Pietro Grandinetti
email: pietro.grandinetti@gipsa-lab.fr

Federica Garin
email: federica.garin@inria.fr

Carlos Canudas-De-Wit
email: carlos.canudas-de-wit@gipsa-lab.fr

Carlos Canudas

Towards scalable optimal traffic control

This paper deals with scalable control of traffic lights in urban traffic networks. Optimization is done in real time, so as to take into account variable traffic demands. At each cycle of the traffic lights, the optimization concerns time instants where each traffic light starts and ends its green phase: this allows to describe both the duty-cycle and the phase shifts. First, we formulate a global optimization problem, which can be cast as a mixed-integer linear program. To overcome the complexity of this centralized approach, we also propose a decentralized suboptimal algorithm, whose simplicity allows online implementation. Simulations show the effectiveness of the proposed strategies.

I. INTRODUCTION

The problem of traffic congestion has always been a crucial aspect for the design of efficient infrastructures, but it is particularly in the second half of the last century that this phenomenon has become predominant, due to the quickly increasing traffic demand and the more frequent congestions. Congestions appear when too many vehicles try to use a common transportation route, which, due to physical reasons, has limited capacity. If it happens, they lead to queueing phenomena or, even worse, to a severe degradation of the available infrastructure's usage. Hence, congestions result in reduced safety, increased pollution and excessive delays. Economic implication of such events are recently widely discussed [START_REF] He | Economic impact of highway investment at the metropolitan level: Empirical analysis with consideration for induced demand and induces supply[END_REF][START_REF] Sweet | Traffic Congestion's Economic Impacts: Evidence from U.S. Metropolitan Regions[END_REF].

Traffic in urban scenarios is mainly regulated by the traffic lights installed at intersections of roads. Even though these devices were first conceived to guarantee avoidance of collision, with steadily increasing traffic demands it was soon realized that they may lead to more or less efficient network operation. Therefore, the idea to increase the efficiency of the infrastructure using smart traffic lights policies has constantly been employed in academic and industrial research. Some existing strategies, called fixed-time techniques [START_REF] Robertson | TRANSYT method for area traffic control[END_REF], have limitations due to their settings, which are based on historical, rather than real-time, data. A survey about the existing techniques can be found in [START_REF] Papageorgiou | Review of road traffic control strategies[END_REF]. More advanced schemes have been presented recently and they refer to different models for the network and for the chosen control actions, such as max pressure control [START_REF] Varaiya | Max pressure control of a network of signalized intersections[END_REF] and cooperative green lights policies [START_REF] Savla | On Maximally Stabilizing Traffic Signal Control with Unknown Turn Ratios[END_REF]. Concerning the control action, we will instead contribute focusing on a more realistic representation of traffic lights, whose mathematical abstraction often produces problems difficult to solve, due to the binary (green-red) nature of these signals. Because of that, scientific works often consider optimization that takes into account either bandwith maximization via phase shift selection [START_REF] Little | The synchronization of traffic signals by mixed integer linear programming[END_REF][START_REF] Gartner | A multi band approach to the arterial traffic signal optimization[END_REF] or duty cycle design [START_REF] Grandinetti | An efficient one step ahead optimal control for urban signalized traffic networks based on an averaged Cell Transmission Model[END_REF]. The control of both parameters is a very challenging problem still open in transportation optimization environments.

In this paper, we decided to deal with the most general and difficult problem: the control of traffic lights, in urban networks, parametrized by two degrees of freedom: such representation (i.e., their description with two variable time instants) includes the most generic scenario, being able to embody phase shift as well as duty cycle.

Our contribution consists of two strategies for traffic control: the first one is a centralized control scheme that provides an optimal solution. Even though it gives the optimal control action, it may be computationally inefficient because based on a mixed integer linear program. Therefore, we also propose a decentralized suboptimal scheme that is able to dramatically reduce the computational burden, and hence is scalable to large networks.

Furthermore, the strategies we elaborated are on-line strategies based on the receding horizon philosophy. This choice is due to the fact that in traffic scenarios demands are continuously changing and, therefore, proactive decisions are needed. Our control schemes are able to accomplish such requirement, i.e., they may improve the quality of the system in the upcoming future taking into account external demands.

The rest of the paper is organized as follows: Section II introduces the model used to describe the traffic lights behavior, the network state evolution and the metrics used to describe traffic's performance; Section III illustrates how to formulate the centralized optimal control using a mixed integer linear formulation; Section IV explains our idea to decentralize the control scheme; Section V shows the numerical results of our software simulations employing the designed algorithms; Conclusions and future works are outlined in Section VI.

II. TRAFFIC NETWORK DESCRIPTION

In this section we first motivate the mathematical representation of traffic lights and, then, describe the network's model and our chosen measures of traffic performance.

A. Traffic lights model

Traffic lights are electronic devices installed at the end point (w.r.t. the flow direction) of roads for the purpose to

r .

avoid any collision between vehicles. A traffic light can either allow of forbid vehicles to continue their route, therefore it can be mathematically described as a map from the set of time instants to the set {0, 1}.

In a typical urban environment a fixed cycle length T is assigned for the traffic lights. This cycle is a slot of time during which every traffic light can switch from green to red (or viceversa) at most once. Several physical reasons justify such a behavior: faster switching may, in fact, give unfavourable consequences regarding pollution generation and drivers comfort, since in this case vehicles will have to stop-and-go with higher frequency. Typical values for T are from 90 seconds up to 2 minutes.

To fully capture the above mentioned dynamics we decide to describe the time trajectory of traffic lights with a two-degree-of-freedom trajectory (see Fig. 1), given by the vector of two natural numbers σ r = σ

(1) r σ (2) r ∈ {T s , 2T s , . . . , T } 2 ,
where T s is the sampling time used to discretize the dynamics, such that T /T s ∈ N. Hence, σ r represents the raising and falling time instants of the signal u r :

u r (t, σ r ) = 1 if σ (1) r ≤ t ≤ σ (2) r 0 otherwise, (1)
where t has to be intended modulo T .

B. Traffic dynamics on signalized networks

To describe traffic's time evolution we use the description given by a mass conservation law [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF] and by its discrete-time representation widely known as Cell Transmission Model (CTM) [START_REF] Daganzo | The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory[END_REF]. In particular, we consider an extension of the CTM for networks with FIFO policy at intersections, similar to the one we already introduced in [START_REF] Grandinetti | An efficient one step ahead optimal control for urban signalized traffic networks based on an averaged Cell Transmission Model[END_REF]. A urban network is a collection of roads entering it (R in ), internal roads (R), and exiting roads (R out ). We use the word intersections to identify locations where two or more roads merge. Such intersections have no capacity storage and they are signalized, in the sense that the traffic flow exiting each road r ∈ R in ∪R is regulated by a traffic light whose value at time instant t will be denoted by u r (t, σ r ) ∈ {0, 1}. Whether two roads r and q are connected to the same intersection in such a way that flow can exit r and enter q we use the notation r q (q is downstream w.r.t. r). Furthermore, to every road q is associated a value β q ∈ (0, 1) (split ratio) that expresses the percentage of the flow upstream q which actually wants to turn in q.

In urban scenarios we consider every road r as a cell of the CTM, hence the value of vehicles' density in it, indicated as ρ r (which we consider as normalized w.r.t. the length L r of the roads itself) depends on the flows f in r and f out r , entering and exiting r, respectively.

Inflow and outflow values are given according to the demand and supply paradigm [START_REF] Lebacque | The Godunov scheme and what it means for firstorder traffic models[END_REF]. Given a road r its demand D r is the flow of vehicles that want to exit r; its supply S r is the flow that r can receive according to its storage capacity.

Every road r is characterized by given parameters: the maximum speed in free flow v r , the speed in congestion phase w r , the maximum density allowed ρ max r and the maximum flow ϕ max r . Whether the system is sampled with step size T s , to ensure stability it must hold T s v r /L r < 1, for every road.

Finally, external traffic demand for every road r entering the network is indicated by D in r , and external supply for every road q exiting the network is indicated by S out q . The model is given by the following set of equations:

ρ r (t + T s ) = ρ r (t) + T s f in r (t) -u r (t, σ r )f out r (t) (2a) f in r (t) = min{D in r (t), S r (t)}, r ∈ R in β r q:q r u q (t, σ q )f out q , oth. (2b) f out r (t) =      min{D r (t), S out r (t)}, r ∈ R out min D r (t), Sq(t) βq q:r q , oth. (2c) D r (t) = min{v r ρ r (t), ϕ max r } (2d) S r (t) = min{ϕ max r , w r (ρ max r -ρ r (t))} (2e) u r (t, σ r ) = 1 if σ (1) r ≤ t ≤ σ (2) r 0 otherwise. mod T (2f)

C. Urban traffic performance metrics

Traffic behavior needs to be evaluated and assessed with respect to properly defined performance indices. There exist several metrics in literature to address traffic performance evaluation; in this paper we focus mainly on two features of the urban network.

1) Service of demand (SoD): A urban traffic network is an highly dynamical environment that continously receives demand from outside. This demand cannot be ignored just to favour the inner quality of the system, because the external request will end up growing with several undesired effects, due to the bigger and bigger queues arising outside.

