Tiphaine Saulnier
email: tiphaine.saulnier@u-bordeaux.fr

Viviane Philipps

Wassilios G Meissner

Olivier Rascol

Anne Pavy-Le Traon

Alexandra Samier Foubert

Cecile Proust-Lima

Anne Pavy-Le-Traon

Alexandra Foubert-Samier

Cécile Proust-Lima

Joint models for the longitudinal analysis of measurement scales in the presence of informative dropout

Keywords: joint model, time-to-event data, longitudinal data, latent process, ordinal data

Introduction

In health cohort studies, markers measured at multiple times are often used to describe the natural history of a disease, monitor patients or predict the risk of clinical progression. Classical examples include T-cell CD4 counts and viral load for the progression of HIV [START_REF] Tsiatis | Joint modeling of longitudinal and time-to-event data : an overview[END_REF] or PSA for prostate cancer evolution [START_REF] Ferrer | Joint modelling of longitudinal and multi-state processes: application to clinical progressions in prostate cancer[END_REF]. Due to the intrinsic intra-subject correlation between the repeated measures of markers, their evolution can not be modelled using classical regression models, and mixed models which include individual random effects to account for this serial correlation are now adopted worldwide [START_REF] Commenges | Dynamical Biostatistical Models[END_REF].

During the follow-up, clinical events (e.g. diagnosis, recurrence, death) may disrupt the progression of the markers and induce a dropout. When this interruption of the follow-up is predictable by the observed marker data, the missing data mechanism is called missing-at-random (MAR), and inference provided by the mixed model is still valid [START_REF] Thomadakis | Longitudinal and time-to-drop-out joint models can lead to seriously biased estimates when the drop-out mechanism is at random[END_REF]. However, in many cases, the dropout is likely to depend on the underlying (unobserved) disease mechanism rather than only on the strictly observed data. The missing data mechanism becomes missing-not-at-random (MNAR), and mixed models may not provide correct inference anymore [START_REF] Little | Modeling the Drop-Out Mechanism in Repeated-Measures Studies[END_REF][START_REF] Rouanet | Interpretation of mixed models and marginal models with cohort attrition due to death and drop-out[END_REF]. During the last twenty years, the statistical community has massively embraced the issue of dropout in longitudinal analysis which lead to the development of joint models for the simultaneous analysis of repeated markers and clinical events [START_REF] Rizopoulos | Joint Models for Longitudinal and Time-to-Event Data[END_REF][START_REF] Henderson | Joint modelling of longitudinal measurements and event time data[END_REF][START_REF] Tsiatis | Joint modeling of longitudinal and time-to-event data : an overview[END_REF][START_REF] Proust-Lima | Joint latent class models for longitudinal and time-to-event data: a review[END_REF]. Joint models combine a mixed model describing the progression of the markers and a survival model for the time of occurrence of the clinical events, while associating the two models through shared random variables, usually the random effects from the mixed model [START_REF] Rizopoulos | Joint Models for Longitudinal and Time-to-Event Data[END_REF]. Beyond the study of the marker progression in the presence of MNAR dropout, this method also more generally assesses the association between a marker trajectory and an associated event of interest [START_REF] He | Joint modeling of multivariate longitudinal measurements and survival data with applications to Parkinson's disease[END_REF][START_REF] Wang | Joint modeling of multiple repeated measures and survival data using multidimensional latent trait linear mixed model[END_REF][START_REF] Ferrer | Joint modelling of longitudinal and multi-state processes: application to clinical progressions in prostate cancer[END_REF]. Despite many developments in joint models in the recent years [START_REF] Alsefri | Bayesian joint modelling of longitudinal and time to event data: a methodological review[END_REF][START_REF] Hickey | Joint Models of Longitudinal and Time-to-Event Data with More Than One Event Time Outcome: A Review[END_REF], most works are dedicated to classical continuous biomarkers stemmed from blood samples, MRI, etc. Yet, in an ever-increasing number of health studies, the actual quantity of interest is not directly measured. It is a latent construct which is assessed using a set of indicators measured with error, usually stemmed from questionnaires or measurement scales. Examples include health related quality of life in Cancer research and beyond [START_REF] Fayers | EORTC Quality of Life Group, Quality of Life Unit, Quality of life research within the EORTC-the EORTC QLQ-C30. European Organisation for Research and Treatment of Cancer[END_REF], cognitive functioning and functional dependency in neurodegenerative diseases [START_REF] Edjolo | Natural History of Dependency in the Elderly: A 24-Year Population-Based Study Using a Longitudinal Item Response Theory Model[END_REF][START_REF] Proust-Lima | Analysis of multivariate mixed longitudinal data: a flexible latent process approach[END_REF], or many other constructs such as anxiety or depressive symptomatology [START_REF] James | Exploring the psychometric properties of the CES-D-10 and its practicality in detecting depressive symptomatology in 27 low-and middle-income countries[END_REF][START_REF] Kruyen | Shortening the S-STAI: consequences for research and clinical practice[END_REF]. The analysis of latent constructs stemmed from measurement scales and questionnaires requires a specific attention. Measurement scales usually translate into multiple categorical and/or continuous items that measure different aspects of the underlying construct of interest. Furthermore, when aggregating the item information into a sumscore, the resulting univariate marker may not have classical Gaussian properties for continuous markers; they are usually bounded with floor/ceiling effects and possible unequal-interval scaling [START_REF] Proust-Lima | Analysis of multivariate mixed longitudinal data: a flexible latent process approach[END_REF][START_REF] Proust-Lima | Are latent variable models preferable to composite score approaches when assessing risk factors of change? Evaluation of type-I error and statistical power in longitudinal cognitive studies[END_REF].

In this contribution, we show how the joint modeling methodology can be extended to handle repeated data from measurement scales in the presence of an informative clinical event, and provide a software solution with the R package JLPM. We illustrate the methodology through simulations and in Multiple-System Atrophy (MSA), a rare and deadly neurodegenerative disease, which progression is almost exclusively described by measurement scales [START_REF] Foubert-Samier | Disease progression and prognostic factors in multiple system atrophy: A prospective cohort study[END_REF].

The article is organized as follows: Section 2 presents the methodology developed to handle previously mentioned challenges (multiple markers of different nature, competing risks, delayed entry), Section 3 reports simulation studies aiming at validating the inference procedure. Section 4 illustrates the methodology with the analysis of dysphagia progression in MSA, and Section 5 concludes.

Methodology

Let consider a cohort of N individuals followed up over time. We define T * i the time of occurrence of an event of interest for subject i (i = 1, ..., N ). This event may be due to P different causes, defining a competing risk setting. As some individuals may be censored before this occurrence, we observe T i = min(T * i , C i ) where C i is the censoring time, and δ i the event indicator such that δ i = p when the event of cause p occurs first and before censoring (T * i ≤ C i ), and δ i = 0 otherwise. We also collect repeated measures of K ≥ 1 markers measuring the same construct of interest (for instance K items of a measurement scale or only 1 isolated item, or 1 sumscore). The marker values are noted Y ikj for subject i (i = 1, ..., N ) and marker k (k = 1, ..., K) at time t ikj with j the repeated occasion (j = 1, ..., n ik ). The event of interest interrupts the collection of the repeated markers so that necessarily t ikn ik ≤ T i . In this methodology, the markers do not need to be measured at the same visit times across markers and across individuals.

The joint model aims to analyze the trajectory of the construct of interest over time and the risk of event by accounting for their correlation. As shown in Figure 1, the joint model includes two submodels, one for the longitudinal process (on the left) and one for the time-to-event process (on the right). They are detailed below.

[Figure 1 about here.]

Longitudinal model

When analyzing data from measurement scales or questionnaires, and more generally psychometric data, one makes the distinction between the construct of interest which is a latent process, denoted ∆ i (t) and defined in continuous time (t ∈ R), and its observations, denoted Y ikj , measured with error at discrete time visits t ikj . As emphasized in Figure 1, the longitudinal model thus consists of a structural model describing the latent process over time, and measurement models defining its link with each marker [START_REF] Proust-Lima | Analysis of multivariate mixed longitudinal data: a flexible latent process approach[END_REF].

Structural model

The trajectory over time of ∆ can be described using a linear mixed model [START_REF] Laird | Random-effects models for longitudinal data[END_REF] as follows :

∆ i (t) = X i (t) L β L + Z i (t) b i (1)
with X L i (t) and Z i (t) two vectors of covariates associated to β L the vector of fixed effect parameters, and b i the q-vector of normally distributed individual random effects (b i ∼ N (0, B)), respectively. We note that, as the quantity described here is not directly observed or measured, we do not consider any measurement error in the equation. The term X i (t) L β L defines the mean trajectory of the latent process at the population level while Z i (t) b i captures the individual deviation to the mean trajectory.

