Keywords: total cost of ownership, optimization based design, plug-in hybrid electric vehicles, component sizing, optimal energy management

This paper introduces a novel methodology for minimizing the total cost of ownership (TCO) of plug-in Hybrid Electric Vehicles (pHEVs) in the context of heavy duty application. This approach enables the best powertrain components sizing to be determined coupled with the optimal energy management. A detailed financial cost model of the truck is developed to take into consideration the differences in spending from a conventional Diesel vehicle. The introduction of eight design variables enables to explore both internal combustion engine, electrical motor and battery alternative designs. Then, a coupled optimization problem is formulated as a bi-level form with powertrain optimal energy management based on a combinatorial problem formulation solved by Simplex algorithm and Branch & Bound in the inner loop and exhaustive evaluation of the powertrain designs in the outer loop. The results obtained from this new optimization framework show a 2% potential financial savings for a pHEV operating by the end of the decade on a regional haul application while decreasing CO2 emissions by more than 38% compared to a conventional Diesel truck.

Total Cost of Ownership Optimization of a plug-in Hybrid Electric Truck Operating on a Regional Haul Cycle X. Huin * , * * M. Di Loreto * E. Bideaux * H. Benzaoui * *

INTRODUCTION

New European CO2 emission standards for heavy-duty vehicles sets at least 30% average reduction from new lorries in 2030 compared to a 2019-2020 reference period. In addition zero-emission zones are planned to be introduced in large European cities such as Paris in 2024 to restrict or ban Diesel internal combustion engine (ICE) usage with the aim of improving air quality.

In the meantime, minimizing the vehicles total cost of ownership (TCO) is constantly sought in the transportation industry in order to improve the profitability of operations. Plug-in Hybrid Electric Vehicles (pHEVs) can certainly improve fuel economy benefits compared to a conventional Diesel truck [START_REF] Stockar | Optimal control for Plug-in Hybrid Electric Vehicle applications[END_REF], but the electrical charging costs from the grid combined with the additional components of the electric propulsion including the motor drive systems, the battery pack and the on-board charger are increasing the TCO of the vehicle. In order to minimize the TCO of the pHEVs, the objective therefore consists in optimizing the powertrain design as well as its energy management strategy. [START_REF] Silvas | Integrated Optimal Design for Hybrid Electric Vehicles[END_REF].

However, these studies combining design and energy management optimization often consider simplified total cost of ownership formulation. [START_REF] Pourabdollah | Optimal Sizing of a Parallel PHEV Powertrain[END_REF][START_REF] Silvas | Comparison of Bi-Level Optimization Frameworks for Sizing and Control of a Hybrid Electric Vehicle[END_REF]. Nevertheless, other financial costs not included in these papers can also differ between the two types of vehicles such as the insurance, the maintenance, the funding cost or the resale value which must also be taken into account to better reflect the real expenses of owning such a vehicle and to be able to draw a fair comparison with a purely Diesel truck.

In Huin et al. (2021a), we developed an optimization based methodology for the design of hybrid dedicated internal combustion engines. The introduction of five design variables were used to perform parametric transformations of the engine static efficiency and maximum torque characteristics. This methodology was however limited to the study of alternative engine designs. In order to optimize even more the pHEVs powertrain, [START_REF] Pourabdollah | Optimal Sizing of a Parallel PHEV Powertrain[END_REF] and [START_REF] Silvas | Comparison of Bi-Level Optimization Frameworks for Sizing and Control of a Hybrid Electric Vehicle[END_REF] proposed one additional design variable for scaling linearly in torque the electric machine (EM) static efficiency and maximal torque and one for optimizing the number of battery cells. However, this last sizing variable only allows to explore batteries of the same family with identical ratio of maximum power output to capacity performances (C-rate). In this paper, we introduce a third design variable which, combined with the second one, offers the opportunity of scaling independently the battery number of cells and its C-rate. [START_REF] Fathy | On the coupling between the plant and controller optimization problems[END_REF] studied the different strategies that can be used to solve design and control optimization problems. Among the four possible approaches, only the bi-level (or nested) and the simultaneous strategies can guarantee optimality. However, this second strategy requires to heavily re-formulate the problem as in [START_REF] Pourabdollah | Optimal Sizing of a Parallel PHEV Powertrain[END_REF] where the entire problem is formulated as a convex optimization problem. This leads to a lower fidelity of component modeling and simplified energy management without discrete variables such as the transmission selected gear for example. Since this strategy strongly affects the vehicle model fidelity, a bi-level approach is chosen in the present work.

Review and comparison of different optimal energy management strategies were performed. [START_REF] Silvas | Comparison of Bi-Level Optimization Frameworks for Sizing and Control of a Hybrid Electric Vehicle[END_REF]. Dynamic programming developed by Richard Bellman is often implemented and used because it offers a global optimum and deterministic solution to this mixed and non-convex problem. However, this technique requires a lot of storage memory for large scale problems and the computation time increases exponentially with the number of state variables. [START_REF] Bellman | Dynamic programming[END_REF]. In addition it can be difficult to integrate some of the constraints without introducing penalty coefficients [START_REF] Murgovski | Hybrid powertrain concept evaluation using optimization[END_REF] that require calibration leading to a lack of robustness. [START_REF] Chauvin | Cost Optimization for Plug-In Integration in a Hybrid Electric Mini-Excavator with Mixed-Integer Linear Programming[END_REF] proposed a re-formulation of the original problem under several combinations of mixedinteger linear problems which can be solved using Branch & Bound algorithm. Commercial solvers such as Cplex or Gurobi have good computational performances for this type of problem ensuring acceptable resolution time. With the same ability to find a global optimum as dynamic programming, this approach enables to easily integrate additional constraints in the problem formulation without requiring calibration. Hence, we select this method as it provides a fair evaluation of the potential achievable TCO for each powertrain design. However, the proposed approach have still some limitations regarding the computation time for large problems. With the introduction of all the different design variables previously mentioned, the development of an improved optimization framework has been conducted and is also one of the contributions of this paper. Some integers constraints of the problem have been relaxed in order to solve linear sub-problems faster and heuristically sort the design to converge faster towards the best solution.

This paper is organized as follows: Section 2 presents the TCO calculation. Then, Section 3 introduces the problem formulation and Section 4 presents the new optimization framework. Section 5 describes the results achieved by the methodology and the last section summarizes conclusions.

TCO CALCULATION

In this paper the optimization problem is defined as follows: from a predefined regional haul mission and a parallel hybrid topology, we want to find the best powertrain component sizes and the optimal energy management strategy to minimize the total cost of ownership (TCO) also called the objective function of our problem. It consists of operational costs which are function of the mileage of the vehicle: fuel and electricity costs but also fixed costs which includes truck purchase, maintenance, insurance, funding and resale costs.