For this reason we consider as quality of the service for a road r ∈ R in the number of vehicles (users) served by that road:

SoD r (t) = f in r (t)| r∈R in = = min D in r (t), ϕ max r , w r (ρ max r -ρ r (t)) , (3)
To improve performance we would like to maximize the sum of (3) over all roads in R in .

2) Optimization of the infrastructures usage: In urban networks there are roads preferred by the users. The civil authority would like to set traffic lights as to diminuish this usage disparity, to guarantee a more equilibrate diffusion of vehicles, thus reducing hard congestions in main streets as well as the possibility of accidents.

A standard metric that takes into account this behavior is the Total Travel Distance [START_REF] Gomes | Optimal freeway ramp metering using the asymmetric cell transmission model[END_REF], defined as follows:

TTD r (t) = min v r ρ r (t), w r (ρ max r -ρ r (t)) . (4)
We want to maximize the sum of ( 4) over all roads in R ∪ R out (because boundary flows are considered by SoD).

III. CENTRALIZED OPTIMAL CONTROL

The control strategy we designed consists in solving an optimization problem at the beginning of every cycle (i.e., every t 0 = kT , k ∈ N), in order to decide the optimal traffic lights in the upcoming cycle. Such procedure optimizes the traffic behavior using a receding horizon philosophy that predicts densities ρ(t 0 + nT s ), n = 1, . . . , T /T s . Predictions are carried out assuming that a measure of densities ρ(t 0 ) is available for the controller.

The optimal values σ * , i.e., the optimal activation time instants for the traffic lights, are given by:

σ * = arg max σ T /Ts n=1 a 1 r∈R∪R out TTD r (t 0 + nT s )+ + a 2 r∈R in SoD r (t 0 + nT s ) under network dynamics (2) ∀ t = t 0 + nT s , n = 1, . . . , T /T s , (5)
where a 1 , a 2 ∈ R are weights for the two involved objectives.

In the rest of this section we show that equations (2) and the objective function can be reformulated using logical constraints and variables, obtaining a mixed integer linear problem (MILP). To this aim, we use the reasoning outlined in [START_REF] Bemporad | Control of systems integrating logic, dynamics, and constraints[END_REF]. At the end of the reformulation, the problem's constraints, instead of (2), will be given by ( 7)-( 11),( 14),( 16)-(18).

Such final formulation of the problem is more convenient because MILPs are extensively studied, and there exist good numerical solvers to deal with them, e.g. [START_REF]Gurobi Optimization[END_REF].

A. Traffic lights constrained trajectory

Introducing the binary variables δ

(1)

r (t), δ (2)
r (t), the constraint expressed by [START_REF] He | Economic impact of highway investment at the metropolitan level: Empirical analysis with consideration for induced demand and induces supply[END_REF], for a given time instant t, is equivalent to the following:

δ (1) r (t) = 1 ←→ σ (1) r ≤ t (6a) δ (2) r (t) = 1 ←→ t ≤ σ (2) r (6b) u r (t) = 1 ←→ δ (1) r (t) = 1 ∧ δ (2) r (t) = 1 . (6c)
Let M (1) and m (1) upper and lower bounds 1 such that m (1) < σ

(1) r -t < M (1) , for every σ

r ; then (6a) is 1 Here and in what follows the numerical values for these bounds are omitted. Notice that flows in the network are always in [0, ϕ max ], while time instants indicated by σ are in [Ts, T ]. Bounds can be manually set even if some software, like Yalmip [START_REF] Löfberg | YALMIP : A Toolbox for Modeling and Optimization in MATLAB[END_REF], is able to determine them while parsing the problem formulation.

equivalent to the following constraints:

σ (1) r -t ≤ M (1) (1 -δ (1) r (t)) (7a) σ (1) r -t > m (1) δ (1) r (t). (7b)
Similarly, let M (2) and m (2) such that m (2) < σ

(2) r -t < M (2) , for every σ (2) r ; then (6b) is equivalent to the following constraints:

σ (2) r -t ≤ M (2) (1 -δ (2) r (t)) (8a) σ (2) r -t > m (2) δ (2) r (t). (8b)
Finally, the constraint (6c) is equivalent to the following:

u r (t) -δ (1) r (t) ≤ 0 (9a) u r (t) -δ (2) r (t) ≤ 0 (9b) δ (1) r (t) + δ (2) r (t) -u r (t) ≤ 1 (9c) δ (1) r (t), δ (2) r (t) ∈ {0, 1}.
Depending on the physics of system it may be useful to impose another constraint: we can require that the rising instant σ

(1)
r and the falling one σ

(2) r are sufficiently separated in time, so ensuring that too short green slots are avoided. This can be obtained with the following constraints:

σ (1) r + σ min r ≤ σ (2) r (10a)
T s ≤ σ (1) r , σ (2) r ≤ T, (10b)
where σ min r are consistently assigned to every traffic light.

B. Collision avoidance constraints

To guarantee the safe crossing we impose an hard constraint such that at every time instant only one among roads entering the same intersection has the right of way. This condition is expressed by the following linear constraint:

r:r q u r (t, σ r ) ≤ 1, (11)
for every road q. Notice that constraint [START_REF] Richards | Shock waves on the highway[END_REF] let the further freedom to assign red to all traffic lights at the same intersection, if it is required.

C. State constraints

The dynamics defined in ( 2) is non linear due to the min operator and to the product between flows and traffic lights' values. Our aim is to show how such a dynamics may be reformulated with mixed integer linear constraints.

Let σ be a vector containing σ r 's for every road r, and let f in r (t, σ) and f out r (t, σ) be the following modified flows:

f in r (t, σ) = β r q:q r f out q (t, σ), r ∈ R ∪ R out f in r (t), r ∈ R in (12a) f out r (t, σ) = u r (t, σ r )f out r (t), r ∈ R in ∪ R f out r (t), r ∈ R out . (12b)
We now show a scheme to rewrite (12b), i.e., min of several functions multiplied by a binary variable. For every road r ∈ R in ∪ R let d (r) (t) ∈ {0, 1} h be a vector of binary variables, where h = 1 + #{q : r q}. Then the definition of outflow (2c) is equivalent to the following constraints:

d (r) 0 (t) = 1 -→ f out r (t) = D r (t) (13a) d (r) q (t) = 1 -→ f out r (t) =
S q (t) β q ∀ q : r q (13b)

d (r) 0 (t) = 1 -→ D r (t) ≤ S q (t) β q ∀ q : r q (13c) d (r) i (t) = 1 -→ S q (t) β q ≤ S i (t) β i ∀ i = q (13d) h i=1 d (r) i (t) = 1. (13e)
Given the following upper and lower bounds: l 0 < f out r (t) -D r (t) < L 0 , l q < f out r (t) -S q (t)/β q < L q , ψ 0 < D r (t) < Ψ 0 , ψ q < S q (t)/β q < Ψ q , logical constraints (13a)-(13d) are equivalent to the following linear ones:

f out r (t) -D r (t) ≥ l 0 (1 -d (r) 0 (t)) (14a)
f out r (t) - S q (t) β q ≥ l q (1 -d (r) q (t)) (14b) f out r (t) -D r (t) ≤ L 0 (1 -d (r) 0 (t)) (14c)
f out r (t) - S q (t) β q ≤ L q (1 -d (r) q (t)) (14d) D r (t) ≤ S q (t) β q + (Ψ 0 -ψ q )(1 -d (r) q (t)) (14e)
S q (t)

β q ≤ S i (t) β i + (Ψ q -ψ i )(1 -d (r) q (t)) (14f) ∀ q : r q, ∀ i = q d (r) (t) ∈ {0, 1} h ,
for every road i and j = i. Finally, given u r (t, σ) ∈ {0, 1} setting

f out r (t, σ) = u r (t, σ)f out r (t) = f out r (t) if u r (t, σ) = 1 0 otherwise (15) is equivalent to g(1 -u r (t, σ)) + f out r (t, σ) ≤ f out r (t) (16a) -G(1 -u r (t, σ)) -f out r (t, σ) ≤ -f out r (t) (16b) -Gu r (t, σ) + f out r (t, σ) ≤ 0 (16c) gu r (t, σ) -f out r (t, σ) ≤ 0, (16d)
where g < f out r (t) < G.

D. Objective functions

The metrics illustrated in Section II-C are nonlinear functions of roads' density. Therefore, we use the same technique previously employed, which allows us to transform their expression as linear constraints.

1) SoD: Using the scheme illustrated by ( 13)-( 16), expression ( 3) is equivalent to:

π 1 (1 -b 1 (t)) ≤ SoD r (t) -D in r (t) ≤ Π 1 (1 -b 1 (t)) (17a) π 2 (1 -b 2 (t)) ≤ SoD r (t) -ϕ max r ≤ Π 2 (1 -b 2 (t)) (17b) π 3 (1 -b 3 (t)) ≤ SoD r (t) -w r (ρ max r -ρ r (t)) (17c) SoD r (t) -w r (ρ max r -ρ r (t)) ≤ Π 3 (1 -b 3 (t)) (17d) D in r (t) ≤ ϕ max r + (P 1 -p 2 )(1 -b 1 (t)) (17e) D in r (t) ≤ w r (ρ max r -ρ r (t)) + (P 1 -p 3 )(1 -b 1 (t)) (17f) ϕ max r ≤ D in r (t) + (P 2 -p 1 )(1 -b 2 (t)) (17g) ϕ max r ≤ w r (ρ max r -ρ r (t)) + (P 2 -p 3 )(1 -b 2 (t)) (17h) w r (ρ max r -ρ r (t)) ≤ D in r (t) + (P 3 -p 1 )(1 -b 3 (t)) (17i) w r (ρ max r -ρ r (t)) ≤ ϕ max r + (P 3 -p 2 )(1 -b 3 (t)) (17j) b(t) ∈ {0, 1} 3 ,
where Π, P (π, p) are upper (lower) bounds consistently chosen.