Measurement model

The measurement models define the nature of the link between the latent process ∆ and each marker, considered as a noisy measure of ∆, that is a marker of ∆ prone to measurement error. Depending on the study to be conducted and the data available, the markers may vary in number and nature (e.g., continuous, ordinal). We review here different measurement models according to the nature of the markers.

In any case, one measurement model is to be defined for each marker k ∈ {1, ..., K}.

Continuous Gaussian marker

Classically, continuous markers are considered as having a multivariate normal distribution with Gaussian errors. In this case, the measurement model for marker k is:

Y ik (t ikj ) = η k1 + η k2 × (∆ i (t ikj ) + ikj ) (2)
where η k1 and η k2 are two marker-k-specific parameters transforming the continuous marker into the latent process scale, and is the independent Gaussian measurement error ( ikj ∼ N (0, σ 2 k )).

In the case of one single Gaussian marker (K = 1), η k1 = 0 and η k2 = 1 reduces the equation to the standard definition of a linear mixed model:

Y i (t ij ) = ∆ i (t ij ) + ij = X i (t ij ) L β L + Z i (t ij ) b i + ij (3)

Continuous non-Gaussian marker

When analyzing data from measurement scales, the sum-score of all or a part of the items is frequently considered as a relevant marker of the underlying process of interest. This score is often treated as continuous because of its large number of possible levels. However its distribution is very rarely Gaussian due to ceiling and floor effects induced by its lower and upper bounds, and due to the fact that, as based on the sum of ordinal items, a one-point change does not necessarily have the same meaning depending on the level of the score. This phenomenon, called curvilinearity, is not compatible with the assumptions of the linear model and linear mixed model [START_REF] Proust-Lima | Misuse of the linear mixed model when evaluating risk factors of cognitive decline[END_REF][START_REF] Proust-Lima | Are latent variable models preferable to composite score approaches when assessing risk factors of change? Evaluation of type-I error and statistical power in longitudinal cognitive studies[END_REF]. In this context, a continuous but non-gaussian marker k can be described using a curvilinear measurement model where a parameterized transformation function

H k normalizes marker k: H k (Y ik (t ikj ); η k ) = ∆ i (t ikj ) + ikj (4)
with ikj ∼ N (0, σ 2 k ) and H k (.; η k ) the marker-k-specific parametric link function. This function is monotonically increasing and is often approximated by splines [START_REF] Proust-Lima | Analysis of multivariate mixed longitudinal data: a flexible latent process approach[END_REF]. We can notice that if H k (.) is a linear function, this model turns into the classical linear mixed model defined in the previous subsection.

Discrete marker

Ordinal items (including binary items as 2-level ordinal items) are frequently considered for measuring complex constructs. They can be described using proportional odds logistic models or cumulative probit models [START_REF] Commenges | Dynamical Biostatistical Models[END_REF][START_REF] Proust-Lima | Continuous-time modeling of selfreported outcome data: a dynamic Item Response Theory model[END_REF]. Such models are based on the assumption of increasing monotonicity indicating that a higher level of the item reflects a higher degradation of the latent dimension that it measures. This is formalized as follows:

Y ik (t ikj ) = m ⇔ η k,m < ∆ i (t ikj ) + ikj ≤ η k,m+1 with m ∈ {0, 1, ..., M k } and -∞ = η k,0 ≤ η k,1 ≤ ... ≤ η k,m ≤ η k,m+1 ≤ ... ≤ η k,M k +1 = +∞ (5)
where ikj is an additional noise either Logistic in the proportional odds logistic model or Gaussian in the cumulative probit [START_REF] Commenges | Dynamical Biostatistical Models[END_REF]. We consider here the latter with ikj ∼ N (0, σ 2 k ). Parameters η k = (η k,m ) m=1,...,M k are to be estimated. Also called thresholds or locations, they correspond to the level of the latent construct at which the probability of observing Y ik (t ikj ) lower/higher m + 1 is 0.5. These measurement models are usual in Item Response Theory (IRT) where they are called Graded Response Models (GRM) [START_REF] Baker | Item Response Theory | Parameter Estimation Techniques[END_REF].

Caution note

In the case of multiple markers, this framework relies on important assumptions:

• unidimensionnality of the latent construct. The model assumes that all markers should measure the same phenomenon, the latent dimension of interest. In the case of a multidimensional measurement scale, items should be grouped by unidimensional underlying dimensions, and different dimensions analyzed separately.

• inter-marker and intra-marker conditional independence given the latent construct. The latent construct should constitute the only source of shared information across pairs of markers and within repeated measures of the same marker. In the case of residual dependency across markers, redundant markers may be deleted or their correlation modeled in the measurement models. In the case of residual dependency within marker, additional random-effect specific to the marker and/or markerspecific association with covariate may be added [START_REF] Proust-Lima | Analysis of multivariate mixed longitudinal data: a flexible latent process approach[END_REF].

These assumptions need to be verified in preliminary analyses to prevent any bias in the results. For this purpose, we recommend to follow the PROMIS strategy [START_REF] Reeve | Psychometric evaluation and calibration of health-related quality of life item banks: plans for the Patient-Reported Outcomes Measurement Information System (PROMIS)[END_REF]. Note that intra-marker conditional independence given the latent construct may be assessed in sensitivity analyses.

Identifiability Constraints

Except for the specific case of equation ( 3), the longitudinal submodel, as defined in this section, is not identifiable. Due to the introduction of a latent process, identifiability constraints must be added to determine the dimension of the latent process; this is a requirement in any latent variable model. In this work, we chose to center and reduce the latent process ∆ at time 0 in the reference category. This translates into an intercept fixed to β L int = 0 (location constraint) and a variance of the first random-effect (usually the random intercept) fixed to B 1,1 = 1 (dispersion constraint) [START_REF] Commenges | Dynamical Biostatistical Models[END_REF]. This implies that the unit of the latent process corresponds to the residual inter-individual standard deviation of the latent quantity when time t = 0.

Survival model

The survival model describes the risk of occurrence of each cause of event, usually using a cause-specific proportional hazard model [START_REF] Commenges | Dynamical Biostatistical Models[END_REF]. In shared random effect joint models, the instantaneous risk function depends on a (possibly time-dependent) function of the elements of the structural model (1) noted g ip (b i , t). This element, which notably depends on the random effects b i , is added as a linear predictor in the survival model to quantify the intensity of the association between the latent dimension and the considered event p.

The cause-specific hazard model is described in continuous time t (∈ R) for each cause p = 1, ..., P and each subject i (∈ {1, ..., N }) as follows:

λ ip (t) = λ 0p (t; ψ p ) exp(X S pi β S p + g ip (b i , t) α p ) (6)
with λ 0p (.) the parametric baseline risk function associated to the vector of parameters ψ p for cause p. This function is usually chosen among Weibull hazards, piecewise constant hazards or approximated using spline functions. The vector of exogenous time-independent covariates X S p is associated with the vector of parameters β S p , and the function g ip (b i , t) is associated with the vector of parameters α p which quantifies the association between the two processes. Many different functions g ip (b i , t) can be specified [START_REF] Sène | Shared random-effect models for the joint analysis of longitudinal and time-to-event data: Application to the prediction of prostate cancer recurrence[END_REF]. We focus here on two frequently used structures encountered in the joint modeling literature [START_REF] Rizopoulos | Joint Models for Longitudinal and Time-to-Event Data[END_REF][START_REF] Henderson | Joint modelling of longitudinal measurements and event time data[END_REF] :

-the vector of random effects (dimension q): the survival model is adjusted on the individual deviations to the mean latent process trajectory with

g ip (b i , t) = g ip (b i ) = b i (7)
-the current latent process level (dimension 1): the instantaneous risk is adjusted on the level of the underlying process (defined in Equation ( 1)) at the same time with

g ip (b i , t) = ∆ i (t) (8)

Estimation

Let Θ denote the vector of all sub-model parameters including :

-for the structural model: vector β L of fixed effects, parameters of the random-effects variancecovariance matrix B;

-for the measurement model: for all marker k ∈ {1, ..., K}, variance σ 2 k of the measurement error and vector η k of parameters, it can be regression parameters η k = (η k1 , η k2 ) if marker k is Gaussian, link function parameters η k if marker k is continuous but not necessarily Gaussian, or thresholds

η k = (η k,m ) m∈{1,...,M k } if marker k is ordinal;
-for the survival model: for all cause p ∈ {1, ..., P }, vector ψ p of parameters in the baseline hazard function, vector of parameters β S p for all covariates included in the survival model, and α p the association parameters.