Operational costs

Operational costs include both fuel and electrical grid charging costs which form the energy consumption of the pHEV truck. For the conventional vehicle, only fuel cost is obviously considered. These financial charges are calculated for the predefined vehicle duty cycle discretized in N steps and then we assume to repeat this mission M times over a total period of ownership. It can be expressed as:

J o = N k=1 M c 1 ṁfuel (k) + c 2 i c (k) ∆t(k) (1)
with c 1 and c 2 constants to convert fuel and electricity consumption to financial costs, ṁfuel the engine fuel mass flow, i c the battery cell current and ∆t the sampling time between two discretized steps. It should be noted that the Diesel and electric energy consumption of the truck is highly dependant of the sizing selected as well as the energy management strategy defined.

Fixed costs

Fixed costs regroup the other vehicle expenses which are assumed to be independent of the transportation missions. First of all, the initial purchase of the vehicle consists of the truck frame, cabin and combustion driveline for the reference truck and is denoted as

J ref v .
In addition the pHEV truck powertrain is also equipped with a motor drive system, a battery pack and an on-board charger. Its combustion driveline and these components sizing are to be found during the optimization. While the electric machine cost is chosen linear with respect to its maximum power, the battery price model is of the following form:

J bat = a 1 P bat + a 2 E bat (2)
with a 1 and a 2 constant values to calculate with reference battery costs, P bat and E bat the maximal power and capacity of the battery considered. The total purchase cost of the pHEV is designated as J pHEV v

. A loan with a rate of r% must also be included in the TCO to finance the initial purchase. This additional cost can be expressed as:

J f i = J v rT (3)
where T corresponds to the total period of ownership in years. Insurance cost for the reference vehicle is described by J ref i and a linear relationship is assumed to estimate the pHEV insurance cost:

J pHEV i = J ref i J pHEV v J ref v . (4)
Vehicle resale percentage at the end of ownership with respect to the purchase cost are denoted by r s and assumed to be zero for the battery pack. Finally, maintenance costs are assumed to be equivalent between the two vehicles since the additional electric propulsion requires barely no servicing during its lifetime. They are integrated into J e which regroup all other equivalent costs such as the driver salary, the toll and taxes or the trailer. The fixed costs can then be summed up as:

J ref f = J ref v (1 + rT -r s ) + J ref i + J e J pHEV f = J pHEV v (1 + rT -r s ) + J bat r s + J pHEV i + J e
(5) with J bat the battery cost of the pHEV truck. In the end the total cost of ownership or the objective function of our problem consists in minimizing:

J = J o + J pHEV f . (6)

PROBLEM FORMULATION

In order to find the sizing that minimizes the total cost of ownership of the truck, an accurate evaluation of the vehicle fuel consumption for each powertrain design must be performed. Indeed, there is a strong coupling between the sizing and the energy management strategy. [START_REF] Fathy | On the coupling between the plant and controller optimization problems[END_REF]. It thus requires a simultaneous resolution of the complete problem. The assumptions made and the chosen reformulation of this optimization problem to ease the resolution are described in the next section and are similar to Huin et al. (2021a).

Vehicle powertrain description modeling

The considered topology is a parallel plug-in hybrid electric truck as depicted in Fig. 1. The powertrain is mainly composed of the internal combustion engine connected to the transmission system by a clutch and the electric propulsion which consists in the energy storage system (battery), an on-board charger, the power electronics and the electrical motor. The latter is connected to the counter shaft of the gearbox by an additional reduction gear in order to operate in higher efficiency areas while driving. Using a backward modeling approach, we can deduce the

F t (k) = (m v + I v (k))a(k) + F r (k) + F g (k) + F a (k) = (m v + I v (k))a(k) + m v gc rr cos α(k) + m v g sin α(k) + 1 2 ρ a A f c d v 2 (k), (7)
with m v the vehicle mass which is dependent of the powertrain sizing, I v the powertrain rotating parts inertia which changes with engaged gear, a the vehicle acceleration, F r the rolling resistance, F g the gravitational force, F a the aerodynamic drag force, g the gravitational constant, c rr the rolling resistance coefficient, α the road angle with horizontal, ρ a the air density, A f the frontal area, c d the aerodynamic coefficient and v the vehicle speed. From the traction force and the vehicle speed we can respectively calculate the torque demand at the output shaft T d of the transmission and its rotational speed ω d that the engine and the electrical motor must meet in order to follow the input trajectory:

         ω d (k) = v(k)R f g R w , T d (k) = F t (k)R w + S f (v(k)R f g ) e f + D R f g η sgn(Ft(k)) f g (8)
with R f g the final gear ratio, R w the wheel radius, S f and e f speed dependent losses coefficients, D dry force coefficient and η f g the final gear efficiency. sgn refers to the mathematical sign function. The 12-speed automated manual gearbox is represented by gear-dependent ratios γ i for i ∈ [1, ..., 12]. With the choice of the following command variables : the engine torque T ice , the gear selected γ, the engine state st ice and the clutch state st clu , we can then deduce the relation of the power split between the two actuators and write the expression of the electrical motor (EM) torque T em :

T em (k) = T d (k) R em (γ(k))η em (γ(k)) sgn(Tem(k)) + S g ω eg d (k) -T ice (k)R ice (γ(k))η ice (γ(k)) sgn(Tice(k)) , (9)
with R em and R ice the ratios of the EM and the ICE to the output shaft of the gearbox, η em and η ice the efficiencies through the gearbox, S g and e g speed dependent losses coefficients. The Diesel engine and the electrical machine are characterized by their static efficiency maps and maximum torque curves. Engine rotation speed is function of the engine state (on or off), clutch state (engaged or disengaged) and the gear selected:

ω ice (k) =    0 if st ice (k) = 0 ω idle if st ice (k), st clu (k) = (1, 0) ω d (k)R ice (k) otherwise, ( 10
) where ω idle is the engine idle speed. The electrical machine rotation speed is dependent of the gear selection:

ω em (k) = ω d (k)R em (γ(k)) (11)
The following relation establishes the power demand on the battery P bat :

P bat (k) = ω em (k)T em (k) + P aux (k) + P losses em ω em (k), T em (k), s em , ( 12
)
with P losses em the electric machine and power electronics losses map, P aux the auxiliary power demand of the vehicle and s em the EM sizing variable detailed further. The battery is modeled using an equivalent circuit including an ideal open-circuit voltage source V oc in series with an internal resistance r c as in Guzzella and Sciarretta (2007) expressing the cell current i c as:

i c (k) = s P bat V oc -V 2 oc -4rcP bat (k) s P bat nc 2r c , (13)
where n c represents the number of cells in the battery pack and s P bat is a battery sizing variable described in the next section. Finally the state of charge SoC of the battery can be expressed as:

dSoC dt = s C bat i c (k) s P bat Q c (14)
with Q c the cell capacity and s C bat a second battery design variable.