2) TTD: Similarly, expression ( 4) is equivalent to the following:

γ 1 (1 -c 1 (t)) ≤ TTD r (t) -v r ρ r (t) (18a) TTD r (t) -v r ρ r (t) ≤ Γ 1 (1 -c 1 (t)) (18b) γ 2 (1 -c 2 (t)) ≤ TTD r (t) -w r (ρ max r -ρ r (t)) (18c) TTD r (t) -w r (ρ max r -ρ r (t)) ≤ Γ 2 (1 -c 2 (t)) (18d) v r ρ r (t) ≤ w r (ρ max r -ρ r (t)) + (Θ 1 -θ 2 )(1 -c 1 (t)) (18e) w r (ρ max r -ρ r (t)) ≤ v r ρ r (t) + (Θ 2 -θ 1 )(1 -c 2 (t)) (18f) c(t) ∈ {0, 1} 2 ,
where Γ, Θ (γ, θ) are upper (lower) bounds consistently chosen.

IV. DECENTRALIZED SUBOPTIMAL CONTROL

The control strategy illustrated in the previous section, while providing the optimal values for traffic lights, requires to solve an intractable (NP-hard) problem. In this section we therefore propose a decentralized realization of such strategy, which reduces the computational burden substantially. It is based on two main ingredients: suboptimal solutions for local problems and agreement policy between local solutions.

A. Local problems

The optimization procedure is decentralized among intersections, each of which solves a local MILP. For every local problem, i.e., over intersection A, a receding horizon approach is used and the predicted densities belong to the set

R A = r ∈ R in ∪R∪R out : r entering or exiting A , (19)
while the optimization variables are all traffic lights in the following set:

Ω A = σ r : r ∈ R A ∪ σ q : q r, r entering A . (20)
Furthermore, densities for roads in R in ∪ R ∪ R out \ R A are considered constant (equal to the last measured values). This is the main reason of suboptimality, since the objective function maximized in every subproblem is computed over the set R A , i.e.,

J A = T /Ts n=1 a 1 r∈RA\R in TTD r (t 0 + nT s )+ + a 2 r∈RA∩R in SoD r (t 0 + nT s ) . (21)
As illustrative example look at Fig. 2, where we consider the optimization solved by intersection A. In such a scenario density predictions are carried out only for roads 1-4, and the problem is solved with respect to the variables σ 1 , . . . , σ 8 (notice that σ 5 , . . . , σ 8 are needed to compute inflows for roads 1 and 2).

It is worth to note that, among all σ's considered by an intersection, only a subset of them fulfills hard constraints for collision avoidance within the local optimization: referring again to Fig. 2, intersection A guarantees such fulfillment only for σ 1 and σ 2 . For the other variables considered by A (marked in red in the figure) there might be some hard constraint which is not included within the local problem (in Fig. 2 this happens for σ 3 and σ 4 ); Therefore, the values computed by A for these variables can be interpreted only as suggestions that A would like to advice its neighbor intersections, to optimize its own objective. How such suggestions are considered is explained in the next section.

B. Agreement policy

To let local problems taking care of advices provided by neighbors we save the result of every local optmization. For instance, when A solves its subproblem, we save the variables σ r,A , for every r involved in such problem. Once these informations are available we can identify, for every road r, the set S r of all intersections whose local problem involves σ r . Then the average suggestion given by intersections about σ r is computed as:

σr = 1 |S r | I∈Sr σ r,I . (22)
We now use the values σr to modify the cost function of every local problem, as the following:

ĴA = J A -a 3 σr∈ΩA σ r -σr 1 , (23)
where a 3 is a real number used to weight neighbors' suggestions w.r.t. A's own objective. Notice also that the 1-norm can be turned in a mixed integer linear formulation adding two binary variables for every σ r ∈ Ω A . This is the computational prize to pay in order to take into account suggestions by neighbors, rather then ignoring them; the problem, however, is still significantly numerically more efficient than its centralized version. This idea can be iterated N it times, in order to let suggestions spread among intersections. The scheme we implemented is the following: 0) Let N it be assigned; 1) For every intersection A solve the local MILP with cost function J A and save the resulting σ r,A ; 2) If N it = 0 then stop; 3) For every signalized road r compute the average suggestions σr ; 4) For every intersection A solve the local MILP with cost function ĴA , and use the result to update σ r,A ; 5) If for every r and for every A there was no update in σ r,A then stop; 6) Decrement N it by 1, goto 2. Whenever the algorithm stops, the value assigned to every traffic light is

σ * r = σ r,A , where r enters A, (24)
since every intersection guarantees the fulfillment of collision avoidance (hard) constraints only over roads entering it.

Notice that if N it = 0 is given, this means that suggestions from neighbors are ignored. An important feature of our scheme is that the subproblems' complexity does not depend on the network size; therefore the algorithm complexity is linear in the number of intersections and in the chosen number of iteration, which is a tunable parameter. Our numerical results, presented in the next section, show that the algorithm stops already after three iterations as no changes appear in the solution.

Another benefit of this decentralized strategy is that step 1 (the same applies to step 4), requires to solve a set of optimization problems that are completely indipendent of each other. Therefore the procedure can be implemented even more efficiently in a parallel architecture, where there is a controller at every intersection that exchanges σ's values with the others.

V. SIMULATIONS AND COMPARISONS

The strategies illustrated in the previous sections have been tested via software simulations in MatLab environment, using software [START_REF]Gurobi Optimization[END_REF][START_REF] Löfberg | YALMIP : A Toolbox for Modeling and Optimization in MATLAB[END_REF] to solve the mixed integer linear programs. The simulated scenario is the following: all roads in the network have same physical properties (v,w,ϕ max ,ρ max ). Each simulation is run for a virtual time of 45 minutes, when traffic lights' cycle is 90 seconds and sampling time is 10 seconds. As explained in Section III, the optimization problem is solved at the beginning of each cycle, therefore in this set-up the control values are computed 30 times. The constant σ min is set equal to 2 sample steps, so the minimum green slot is 20 seconds. Outside the network, time-varying demands and supplies are randomly uniformly generated in the interval [0.5, 1]ϕ max for the entire simulated time; by doing so, the network's state changes during the simulation (from overall free to overall congested, and viceversa), and the controllers are then tested in different circumstances. The numerical results here presented are obtained as mean values over the entire simulation time.

We would like to stress the fact that the centralized strategy we propose is able to solve a very general problem, despite the computational inefficiency. The proposed traffic lights' representation, along with the numerical optimization, guarantees optimal behavior by means of the chosen objective index, as it may cleverly choose phase shifts between lights as well as green time for each of them. Therefore, the results obtained applying this techinique are considered as benchmark (upper bounds) to evaluate performances of the decentralized strategy, from traffic and computational performance point of view. We generate the afore mentioned scenario for two sample networks, shown in Fig. 3.

Representative results of the simulations are reported in Table I. Our numerical results are encouraging:

• The decentralized strategy obtains performance around 80-85% of the ones obtained by the centralized optimization;

• Computational time is dramatically reduced, especially with growing network's size (as expected from the NPhardness of the problem). Notice also that in these simulation we did not make use of possible multithread implementation.

VI. CONCLUSIONS AND FUTURE WORKS

In this paper we presented an optimal control scheme for urban signalized traffic networks that schedules traffic lights taking into account upcoming external demands. This strategy, on one hand, ensures optimal performances, on the other, requires to solve a mixed integer linear program. To overcome the hardness of this problem, we proposed a decentralized realization of the same control scheme, that allows parallel computation. Our software simulations have shown that the computational load is extremely reduced by the decentralized scheme, which moreover achieves quite good performances from traffic point of view.

Future research will aim to investigate the scalability properties of the depicted techniques, combined with improvements on the gap with the upper bound provided by the centralized algorithm.

Fig. 1 :

 1 Fig. 1: Traffic lights described as a time trajectory with two degrees of freedom, represented by the timed-controlled variables σ (1) r , σ

Fig. 2 :

 2 Fig. 2: Illustrative example of local subproblems.

Fig. 3 :

 3 Fig. 3: Networks used for simulations and comparisons.

TABLE I :

 I Normalized performance comparison between the centralized strategy (MILP), considered as benchmark, and the decentralized one (Dec-MILP).

			Network 3a		Network 3b
		MILP Dec-MILP MILP Dec-MILP
	TTD SoD	1 1	0.82 0.86	1 1	0.83 0.85
	cpu time	1	0.6	1	0.03

This work has been partly funded by the EU FP7 project SPEEDD (619435).