The joint model parameters Θ can be estimated by maximizing the log-likelihood

L(Θ) = N i=1 log L i (Θ)
where L i (Θ) is the individual contribution to the likelihood.

Individual contribution to the likelihood

Thanks to the assumption of independence between the latent dimension ∆ i , measured by the markers Y i = (Y ik ) k=1,...,K , and the time-to-event (T i , δ i ) conditionally to the random effects b i , the individual contribution to the likelihood for subject i ∈ {1, ..., N } can be developed as follows:

L i (Θ) = f Y i ,(T i ,δ i ) (Y i , (T i , δ i ); Θ) = R q K k=1 {f Y ik |b i (Y ik |b; Θ)}f (T i ,δ i )|b i ((T i , δ i )|b; Θ)f b i (b; Θ)db (9)
with f (.) the generic notation for a density function.

The density of the repeated marker depends on its nature. The density of a continuous marker can be expressed using the change of variable theorem:

f Y ik |b i (Y ik |b; Θ) = φ( Ỹik |b; Θ) × n ik j=1 J( Ỹikj |b) (10)
where Ỹk = H -1 k (Y k ) designates the value of marker Y k transformed by its link function. Functions φ(.) and J(.) are the density function of a normal distribution and the jacobian associated to the link function, respectively.

For a discrete marker, it can be expressed as follows:

f Y ik |b i (Y ik |b; Θ) = n ik j=1 M k m=0 P(Y ik (t ikj ) = m|b) 1 Y ik (t ikj )=m = n ik j=1 M k m=0 [Φ( 1 σ k (η k,m+1 -∆ i (t ikj ))) -Φ( 1 σ k (η k,m -∆ i (t ikj )))] 1 Y ik (t ikj )=m (11)
where Φ(.) is the standard normal distribution function.

The density of the time-to-event is:

f (T i ,δ i )|b i ((T i , δ i )|b; Θ) = S i (T i |b; Θ) p p=1 λ ip (T i |b; Θ) 1 δ i =p (12)
with λ ip (.) the hazard function specific to cause p given in Equation ( 6) and S i (.) the survival function combining all P causes:

S i (t|b; Θ) = P p=1 S ip (t|b; Θ) = P p=1 exp(- t 0 λ ip (s|b; Θ)ds) (13)
Finally, the density of the random effects is the density of a centered multivariate normal distribution:

f b i (b; Θ) = φ N (O,B) (b) (14)
In the presence of delayed entry, that is when a subject is included in the study some time after time zero, the time-to-event is left-truncated. This is accounted for by considering instead the individual contribution to the likelihood for truncated data L trunc,i (Θ) in which the naive individual contribution L i (Θ) is divided by the probability to survive until the entry time T 0i :

L trunc,i (Θ) = L i (Θ) S i (T 0i;Θ ) (15)

Software

Maximum likelihood estimation of the joint model parameters is implemented in the R package JLPM (https://github.com/VivianePhilipps/JLPM). The optimization of the log-likelihood is numerically car-ried out by a Marquardt-Levenberg algorithm with stringent convergence criteria based on the first and second derivatives of the log-likelihood (see [START_REF] Philipps | Robust and efficient optimization using a Marquardt-Levenberg algorithm with R package marqLevAlg[END_REF] for details). Example scripts are given in the package.

To reduce the computation time, optimization can be carried out in parallel mode (i.e., using multiple cores).

As with any complex model, we recommend to proceed step-by-step by estimating submodels with increasing complexity (e.g., survival submodel without dependency on the latent process, longitudinal model with a reduced number of random effects) to determine plausible initial values, and as such, reduce the number of iterations of the optimization algorithm applied to the final model.

The integral on the q random effects in Equation ( 9) does not have analytical solution so that the computation of the likelihood involves numerical integrations. This can be carried out with a Monte-Carlo algorithm which consists in simulating many draws of the random effects (generally around 100000) and to compute and then average the results of the function to integrate. In this package, we used instead a quasi Monte-Carlo algorithm [START_REF] Pan | Quasi-Monte Carlo estimation in generalized linear mixed models[END_REF]. This integration method uses a deterministic sequence rather than a random one. Then, smaller integration errors are obtained due to the low discrepancy of the chosen deterministic sequence, so that the number of required points is usually reduced to 1000.

Moreover, when the association structure g(b i , t) is time-dependent (not the case for the dependence on the random effects), the survival function in Equation ( 13) also involves an integral with no analytical solution; it is approximated by a Gauss-Kronrod gaussian quadrature with 15 points [START_REF] Gonnet | A review of error estimation in adaptive quadrature[END_REF].

Simulation studies

Two simulation studies were performed to illustrate the methodology and validate the estimation procedure implemented in the R-package JLPM according to the the type of markers and times-to-event included, and the nature of the dependency structure between the longitudinal and the survival submodels.

Simulation design

The first simulation study, described below, included 2 ordinal and 2 Gaussian markers measuring the same underlying process, 2 competing causes of event and a dependence on the shared random effects as in [START_REF] Rizopoulos | Joint Models for Longitudinal and Time-to-Event Data[END_REF]. The second one, reported only in supplementary materials, included 4 ordinal outcomes, 1 cause of event and a dependence on the current process level as in [START_REF] Henderson | Joint modelling of longitudinal measurements and event time data[END_REF].

We simulated 500 samples of 300 subjects each. The structural model for the underlying process consisted in a linear function of time at the population (β L = (0, 1) ) and individual level (B = 1 0 0 0.2 ). For each cause of event p (p = 1 or 2), the time-to-event was defined by a Weibull baseline risk function

λ 0p (t; ψ p ) = ψ ψ p2
p1 ψ p2 t ψ p2 -1 with ψ 11 = 0.2, ψ 12 = 5 for cause 1 and ψ 21 = 0.198, ψ 22 = 8 for cause 2. The unique linear predictor was the random effects of the underlying process with association parameter fixed to α 1 = (0.1, 0.2) for cause 1 and to α 2 = (0.3, 0.2) for cause 2.

Visit times were generated every year (or time unit) from year 0 and up to the minimum between year 4 (administrative censoring) and the time-to-event. A delayed entry was generated to translate an individual-specific time of entry (year 0). To do so, we generated individual time-of-entry from a continuous uniform distribution defined on the interval [0,2]. This lead to a time-of-entry mean around 1.00 (SD=0.58), 21.09% of censoring on average and 3.64 repeated measures of each marker on average. The 2 discrete markers had 4 ordinal levels each. Those marker data were generated according to equation [START_REF] Little | Modeling the Drop-Out Mechanism in Repeated-Measures Studies[END_REF] with thresholds η 1 = (0.5, 1, 1.5) for marker 1 and η 2 = (0.25, 0.75, 0.8) for marker 2. The 2 Gaussian marker data were generated according to equation (2) with parameters η 3 = (1, 0.4) for marker 3 and η 4 = (2, 0.2) for marker 4. Measurement error variances were fixed to σ = 1 for all markers.

Results

We report in Table 1 the mean estimate, relative bias, variance estimate (empirical or asymptotic) and the coverage rate of the 95% confidence interval for each parameter. The estimation procedure provides very good results on this example: for all parameters, the bias is negligible, the mean asymptotic variance is close to the empirical variance, and the coverage rate of the 95% confidence interval is very close to the nominal value.

[Table 1 about here.]

Illustration in Multi-System Atrophy

We illustrate our methodology to the Multiple-System Atrophy (MSA), a rare neurodegenerative disease characterized by various combinations of parkinsonism, cerebellar ataxia and dysautonomic symptoms.

The disease progresses very fast and is fatal with a median survival between 8 and 10 years after the first symptoms onset [START_REF] Foubert-Samier | Disease progression and prognostic factors in multiple system atrophy: A prospective cohort study[END_REF]. The occurrence of death suddenly interrupts the follow-up of MSA patients who are usually the most affected. This constitutes an informative dropout that needs to be accounted for in the statistical analyses to avoid biasing the model estimates.