Sizing and command variables

As previously mentioned, the sizing optimization of the powertrain components of the pHEV truck is one of the levers that enables the total cost of ownership to be minimized. In a previous paper we develop an optimization based methodology for hybrid electric vehicle dedicated internal combustion engine design. (Huin et al., 2021a). With the introduction of five sizing variables, we can explore alternative engine designs through parametric transformations of engine static efficiency map and maximum torque curve. These variables are then optimized in the framework presented in the next section to find the most suitable engine design for our problem.

In addition to this work, we introduce in this paper three extra sizing variables for the electrical machine and the battery designs. The first one denoted s em consists in scaling linearly with respect to the torque a reference electric machine efficiency map and its maximum torque curve (both in propulsive and regenerative operating modes) as in [START_REF] Pourabdollah | Optimal Sizing of a Parallel PHEV Powertrain[END_REF] and [START_REF] Silvas | Comparison of Bi-Level Optimization Frameworks for Sizing and Control of a Hybrid Electric Vehicle[END_REF]. A similar approach is used for the battery with s P bat which scales the number of cells connected in parallel in the battery pack at a given voltage. This variable enables to explore different power outputs and capacity of the battery.

Finally, the last design variable s C bat enables to investigate different battery C-rates which correspond to the maximum power the battery can discharge relatively to its maximum capacity. In other words, it measures the ratio between the battery maximum discharging power and capacity. From a reference battery characteristics, s C bat can be scaled to explore new battery capacity with a fixed maximum power output:

with δω, δT , δdω, δdT and s eng the five engine design variables whereas the vector of command variables u(k) is expressed as:

u(k) = {T ice (k), γ(k), st eng (k), st clu (k)}. ( 17
)

Constraints and performance requirements

As in Huin et al. (2021a), not only static operating constraints of the powertrain components such as maximum and minimum speed and torque for the engine or the electrical machine, limits on the current and the state of charge of the battery are applied to our problem but also additional constraints on the engine. Due to the air management system and in particular the presence of a turbocharger, the response dynamics of the engine is not neglected and therefore the torque produced by the ICE is constrained as:

T ice (k + 1) -T ice (k) ∆t(k) -∆T ice ≤ 0 (18)
with ∆T ice the engine maximum positive change in torque over time. Moreover, a constraint on the number of engine state changes is added to the optimization problem to prevent early degradation of ICE internal components coming from excessive occurrences of ignition and shutdown phases. This is expressed as:

1 2 N k=1 |st ice (k + 1) -st ice (k)| -N stice ≤ 0 (19)
where N stice corresponds to the maximum number of engine state changes possible during the cycle.

Vehicle performance requirements both in full electric and hybrid modes are also taken into account in this study. Gradeability, acceleration and range in zero-tailpipe emission requirements are used to determine minimal battery power and capacity as well as engine and electrical power through backward approach and the vehicle model presented in the previous section. These requirements then act as a constraint on the values of the design variables and help reducing the search space while ensuring that the powertrain found will meet the expected performances.

OPTIMIZATION FRAMEWORK

As presented in the previous section, the optimization problem has a strong coupling between the sizing and the command variables and requires a simultaneous resolution of the complete problem. In addition energy management is written using mixed-integer and non convex mathematical expressions making the problem difficult to solve.

A chosen reformulation of the problem based on a bilevel strategy and derived from operational research is selected in this study. [START_REF] Gaoua | A Combinatorial Optimization Approach for the Electrical Energy Management in a Multi-Source System[END_REF][START_REF] Chauvin | Cost Optimization for Plug-In Integration in a Hybrid Electric Mini-Excavator with Mixed-Integer Linear Programming[END_REF]. After discretizing sizing and continuous com- In order to accelerate the resolution time, integer constraints of these problems are relaxed in a first step to enable a fast linear problem (LP) resolution. The algorithm finds the command inputs u * LP that globally minimizes the objective function for each sizing combination and for the given drive cycle. This optimal solution is found using a Simplex algorithm from a commercial solver such as [START_REF] Gurobi Optimization | Gurobi optimizer reference manual[END_REF]. It should be noted that the parallelization of this task enables to speed up even more the solving process. Then the solutions of the different LPs are sorted with respect to their objective function values in ascending order. The MILPs are then solved starting by the lowest potential TCO obtained from the LP resolution using a Branch & Cut algorithm and the best value sol from u * is stored. Once a mixed-integer solution has a lower objective value than the next LP, the algorithm is stopped and the best solution found so far corresponds to the optimal one denoted J * (p * , u * ). The optimization approach is depicted in Fig. 2.

The optimization process has been improved compared to the work of [START_REF] Chauvin | Cost Optimization for Plug-In Integration in a Hybrid Electric Mini-Excavator with Mixed-Integer Linear Programming[END_REF] in order to enable the resolution of larger problems in acceptable computation time. This approach has been selected and developed because it has better computational performances than dynamic programming for optimization problems with multiple command variables and enables to more easily take into account additional constraints.

SIMULATION AND RESULTS

The mission profile used in this study is the regional delivery cycle (see Fig. 3) developed by the European Commission to measure the CO2 emissions and fuel consumption of heavy-duty vehicles in Vehicle Energy Consumption Calculation Tool (VECTO). In order to take into account possible future regulations in cities, we have included zero-emission zones at the beginning and at the end of the cycle where the vehicle operates at relatively low speeds. This additional constraint compels the truck to operate in fully electric mode. Since the vehicle has plug-in capability the cycle is assumed to be performed four times in a row, i.e. 400km, every day without charging events in order to simulate real depleting scenario. Moreover this repetition of operating conditions enables a faster resolution of the problem while providing a representative duty cycle. In addition a variable time discretization of the cycle based on torque demand and vehicle speed temporal variations is performed. It enables to reduce even more the computational time and maintain the fidelity level of the new discrete cycle. This pre-processing of the duty cycle is necessary to obtain a good trade-off between the resolution time and the results quality and is similar as the one developed in Huin et al. (2021a). In order to evaluate the total cost of ownership of the reference Diesel vehicle, we perform a first optimization of the powertrain energy management. In this case the optimization is limited to the selection of the transmission gear at each time step. From this fuel consumption result, the TCO of this vehicle can then be easily calculated. Then the coupled optimization of the 8 design variables and energy management is performed leading to a search space of more than 32000 different pHEV configurations since between 3 and 5 values are considered for each design variable. This exhaustive design research ensures that the best solution will be found among these different combinations. At the end of the process, the couple (p * , u * opt ) minimizing the pHEV's TCO is identified. The problem with a size of 8 design variables, 444 command variables for each of the 2000 steps of the cycle is solved using Matlab 2017b in 6.5 hours on a personal computer with 16-GB random access memory and a 2.90 GHz processor. The TCO results are detailed in Fig. 4. As anticipated the pHEV has higher vehicle ownership and fixed costs compared to the reference Diesel truck of about 6% because of the additional electric propulsion that increases both the insurance, the truck purchase and its financing. However, the operational costs which consists in the fuel and electricity consumption are much lower for the electrified vehicle. From a contribution of 25% of the energy consumption in the Diesel vehicle, the optimized pHEV decreases these expenses to less than 17% by globally increasing the powertrain efficiency from 37% to 62 % thanks to the electric propulsion and reducing the fuel consumption with the battery charging from the grid. As a result, the hybridation of this vehicle can lead to a potential gain on total cost of ownership of almost 2% by the end of the decade on this regional haul cycle for a five years ownership while decreasing tank to wheel CO2 emissions by over 38%.