Dr Agata Krasny-Pacini

MD Jonathan Evans

PhD Mckay Moore Sohlberg

PhD Mathilde Chevignard

Keywords: goal attainment scaling, outcome measures, goal setting, methodology, scale validity, scale reliability, standards, quality appraisal, guidelines Goal Attainment Scaling ICF: International Classification of Functioning IRR: Inter-rater reliability

Goal Attainment Scaling (GAS) is a method for writing personalized evaluation scales in order to quantify progress toward defined rehabilitation goals. In published literature, GAS methodology is used with different levels of rigor, ranging from precisely written GAS scales, that ensure minimal bias, explicitly describing five levels of goal attainment; to subjective rating of goal attainment by adjectives such as "worse/better than expected", which are transformed into a T-score, wrongly giving the reader the impression of a truly standardized, interval scale. A drawback of GAS methodology is that it is highly dependent on the ability of the GAS setting team/person to generate valid, reliable and meaningful scales, therefore reliability and validity of GAS scales are idiosyncratic to each study. The aims of this article were to: (1) increase awareness of potential sources of bias in GAS processes; (2) propose Goal Attainment Scaling quality appraisal criteria, allowing judgment of the quality of GAS methodology in individual rehabilitation studies; and (3) propose directions to improve GAS

Introduction

Goal Attainment Scaling (GAS) 1 is a method for writing personalized evaluation scales in order to quantify progress towards defined goals (see practical guidelines 2,3,4,5 and literature reviews on GAS 6,7,8,9,10 ). Goal Attainment Scaling produces an individualized, criterionreferenced measure of a client's goal achievement. Scores can be aggregated to quantify the extent to which a group of clients who are receiving the same type of intervention achieve their personalized rehabilitation goals. One GAS scale is written for each identified rehabilitation goal, with an emphasis on the client's participation in goal selection when possible. Success of the intervention is then quantified on an ordinal scale, typically ranging from -2 (or -3) to +2.

GAS has therefore two intertwined components: (1) GAS methodology is a personcentered approach in rehabilitation that emphasizes collaborative goal setting with the establishment of goals and levels of progress that are meaningful to the client; (2) GAS is an outcome measure that can be used both in clinical work and research to assess the effectiveness of an intervention based on personally relevant goals. This paper focuses on the use of GAS as an outcome measure specifically for rehabilitation efficacy research. The reader is referred elsewhere to reviews of the literature on the clinical aspects of collaborative goal setting 10,9,11,12 .

Writing personalized scales through GAS methodology is useful in measuring rehabilitation outcomes, and use of GAS methodology is expanding in research settings, especially in areas where standard scales do not adequately capture a study participant's progress or when a standardized assessment does not exist to measure the construct. GAS methodology offers benefit in the provision of individualized, dependent variables, a critical characteristic for measuring rehabilitation effects. GAS allows use of the same 5-point scale

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

4 method for all clients and therefore aggregation of results independent of goal type. Further, the goal of rehabilitation is to improve clients' activity and participation in natural contexts, but very few measures are designed to ecologically assess performance. By contrast, GAS allows the transformation of goals related to the International Classification of functioning (ICF) activity domains into participation goals in defined contexts where the activities occur 13,14 . Feasibility of GAS has been shown across a variety of rehabilitation fields 15,16,17,18,19 . GAS scales are sensitive to change when testing an intervention in rehabilitation 20,21,15,16,22,23 . GAS characteristics in terms of safety, utility and responsiveness are therefore encouraging.

However, in published literature, GAS methodology is used with different levels of rigor, ranging from precisely written GAS scales that ensure minimal bias, explicitly describing five levels of goal attainment; to subjective rating of goal attainment by adjectives such as "worse/better than expected", which are transformed into a T-score, wrongly giving the reader the impression of a truly standardized, interval scale. Although the less rigorous form of GAS methodology can be convenient, useful, fast and practical to use in clinical practice, there is growing concern for its use as an outcome measure in clinical trials 24,25 and mixed findings as to the reliability 24,26 and validity of GAS as an outcome measure 25 .

The aims of this paper are to (1) increase awareness of potential sources of bias in GAS processes; (2) propose Goal Attainment Scaling quality appraisal criteria, allowing judgment of the quality of GAS methodology in individual rehabilitation studies; and (3) propose directions to improve GAS implementation in order to increase its reliability and validity as a research measurement tool. This paper is not addressing use of GAS in clinical setting outwith research.

Methods

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

5

A literature search using PubMed data base was conducted to ensure that our critical appraisal of the research was inclusive. The keywords "goal attainment scaling" and "rehabilitation OR therapy" were utilized to identify articles published between 1990 and 2014.

The search returned 179 articles. Twelve articles were excluded because an abstract was not available or because the article was not written in English. A title and abstract review was conducted to identify those articles that evaluated GAS methodology as an outcome measure.

Included papers were: (1) Literature reviews on GAS; (2) GAS clinical guidelines; (3) Papers relating to GAS validity and reliability; (4) Papers relating to training in GAS. We purposefully included papers referring to fields outside Physical Medicine and Rehabilitation (PMR), that face the same challenges in evaluating treatment efficacy as rehabilitation does (especially cognitive interventions from the field of psychiatry and developmental disorders). Papers were excluded if they assessed only GAS feasibility or sensitivity to change/responsiveness, without references to its validity and reliability as an outcome measure. This yielded 36 relevant full text papers that were reviewed in order to identify bias in GAS and generate the quality appraisal criteria.

Potential sources of bias in GAS processes and published recommendations for constructing goal attainment scales.

Usual criticisms of how GAS methodology has been used include: (1) unknown clinimetric qualities of GAS scales used in a given study due to their idiosyncratic nature 25 ; (2) subjective scoring, especially if not all levels of the scale are formulated or if descriptions are not precise enough; (3) risk of choosing goals that are not clinically relevant or too easy/too difficult to attain 27 and therefore do not represent a meaningful or realistic change in function; (4) ordinal (rather than interval) nature of GAS scales 28 and the lack of equidistance

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 6
between GAS levels which cannot be controlled for 24 ; (5) the use of a T-score that uses subjective values, especially a subjective weighting of GAS scores and a ρ coefficient assumed to be 0.3 which has not been confirmed in the literature 9,28,6 .

A major drawback of GAS methodology is that it is highly dependent on the ability of the GAS setting team/person to generate valid, reliable and meaningful scales. It has even been proposed that GAS is more a measure of how adequately a therapist can foresee outcome than an outcome measure itself 6,29,[START_REF] Cytrynbaum | Goal Attainment Scaling A Critical Review[END_REF][START_REF] Simeonsson | Scaling and attainment of goals in family-focused early intervention[END_REF][START_REF] Ertzgaard | Practical considerations for goal attainment scaling during rehabilitation following acquired brain injury[END_REF] . A group of clients may show progress on their GAS scale due to a measurement error, on a GAS scale that is not reliable because of poor interrater reliability, too easy goals, unequal distances between GAS levels or use of subjective criteria for goal attainment. This issue has been raised by Ruble et al. 27 : "If GAS scores are higher in the experimental conditions […]one could argue that the targeted outcomes as scaled using GAS were less difficult and easier for [clients] in the experimental group to achieve compared to the control group; that skills were written in more measureable terms and thus easier to be observed and coded in the experimental groups; or that the intervals between each scaled description were unequal and favored the experimental group." (p3).

Because these potential biases can threaten reliability of results obtained through GAS, Kiresuk et al. 1,[START_REF] Kiresuk | Goal attainment scaling: Applications, theory, and measurement[END_REF] recommended the review of GAS scales by an independent third party, and even suggested that clients should be evaluated on two different sets of GAS scales, developed by two independent research groups 1 (i.e. treatment success should be independent of how the goals were formulated) 1,[START_REF] Kiresuk | Measurement of goal attainment in clinical and health care programs[END_REF] to minimize bias. Although few publications address this demanding recommendation [START_REF] Steenbeek | Interrater reliability of goal attainment scaling in rehabilitation of children with cerebral palsy[END_REF][START_REF] Ra | The use of goal attainment scaling in a geriatric care setting[END_REF] it seems crucial that authors using GAS as a research outcome measure provide the reader with information on how the scales were generated and verified (and/or compared between groups on items that may impact

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 7
on GAS scoring as suggested by Ruble et al. 27 ), in order to provide information on reliability and validity. Some authors have found encouraging values of GAS reliability and validity 25,15,[START_REF] Palisano | Validity of goal attainment scaling in infants with motor delays[END_REF]22,[START_REF] Steenbeek | Interrater reliability of goal attainment scaling in rehabilitation of children with cerebral palsy[END_REF] (see recent systematic review by Vu et al 8 ). However the validity and reliability of GAS set by one team (especially an experienced one) does not presume that other GAS scales set by other teams, in other rehabilitation contexts, are valid and reliable 27 . GAS clinimetric qualities depend mainly on how experienced the team is in GAS writing. Grant et al. reported the problems encountered when GAS are used by inexperienced teams, without an independent experienced judge checking the scales 12 .

