe situation to begin with. Once it's been out maybe 10 years, I might consider it then when all the effects are known. But right now it's just too new. I don't want to be one of the guinea pigs.

As noted above, PrEP users were sometimes perceived as responsible individuals who care about their health. In other cases, they were seen as irresponsible.

Participant S (Pos, <35, V2): And then also from a health standpoint if PrEP is available to anyone who wants to take it, does that mean you've got a bunch of guys who think they're totally invincible and they're going to go fuck their brains out? Or are they going to do that anyways? I wish that we had PrEP for my personal situation.

In this case, there is evident ambivalence. In this instance, PrEP is understood as an incentive for risk that could potentially weaken the use of condoms. Yet at the same time, Participant S, an HIV-positive man under 35 years of age in a serodiscordant relationship, voices regret about not having had access to it. In this case, PrEP represents the possibility of greater intimacy (and/or less anxiety) with his negative partner. It opens up the possibility of imagining a sex life without condoms and without risk, which calls into question the many years of preventive socialisation.

Undetectability: Limits and pitfalls

In discussions about undetectability, a number of participants expressed skepticism or doubt. This resistance is partly explained by the lack of clarity of community and public health organisations in their public communications about UVL. This fact is clearly illustrated in the following excerpt: Participant T (Neg, 35-49, M4): There has been information about it going around for a few years now and at first I was skeptical and wondering 'Is that a lie? You can't find anything written about it almost anywhere else. Is this information boycotted? Is it censored?' Because it started to come out in late 2011, these stories. It came from Switzerland. They were saying that 'for people who have an undetectable viral load and take their meds regularly, the risk of transmission is almost nil.' Well, this information was well kept. We were wondering how can we even know if it's true? No one is passing it on. Where is this coming from? Is this reliable? So that's still the situation today.

But other forms of reluctance are expressed, around the biomedical limits -real or assumed -of this strategy. This is particularly the case for issues of superinfection by strains that are resistant to treatments. For the participant below, the discussion about superinfection is the opportunity to bring up his difficulty navigating through medical information that is often alarmist: Participant L (Pos, <35, T1): Even as a positive person we used to… you know the scare was put into us by our doctors that if you have unsafe sex you're going to get another strain probably. Superinfection is an old concern in the gay community [START_REF] Adam | Risk construction in reinfection discourses of HIV-positive men[END_REF]. Since the Swiss Statement, it garners only a low level of attention in the medical literature, but it remains in the discourse of HIV-positive men. Lastly, a number of participants shared their concerns about the potential for "trivialising" risk with new prevention technologies, and particularly with UVL.

Participant B (Pos, <35, M1):

The term undetectable, I agree with you, that trivialises the risks of transmission, enormously.

Participant I (Pos, <35, M3):

So what also happens with this famous undetectable status, it's that it becomes the key to have any kind of sex when and where you want. But did I take a blood sample and test it to find out if you were truly undetectable or if you had your viral load was through the roof? . . . We take that at face value, but unless you really know the person and then know that you can have a relationship of trust with him, if the person says the magic words, it's not a reason for having sex without a condom, you know.

Here, managing HIV prevention in everyday life becomes a problem of verifiability: can one rely on the other's claim to undetectability? Although UVL is a factor in alleviating the anxiety about transmission, this information did not result in major changes in prevention practices with these two HIV-negative participants:

Participant U (Neg, <35, V4): It hasn't changed my practice but it's planted some confusion, or not confusion, but it's kind of like, well maybe it would be safe. But I think in my mind first and foremost it's highly engrained in me to always use a condom. So it hasn't changed my practice but the messaging makes me wonder a little bit.

Participant V (Neg, <35, V4): For me the same. It didn't change my practice but at the same time it gives me a bit more peace of mind because when we talk about the risk there is not any definite number. It's always like a range. It minimises the risk so it's good. But I still have my practice. I still use the condom but well it is less risk.

Critical discussion around undetectability highlighted the social and cultural conditions for the adoption of scientific data [START_REF] Hacking | Genetics, biosocial groups & the future of identity[END_REF]). In the various groups, participants voiced their concerns about a "paradigm shift" that is underway, a shift they have perceived in online hook-up sites, where undetectability now appears in user profiles. "Paradigm shift", a term largely used in recent HIV prevention discourses, refers here to the transition from condom-centred prevention to a context where new highly effective biomedical tools are also available. But, this new context also rekindles deep-seated fears about risk and verifiability. In this respect, the residual risk seems off-putting to a number of participants who also express distrust of biomedicine.