The best sizing obtained by the optimization process is depicted in Fig. 5. The first observation we can make is that the internal combustion engine is downsized by almost 40%. Indeed, the additional source of power from the electrical motor can be used when the power demand is high making the reference engine inappropriate in this new powertrain. More detailed results on the new engine design are presented in Huin et al. (2021a). The obtained EM sizing is mainly due to the full electric operating points in the zero-emission zones in addition to the regenerative abilities to maximize energy recovery during braking phases. Conclusions are similar for the battery maximal power as this component operates in tandem with the electric machine. However, the optimal battery capacity is a compromise between weight, cost and fuel saving as it fulfills extensively the minimal range in full electric requirements by over 20%.

These results clearly highlight the potential financial savings of a plug-in hybrid electric truck operating by the end of the decade on a regional haul application while significantly decreasing CO2 emissions and energy consumption as well as having the ability to run in full electric mode.

The methodology developed provides a robust optimization framework for minimizing total cost of ownership of hybrid electric trucks.

CONCLUSION

In this paper, a detailed total cost of ownership minimization based-methodology has been developed for sizing the powertrain of a plug-in hybrid electric truck. The good computational performances of this approach enable to address a large problem with multiple design variables with an acceptable resolution time. The right sizing of the internal combustion engine, the electrical motor and the battery of the vehicle could provide a potential TCO reduction of almost 2% for the pHEV compared to a reference Diesel vehicle by the end of the decade on a regional haul mission with a five years possession time.

In addition, the electrified vehicle offers a CO2 emissions reduction of over 38% while bringing the possibility of a full electric mode for urban operations.

Further work will involve the development of a sensitivity analysis methodology to measure the robustness of the solution resulting from the optimization but also to understand the impacts of input parameters variations on the results such as the drive cycle, the powertrain design or the vehicle characteristics (Huin et al., 2021b).

Fig. 1 .

 1 Fig. 1. Parallel plug-in hybrid topology considered required traction force F t to propel the vehicle at the desired velocity and to overcome external forces:F t (k) = (m v + I v (k))a(k) + F r (k) + F g (k) + F a (k) = (m v + I v (k))a(k) + m v gc rr cos α(k)

  bat are respectively the scaled and reference battery maximum power output, E new bat and E ref bat the scaled and reference battery capacity. The vector of design variables p is therefore set as: p = {δω, δT, δdω, δdT, s eng , s em , s P bat , s C bat },

Fig. 2 .

 2 Fig. 2. Optimization framework developed mand variables, n combinatorial mixed integer linear subproblems (MILP) are formulated. In order to accelerate the resolution time, integer constraints of these problems are relaxed in a first step to enable a fast linear problem (LP) resolution. The algorithm finds the command inputs u *LP that globally minimizes the objective function for each sizing combination and for the given drive cycle. This optimal solution is found using a Simplex algorithm from a commercial solver such as[START_REF] Gurobi Optimization | Gurobi optimizer reference manual[END_REF]. It should be noted that the parallelization of this task enables to speed up even more the solving process. Then the solutions of the different LPs are sorted with respect to their objective

Fig. 3 .

 3 Fig. 3. Regional delivery cycle: vehicle speed (in blue), road altitude (in red) and zero-emission zones (in green)

Fig. 4 .

 4 Fig. 4. Comparison of Total Cost of Ownership for oper-ating Diesel reference vehicle and optimal pHEV on Regional Haul application.

Fig. 5 .

 5 Fig. 5. Comparison of tank to wheel CO2 emissions and powertrain sizing between the reference Diesel vehicle and the optimal pHEV.

Table 1

 1 presents a non-exhaustive list of the parameters values used for this numerical application. The cost assumptions used in this study are based on Volvo internal projections for the end of the decade.

Table 1 .

 1 Cost and vehicle parameters.

	Parameter	Value	Unit
	Possession time	5	Years
	Price of Diesel	1.30	EUR/L
	Price of electricity	0.16	EUR/kWh
	Reference vehicle mass	28 146 kg
	Reference engine maximum power 370	kW
	Reference EM peak power	350	kW
	Reference battery pack capacity	65	kWh
	Minimal ZEZ range requirements	50	km

ACKNOWLEDGEMENTS

The authors would like to thank the ANRT and Renault Trucks for funding this research.


Alexia Carrizales
email: alexiacarrizales1@gmail.com

Susan Branje
email: s.branje@uu.nl

Lyda Lannegrand
email: lyda.lannegrand@u-bordeaux.fr

Disentangling between-and within-person associations between empathy and prosocial behaviours during early adolescence

Keywords:

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Disentangling Between-and Within-person Associations between Empathy and Prosocial Behaviour during Early Adolescence

Prosocial behaviour has been defined as voluntary, intentional actions that result in benefits for another individual or group (e.g., [START_REF] Caprara | Why and how to promote adolescents' prosocial behaviors: Direct, mediated and moderated effects of the CEPIDEA school-based program[END_REF] and are considered as fundamental to the development of social competence and morality during childhood and adolescence [START_REF] Eisenberg | Prosocial development[END_REF]. A large body of empirical research has shown the beneficial role of prosocial behaviour not only for the target but also for the provider of prosocial behaviour (e.g., [START_REF] Eisenberg | Empathy-related and prosocial responding: Conceptions and correlates during development[END_REF]. Adolescents who engage in prosocial behaviour have a range of positive personal outcomes such as higher selfesteem, self-efficacy, self-concept clarity, well-being [START_REF] Fu | Longitudinal relations between adolescents' self-esteem and prosocial behavior toward strangers, friends and family[END_REF][START_REF] Zuffianò | The relation of pro-sociality to self-esteem: The mediational role of quality of friendships[END_REF], agreeableness [START_REF] Hilbig | Personality and prosocial behavior: Linking basic traits and social value orientations[END_REF] and academic achievement [START_REF] Gerbino | Adolescents' prosocial behavior predicts good grades beyond intelligence and personality traits[END_REF]. At the relational level, adolescents who engage in prosocial behaviour are more likely to have positive peer relationships and high friendship quality, are capable of managing social challenges, are more socially competent and have a higher sense of community and civic engagement [START_REF] Boele | Linking Parent-Child and Peer relationship quality to empathy in adolescence: A Multilevel Meta-Analysis[END_REF]. Moreover, prosocial behaviour is considered as a protective factor against externalising (e.g., aggression, delinquency), internalising (e.g., depressive symptoms, suicidal ideation) and relational problems (loneliness, discrimination, peer victimisation) (Davis et al., 2016;[START_REF] Griese | Prosocial behavior as a protective factor for children's peer victimization[END_REF][START_REF] Mcdonald | Prosocial behavior moderates the effects of aggression on young adolescents' friendships[END_REF]. It is therefore important to understand the factors that promote prosocial behaviour.