A series of criteria for writing GAS has been proposed [START_REF] King | Goal Attainment Scaling: its use in evaluating pediatric thearpy programs[END_REF]1,2,[START_REF] Steenbeek | Interrater reliability of goal attainment scaling in rehabilitation of children with cerebral palsy[END_REF] : (1) each GAS level must be described accurately enough to allow a person who was not involved in the GAS-writing process to easily classify the client at one of the GAS levels described 1 , with no "blank levels" [START_REF] Smith | Introduction and overview[END_REF] (levels not precisely described, which content is inferred from adjacent levels); (2) each scale must represent a single dimension of change 12 ; (3) the levels must be measurable and thus defined in terms of observable behaviors 6,9,[START_REF] Stolee | An individualized approach to outcome measurement in geriatric rehabilitation[END_REF] ; (4) the scales must correspond to goals that are important/meaningful to the client; (5) all the levels must be realistic and attainable (in particular, the +2 level must not correspond to an unexpected or miraculous goal attainment level) 1 ; (6) the time scale within which goals must be attained and scales must be scored should be defined in advance; (7) the inter-level differences in difficulty must all be the same [START_REF] Steenbeek | Goal Attainment Scaling in paediatric rehabilitation: a report on the clinical training of an interdisciplinary team[END_REF][START_REF] Jones | Using goal attainment scaling to evaluate a needs-led exercise programme for people with severe and profound intellectual disabilities[END_REF] , i.e. it must be as difficult to progress from -2 to -1, as from -1 to 0 or from 0 to +1, etc… and there should be no overlapping and no gap between the levels [START_REF] Smith | Introduction and overview[END_REF] . Part of these criteria are reflected in the ''SMART'' acronym [START_REF] King | Goal Attainment Scaling: its use in evaluating pediatric thearpy programs[END_REF] i.e. a goal should be Specific, Measurable, Achievable, Relevant and Time-determined. Although all authors acknowledge the need for GAS to be "SMART", few report precautions taken to ensure GAS scales are

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 8
actually "SMART", and virtually none assesses GAS quality when using it as an outcome measure. Some authors proposed additional recommendations for GAS when it is used in research:

(1) including a training program 43,25,15,44 ; (2) establishing all goals prior to randomization [START_REF] Palisano | Goal attainment scaling as a measure of change in infants with motor delays[END_REF][START_REF] Ottenbacher | Discriminative Versus Evaluative Assessment: Some Observations on Goal Attainment Scaling[END_REF] or blinding the goal-setter to the patient's treatment/control status 47 ; (3) testing of interrater reliability for initial and post intervention GAS rating [START_REF] Mailloux | Goal attainment scaling as a measure of meaningful outcomes for children with sensory integration disorders[END_REF][START_REF] Ottenbacher | Discriminative Versus Evaluative Assessment: Some Observations on Goal Attainment Scaling[END_REF] ; (4) GAS scoring by a blind examiner [START_REF] Ottenbacher | Discriminative Versus Evaluative Assessment: Some Observations on Goal Attainment Scaling[END_REF] who is independent from the team that set the goals [START_REF] Steenbeek | The effect of botulinum toxin type A treatment of the lower extremity on the level of functional abilities in children with cerebral palsy: evaluation with goal attainment scaling[END_REF][START_REF] Sakzewski | Clinimetric properties of participation measures for 5-to 13-year-old children with cerebral palsy: a systematic review[END_REF][START_REF] Ottenbacher | Discriminative Versus Evaluative Assessment: Some Observations on Goal Attainment Scaling[END_REF][START_REF] Jones | Using goal attainment scaling to evaluate a needs-led exercise programme for people with severe and profound intellectual disabilities[END_REF] and independent from the therapist providing intervention [START_REF] Mailloux | Goal attainment scaling as a measure of meaningful outcomes for children with sensory integration disorders[END_REF][START_REF] Shefler | Psychometric properties of goal-attainment scaling in the assessment of mann's time-limited psychotherapy[END_REF] ; (5) the use of ''control goals'' that are not targeted by the intervention 6 ; (6) evaluation of the patients on two different GAS scales developed by independent therapists (i.e. treatment success must be independent of how the goals were formulated) 1,[START_REF] Steenbeek | Interrater reliability of goal attainment scaling in rehabilitation of children with cerebral palsy[END_REF][START_REF] Ra | The use of goal attainment scaling in a geriatric care setting[END_REF] ; (7) goal-setting by a group (rather than a single therapist or the patient alone), in order to avoid overly simple or unrealistic goals [START_REF] Kiresuk | Measurement of goal attainment in clinical and health care programs[END_REF] . To our knowledge, the impact of those recommendations on GAS validity and reliability has not yet been studied and few studies follow these guidelines.

Their utility and applicability will be discussed in the discussion section of this paper.

Proposed criteria for appraising Goal Attainment Scales

Because GAS is a relevant and responsive outcome measure in rehabilitation research, but used with great variability that weakens the confidence in the results of trials that use this methodology, there is an urgent need for standards relating to GAS use in rehabilitation efficacy research. Our aim was therefore to propose GAS quality appraisal criteria, that would allow judgment of the quality of GAS methodology in individual rehabilitation studies, that could be used as guidelines to reduce bias, and strengthen GAS validity and reliability.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 9

Based on our review of the literature, items for the quality appraisal were included if they met one of the following: (1) historically or traditionally recognized quality criteria (such as the "SMART" criteria and Kiresuk et al.'s 1,[START_REF] Kiresuk | Goal attainment scaling: Applications, theory, and measurement[END_REF] rules for writing GAS scales); (2) criteria used by teams who obtained and published a good level of inter-rater reliability of their GAS data;

(3) criteria used in rehabilitation trials to compare GAS quality across experimental groups;

(4) items judged consensually by all authors of this paper as potential key candidates for increasing GAS validity and reliability (even in the absence of literature showing their impact on GAS clinimetric quality). Disagreements between authors on included items are developed in the discussion section.

In selecting criteria, the publications of two teams were particularly useful. In Steenbeek et al.'s methodology [START_REF] Steenbeek | Interrater reliability of goal attainment scaling in rehabilitation of children with cerebral palsy[END_REF][START_REF] Steenbeek | The effect of botulinum toxin type A treatment of the lower extremity on the level of functional abilities in children with cerebral palsy: evaluation with goal attainment scaling[END_REF]23 , eight GAS characteristics can be identified to ensure the construction of reliable scales: (1) all five levels of the GAS are precisely described; (2) GAS scales use objective and observable measures based on performance; (3) context of measurement is precisely described and factors that might influence performance are controlled for; (4) initial level is systematically verified after scale is set; (5) an independent, blind assessor scores GAS after intervention; (6) GAS data analysis respects the ordinal nature of GAS, using only raw scores and non-parametric statistics; (7) inter-rater reliability of GAS used in each study is reported; (8) teams are specifically trained to write reliable GAS scales (refer to Steenbeek et al. [START_REF] Steenbeek | Goal Attainment Scaling in paediatric rehabilitation: a report on the clinical training of an interdisciplinary team[END_REF] for an example of training).

Ruble et al. describe criteria for using GAS as a main outcome measure in a randomized controlled trial (RCT) 27 . They suggest the following three key questions to ensure comparability of GAS in different experimental groups: "(a) are the goal and the associated benchmarks relative to each goal described in measurable terms that are comparable between groups (measurability criteria); (b) is the distance between each of the benchmarks

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 10
for each scale of equal intervals and comparable between groups (equidistance criteria); and (c) is the level of difficulty between the baseline or starting levels of performance and the targeted outcome goal comparable between groups (difficulty criteria)?" 27 . In a recent controlled trial 50 , all GAS scales were compared across the two experimental groups on these criteria using a three-point Likert scale, showing the feasibility of comparing GAS scales across groups in RCTs.

We propose 17 GAS quality criteria and these are presented in Table 1. They are broadly grouped into criteria that relate to the content validity of scales (4 items), the reliability of scale construction (4 items), reliability of scale rating (5 items), and an additional four items relating to training, examiner bias, statistical analysis and provision of a sample scale. If GAS is used in a controlled trial, we propose that GAS scales should not only be checked, but also compared between groups, similarly to the methodology proposed by Ruble et al. 27 for relevant items. For large trials, we propose that at least 20% of GAS scales be checked/compared.

INSERT TABLE 1 ABOUT HERE

GAS validity

GAS should be used when standardized assessment does not exist to measure the construct. Content validity of GAS scales is commonly thought to be high if the goal has been collaboratively set with the client wherever possible and this is the first criterion. Goals should be relevant and reflect clinically meaningful change, and this needs to be independently verified. To document the functional relevance of goals, the ICF domain that the goals reflect should be documented. Specificity is core to SMART goal-setting but with several definitions. Our specificity criteria relates to whether goals set specifically relate to

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

11 the intervention being tested -in research it is important to be able to articulate how a particular intervention will lead to achieving the desired goal.

GAS reliability

There are two types of reliability that are particularly important in GAS:

(1) Reliability in the way the scale was constructed (i.e. even with an excellent inter-rater reliability, the scale may not be reliable because of non equidistant levels, erroneous starting pre-intervention levels, too easy goals/GAS levels, and an unspecified time frame for goals' attainment influencing the relative difficulty of attaining a specific goal at the generic postintervention assessment time point). The first four reliability criteria reflect these issues.