As many other neurodegenerative diseases, MSA clinical progression is studied almost exclusively using scales that measure the motor degradation, the dysautonomic dysfunction or the pathology-related quality-of-life. In particular, the Unified Multiple-System Atrophy Rating Scale (UMSARS) assesses disease severity in MSA patients in 4 sub-scales. The first one, reported by the patient, assesses functional impairments. In this illustration, we focus essentially on one item, the 2 nd item of the first UMSARS subscale, which measures dysphagia, the difficulty to swallow food and saliva. Clinical assumptions are that, among the motor and functional impairments of MSA, dysphagia could represent an important aspect of the prognosis and have a substantial impact on the risk of death. This ability progressively deteriorates during the course of the disease and most severe cases require nasogastric tube or gastrostomy feeding. Dysphagia is measured as a 5-level likert scale (from 0 Normal state to 4 Severe incapacity) leading to repeated ordinal data.

Finally, the MSA diagnosis can be made several years after the first symptoms since the first MSA symptoms may not be specific (e.g., some are similar to those a Parkinson's disease). This delayed diagnosis yields to a delayed entry in the cohort. By accounting for these three challenges (ordinal repeated data, informative dropout and delayed entry), our methodology is particularly useful to study MSA progression.

Multiple-System Atrophy French cohort

Since 2007, the university hospitals of Bordeaux and Toulouse, the two French reference centres for MSA, have constituted the MSA French cohort which includes all the patients diagnosed with a MSA, and follows them every year with a complete clinical examination including the UMSARS completion. For this application, we considered all the MSA cases included in the cohort who had at least one completed dysphagia item during the follow-up before the administrative censoring on December 31 st 2019. A total of 634 MSA patients were included with a total of 1819 dysphagia assessments (see description in Table 2). Patients were included in the cohort 4.58 (SD=2.61) years on average after their first symptoms and were 65.1 (SD=8.2) years old in mean at entry. Two subtypes of MSA are distinguished: MSA-P for predominant parkinsonism and MSA-C for predominant cerebellar impairment. MSA-P (66.1%) were more frequent than MSA-C in the cohort (33.9%) as reported in Caucasian cohort [START_REF] Wenning | European Multiple System Atrophy Study Group, The natural history of multiple system atrophy: a prospective European cohort study[END_REF]. As for many neurodegenerative diseases, the definite diagnosis of MSA is based on post-mortem neuropathologies. In clinical practice, the MSA diagnosis is therefore based on criteria established in 2008 by Gilman et al. [START_REF] Gilman | Second consensus statement on the diagnosis of multiple system atrophy[END_REF] and giving two degrees of diagnostic certainty: probable and possible. At first visit, 151 (23.8%) and 483 (76.2%) met consensus for possible MSA and probable MSA, respectively. Patients had in mean 2.87 (SD=2.09) clinical visits, and remained in the cohort up to 6.94 (SD=3.34) years after first symptoms onset on average. They entered the cohort about 4.58 (SD=2.61) years after first symptoms onset. More than half of the patients (51.89%) died and 17.82% were not seen in the 18 months preceding administrative censoring. As shown in Figure 2 (bottom right), the survival probability dropped very quickly from approximately 2 years after the first symptoms onset reaching a survival probability of 0.19 [0.16 -0.24] after 10 years, 0.04 [0.02 -0.06] after 15 years and 0 after 20 years. This illustrates the importance of the truncation of the follow-up by the occurrence of death in MSA. Previous analyses suggest that this truncation is related to the course of the disease [START_REF] Foubert-Samier | Disease progression and prognostic factors in multiple system atrophy: A prospective cohort study[END_REF].

Regarding dysphagia, at entry, 180 (28.39%) patients had a normal state (no particular difficulty to swallow), 242 (38.17%) a mild impairment (choking less than once a week), 139 (21.92%) a moderate impairment (occasional food aspiration with choking more than once a week), 62 (9.78%) a marked impairment (frequent food aspiration) and 6 (0.95%) a severe state requiring assistance to be feeded.

Note that even if the severe state is under-represented at entry, the frequency increases to 31 during the follow-up so that it does not pose any problem of sparse category in the estimation process (in a case where a category would not be represented at all, it could be grouped with another one). Dysphagia item degradation substantially progressed over disease time as illustrated with the individual observed trajectories (Figure 2 -bottom left).

[Table 2 about here.]

Specification of the model

This study aimed at better understanding progression of swallowing difficulties over disease time and at quantifying its association with death. To do so, we built a shared random effect joint model as illustrated in Figure 2, and considered time since first symptoms as the study timescale to describe the natural history of the item progression. The model handles delayed entry due to the fact that patients may enter the study years after the first symptoms occurred.

[Figure 2

about here.]

The observed 5-level item probability was linked to the underlying continuous dysphagia process at the exact same time by a cumulative probit model (cf. Equation ( 5)). The trajectory of the underlying dysphagia process was simultaneously described over time by a latent linear mixed model (cf. Equation ( 1)) with individual random effects on each function of time considered to take into account the within patient correlation. After preliminary analyses comparing a linear trajectory, a quadratic trajectory and a trajectory approximated by natural cubic splines with 1 internal knot, we selected the quadratic trajectory as the one providing the best fit to the data according to the Akaike Information Criterion (AIC) (AIC=5944.53, AIC=5902.23, AIC=6021.21 for the models with linear, quadratic and splines trajectories, respectively). The association between dysphagia and death was simultaneously modelled using a survival model where the instantaneous risk of death was a function of the current underlying level of dysphagia (cf. Equation ( 6) with option ( 8)). The baseline hazard function was defined by cubic M-splines with 3 internal knots placed at the quantiles of the observed times of event, that is 4.65, 6.37, 8.71 years. The boundary knots were placed at 0 and 24 years. The structural longitudinal model and the survival model were adjusted for sex (male or female), diagnosis (MSA-C or MSA-P), certainty of the diagnosis (possible or probable) and age at the first symptoms onset.

Results

The estimates of the joint model are provided in supplementary table. The predicted trajectories of dysphagia item are displayed in Figure 3 according to profiles of patient differing by the four main covariates of interest (sex, age at first symptoms onset, MSA subtype and certainty degree of diagnosis).

The reference profile corresponded to a male patient, 60 years old at first symptoms onset, diagnosed MSA-C with possible certainty (black line).

Overall, the trajectories of dysphagia did not significantly differ according to sex, age and MSA subtype. Male and female patients seemed to endure the same dysphagia progression over at least the first 5 years of the disease. Female patients seemed to face a more important degradation afterwards although the dysphagia progression was not statistically different by sex (interaction on time and quadratic time, p=0.624). Dysphagia evolved slighly faster for older patients with substantial differences during the followup. At 5 years of disease, the dysphagia level was 1.662 (95%CI=1.400,1.956), 1.419 (95%CI=1.184,1.671) and 1.173 (95%CI=0.943,1.445) for patients aged 70, 60 and 50 at the beginning of the disease, respectively. This highlights an approximate gap of 0.5 points between patients with 20 years difference, stable between 5 and 10 years of disease.

No difference in dysphagia progression was observed between MSA-C and MSA-P patients. At the beginning of the disease, MSA-P patients seemed to have a little higher level of dysphagia than MSA-C patients, but after 6 years of disease, this trend reversed and MSA-C patients surpassed MSA-P patients in terms of degradation degree. This could be explained by the fact that MSA-P patients are initially more affected on the motor level as opposed to MSA-C patients who suffer from non-motor and dysautonomia symptoms [START_REF] Foubert-Samier | Disease progression and prognostic factors in multiple system atrophy: A prospective cohort study[END_REF].

Trajectories of dysphagia substantially differed according to the diagnosis certainty. Patients diagnosed with possible MSA had a significantly higher level of dysphagia at the beginning of the disease compared to patients diagnosed with more confidence (probable) with a mean level of 0.600 (95%CI=0.344,0.986) and 0.334 (95%CI=0.192,0.542) at onset for possible and probable, respectively. However, patients with probable MSA progressed much more rapidly reaching a mean level of 3.034 (95%CI= 2.711,3.272) after only 10 years while possible MSA patients still had in mean 2.377 (95%CI=1.995,2.722).