The reciprocal effects of empathy and prosocial behaviour

The bulk of theoretical literature suggests that prosocial behaviour is motivated by empathy, and several empirical studies have shown empathy and prosocial behaviour are associated (e.g., [START_REF] Davis | Empathy as a driver of prosocial behaviour: Highly conserved neurobehavioural mechanisms across species[END_REF][START_REF] Hoffman | Empathy and prosocial behavior[END_REF]. Nevertheless, still little is DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 3 known about the direction of effects between empathy and prosocial behaviour.

Longitudinal research has identified adolescence as a sensitive period for the development of empathy and prosocial behaviour, yet longitudinal research on early adolescents is scarce. The aim of this study was to add to the extant literature by examining the effects between empathy and prosocial behaviour during early adolescence at the within-person level, while controlling for stable between-person differences.

Empathy should be seen as a complex multidimensional construct. The most common distinction is between 'cognitive' and 'affective' empathy [START_REF] Cuff | Empathy: A review of the concept[END_REF].

Cognitive empathy refers to the understanding of another person's feelings and relies on perspective taking or mentalising processes [START_REF] Davis | The effects of dispositional empathy on emotional reactions and helping: A multidimensional approach[END_REF]. Affective empathy includes sharing similar emotions (emotional contagion) or feeling sorrow or concern for the observed other (empathic concern) and to a lesser extent, emotional disconnection [START_REF] Carré | The basic empathy scale in adults (BES-A): Factor structure of a revised form[END_REF][START_REF] Singer | The social neuroscience of empathy[END_REF]. Affective and cognitive empathy are related (e.g., [START_REF] Jolliffe | Development and validation of the Basic Empathy Scale[END_REF][START_REF] Van Der Graaff | Prosocial behavior in adolescence: Gender differences in development and links with DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 35 empathy[END_REF] indicating that adolescents with a better understanding of others' emotions are likely to also experience more shared feelings or empathic concern.

Research has shown correlations between empathy and prosocial behaviour [START_REF] Eisenberg | Empathy-related and prosocial responding: Conceptions and correlates during development[END_REF][START_REF] Hoffman | Empathy and prosocial behavior[END_REF]. Some theoretical and empirical research, mainly cross-sectional, support that empathy is related to prosocial behaviour, these studies suggest that the affective and cognitive processes of empathy should provide the motivation to empathic action, i.e., helpful or prosocial actions, because in general, the individual helps someone who is in distress, pain, danger or other type of suffering [START_REF] Davis | Empathy as a driver of prosocial behaviour: Highly conserved neurobehavioural mechanisms across species[END_REF][START_REF] Siu | Predictors of prosocial behavior among Chinese high school students in Hong Kong[END_REF]. However, social cognitive theory posits that DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 4 exhibiting prosocial behaviour may also influence sociocognitive and socio-emotive tendencies, including empathy. First, engaging in prosocial behaviour provides adolescents with opportunities to take others' perspectives and show empathic concern [START_REF] Malti | Children's moral motivation, sympathy, and prosocial behavior[END_REF]. Second, prosocial behaviour elicit positive feedback from the environment, which may strengthen their social relationships and the image of themselves as a caring and understanding person, and may reinforce them to behave accordingly [START_REF] Crocetti | The interplay of Self-Certainty and prosocial development in the transition from late adolescence to emerging adulthood[END_REF].

Research has shown that the transition from childhood to adolescence engenders physical, cognitive, socio-emotional and contextual changes [START_REF] Steinberg | Adolescent development[END_REF] along with important social changes including the expansion of peer networks, interest in romantic relationships, and autonomy from families. It is a period of enhanced sensitivity for social experiences that might motivate certain behaviours, such as empathy-related responding [START_REF] Blakemore | Is adolescence a sensitive period for sociocultural processing[END_REF]. Indeed, studies have shown mean level increases in both empathy and prosocial behaviour during adolescence [START_REF] Caprara | Why and how to promote adolescents' prosocial behaviors: Direct, mediated and moderated effects of the CEPIDEA school-based program[END_REF][START_REF] Davis | Empathy as a driver of prosocial behaviour: Highly conserved neurobehavioural mechanisms across species[END_REF][START_REF] Eisenberg | Age changes in prosocial responding and moral reasoning in adolescence and early adulthood[END_REF][START_REF] Van Der Graaff | Perspective taking and empathic concern in adolescence: Gender differences in developmental changes[END_REF][START_REF] Van Der Graaff | Prosocial behavior in adolescence: Gender differences in development and links with DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 35 empathy[END_REF]. Thus, early adolescence might represent a unique window into developmental changes that might have long-term consequences for the individual well beyond the adolescent years [START_REF] Allemand | Empathy development in adolescence predicts social competencies in adulthood[END_REF].

To the best of our knowledge, few studies have examined the directionality of the effects between empathy and prosocial behaviour during adolescence. These studies showed that there are bidirectional relations between empathy and prosocial behaviour [START_REF] Caprara | Prosociality: The contribution of traits, values, and self-efficacy beliefs[END_REF][START_REF] Carlo | Longitudinal bidirectional relations between adolescents' sympathy and prosocial behavior[END_REF]. A recent longitudinal study found that earlier prosocial behaviour was related to empathy-related traits, but only for females and that empathic concern mediated the relation between perspective taking and prosocial DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 5 behaviour [START_REF] Van Der Graaff | Prosocial behavior in adolescence: Gender differences in development and links with DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 35 empathy[END_REF].