(2) Reliability in scoring a given GAS scale. Measuring inter-rater reliability (IRR) provides a check on measurement accuracy. The following four items are thought to impact IRR: GAS scales where each level is not precisely described, that use subjective criteria for goal attainment with poor measurability or multidimensional scales, and which do not control for context of measurement, are likely to show lower IRR. Future research is needed to evaluate if respecting those criteria allows better IRR. An adequately measured and reported IRR could release authors from checking all GAS for those items (as training in GAS on those items is likely to generate scales with greater IRR). Because measuring their IRR is time consuming and requires an additional staff member, for large trials, we propose that at least 20% of GAS scales be tested for IRR (similar to the 20% of measurement criteria of N-of-1 trial standards [START_REF] Tate | Revision of a method quality rating scale for single-case experimental designs and n-of-1 trials: the 15-item Risk of Bias in N-of-1 Trials (RoBiNT) Scale[END_REF] and Ruble's study 27 ).

Additional items

The final four criteria are included to further reduce potential for bias and increase confidence in GAS in research reports. They relate to: training of staff writing GAS scales;

Discussion

The proposed criteria are indisputably challenging to meet. Most research using GAS does not meet all criteria. However it seems important to set out the highest possible criteria for all forms of research, even though few studies can meet all criteria. The proposals here are intended for research, where outcome measures must be valid and reliable to prevent erroneous conclusions about the effectiveness of an intervention. They do not discredit less rigorous but more user-friendly, more practical and less time-consuming uses of GAS in clinical practice.

Controversial criteria

Should GAS use be restricted to collaboratively set goals ? In the literature, goals are often set in collaboration between client/family and therapist 52 53 54 55 56 , but may also be chosen by the therapist alone 57 58 , or by the client/ family alone 57 59 . Initially, the GAS methodology was invented to assess any goal-directed enterprise, including the functioning of a Crisis Intervention Center [START_REF] Kiresuk | Measurement of goal attainment in clinical and health care programs[END_REF] or a hospital-based pharmacy project [START_REF] Kiresuk | Measurement of goal attainment in clinical and health care programs[END_REF] . Initially goals could therefore be chosen by any professional, with or without participation of the client. In the last twenty years, as rehabilitation moved towards a more person-centered approach [START_REF] Leach | Patient centered goal-setting in a subacute rehabilitation setting[END_REF] , GAS has become increasingly used as a method for collaborative goal setting, as well as an outcome measure. Literature on brain injury [START_REF] Malec | Goal attainment scaling and outcome measurement in postacute brain injury rehabilitation[END_REF] [START_REF] Ertzgaard | Practical considerations for goal attainment scaling during rehabilitation following acquired brain injury[END_REF] 62 9 10 , and linked use of GAS as an outcome measure with active client participation in goal setting. However GAS may be valuable in PMR domains where client participation in goal selection is not possible (e.g. patients with minimally conscious state, clients with severe intellectual and behavioral impairment…) or is not essential (e.g. early interventions after stroke to prevent contractures and shoulder pain; motor development in infants with attainment of developmental stages that may not be relevant for the family immediately but that are believed to be crucial for future development and future more functional goals…). GAS may also be used to assess effectiveness of an intervention at a health provider level (e.g. goal of reducing pressure ulcer incidence/need for surgery after a group therapeutic education in a spinal cord injury unit). Because collaborative goal-setting is time-consuming for therapists and cognitively demanding for clients (especially those with brain injury), therapists may use collaborative goal setting for some goals, while choosing themselves goals of other domains that are indisputably useful for this client and the focus of the intervention being tested. Therefore we propose here a less restrictive requirement (in the collaborative goal setting item), allowing therapistchosen goals. In all cases, authors should report if goals were not collaboratively set and provide reasons of their choice.

Most often, GAS scales are set for goal-focused rehabilitation where the goal is directly trained and GAS represents the degree of progress towards a goal. However it has also been proposed that GAS could be a measure of generalization [START_REF] Krasny-Pacini | Context-Sensitive Goal Management Training for Everyday Executive Dysfunction in Children After Severe Traumatic Brain Injury[END_REF] : i.e. after training relating to a cognitive function (e.g. executive functions, memory), the use of GAS scales can help assess if training lead to gains in daily life (e.g. GAS relating to activities relying on executive functions such as preparing a schoolbag, GAS relating to memory such as taking medicines on time), without specifically addressing these goals. A valuable approach is to use both a set

M A N U S C R I P T A C C E P T E D

ACCEPTED MANUSCRIPT 14 of trained goals (and corresponding GAS) and a set of untrained goals (and corresponding GAS) and then to focus the intervention on training the former while using the latter as an untrained ecological generalization measure. Further it is has been proposed that "control" goals 6 (and related GAS scales) that are not expected to show progression are used, in order to demonstrate the specificity of an intervention (i.e. the client does not just progress on all goals because of general cognitive stimulation or goal-driven motivation, but progresses on the specific goal that is trained or that relates to the trained function needed to achieve an untrained generalization goal).

Authors should report the types of goals chosen in their study, using the ICF. GAS types vary considerably between studies, and do not always measure a meaningful goal, but may remain in the body structure and function (gait pattern [START_REF] Mall | Treatment of adductor spasticity with BTX-A in children with CP: a randomized, double-blind, placebo-controlled study[END_REF] , range of motion 29 ). A clear demonstration of functional benefit to the client in terms of activity and participation is increasingly required in order to show an intervention is effective, and GAS should relate to activity and participation domains as much as possible. When using GAS for body structures, there is a risk that GAS be used as a way of getting round the need for standardized measures. A methodological error often seen with GAS is to convert existing (or even standardized) scales into GAS scales. This is done for two practical reasons: (1) to help measure a particularly complex goal (e.g. if cognitive restructuring […] is an important treatment component, it may be advantageous to include a standardized pain beliefs scale as a goal area 47 , p62); (2) to transform a meaningless number into a relevant and meaningful goal; (3) in order to obtain the same outcome measure for all clients. For example for botulinum toxin treatment, the goal of one client may be to decrease pain, measured on a visual analogue scale, then transformed into a GAS scale depending on the pain level considered as meeting the goal of a treatment; the goal for another client may be to However such a conversion is done at the expense of losing the linearity of the original measure and although very useful in clinical practice, it is scientifically acceptable only if data is analyzed as truly ordinal (therefore not using T-scores, nor means nor any arithmetical operation) [START_REF] Merbitz | Ordinal scales and foundations of misinference[END_REF] 66 28 67 .

Can GAS levels be equidistant and GAS data be interval in form? This is possible if a calibration by Rasch Analysis on an "item bank" is carried out as proposed by Tennant et al. 24 , but at the expense of losing GAS adaptability to any goal 12 . Otherwise, GAS levels are very unlikely to be interval despite all precautions, corrections and comparisons used to minimize level inequality bias. Therefore ordinal interpretation of GAS, using rank tests (see Steenbeek et al for an example [START_REF] Steenbeek | The effect of botulinum toxin type A treatment of the lower extremity on the level of functional abilities in children with cerebral palsy: evaluation with goal attainment scaling[END_REF] ) and excluding all arithmetic procedures 66,68,24 on GAS scores seems the most reasonable option for GAS data analysis. It is indisputably difficult to set 5 equidistance GAS levels but simple rules can be postulated to facilitate choice of levels (both for goal setters and the external judge who checks GAS levels and compares equidistance between two groups): (1) avoid setting "half levels" ( e.g. -0.5 as proposed by Turner Stokes 69 ); (2) have all clients start from the same initial score and therefore have all clients assessed on the same number of levels of goal attainment (see 5 for a discussion on advantages of scoring initial level at -2 or at -1).

What type of staff training should be required in order to use GAS in a research protocol? Kiresuk, Smith and Cardillo had proposed that a minimum of one-year experience is required to develop relevant and realistic scales [START_REF] Smith | Introduction and overview[END_REF] . Basic knowledge of GAS, and experience in collaborative goal setting is not sufficient, as shown by Grant et al [START_REF] Grant | The application of Goal Management Training to aspects of financial management in individuals with traumatic brain injury[END_REF]12 . Reading a practical guide 2,3,5,4 may be sufficient for GAS clinical use but not for studies aiming at producing valid

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

16 and reliable GAS. A number of authors [START_REF] Cardillo | Goal Setting, Follow-up and Goal Monitoring[END_REF][START_REF] Steenbeek | Goal Attainment Scaling in paediatric rehabilitation: a report on the clinical training of an interdisciplinary team[END_REF] propose practical training that is largely based on formulating and correcting GAS scales, based on clients' real goals. We recommend that such a practical training be used. This should also raise awareness on GAS quality criteria the study will be judged on, and make GAS scoring easier after the intervention. The quality appraisal here proposed could be the focus of such training.

Challenges in GAS methodology

GAS was intended to be a person-centered measure. Is there a risk of losing its personcentered nature when trying to meet measurement criteria for controlled efficacy trials? The risk of insisting upon an observable and measurable goal is to exclude family and client appreciation of goal attainment. Appreciation of goal attainment by clients may be subjective and is affected by a series of factors (self-awareness, denial, memory, high involvement in goal pursuit) that can bias the perceived attainment of a goal; however goal identification and measuring its attainment should be person-centered. The challenge for clinicians and researchers is therefore to understand and analyze the client's (and/or his /her families) goal and transform it into an observable, objective, performance-based measure that can then be used to discuss goal attainment with the client. For example a general goal of "improving my memory" expressed by a client can be transformed into a GAS scale that measures memory functioning in real life situations agreed with the client (e.g. number of medicines taken on time without prompting; number of failed-to-deliver messages in one week…). Scoring GAS based on simple interview should be avoided when possible and creativity used to link subjective goals in difficult domains to more objective goal attainment indicators. For example for a GAS scale on anger management or use of social skills, rather than asking the client to recall how often she/he felt he could cope with his anger or

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

17 effectively use social/language skills, an effort should be made to choose indirect indicators of goal attainment such as a review of controlled versus overt anger at the end of a day by the client or proxy rating of social/language skill (how many times he/she initiated/contributed to conversation or was understood) after regular naturally occurring events (outing with friends, family dinner). It is probably the most challenging part of GAS methodology but it has been demonstrated that it is feasible to have GAS that are both person-centered in the choice of goal and objective in the formulation of GAS scales (see Steenbeek et al. for an example [START_REF] Steenbeek | Interrater reliability of goal attainment scaling in rehabilitation of children with cerebral palsy[END_REF] ).