Discussion

The data from the Resonance Project shed a unique light on the dynamics of the diffusion of new HIV prevention technologies in the gay community. The shift in prevention norms has already been addressed in various reviews that highlight growing individualisation, but also the continuity of forms of collective solidarity in the gay community [START_REF] Rowe | Sex, love, friendship, belonging and place: is there a role for "gay community" in HIV prevention today?[END_REF][START_REF] Davis | The "loss of community" and other problems for sexual citizenship in recent HIV prevention[END_REF][START_REF] Girard | HIV risk and sense of community, French gay male discourses on barebacking[END_REF]). An analysis grounded in biosociality offers a different framework, by focussing on how new prevention technologies contribute to the redefinition of these social relationships.

The analyses outlined in this article have a number of limitations. First, the Resonance Project data were collected in late 2013 and early 2014. Since then, scientific and community debates surrounding topics such as PrEP and undetectability have continued to evolve, landing on much broader consensus about their high level of efficacy. Second, while the focus groups allow for the dynamics of the debates to be recorded, they also reproduce the power relationships between the speakers. Despite attempts to ensure that the voices of younger and non-white participants were heard -through facilitation techniques -overall the data collected nevertheless generally reflect the predominant voices of older white men. Due to this limitation, we need to be careful about generalising the results. There were three key learnings that came out of the data.

• First of all, the respondents seem to view UVL and PrEP through the prism of their generational experience of HIV prevention. Within the Resonance Project, disparities arose according to when the participants initially became sexually active. This finding is in line with the AIDS generations analysis developed by [START_REF] Hammack | Gay men's health and identity: social change and the life course[END_REF]. Having experienced the emergence of HIV and the harm it caused in gay communities does not necessarily lead to a critical position with respect to treatment as prevention. As we have observed, these positions vary. However, the data point to the weight of sexual and preventive socialisation. For some respondents, developing their sexuality in an environment where condoms were a central prevention tool for several decades has conditioned the acceptability of alternative risk reduction techniques. In this respect, biosocialities highlight an experiential dimension that is tied to the context of the HIV epidemic.

• The biosocialities of HIV prevention are also built around serological status. However, our study shows the diversification of these positions. Testing draws a biomedical boundary between HIV-positive men and HIV-negative men. Greater complexity is introduced with treatment as prevention. Some men living with HIV are claiming undetectability as a new means of self-identification. Likewise, new biological socialities are being developed in light of PrEP. A number of respondents who are reticent about this new tool consider that those taking PrEP are making light of or are unaware of the adverse effects of treatments. This critical view touches on an underlying commonality among HIV-positive respondents taking antiretroviral treatments and HIV-negative respondents who have someone close to them taking antiretroviral treatments. This pharmaceutical solidarity may be situational and tied to the time the data were collected, when the effectiveness of PrEP was being widely discussed. It nevertheless reveals a shift in biological socialities. • Analysis grounded in biosocialities is also useful for better understanding how scientific information circulates, is made sense of, and generates debate among gay men. These issues around information flow are particularly evident when respondents discuss remaining areas of uncertainty. For instance, for some, we noted the lack of knowledge of biomedical information related to geographic or symbolic boundaries: the fact of living outside of urban centres or outside of the "Gay Village". However, the disagreements expressed in the focus groups draw attention to reluctance or distrust with respect to new prevention technologies as a foundational element of biosocialities. While biosocialities are often studied within the framework of integrating and adhering to scientific innovations, this research shows the usefulness of having a better understanding of the potential solidarities expressed in the resistance to prevention biotechnologies. The idea is not to take sides but to consider properly how references to "old" approaches (such as the condom) can be the basis for social relationships, just as enthusiasm for new technologies can be. This is especially important to consider in a context where condom use has been such a pervasive and ubiquitous HIV prevention message, and the emergence of PrEP and UVL presents a major cultural shift that forces an expansion of moral reasoning around responsibility, risk-taking and risk-reduction.