The interplay among empathy and prosocial behaviour: disentangling betweenand within-person associations

Although the few studies so far provide evidence of the expected bidirectional relations between empathy and prosocial behaviour, it should be noted that these studies have been based on statistics for between-person effects or aggregated within-and between-person effects. Thus, the practical implications of their findings to development at the individual level are unclear, considering that one should not equate between-person differences (i.e., whether group-level change in empathy is related to group-level change in prosocial behaviour) with within-person processes (i.e., whether individual-level change in empathy is related to individual-level change in prosocial behaviour) as these might differ in strength and direction [START_REF] Curran | The separation of Between-Person and Within-Person components of individual change over time: A latent curve model with structured residuals[END_REF][START_REF] Hoffman | Persons as contexts: Evaluating between-person and within-person effects in longitudinal analysis[END_REF]. For example, the fact that adolescents who report high levels of empathy than other adolescents are also likely to report higher prosocial behaviour does not necessarily imply that individual adolescents will experience higher prosocial behaviour when they improve their empathy. Only statistical analysis separating withinperson effects from between-person allow one to make valid statements regarding within-person processes. This information could lead to practical implementations such as the innovation of prevention and intervention efforts in order to support optimal empathy and prosocial behaviour development in adolescents. Only within-person change can allow us to answer questions such as when individuals increase in empathy compared to before, does their prosocial behaviour subsequently change compared to their own prosocial behaviour before?

The present longitudinal study aimed to explore the effects between empathy and prosocial behaviour in early adolescence at the within-person level while controlling for stable, between-person differences. Using state-of-the-art methodologies (i.e. Random-Intercept Cross-Lagged Panel Model; RI-CLPM, [START_REF] Hamaker | A critique of the crosslagged panel model[END_REF], which allows to partition the variance of the observed scores into between and withinperson variance and to examine cross-lagged relations at the within-person level. This approach is powerful considering that each adolescent acts as his/her own control, and any association between variables that remains results from adolescents' deviating from their own stable levels.

Gender differences

In addition, although gender differences in empathy and prosocial behaviour are consistently reported, there is a relative dearth of research on the role of gender in the bidirectional relations between empathy and prosocial behaviour [START_REF] Allemand | Empathy development in adolescence predicts social competencies in adulthood[END_REF]. Previous research suggests that gender was significantly related to the level of adolescents' prosocial behaviour and empathy, with males reporting lower levels of prosocial behaviour and empathy than females (e.g., [START_REF] Carlo | Longitudinal bidirectional relations between adolescents' sympathy and prosocial behavior[END_REF][START_REF] Crocetti | The interplay of Self-Certainty and prosocial development in the transition from late adolescence to emerging adulthood[END_REF][START_REF] Eisenberg | Age changes in prosocial responding and moral reasoning in adolescence and early adulthood[END_REF]. These gender differences in empathy might support the notion that the interplay between empathic concern and prosocial behaviour has a stronger reinforcing quality for females relative to males [START_REF] Carlo | A latent growth curve analysis of prosocial behavior among rural adolescents[END_REF][START_REF] Van Der Graaff | Perspective taking and empathic concern in adolescence: Gender differences in developmental changes[END_REF][START_REF] Van Der Graaff | Prosocial behavior in adolescence: Gender differences in development and links with DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 35 empathy[END_REF]. Considering social expectations, gender stereotypes and gender-specifi c socialisation practises, it is possible that females are more encouraged to develop empathic skills and display greater empathy than males (e.g., [START_REF] Christov-Moore | Empathy: Gender effects in brain and behavior[END_REF][START_REF] Kite | Gender stereotypes[END_REF]. This may not only result in differences in mean levels of prosocial behaviour and empathy but may also affect their bidirectional longitudinal relations. Thus, in the current study, we also examined DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 7 whether the pattern of longitudinal associations between empathy and prosocial behaviour differed for adolescent females and males.

The current Study

The purpose of the current study was to examine the longitudinal, within-person bidirectional relations between empathy and prosocial behaviour in early adolescence while controlling for trait-like differences at the between-person level. In this study when we use the term prosocial behaviour, we refer to a general prosocial disposition.

To disentangle the direction of effects between empathy and prosocial behaviour, we explored cross-lagged relations, thereby controlling for concurrent associations and stability of empathy and prosocial behaviour when estimating bidirectional effects over time. We expected that empathy would be associated with subsequent prosocial behaviour, and that earlier prosocial behaviour would also be associated with subsequent empathy. In addition, we examined within-person associations between prosocial behaviour and empathy employing an innovative within-person analytical approach (RI-CLPM; [START_REF] Hamaker | A critique of the crosslagged panel model[END_REF]. It was expected that within-adolescents changes in empathy would be related to within-adolescent changes in prosocial behaviour and that within-adolescents changes in prosocial behaviour would be related to within-adolescent changes in empathy. Last, because of well-known gender differences in empathy and prosocial behaviour [START_REF] Stuijfzand | Gender differences in empathic sadness towards persons of the sameversus other-sex during adolescence[END_REF][START_REF] Van Der Graaff | Prosocial behavior in adolescence: Gender differences in development and links with DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 35 empathy[END_REF], multigroup analyses across gender were performed. We expected that the bidirectional relations between empathy and prosocial behaviour would be stronger for females than for males.

DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 8

Method

Participants and procedure

The data used in this study were part of a three-wave study (2014)(2015)(2016)(2017) on the role of different contexts in empathy and prosocial behaviour development in adolescence. The waves were relatively equidistantly spaced, with the majority of the data collected annually in the same month as the first measurement time at each Junior High School participating in the study.

The sample consisted of 383 adolescents, 50.4% were males, the adolescents were in their 1 st and 2 nd year of Junior High School (n = 383; Mage = 12.15; SD = 0.35 at Time 1 (T1). There were no exclusionary criteria with respect to participant selection.

The schools were recruited from the pool of all the junior high schools in Bordeaux (the capital of Gironde, on the 6 th place among 34971 municipalities in France by demographic size). Access to this pool is available from the website of the French Ministry of Education. In order to broaden the population of interest, five junior high schools from different parts of Bordeaux metropolitan area were selected.

Adolescents and their parents received written information about the research prior to the study; the possibility of not participating was also explained in this letter.

Parents and adolescents provided written informed consent, and written informed consent was also obtained from the schools. Adolescents were assessed three times, respectively at T1, T2 and T3 with 1-year intervals by filling out a battery of questionnaires at school after school hours. Confidentiality was guaranteed explicitly.

Research assistants gave verbal instructions to the adolescents to complement the written instructions printed above each questionnaire. Completing the questionnaires lasted an hour. Adolescents received no reward for any wave they participated in.

In total, the percentage of missing data across waves and variables was 6.88%.

The percentage of missing values across the prosocial behaviour and empathy variables ranged between 1.1% and 12.5% respectively. The patterns of missingness were related to the fact that of the 383 adolescents, 47 were not present the day of the first measurement wave. Most of these 47 adolescents were males (n = 33) and in 7 th grade at the first wave. In our analyses these two variables have been considered, specifically, gender in the multigroup analyses and grade at first wave in the sensitivity analyses.