[Figure 3 about here.] The trajectory of dysphagia was also substantially associated with the risk of death through its current underlying level (HR = 3.773, 95%CI =2.588,5.501) after adjustment on sex, age at first symptoms onset, MSA subtype and certainty degree of diagnosis (Table 3, right column). This means that the level of dysphagia at a certain time t is strongly associated with the instantaneous risk of death at the exact same time: the higher the level of dysphagia, the higher the risk of death. A second model was estimated without accounting for dysphagia progression (i.e., fixing its association parameter to α = 0) to assess whether some covariate association with death were changed when considering the association between dysphagia and death. Results are reported in Table 3 (left-column). The association with age at first symptoms was not impacted by the adjustment on dysphagia (HR = 1.215, 95%CI = 1.063,1.388 for a 10-year difference and HR = 1.160, 95%CI = 1.007,1.337 for the model without and with dysphagia, respectively). This suggests that the effect of age on the risk of death in this pathology may be independent of the level of clinical progression. Older patients at first symptoms have a higher risk of death. As well, with or without adjustment for dysphagia, there was no evidence of difference of death risk according to MSA diagnosis subtype (HR = 1.081, 95%CI = 0.857,1.363 without dysphagia, HR = 1.132, 95%CI = 0.889,1.441 with dysphagia). Furthermore, the higher risk of death of probable cases was slightly attenuated when accounting for the level of dysphagia impairment (from HR = 1.864, 95%CI = 1.398,2.485 without dysphagia to HR = 1.376, 95%CI = 1.026,1.844) with adjustment for. This was expected as we saw that probable patients had a significantly higher level of dysphagia than possible patients (Figure 3).

Finally, the association between sex and risk of death was not substantially changed when adjusting for the underlying level of dysphagia (HR = 0.887, 95%CI = 0.713,1.103 without adjustment to HR = 0.827, 95%CI = 0.660,1.036 with adjustment for dysphagia).

[Table 3 about here.]

Goodness-of-fit assessment

The goodness-of-fit of the final model was investigated using two graphical tools:

• for the survival submodel, we generated 500 samples from the fitted joint model and compared the distribution of the corresponding predicted survival curves with the one observed in the original sample. As shown in Figure S1 in supplementary materials, the survival curves predicted by the model were very close to the observed one, showing how the joint model adequately fitted the time-to-event data.

• for the longitudinal submodel, we computed the predicted item values derived from the estimated joint model, and compared the mean trajectory of these predictions with the mean trajectory of the observations by splitting time into 1-year intervals. The predicted item values were obtained by (i) predicting the patient-specific vector of random effect as the mode of the posterior distribution of the random effects given the patient observations, and (ii) computing the expected item level according to the predicted patient-specific random-effect. As shown in Figure S2 in supplementary materials, the predicted item mean trajectory was close to the observations, illustrating the adequacy of the estimated model to the longitudinal data.

Concluding remarks

The joint modelling of longitudinal data and time-to-event data has become a standard methodology to address the problem of informative dropout and more generally investigate the association between longitudinal markers and clinical events [START_REF] Rizopoulos | Introduction to the special issue on joint modelling techniques[END_REF]. In this work, we have shown how this methodology could be extended to longitudinal data of different natures by separating the longitudinal model for the quantity of interest defined as a latent process from the measurement model that links the quantity of interest to its repeated observations. This model thus naturally handles univariate or multivariate markers of the same underlying quantity but also continuous (Gaussian or not) or discrete markers. Specific examples include the case of a unique continuous Gaussian marker which corresponds to the classical joint model framework [START_REF] Rizopoulos | Joint Models for Longitudinal and Time-to-Event Data[END_REF] or the case of various ordinal items measuring the same underlying construct. In the latter case, the longitudinal submodel is a dynamic Item Response Theory (IRT) submodel (see [START_REF] Proust-Lima | Continuous-time modeling of selfreported outcome data: a dynamic Item Response Theory model[END_REF][START_REF] Barbieri | Méthodes longitudinales pour l'analyse de la qualité de vie relative à la santé en cancérologie[END_REF] in this special issue). Although not of importance in the application to MSA, the methodology also handles competing risks of events, a situation which is often encountered in cohorts when different types of clinical progression and/or dropout may occur (e.g., types of recurrence in cancer research, multiple causes of death).

All joint models assume a given structure of correlation between the longitudinal data and the timeto-event data. In the literature, two main structure types can be found, the shared random-effect models which consider that the random effects from the longitudinal model capture all the correlation between the longitudinal process and the time-to-event process, and the joint latent class models which assume that a latent group structure captures all the correlation between the two processes [START_REF] Proust-Lima | Joint latent class models for longitudinal and time-to-event data: a review[END_REF]. In this work, we chose to focus on the extension of the shared random effect joint model which enables an explicit quantification of the association with hazard ratios, as illustrated in MSA, and which is more often used in practice, and has been usually favored for treating informative dropout [START_REF] Thomadakis | Longitudinal and time-to-drop-out joint models can lead to seriously biased estimates when the drop-out mechanism is at random[END_REF][START_REF] Dantan | Pattern mixture models and latent class models for the analysis of multivariate longitudinal data with informative dropouts[END_REF]. We refer to Proust-Lima et al. [START_REF] Proust-Lima | Joint latent class models for longitudinal and time-to-event data: a review[END_REF] and R package lcmm [START_REF] Proust-Lima | Estimation of extended mixed models using latent classes and latent processes: the R package lcmm[END_REF] for further details on joint latent class models incorporating multivariate outcomes measuring the same underlying process.

Within the framework of shared random effect joint models, we considered a dependence structure on the current level of the latent process or on the individual random-effects of the structural longitudinal model. However, other dependence structures could also be worth exploring such as the current slope of the latent process, an accumulation of the latent process over time [START_REF] Andrinopoulou | Bayesian shrinkage approach for a joint model of longitudinal and survival outcomes assuming different association structures[END_REF] or nonlinear functions of them [START_REF] Sène | Shared random-effect models for the joint analysis of longitudinal and time-to-event data: Application to the prediction of prostate cancer recurrence[END_REF].

The choice of the dependence structure depends on the application framework and is part of the model specification and building. The R-package JLPM is still under development to provide more options for the dependency structure between the two processes.

For simplicity, we described the methodology of the latent process model under the assumption of measurement invariance. A lack of measurement invariance can occur when markers present a different functioning according to a specific covariate (called Differential Item Functioning in IRT literature) or according to time (called response shift in IRT literature). Such lack of invariance can be explored with our methodology and is implemented in the software solution. We refer to another paper in this special issue for further details [START_REF] Proust-Lima | Modeling repeated self-reported outcome data: a continuous-time longitudinal Item Response Theory model[END_REF].

To conclude, measurement scales and questionnaires become more and more central in health studies to measure the physical, mental or psychological state of patients. With this joint model implemented in the R-package JLPM, we provide a new and relevant solution to study their trajectories over time and their association with events of interest.

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Schematic diagram of the joint model structure for K repeated markers measuring the same construct and a time of event (possibly of multiple causes)

Table S 1

 S : Summary on 500 replicates of the estimation of a joint model with 4 repeated ordinal markers and 1 event with latent process current level dependency structure on samples of 300 individuals.

	parameters	true	mean	relative empirical mean asymptotic	95%CI
		value estimate	bias	standard	standard	coverage
				(in %) deviation	deviation	rate (in %)
	survival model
	baseline risk function (Weibull) scale √ ψ 1 0.447 shape √ ψ 2 2.236	0.448 2.252	0.1 0.7	0.008 0.069	0.008 0.070	93.6 95.2
	association parameter, α 0.100	0.099	-1.2	0.044	0.044	93.8
	structural model
	adjustment covariates, β L
	intercept	1.000	-	-	-	-	-
	time	1.000	1.023	2.3	0.137	0.131	93.8
	random effect covariance parameters, B
	choleski 0	1.000	-	-	-	-	-
	choleski 1	0.000	0.013	-	0.078	0.076	92.6
	choleski 2	0.447	0.445	-0.6	0.073	0.068	90.6
	outcome-specific measurement model
	thresholds for items
	item 1
	η 11	0.500	0.516	3.3	0.138	0.130	94.4
	η 12 *	0.707	0.712	0.7	0.053	0.052	94.6
	η 13 *	0.707	0.711	0.6	0.054	0.051	94.0
	item 2
	η 21	0.250	0.262	4.8	0.126	0.124	95.0
	η 22 *	0.707	0.714	0.9	0.053	0.052	94.4
	η 23 *	0.224	0.220	-1.5	0.041	0.041	94.0
	item 3
	η 31	0.100	0.111	10.6	0.126	0.125	95.0
	η 32 *	0.316	0.320	1.1	0.047	0.046	94.2
	η 33 *	0.447	0.449	0.3	0.049	0.047	93.2
	item 4
	η 41	0.200	0.206	3.0	0.126	0.123	93.6
	η 42 *	0.447	0.454	1.6	0.046	0.047	94.0
	η 43 *	0.632	0.638	0.8	0.051	0.050	95.0
	standard deviation from measurement error, σ
	item 1	1.000	1.017	1.7	0.132	0.126	93.8
	item 2	1.000	1.018	1.8	0.132	0.125	93.4
	item 3	1.000	1.016	1.6	0.136	0.128	93.0
	item 4	1.000	1.024	2.4	0.125	0.126	95.0
		* squared root of the increment