Our use of the concept of biosociality differs in many respects from its classical uses. In this paper, relations and identities are observed "in the wild" [START_REF] Callon | Research "in the wild" and the shaping of new social identities[END_REF], outside the context of clinical or laboratory research, so the issues of biosocialities raised here engage more directly with the research field on the pharamaceuticalisation of society [START_REF] Williams | The pharmaceuticalization of society?, A framework for analysis[END_REF]. Our work highlights how relationships between service providers and users/patients, and relationships among gay men, are being reshaped by pharmaceutic interventions. These are processes that contribute to a "colonisation" of everyday life by pharmaceutical solutions, where biosocialities shed light on the heterogeneous processes of the reconfiguration of HIV prevention in Western gay communities.

In a cross-sectional view, this article reveals the extent to which issues of individual responsibility regarding transmission risk are the backdrop for debate among the focus group participants. Taking risks (or having another take them) remains a central aspect of the moral framework that structures discourse around HIV prevention [START_REF] Race | The Gay Science. Intimate Experiments with the Problem of HIV[END_REF]. It is difficult to say at what point new prevention technologies will reconfigure this situation but these technologies already allow some individuals to renegotiate their place in this moral context. By providing the possibility of safe and condomless sex, TasP and PrEP drastically change how HIV prevention is perceived. There is enthusiasm along with resistance, suspicion, criticism and concern with respect to long-term prevention changes. Individual history, experience with HIV, generation or the level of trust in the expert discourse are some of the factors that explain these differing attitudes. In light of TasP and PrEP, HIV prevention remains a controversial issue, as it affects the normative dimensions of intimacy.

Figure 1 :Figure 2 :

 12 Figure 1: Demographic characteristics of participants : City, serostatus an age

Facilitator

  If you met somebody or you heard that somebody was on PrEP, what comes to mind about that person? What are your preconceptions maybe about that person? Participant E (Pos, 50 + , V2): They're probably a bit like me. F: What does that mean? Participant E: It means that they want to have the possibility of unprotected sex, a bit like me. S-L-U-T.

F:

  If you hear somebody is on PrEP, does it come with any. . . you put it in a positive light -that they're being responsible. Does it come with any negative baggage at all for anyone? Participant G (Pos, 35-49, V3): I kind of wonder how much sex they're having. But it really doesn't matter. Who am I to judge? I'd like to have a lot of sex too. (. . .) Participant H (Pos, 35-49, V3): I just think that they are being very responsible with their health.For some reason I'd also feel more comfortable with them because they're knowledgeable and they know about prevention and they're aware, they're up to date, they're modern.

F:

  Had everyone seen that term before and knew what it was before tonight? Or is this something new tonight? Participant J (Neg, 35-49, V4): I'm a bit embarrassed to say that I never heard of it. Participant K (Neg, <35, V4): I never heard about undetectable. P: I heard of it but I don't necessarily understand it. P: I'll admit too, I had no idea.

(

  There are still a lot of people out there that don't know what undetectable means. Participant N (Pos, 35-49, M2): Most people don't know this. When you say undetectable viral load, people ask, is that a recipe for baking a cake? Participant D (Neg, 35-49, M2): Yeah, it's true that not everyone knows about it. That's true.

Table 1 : Inclusion Criteria for Focus Group Inclusion Criteria for Focus Groups Focus Group 1: gay men connected to HIV organizations

 1

		• Attended a workshop, training or conference on HIV
		in the past year
	Focus Group 2: gay men in serodiscordant relationships	• HIV-negative or HIV-positive • Currently in relationship of more than 6 months • Partner's HIV status is different than theirs

Focus Group 3: sexually active HIV-positive gay men

	• Has ever received a positive HIV test result
	• Has had sex with >1 man in past 3 months

Focus Group 4: HIV- negative gay men at 'high risk'

	• Has never received a positive HIV test result
	• Has been tested for HIV more than twice in last year
	and/or has used recreational drugs in past 3 months
	• Has had sex with >1 man in past 3 months

Table 2 : Total Number of Participants in Focus Groups

 2

	Total Number of Participants in Focus Groups
		(FG1, FG2, FG3, FG4)
	City	FG1	FG2	FG3	FG4
	Vancouver	8	6	5	12

Post-Exposure Prophylaxis is a preventative tool consisting of taking anti-HIV drugs for 1 month after a potential exposure to HIV.

Health Initiative for Men (HIM) is a community-based organization in Vancouver.

Acknowledgements

The authors would like to thank all the participants of the project, CATIE and the Resonance Project