N-of-1 trial literature faces the same challenges and offers growing ingenuous methods for assessment of domains not directly accessible to classical performance-based objective measure. These methods include use of smartphone reminders to self-assess goal attainment at regular periods to decrease memory bias (see [START_REF] Arco | Neurobehavioural treatment for obsessive-compulsive disorder in an adult with traumatic brain injury[END_REF] for an example), use of naturally occurring situations monitored in real life by family/proxy to decrease selfawareness bias (see [START_REF] Baldwin | Google Calendar: A single case experimental design study of a man with severe memory problems[END_REF] for an example), use of objective behavioral measures that are thought to reflect the underlying psychological (e.g.: happiness [START_REF] Logan | The Impact of Typical Peers on the Perceived Happiness of Students with Profound Multiple Disabilities[END_REF] ) or cognitive (e.g.: functioning at school [START_REF] Feeney | Context-sensitive behavioral supports for young children with TBI: short-term effects and long-term outcome[END_REF][START_REF] Feeney | Context-sensitive cognitive-behavioural supports for young children with TBI: A replication study[END_REF] ) target goals. Future research should extend "goal menus" such as those proposed by Turner-Stokes et al. [START_REF] Ashford | Goal setting, using goal attainment scaling, as a method to identify patient selected items for measuring arm function[END_REF] to more challenging domains (such as goals relating to social and psychological functions, as well as goals focusing on performance in ecological setting), using also N-of 1 trial literature. When goal attainment indicators are ecological and monitored by the client or proxies, calculating an IRR is impossible (or difficult) and GAS should be checked for items: precise description of all GAS levels, measurability, unidimensionality and context of measurement to increase reliability in scoring. A reasonable compromise between scientific rigor and person-centered approach could be to have for each client at least one ecological client-centered GAS (with the risk of being less

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

18 reliable) and at least one performance-based GAS (with the risk of being less ecological). In all cases, Kiresuk recommended to "anchor scale points with behavioral or other evidence that will be meaningful to the client and readily scored by the rater" 33 (p 31).

Limitations

The present article has a number of limitations. Choice of included criteria was not based on a consensual agreement of all major teams using GAS in research but of four teams, from three different countries. The aim of this paper was to raise awareness about the variability of GAS use in published research and the need to build, in future, a consensus on the use of GAS in efficacy research in rehabilitation. Although it may be viewed as a limitation, we purposefully did not validate externally an appraisal score, so that the present guide acts only as a starter for discussion and not a validated tool imposed on other teams that were not included in the writing of the paper.

Conclusion

Goal Attainment Scaling has the potential to be sensitive to change following treatment and applicable across divergent domains of rehabilitation, making it a useful rehabilitation efficacy research outcome measure. However, GAS is used in studies with variable rigor that impacts its validity and reliability, and therefore reduces the confidence one can have in the results of a trial using GAS as an outcome measure. Clinimetric qualities of GAS are highly dependent on the way GAS scales are written and therefore clinicians, researchers and reviewers cannot rely on published studies of metrological qualities of GAS obtained in different research studies. Researchers should be aware of the risk of bias related to the use of imprecisely written GAS scales in research and make all possible efforts to minimize this

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

19 bias by constructing high quality GAS scales for their clients, following recommendations described here and previously published examples 27,[START_REF] Steenbeek | The effect of botulinum toxin type A treatment of the lower extremity on the level of functional abilities in children with cerebral palsy: evaluation with goal attainment scaling[END_REF] . There is a need to develop in the future a GAS quality appraisal score, similarly to standards used in other rehabilitation controversial fields such as N-of-1 trials [START_REF] Tate | Revision of a method quality rating scale for single-case experimental designs and n-of-1 trials: the 15-item Risk of Bias in N-of-1 Trials (RoBiNT) Scale[END_REF] . In this paper, we propose 17 criteria for appraising GAS quality in trials using GAS as an outcome measure. We recommend that authors using GAS report accurately how GAS methodology was used based on these criteria. The reliability of these criteria needs to be established, but in the meantime we invite comment and discussion of these proposals as we move towards a consensus on standards for use of GAS in rehabilitation research.

Quality appraisal items

Item description

Examples of reported criteria, extracted from rehabilitation effectiveness studies and/or published methods that satisfy criteria.

Potential for bias arising from failure to report criteria and/or utility of reporting the criterion. Content Validity

Collaborative goal setting

The client/family is included in goal selection, when possible and appropriate. If goals are therapistchosen, they rely on a comprehensive assessment (and when possible, a client/family interview), identifying key target domains for intervention.

"GAS was used to assess functional and participation changes from both a parent and therapist perspective"

1 "Using a semi-structured interview […] 3 performance goals were identified at baseline by parents and child with the research physical therapist. The same 3 goals were then structured for GAS through semi-structured interview and by collaboration between research therapist, parent and child."

2 .
Collaborative goal setting allows evaluation of intervention efficacy for personally meaningful goals, rather than generic goals and is a core component of GAS methodology. Involvement of the client in goal setting is considered to increase the likelihood that the intervention has direct impact on client's daily life.

Relevance/ importance GAS scales have been verified by an external judge to check for the relevance of chosen goals and to check if GAS levels represent clinically meaningful change.

In a study of infants with motor delays, Palisano 3 used a 5 point scale to assess: (1) importance of goals for motor development ranging from "unimportant or inappropriate" to "important for development and for function"; (2)extent each level represents an important progress based on number of paired levels that represent important change ("none" to "all four paired levels").

Cardillo 4 proposed a 5 point scale ranging from "1: No relevance" to "5: Total relevance".

If the target goal is unimportant to the client, irrelevant for function or does not correspond to a clinically meaningful change, progress on the GAS scale has no clinical relevance. At the extreme, an intervention could be proven to be effective, by writing clinically and personally irrelevant goals but showing statistically significant progress on the corresponding GAS scale.

ICF classification of goal types

GAS themes correspond to functional domains.

Authors report the ICF domain the GAS relate to.

In a study of botulinum toxin effectiveness, Turner-Stokes 5 et al. report exhaustively the types of goals for treatment, categorizing them into ICF domains (Body function: 46 GAS, including 12 GAS on passive movement/range; Activities and Participation: 119 GAS, divided into: Upper limb activities: 30 GAS; Mobility: 11 GAS; Self care: 57 GAS; Domestic and Community: 21 GAS).

Phillips et al. 6 report precise examples of target goals in different ICF domain (e.g.: "driving for 40 minutes without feet going floppy"; "standing in supermarket queue for 3 minutes without support").

If GAS scales assess change in body structures (e.g. RoM, spasticity), the reader may wrongly conclude that the intervention had an impact on meaningful activity and participation, because most readers associate GAS with functional daily life goals. It is therefore crucial that authors report the proportion of goals in each ICF domain, especially if some of goals do not correspond to functional domains.

Specificity

GAS scales have been verified by an external judge to check for specificity to the aim of the intervention.

If GAS is used as a generalization measure to untrained goals, GAS should be specific to the function the In an intervention for executive dysfunction in children 7 , a specific goal of intervention was to manage a cooking recipe unaided, which was trained on different recipes; a generalization goal was to be able to prepare school bag which was not trained (but stepwise processing taught was expected to generalize to this untrained goal). Although important to the children, goals such as "have more friends", were not included as they were not specific to the aim of Goals that are relevant to the client, but unrelated to the specific intervention, are unlikely to show progress and may erroneously lead to the conclusion that the intervention is not effective (this is especially a risk in replication studies).

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

intervention is aiming to improve.

intervention, but they might have been used as "control" goals, not expected to be attained.

Reliability

Reliability of scale construction

Equidistance of levels GAS scales have been verified by an external judge to check if difficulty from one level to the next is roughly equal.

Equidistance of GAS levels was assessed and compared statistically between two experimental groups using a 3 point scale by Ruble et al. : "1: None or only one of the descriptions are equilibrated appropriately in reference to the goal; 2: Two of the descriptions are equilibrated appropriately in reference to the goal; 3: All of the descriptions relative to the goal are equilibrated and scaled appropriately)".

If GAS scores are higher in the experimental conditions, one could argue that the intervals between each scaled description are unequal and favor the experimental group. This problem is particularly serious if parametric statistics and T-scores are used.

Pre-intervention performance

Pre-intervention performance has been verified and corresponds to initial level described in the scale.

Pre-intervention score is comparable across groups (same number of clients starting from -2 and -1).