Therefore, although [START_REF] Little | A test of missing completely at random for multivariate data with missing values[END_REF] missing completely at random test was significant, χ² (37) = 75.44, p = <.001, the normed chi-square (χ²/df ) of 2.04 showed a good fit between the imputed and non-imputed sample scores, suggesting the pattern of missing data values was completely random [START_REF] Bollen | Structural equations with latent variables[END_REF]. In order to deal with the missing data, full information maximum likelihood (FILM) with robust estimates (MLR) in Mplus version 8.1 was used [START_REF] Enders | The relative performance of full information maximum likelihood estimation for missing data in structural equation models[END_REF][START_REF] Muthén | Mplus. Statistical Analysis with Latent Variables: User's Guide[END_REF]. The full-information maximum likelihood method is regarded as the method of choice for estimating missing data because it yields more accurate results than other approaches by minimising bias in regression and standard error estimates for all types of missing data when the amount of missing data is not larger than 20% [START_REF] Schlomer | Best practices for missing data management in counseling psychology[END_REF].

Measures

Empathy

In order to assess adolescents' perception of their own empathy, we used the Basic Empathy Scale (BES, [START_REF] Jolliffe | Development and validation of the Basic Empathy Scale[END_REF], a 20-item questionnaire rated on a 5-point Likert scale ranging from 1 (strongly disagree) to 5 (strongly agree). The scale was adapted in French by [START_REF] Carré | The basic empathy scale in adults (BES-A): Factor structure of a revised form[END_REF]. It includes three dimensions: emotional contagion (6 items, e.g. 'I get caught up in other people's feeling easily'), DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 10 cognitive empathy (8 items, e.g. 'I can understand my friend's happiness when she/he does well at something') and emotional disconnection (6 items, 'I am not usually aware of my friend's feelings'). To assess the psychometric validity of the three-factor structure on our sample, a CFA was performed, suggesting a good model fit, χ2 (116, N = 1545) = 879.25, p <.001, RMSEA =.06 [0.06-0.07], CFI=.95, WRMR = 1.81. The three-factor structure was supported except for three items (item 15 from emotional contagion with factor loading lower than .30, and items 6 and 20 from cognitive empathy loading on more than one dimension). All items were mean scored to create a composite to reflect empathy. Omega coefficients for the composite scales can be found in Table 1.

Prosocial Behaviour

Adolescents reported on their own prosocial behaviour toward others using a validated French adaptation [START_REF] Carrizales | Brief report: How many dimensions in the prosocial behavior scale? Psychometric investigation in French-speaking adolescents[END_REF] of the Prosocial Behaviour Scale (PBS) [START_REF] Caprara | A new scale for measuring adults' prosocialness[END_REF]. The French version assesses two types of prosocial behaviours: helping (6 items, e.g., 'I try to help others') and caring (8 items, e.g., 'I try to be close to and take care of those who are in need') on a 5-point Likert scale ranging from 1 (never/almost never) to 5 (almost always/always). In the present study, in order to avoid possible overlap among items of the caring dimension and the empathy scale, only four items1 for the caring dimension were used. All items were mean scored to create a composite to reflect prosocial behaviour. Omega coefficients for these composite scales can be found in Table 1.

Analytic Strategy

The research questions were examined with three types of analyses: (1) standard cross-lagged panel models (CLPM), (2) Random-Intercept cross-lagged panel models (RI-CLPM) and (3) Multigroup analyses (MG) with Mplus Version 8.1 [START_REF] Muthén | Mplus. Statistical Analysis with Latent Variables: User's Guide[END_REF].

To address our primary aim whether adolescents' empathy would be related to later changes in prosocial behaviour and whether adolescents' prosocial behaviour would be related to later changes in empathy, first, only for comparison purposes, a 'standard' cross-lagged panel model with bidirectional paths between empathy and prosocial behaviour was estimated, aggregating the between and within-person source of variance. For the sake of parsimony and to maximise measurement equivalence of the constructs across waves, we tested whether the stability paths, cross-lagged effects and within-time correlations were time invariant and could therefore be constrained over time, reducing the model complexity [START_REF] Hamaker | A critique of the crosslagged panel model[END_REF].

To examine our aim whether adolescents' empathy would be related to later within-person changes in prosocial behaviour and whether adolescents' prosocial behaviour would be related to later within-person changes in empathy, a Random-Intercept Cross-Lagged Panel Model [START_REF] Hamaker | How to run the RI-CLPM with Mplus[END_REF][START_REF] Hamaker | A critique of the crosslagged panel model[END_REF] was used.

This model separates the within-person variance from stable between-person differences through the inclusion of random intercepts. Before conducting these analyses, we examined if there was sufficient variance at the within-person (i.e., fluctuations of the same adolescents over time) and between-person level (i.e., differences between adolescents) by calculating the intraclass correlation coefficients (ICCs). The calculation of the ICC used in this study followed the same steps used in multilevel analyses. 'The individual data are decomposed into two separate models for the within and between groups structures. The total score is decomposed into an individual component (i.e., the individual deviation from the group mean) and a group component (i.e., the disaggregated group mean). This individual decomposition is used to compute separate within-and between-group covariances matrixes' (Heck, 2001, p.98).

For the RI-CLPM, at the between-person level, fixed effects (i.e., random intercepts) were included, which accounted for time-invariant trait-like individual differences. At the within-person level, an individual's deviation from his or her typical level, which was not explained by the between-person latent variable (i.e., random intercept) was accounted for using the within-person structural residuals [START_REF] Curran | The separation of Between-Person and Within-Person components of individual change over time: A latent curve model with structured residuals[END_REF]. Consistent with the random-intercept cross-lagged panel model, we created (a) the within-person centred variables for empathy and prosocial behaviour at all three waves; (b) specified autoregressive and cross-lagged pathways between the withinperson centred variables; and (c) estimated the covariances between the within-person centred variables at each time point (see [START_REF] Hamaker | How to run the RI-CLPM with Mplus[END_REF], for more information on specifying RI-CLPM models). Given the changes in empathy and prosocial behaviour shown by previous studies [START_REF] Eisenberg | Empathy-related and prosocial responding: Conceptions and correlates during development[END_REF][START_REF] Van Der Graaff | Prosocial behavior in adolescence: Gender differences in development and links with DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 35 empathy[END_REF], the means and intercepts were allowed to vary freely across time [START_REF] Hamaker | A critique of the crosslagged panel model[END_REF].

Finally, the measurement error variances of the observed scores were constrained to zero. Thus, all variation in the observed measures was completely captured by the within-person and between-person latent factor structure.

After determining which model best represented the data of the total sample, a multigroup approach was used in all analyses to examine whether CLPM and RI-CLP model significantly differed between males and females. The square difference test (ΔS-Bχ2) was used to compare a multi-group model, which constrained the parameters DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 13 across adolescent gender to be equal with a model that allowed the parameters to vary across adolescent gender. If the setting of different paths as equal did not significantly worsen the model fit, there were no significant differences between the two groups, indicating that interplay between empathy and prosocial behaviour is similar for females and males and we used the constrained models. To assess model fit, several goodnessof-fit indices were used including the Akaike Information Criterion (AIC), the Bentler Comparative Fit Index (CFI), the root mean square error of approximation (RMSEA), and the Standardised Root Mean square Residual (SRMR). Generally, lowest AIC, CFI values larger than .95, RMSEA values smaller than .05 and SRMR values smaller than

.08 indicate good model fi t [START_REF] Akaike | A new look at the statistical model identification[END_REF][START_REF] Kline | Principles and practice of structural equation modeling[END_REF].