Table S 2

 S : Estimates of the shared random effect joint model in the MSA application to describe dysphagia progression (5-level ordinal item) and its strength association with death (more details about the model specification can be found in section 4.2).

	parameters	estimate standard error p-value
	survival model
	baseline risk function, ψ
	splines 1	0.000	0.053	1.00
	splines 2	-0.090	0.149	0.54
	splines 3	0.262	0.062	<0.001
	splines 4	0.464	0.097	<0.001
	splines 5	-0.385	0.133	<0.001
	splines 6	0.261	0.363	0.47
	splines 7	0.389	0.337	0.25
	linear predictors, β S
	sex, female	-0.190	0.115	0.10
	age	0.269	0.072	0.04
	diagnosis, MSA-P	0.124	0.123	0.31
	certainty, probable	0.319	0.150	0.03
	association α	1.328	0.192	<0.001
	structural model
	adjustment covariates, β L
	intercept	0	-	-
	time	0.214	0.062	<0.001
	time 2	-0.005	0.004	0.19
	sex, female	-0.076	0.152	0.62
	age	0.073	0.111	0.51
	diagnosis, MSA-P	0.199	0.157	0.20
	certainty, probable	-0.421	0.216	0.05
	time : sex, female	0.021	0.045	0.64
	time : age	0.031	0.033	0.35
	time : diagnosis, MSA-P	-0.034	0.046	0.47
	time : certainty, probable	0.155	0.064	0.01
	time 2 : sex, female	0.000	0.003	0.97
	time 2 : age	-0.002	0.002	0.23
	time 2 : diagnosis, MSA-P	0.000	0.003	0.97
	time 2 : certainty, probable	-0.006	0.004	0.11
	random effect covariance parameters, B
	choleski 0	1	-	-
	choleski 1	-0.242	0.018	<0.001
	choleski 2	0.091	0.020	<0.001
	choleski 3	0.009	0.001	<0.001
	choleski 4	-0.004	0.002	<0.001
	choleski 5	0.000	0.001	0.15
	outcome-specific measurement model
	thresholds for the item
	η 1	0.402	0.215	0.06
	η 2 *	0.788	0.040	<0.001
	η 3 *	0.761	0.040	<0.001
	η 4 *	0.872	0.051	<0.001
	standard deviation from measurement error
	σ	0.343	0.038	<0.001
	* squared root of the increment

Table 1 :

 1 Summary on 500 replicates of the estimation of a joint model with 4 markers (2 ordinal and 2 Gaussian) and 2 competing events with shared random effect dependency structure on samples of 300 individuals.

	parameters	true	mean	relative empirical mean asymptotic	95%CI
		value estimate	bias	standard	standard	coverage
				(in %) deviation	deviation	rate (in %)
	survival model
	baseline risk functions(Weibull)
	cause 1 scale shape √ √ ψ 11 ψ 12 2.236 0.447	0.447 2.249	0.0 0.6	0.004 0.087	0.004 0.085	94.0 94.8
	cause 2 scale shape √ √ ψ 21 ψ 22 2.828 0.445	0.445 2.856	0.0 1.0	0.003 0.109	0.003 0.112	94.4 95.0
	association parameters
	cause 1
	α 1,intercept	0.100	0.112	12.0	0.111	0.116	96.0
	α 1,slope	0.200	0.183	-8.4	0.276	0.258	93.4
	cause 2
	α 2,intercept	0.300	0.318	6.0	0.143	0.140	95.8
	α 2,slope	0.200	0.200	0.0	0.302	0.294	95.6
	structural model
	adjustment covariates, β L
	intercept	0.000	-	-	-	-	-
	time	1.000	1.005	0.5	0.079	0.078	93.8
	random effect covariance parameters, B
	choleski 0	1.000	-	-	-	-	-
	choleski 1	0.000	0.007	-	0.047	0.039	87.4
	choleski 2	0.447	0.449	0.4	0.044	0.039	92.0
	outcome-specific measurement model
	thresholds for discrete markers
	marker 1
	η 1,1	0.500	0.496	-0.9	0.110	0.104	93.2
	η 1,2 *	0.707	0.708	0.1	0.042	0.042	95.4
	η 1,3 *	0.707	0.711	0.5	0.041	0.041	94.6
	marker 2
	η 2,1	0.250	0.250	-0.1	0.112	0.104	93.8
	η 2,2 *	0.707	0.705	-0.2	0.044	0.044	94.0
	η 2,3 *	0.224	0.221	-1.3	0.038	0.038	94.4
	parameters for Gaussian markers
	marker 3
	η 3,1	1.000	0.101	0.1	0.035	0.034	94.6
	η 3,2	0.400	0.400	-0.1	0.028	0.028	93.8
	marker 4
	η 4,1	2.000	2.001	0.0	0.017	0.017	95.2
	η 4,2	0.200	0.200	-0.1	0.014	0.014	94.0
	standard deviation from measurement error, σ
	marker 1	1.000	1.006	0.6	0.084	0.085	95.2
	marker 2	1.000	0.998	-0.2	0.090	0.090	94.2
	marker 3	1.000	1.008	0.8	0.078	0.077	93.4
	marker 4	1.000	1.005	0.5	0.077	0.077	95.0
			* squared root of the increment

Table 2 :

 2 Description of the MSA sample at entry and over time (N=634)

	Characteristic

Table 3 :

 3 Hazard ratios for the risk of death in proportional hazard models when accounting or not for the progression of dysphagia over time (N=634)

	Hazard ratios [95%CI]

Funding This work was funded by the French National Research Agency 455 (Project DyMES -ANR-18-C36-0004-01).

Abbreviations

Simulation design

We considered the setting of the time to a unique cause of event, and the repeated measures to 4 ordinal markers measuring the same underlying construct. The longitudinal part constitutes a dynamic Item Response Theory model for graded responses (see [START_REF] Proust-Lima | Continuous-time modeling of selfreported outcome data: a dynamic Item Response Theory model[END_REF] in the same special issue). The longitudinal and survival processes were associated through the current latent process level (i.e., dependence structure defined in Equation ( 8)).

We simulated 500 samples of 300 subjects each. The structural model for the underlying process consisted in a linear function of time at the population (β L = (0, 1) ) and individual level (B = 1 0 0 0.2 ). The time-to-event was defined by a Weibull baseline risk function λ 0 (t; ψ) = ψ ψ 2 1 ψ 2 t ψ 2 -1 with ψ 1 = 0.2, ψ 2 = 5. The unique linear predictor was the current level of the underlying process with an association parameter fixed to α = 0.1.

Visit times were generated every year (or time unit) from year 0 and up to the minimum between year 4 (administrative censoring) and the time-to-event. A delayed entry was generated to translate an individual-specific time of entry (year 0). To do so, we generated individual time-of-entry from a continuous uniform distribution defined on the interval [0,2]. This lead to a time-of-entry mean around 1.00 (SD=0.58), 26.07% of censoring on average and 3.45 repeated measures of each marker on average. The 4 markers had 4 ordinal levels each. The marker data were generated according to equation ( 5) with thresholds η 1 = (0.5, 1, 1.5) , η 2 = (0.25, 0.75, 0.8) , η 3 = (0.1, 0.2, 0.4) and η 4 = (0.2, 0.4, 0.8) and measurement error variances fixed to σ = 1.

Results

We report in Table S1 the mean estimate, relative bias, variance estimate (empirical or asymptotic) and the coverage rate of the 95% confidence interval for each parameter. The estimation procedure provides very good results on this example: for all parameters, the bias is negligible, the mean asymptotic variance is close to the empirical variance, and the coverate rate of the 95% confidence interval is very close to the nominal value.