"At first baseline, GAS were created based on parent's, teacher's and school assistant's concerns. After two months (second baseline), the paper versions of the scales were scored by parents, teachers and school assistants who were not aware that the intervention had not started yet. Their answers allowed readjustment of the scales, through the following rules: Scales that scored 0 were reformulated in order to have the pre-intervention level (measured at second baseline) corresponding to -1 by fixing more challenging 0, +1 and +2 scores. Scales scoring -2 or -1 were not reformulated. Scales scoring +1 or +2 pre-intervention were removed as the goal seemed attained without intervention or unreliably scored due to potential enthusiasm effect and motivation driving perceived change independently from intervention that had not started " 7 .

GAS scales are constructed uniquely for each client, according to his/her initial level in relation to the target goal. If pre-intervention level is not verified, the whole scale may be unreliably constructed (erroneous starting point of the scale generating inappropriate next levels).

Attainability/ difficulty

GAS scales have been verified by an external judge to check for their difficulty/attainability. Ruble et al. compared GAS scales for difficulty in a trial of cognitive intervention: "1: Skill is very close to what the child is already described as able to perform (very easy); 2: The child is able to perform the skill in limited ways compared to what is written in the objective (limited people, prompts, or places […]); 3: The child is unable to perform skill with anyone, anywhere, or with any prompts compared to what is written in the objective (difficult)". 8 9 In a study of infants with motor delays, Palisano 3 asked experts to decide which of the 5 GAS level the child was most likely to achieve after 3 months, aiming at a maximum of 0 and a minimum of extreme -2 and +2 scores, if GAS levels were decided correctly.

Cardillo 4 reported that realism of the expected level of outcome An experimental group may falsely present higher GAS scores post intervention because GAS scales were formulated with easier levels of goal attainment.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

"for each goal, for this patient, at this time, in this mental health service" was assessed by a scale ranging from 1: much too difficult to 5: much too easy. He also reviews other methods that use 3 or 5 point scales using "pessimistic/realistic/optimistic" terms. Such scales could be easily used in PMR to compare attainability/difficulty of goals between two groups.

Time-specificity

Authors specify if/how longer-term goals were adapted to the specific time frame of the research study.

In the case of multiple assessment, authors specify which assessment was taken as the target moment for goal achievement.

In the RCT of Lowe et al 1 , children were evaluated at 13 different time points. It is not reported which assessment point was taken as the reference to choose the 0 level (level that will most probably be attained after intervention) -criterion unmet.

Goal difficulty across experimental groups may have been unequal, at a given assessment point, due to differences in time-frames for goal completion.

Reliability of scale rating

Inter-rater reliability (IRR)

Inter-rater reliability of GAS scales is reported.

In a study of Steenbeek et al. in cerebral palsy, inter-rater reliability was reported based on two judges using video-taped performance of each goal 10 .

Reliability of GAS set by one team (especially an experienced one) does not presume that other GAS scales set by other teams, in other rehabilitation contexts, are reliable. Therefore IRR should be reported for the specific GAS scales generated in each study.

Criteria affecting IRR

Precise description of all levels

Five GAS levels have been precisely described pre-intervention for each scale.

"Goal: Reducing weekly shopping expenditure Total weekly food/household shopping expenditure = +2: less than $42.99 +1: $46.99-$43.00 0: $54.00-$47.00 -1: $63.00-$54.01 -2: greater than $63.00" 11 When all levels of the scales have not been precisely decided and described prior to intervention, authors often use adjectives such as "worse than expected", "much better than expected", to score goal attainment. This is a subjective appreciation that may be useful clinically but is too imprecise to objectively determine intervention efficacy. Non-unidimensional goals are impossible to score as progress on one dimension may not be accompanied by progress on another dimension and generate situations where GAS cannot be scored (see Grant 15 for an example). Bi (Multi) dimensional GAS's should be split into two (or more) unidimensional GAS prior to intervention start.

Measurability

Context of measurement

Context of performance measurement is clearly defined (prompts, cueing, support, amount of help/guidance, location…) and is controlled for during GAS rating. OR Changes in context are carefully manipulated across the GAS levels, with one change per level at a time 16 . e.g.: setting/prompting/guidance: "in order to create an irregular surface, a ladder is placed horizontally at a height of 15cm, the girl is asked to walk barefoot without orthosis, as quickly as possible, through the rungs over a distance of 8 meters. Only if she falls a therapist will help her holding one of her hands." e.g.: "GAS level -1: Prepares school bag but requires constant verbal guidance from the parents or teacher; GAS level 0: Manages to prepare the school bag using a check-list of necessary steps and under supervision; GAS level+1: Manages to prepare school bag alone, using a check-list of necessary steps; GAS level +2: No supervision required, child only occasionally forgets items." 14 7 Context of measurement influences performance on a given target goal (environment, fatigue, help provided…). These factors must be controlled in order to increase GAS scoring reproducibility.

Other criteria

Training

Researchers setting the GAS with the client and verifying GAS have received training in writing GAS, have practiced GAS writing, are aware of potential sources of bias in GAS, and are experienced in the goal domain/population.

Although training in GAS writing is reported as being important 18 , and successful training methods have been published 17 , most studies do not report on therapists' training. Some studies report to which practical GAS guide 16 19 they refer to, but without mention of training 20 21 . Those mentioning training do not explain the type of training (e.g.: "Experienced pediatric occupational therapists were trained in and completed the GAS collaboratively with the families, thus enhancing the reliability of the GAS"

22

).

Given the numerous and complex potential sources of bias in GAS processes, a team that is not experienced in using GAS methodology is unlikely to produce valid and reliable GAS. Further, a team without specific experience in the goal domain or the specific population with whom the intervention is tested, will have difficulty in predicting what can be attained in a given time-frame, even if experienced in GAS methodology in another domain (risk unrealistic goals, unequal difficulty across clients, irrelevant goals to the specific population…).

Examiner bias

The person scoring the GAS at the end of the intervention is independent from the team who set the GAS (and independent from the team that provided the intervention although the latter is not a GAS specific criterion). e.g.: "Goals were chosen and set before the patient was allocated to a group […] goal attainment was scored by an independent assessor at post-treatment and at follow-up." 20 . e.g.:"The therapist-GAS was scored from video by blind evaluators. The parent-GAS was scored by two blinded occupational therapists." If the same person sets the GAS and scores them, he/she is likely to be biased towards scoring a maximum of 0 ("attained as expected"). He/she may rely on memory of initial performance and subjective impression of improvement to score ambiguous progress. The independence of assessor should also be

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

respected when goals are client/family chosen when GAS is an outcome measure in research, (in contrast with clinical practice, where GAS scoring by the client may be relevant and appropriate).

Statistical analysis Ordinal nature of GAS scales is preserved using non parametric statistics (rank tests, medians, boxplots). e.g.:"It was decided not to use the popular T score in order to preserve the ordinal nature of the data […] group effects were demonstrated by testing the difference between all medians, […] using a two-tailed Wilcoxon signed ranks test" The performance of arithmetic operations such as T-scores on ordinal data is scientifically not valid 23 and should be discouraged, as it yields erroneous interpretation of data. In GAS, the problem is multiplied by characteristics of the T-score formulae (unknown true value of ρ, T-score variation according to the number of goals per client even at equal degree of attainment, highly subjective weighting of goals which, although clinically meaningful, introduces further potential arithmetic incoherence in the final Tscore).

Example of GAS

One (or more) example of a typical GAS full scale, extracted from the trial, is provided.

A list of chosen goals is reported. Some authors provide an example of full GAS scale in the paper 10 11 24 in the methods or results section.

Examples of goal types can be given (1) by providing examples of goal in the paper 25 26 27 2 28 ; (2) by reporting all chosen goals in appendix 25 11 ;

(3) by reporting goal type and frequency of each type without providing an exhaustive goal list 22 29 10 .

Providing examples allows the reader a quick judgment/idea of goal type, precision of goal and levels description, measurability, unidimensionality of GAS. The lack of GAS examples contributes to make GAS seem like an abstract outcome measure: unlike standardized scales, the reader cannot build a representation of the target goals of the intervention. Therefore, reporting all goals in an appendix and providing example(s) of full GAS scales (representative of different domains measured) should be encouraged to increase interpretability.

TABLE 1: Goal Attainment Scaling (GAS) methodology quality appraisal for rehabilitation efficacy studies

Criteria are grouped in accordance to the clinimetrics they mostly impact, although some items may impact both reliability and validity. If GAS is used in a controlled trial, GAS scales should not only be checked but also compared between groups, similarly to the methodology proposed by Ruble et al. 9 For large trials, we propose (as for IRR) that at least 20% of GAS scales be checked. ICF: International Classification of Functioning; IRR: inter-rater reliability; RoM: Range of motion

  M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 12 independence of the person(s) evaluating goal achievement from those who set the goals; use of appropriate statistical analysis methods; and provision of examples of GAS scales in research reports.

  developed guides and methods of M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 13 collaborative goal setting

  measured in degrees) in a gait analysis laboratory and the range of motion is transformed into a GAS scale presenting a range of ankle positions as the goal.

1 .

 1

10

 10

.

section(s), division(sAcknowledgment of financial support, including grant numbers: Writing this paper was partially supported by SOFMER (French society of PMR)-Allergan prize, won by the clinical team of Dr. Krasny-Pacini in October 2012.

We would like to thank Dr. Roumenoff-Turcant from Lyon for her feedback on the paper.