Additional figures and tables for the MSA application


EUROPE AND THE UNITED STATES: HOW PUTIN REJUVENATED NATO AND "THE WEST" BY GIVING BOTH A COMMON PURPOSE

Simon Reich

Dossier « Guerre en Ukraine », Les Dossiers du CERI, avril 2022 Prior to his appointment as Director of the CIA, William Burns wrote a short piece in which he laid out America's choices when it came to the Biden administration's foreign policy. He characterised three options: restoration, retrenchment or reinvention. Restoration entailed trying to buttress the major foundations of the multilateral Liberal international order, with the United States in a familiar role: the unquestioned "first among equals" as the leader of the "free world." Retrenchment entailed America's strategic military withdrawal along the lines first mooted by Barack Obama and then emphatically endorsed and pursued by Donald Trump. Burns, however, prescribed reinvention: America should completely rethink its strategic principles, priorities and forms of engagement in a rapidly evolving global environment 1 .

American choices and European responses

In the first year of his presidency, the Biden administration, in contrast to Burns' suggestion, chose a combination of restoration and retrenchment. In practice, this meant an awkward coupling of a reassertion of American global leadership and a recommitment to global institutions and treaties (such as the WHO and COP 21) with a chaotic American withdrawal from Afghanistan in August 2021 and a signalling that it was far more interested in the Indo-Pacific and China than Europe and Russia. These choices had major repercussions for Europe. Many Europeans may have been more relieved than enthralled by Biden's election. Trump's presidency had presented unprecedented challenges, and Biden offered the prospect of relief from major European states being depicted as rivals and free riders. So, they initially welcomed America's return to multilateralism. Nonetheless, both the EU's leadership and those of individual member states worried that their interests would be ignored. Their short-term concerns focused on whether the new administration would treat both the European members of NATO and the EU itself with greater respect, as partners and not subordinates in promoting human rights, free trade and security. The longer-term consideration was whether the Biden administration would simply be an interregnum before the return of Donald Trump or one of his protégés. They therefore were (and today remain) concerned about the reliability of the United States as an ally 2 .

Much of their shorter-term apprehension was justified prior to the outbreak of the Ukraine War. The summits to promote democracy, as part of what Biden characterised as a global conflict between democracy and autocracy, for example, were rhetorically supported by allies. But they resulted in few concrete initiatives. There has been no discernible progress on reestablishing a nuclear agreement with Iran, the unilateral American withdrawal from Afghanistan proved to be an operational nightmare for NATO's European members, and there was initially some very public transatlantic squabbling over the future of the Nord Stream 2 pipeline. Then the announcement of the creation of Aukus (the nuclear submarine agreement between Australia, the United Kingdom and the US), and the indelicate notification of the cancellation of the submarine contract with France only two-hours before that announcement, riled French officials. They very publicly expressed their anger. Foreign Minister Le Drian described it as "a stab in the back." EU Commissioner Thierry Breton (another Frenchman) suggested that trust in the US was eroding and "something is broken in our transatlantic relations… Trust is not a given. And after the latest events, there is, I should say, a strong perception that trust between the EU and United States has been eroded." 3 Gérard Araud, who served as French ambassador to the United States between 2014 and 2019, added that "You don't do that to an ally," that French officials should "never underestimate the incompetence of the U.S. administration" and, most tellingly and undiplomatically perhaps, "You know, Obama didn't care about Europe. Trump was hostile to Europe. And here you have Biden. And Biden, to be frank so far, is a big disappointment all over Europe…. there is no European policy. The administration has no European policy because its strategy is so focused on China." 4 To make matters worse, the Aukus announcement coincided with the EU's own announcement of its Indo-Pacific strategy, diminishing the latter's fanfare 5 . Together these developments helped strengthen one longstanding European fear: that it was being marginalised in the Indo-Pacific, the world's most economically dynamic region, despite French assertions that it is a Pacific power because of its overseas territories there. American behaviour, they believed, was consistent with the US' preference that there be a division of labour-with an American security focus on the Indo-Pacific and a European one eastward towards Russia. Europeans, led by France, disagreed. They want a major role in the Indo-Pacific as well. A subsequent belated diplomatic flurry by the US, principally the much-touted establishment of an EU-US dialogue on the Indo-Pacific, was a seeming balm to reassure Europeans worried about being ignored in the region. But that appeared to do little to calm the concerns about Europe's leadership.

One effect of all these areas of tension and missteps was to enhance an increasingly jaundiced European view of a prospective return to US global leadership 6 . Their concerns about whether the Americans were trustworthy, concerns that had carried over from the Trump administration's policies, spread beyond security to other policy domains. These included trade (where the Biden administration was slow to lift Trump-era tariffs) and the flow of people across the Atlantic (where COVID restrictions on European travellers remained in place even after reciprocal constraints had been rescinded by EU member states).

A second predictable effect was to amplify calls in Brussels and Paris for greater European strategic autonomy, an initiative that would potentially undermine US efforts at re-establishing its global leadership 7 . By the opening months of 2022, that concern was exacerbated by questioning European reactions to American intelligence estimates about a military build-up on Ukraine's border. Even American warnings of an impending Russian invasion was not enough to generate an immediate consensus among Europe's NATO members about a collective response 8 . Newlyminted German Chancellor Olaf Scholtz's refusal to commit to a suspension of the opening of the Nord Stream 2 pipeline should Russia invade, even when standing next to Joe Biden at a White House press conference, epitomised the underlying tension between the United States and many of its major European NATO partners 9 . Conversely, American frustration at what they considered the usual European incapacity to unite predictably grew.

But then came the Ukraine war. It changed some things, but not all things, in the transatlantic relationship.

The Ukraine war-and its major consequences Understood from this perspective, Vladimir Putin's invasion shocked European complacency about the prospects for a large-scale war in Europe. Indeed, Julia Ioffe described it as "Europe's 9/11 moment. 10 " There has subsequently been the inevitable speculation about whether the war signifies a fundamental change in the global system, such as towards US bipolarity with China or a multipolarity that includes Russia and the EU. Plenty of academics, commentators and politicians on both sides of the Atlantic have suggested that it is a watershed or what the Germans have labelled a Zeitenwende (a turning point where everything will change, irreparably), and some have drawn parallels with a prior epoch by labelling it "Cold War 2.0." 11 Others, like former US ambassador to NATO Ivo H. Daalder, have invoked the same logic, calling for a return to the familiar strategy of Containment to quell Russian revanchism 12 .

That kind of speculation and prescription is understandable. It achieves its goal of attracting widespread attention. But it may be premature. There are, however, at least four consequences that are reasonably evident, even at this early stage of the war. The first is that Putin has single-handedly revived the concept of "the West." Long since decried as vacuous in the face of transatlantic disagreements and intra-European fissures, many (particularly Asian) observers had proclaimed its demise-both as a meaningful collective concept and as a collective political and economic powerhouse 13 . But the evident shared declaration of purpose and unity of action, albeit after a hesitant start, suggests that the notion of the West retains some utility. In that vein, so does NATO. An organisation whose mission was mocked by Donald Trump as "obsolete" and which was described by Emmanuel Macron as "brain-dead" has now been invigorated, finding a renewed objective-to defend democracy and European sovereignty. The accelerated recommitment of its members to spend the requisite 2% of GDP on defence budgets faces a series of domestic budgetary and operational complications. But the war has nevertheless given fresh impetus to an organisation feared moribund, with Finland and Sweden now contemplating joining as part of an expanded membership. Furthermore, the West's operational resuscitation has surely given China's leadership pause for thought as it contemplates its future strategy in dealing with a transatlantic alliance composed of its two largest trading partnerspartners that seemed so divided on the fundamentals until March of 2022.

Of course, these observations must be qualified with a question about how long a consensus will hold. Viktor Orban's comments in his recent electoral victory speech (in which he decried some member states and the EU's leadership as "opponents") is the first sign that there may be some disagreement among Europeans moving forward. The same is possibly true of the disagreement between Macron and Biden about whether Russian behaviour in Ukraine constitutes a "genocide"-should the idea of an international war crimes trial come to fruition. Clearly, any disunity among EU members could test American patience with its European partners and disrupt efforts at sustaining a transatlantic accord. But at the current time, NATO's consensus regarding sanctions makes Putin's initial effort to drive a wedge between the US and Europe appear self-defeating.

As a corollary to that first consequence, a second is that Putin has inadvertently done much to rehabilitate Biden's efforts to restore the United States' image as the "leader of the free world." Some trust in Biden has been restored. European doubts about Biden's possible successor remain, but have been muted by his administration's willingness to provide over $3 billion in military funding to Ukraine by early April (in comparison to the EU's EURO 500 military support by that time). To this should be added Biden's successful efforts, not to be overstated, in helping the EU to establish a consensus about both economic sanctions and the fate of the Nord Stream 2 pipeline. Certainly, enough loopholes remain in the sanction regime to mitigate their effects on the Russian economy because of Europe's oil and gas purchases. But it would be churlish not to acknowledge Biden's personal success in diplomatically coalescing Europe's leaders towards a position that has, paradoxically, made those leaders themselves look stronger because of their public unity.

The third consequence is that many of the residual questions about German militarisation, if not wholeheartedly addressed, seem to have disappeared from the public agenda. During the Cold War, while it was still divided, François Mauriac famously suggested that "I love Germany so much I'm glad there are two of them." 14 And as Andrei Markovits and I wrote over two decades ago, the Cold War's conclusion did not resolve the moral and political dilemma other states (and the Germans themselves) faced about the prospect of German rearmament 15 . It was, of course, largely the Americans who aggressively tried to get the reluctant Germans to spend more on defence. But their efforts to cajole the Germans into spending greater sums-stretching from George W. Bush to Trump-had largely failed. In 2017, then German Foreign Minister Sigmar Gabriel summed up his country's position in suggesting that "… two percent would mean military expenses of some 70 billion euros. I do not know any German politician who would claim that is reachable nor desirable" … "The U.S. will realise it is better to talk about better spending instead of more spending" 16 Even though Germany's projected defence budget increase in 2021 by 3.2%, to €53 billion, it still only raised spending to 1.57% of GDP, nowhere near the 2% expenditure required of NATO nations. As a result, according to Global Firepower, in 2021 the German military capacity only ranked 15th (below countries such as Egypt and Italy, and just above Indonesia) 17 . That pacifist posture has now been categorically rejected in Germany, with no complaints (and some wholehearted encouragement) from Germany's European neighbours. To put that last point in some perspective, there is no time in the last seventy years where support among its neighbours of such a significant increase in Germany's military budget was conceivable.

Finally, the Ukraine war has put European efforts to pursue strategic autonomy in the security realm in some perspective. Emmanuel Macron's vision that the EU develop an independent, integrated EU Rapid Deployment force totalling 5,000 troops is attractive to many in Brussels 18 . But it took an estimated Ukrainian military of 500,000 personnel, a hundred times larger than the proposed EU force, to even confront-and only partially repel-the invading Russian force of over 190,000 19 . The proposed small EU force might be capable of addressing instability and insurrections on its periphery in South East Europe and North Africa. But it has no significant role to play in the geopolitics of great power competition. Yet, as my current project with Richard Higgott suggests, all is not lost when it comes to the pursuit of European strategic autonomy: there are other valuable policy domains where it is feasible, adding to the security and prosperity of Europeans 20 .

Concluding thoughts-from the American perspective

The war is a human tragedy of massive proportions. From a strictly American perspective, however, this event has some redeeming features. First, it has awoken many in Europe from their unwarranted belief that the continent is immune from the vagaries of great power politics. With a few notable exceptions, it has reminded many of the more reticent among them that their responsibilities extend beyond Europe's promotion of itself as a normative or civilian power when it comes to sustaining the so-called Liberal international order. By necessity, those responsibilities include a militarised commitmentprimarily to NATO and the European theatre, and nominally (if required) to the US in the Indo-Pacific.

Second, for the United States, America and European collaboration regarding Russia largely serves as a prelude to the larger question about whether they can reach any consensus about how to deal with China. To date, European leaders have hedged between China and the US on several issues including climate change and trade, albeit in large part because of the Trump administration's overt hostility towards them. But there remain grounds for concern, even from the Biden administration's perspective: the evidence suggests that it will have a difficult time reaching a policy consensus with Europe about both the EU's Green New Deal or its linkage between trade and human rights. Furthermore, even if some accord is reached, it will be difficult to sustain if the 2022 US Congressional elections or the 2024 presidential election results in Republican victories. Thus, the big challenge that looms for both sides of the Atlantic is whether they can present a unified Western front to China over a range of issues, or if they are susceptible to a long-term Chinese 'divide-and-rule' strategy.

Third, perhaps temporarily, for American's the war has reminded the EU's leadership and NATO's European membership of-hesitatingly invoking Madeleine Albright's famous term-"the indispensability of American power." 21 European ambivalence towards the United States is understandable. The transatlantic relationship comes with a lot of baggage-about the meaning and functioning of democracy, the character of capitalism, how pragmatic states should be when it comes to addressing concerns about human rights, and the prevailing question of whether the whims of the American electorate can be relied upon as the basis for Europe's foreign policies. Yet, evidently, no other country could or would have stepped in so readily to bolster Europe's defences and, at least at this point, so effectively. And no other country outside of Europe would have been so quick to offer refuge to 100,000 fleeing Ukrainians. That transatlantic relationship, Americans believe Europe has been reminded, therefore comes with benefits as well as costs. The abiding question concerns how much Europe's leaders will be willing to compromise in seeking a united position in the remainder of Biden's term of office.

Speculation about the outcome of the war abounds. Mine is that it will become a frozen conflict. Russia will control the Donbas and a land bridge to Crimea, although Ukrainian insurrection force will continue to operate in the east and Russian saboteurs, whether paramilitaries or official forces, will operate in the West. If Ukraine is divided, without a negotiated truce or peace agreement, it will likely fall upon the United States and the EU to continue to subsidize the free areas of Ukraine. It is only then, once the immediate crisis has passed, that we will be better able to judge the durability and depth of the newfound unity of the Western alliance.

• 1. William J. Burns, "The United States Needs a New Foreign Policy," The Atlantic, 14 July 2020. • 10. Julia Ioffe, "Europe's 9/11" Puck News, 3 March 2022.

• 11. See, for example, Guy-Philippe Goldstein, "A 'Cold War 2.0' Between the US and China," RUSI, November 10, 2021.

• 12. Ivo H. Daalder, "The Return of Containment: How the West Can Prevail Against the Kremlin" Foreign Affairs, March 1, 2022.

• 13. Kishore Mahbubani, Has the West Lost it? A Provocation (New York: Penguin Books, 2018). • 20. For an embryonic formulation of this argument, see Richard Higgott and Simon Reich, "Hedging by Default: The Limits of EU "Strategic Autonomy" in a Binary World Order" (with Richard Higgott), IDEAS, February 2021.

• 21. Madeleine K. Albright, Interview with Matt Lauer on NBC-TV "The Today Show", Columbus, Ohio, 19 February 1998; Michael Dobbs and John M. Goshko, "Albright's Personal Odyssey Shaped Foreign Policy Beliefs," The Washington Post, 6 December 1996, p. A25.

  • 2. Daniela Schwarzer, "Will Germany Stay the Course? How Biden Can Keep Berlin Focused on Defense," Foreign Affairs, 30 March 2022. • 3. "EU Commissioner Thierry Breton: Trust in the US 'has been eroded,'" Atlantic Council, 21 September 2021.• 4. Interview with Gérard Arau "'You Don't Do That to an Ally': Longtime French Ambassador to U.S. Scorches Biden," The New Republic, 21 September 21 2021. • 5. European Commission, "Questions and Answers: EU Strategy for Cooperation in the Indo-Pacific," 16 September 2021. • 6. U.S. Department of State, "EU-U.S.: Joint Press Release by the EEAS and Department of State on the High-Level Consultations on the Indo-Pacific," 3 December 2021. • 7. Norbert Neumann, "Why the Aukus submarine pact caused a falling-out with France," Naval Technology, 20 September 2021; Ursula von der Leyen interviewed by Christine Amanpour, CNN, September 20 2021; "USA: EU FMs 'very disappointed' with Aukus defensive alliance -Borrell," 21 September 2021. • 8. Katrin Bennhold, "Where Is Germany in the Ukraine Standoff? Its Allies Wonder" New York Times, 25 January 2022. • 9. "Remarks by President Biden and Chancellor Scholz of the Federal Republic of Germany at Press Conference" The White House, 7 February 2022.

  • 14. Quoted by Gordon McLachlan in The Rough Guide to Germany (New York, London, Delhi: Rough Guides, 2004), p. vii. • 15. Andrei S. Markovits and Simon Reich, The German Predicament: Memory and Power in the New Europe (Ithaca: Cornell University Press, 1997).

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  16. Quoted in Janakan Muthukumar, "The Politics Around NATO's 2% Defense Spending Requirement" 21 February 2020.• 17. "2021 Military Strength Ranking" Global Firepower, undated.

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  18. "A Strategic Compass for Security and Defence" European External Action Service, 24 March 2022. • 19. "Comparison of the military capabilities of Russia and Ukraine as of 2022," Statista, 6 April 2022.