Results

Preliminary Analyses

Means, SDs and correlations of the study variables can be found in Table 1.

Prosocial behaviour and empathy were positively related at each time point (r = .49, r = .54, r = .51) at T1, T2 and T3 respectively.

Insert Table 1 here

Standard Cross-Lagged Panel Model (CLPM)

To examine the bidirectional relations between empathy and prosocial behaviour, a cross-lagged panel model was estimated. The model fit was good S-Bχ 2 (4) 2.

Insert Figure 1 here As seen in Table 1, significant positive cross-lagged effects emerged from empathy at Wave 1 and Wave 2, to prosocial behaviour at Wave 2 and Wave 3, adolescents who reported higher empathy at Wave 1 and Wave 2, compared to their peers, tended to also report higher levels of prosocial behaviour at Wave 2 and 3 respectively. Moreover, in order to test the difference in size of different cross-lagged parameters (i.e., the ones from empathy to prosocial behaviour with the ones from prosocial behaviour to empathy), these cross-lagged paths were constrained to be equal, and the model fit of this constrained model S-Bχ 2 (11) = 23.728, p = .01, CFI= .97; RMSEA=.05, SRMR = .08 was compared with the freely estimated model in terms of their model fit indices. According to the chi-square difference test, the effects of the cross-lagged parameters could not be constrained to be equal, ΔS-Bχ 2 (1) = 5.426, p = .020, suggesting that the effect from empathy to prosocial behaviour was stronger than the effect from prosocial behaviour to empathy.

Within-person effects between empathy and prosocial behaviour

To test whether the statistical effects found with the 'standard' cross-lagged model reflect within-person linkages or between-person differences, first, the extent to which there is sufficient variance at within-and between-person levels were examined by calculating the ICC. The ICCs suggested that a substantial part of the variance in empathy and prosocial behaviour was located at the within-level. The exact estimates were: 40.6% for empathy and 38.6% for prosocial behaviour. Hence, a small portion of the variation is due to stable differences between adolescents. In other words, a substantial part of the variation observed (59.40% to 61.4%) was due to fluctuations over time in adolescents' reports of empathy and prosocial behaviour and not to stable between-person differences.

The fit of the initial, unconstrained RI-CLPM showed good fit S-Bχ 2 (2) = .105, p = .95, RMSEA =.00, CFI=1.0, SRMR = .01. Model comparisons indicated that the stability paths (except for empathy), cross-lagged effects, and within-time correlations were time invariant and could be constrained over time. The RI-CLPM is depicted in Figure 2. The model demonstrated excellent model fit: S-Bχ 2 (5) = 5.867, p = 0.32, RMSEA =.02, CFI = 1.0, SRMR = .03.

To evaluate whether the differentiated RI-CLPM fit our data better than the CLPM, the fit of both models was compared using the Akaike Information Criterion (AIC). The lower AIC suggested that the inclusion of the random intercept improved the model fit (AIC = 3678.760) compared with the cross-lagged panel model (AIC = 3685.998). The AIC favoured the more complex model, suggesting it provided a better representation of the data.

Insert Figure 2 here

The results revealed that, at the between-person level, there was a strong positive association between empathy and prosocial behaviour, r = .86, p <.05, indicating that DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 16 adolescents who reported higher empathy compared to other adolescents, tended to also report higher prosocial behaviour compared to their peers.

At the within-person level, only the autoregressive paths from empathy across time were statistically significant. Adolescents who's scored higher than their expected score on empathy were likely to subsequently score higher than their expected score as well, providing support for within-person carry-over effects. The cross-lagged parameters suggested that adolescents who reported more empathy than their expected/average score were related to higher than average scores on adolescents' prosocial behaviour one year later. Meanwhile, an adolescent's deviation in prosocial behaviour was not related to the deviation in adolescent's empathy. The detailed parameter estimates are reported in Table 2.

Insert Table 2

Multi-Group Analysis

We conducted a multigroup analysis to examine the group differences (female vs male) in reciprocal relations between empathy and prosocial behaviour.

A fixed CLPM with both autoregressive paths and cross-lagged paths constrained to be equal across groups, was compared with the freely estimated model in terms of their model fit indices. The two groups did not demonstrate a significant difference according to the chi-square difference test ΔS-Bχ 2 (17) = 22.26, p = .175. The fit of the CLPM gender constrained model was good S-Bχ 2 (25) = 33.625, p = .12, CFI= .97; RMSEA=.04, SRMR = .13.

DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 17

The unconstrained RI-CLPM fit was good S-Bχ 2 (4) = 1.70, p = .79, CFI= 1.0, RMSEA=.01, SRMR = .03. All the paths could be constrained without worsening the model fit. The two groups did not demonstrate a significant difference according to the chi-square difference test ΔS-Bχ 2 (10) = 12.46, p = .255. The fit of the RI-CLPM gender constrained model was good S-Bχ 2 (14) = 14.01, p = .45, CFI= 1.0, RMSEA=.00, SRMR = .05. In both CLPM and RI-CLP models, constraining the parameters across gender did not contribute to significant model fit change, reflecting non-significant differences in effects between females and males. Adolescent gender did not appear to moderate any associations that were investigated.

Sensitivity Analyses

Sensitivity analyses were conducted to explore whether results are robust when controlling for adolescent grade at first wave. Table A.1 in the Appendix A presents the fit indices for the initial models (CLPM, RI-CLPM) and the best fitting alternative models (either standard CLPM or RICLPM-constrained), after controlling for adolescent grade at first wave, by regressing the observed scores of empathy and prosocial behaviour on grade at first wave. As can be seen by comparing those estimates with the models without covariates, only few minor changes emerged on the CLPM and they mostly referred to the autoregressive stability of prosocial behaviour, which decreased between the first and second waves in the models with covariates (Table A. 2, Appendix A). All other parameter estimates remained largely unchanged, which indicates that the substantive results of this study also hold when controlling for adolescent grade at first wave.

Discussion

DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 18

This study examined how empathy and prosocial behaviour are associated to each other over time at the within-person, state-like level, while controlling for trait-like differences at the between-person level. We did so by using a RI-CLPM [START_REF] Hamaker | A critique of the crosslagged panel model[END_REF] and compared the results with the most commonly used method so far, the standard CLPM [START_REF] Mund | Beyond the Cross-Lagged Panel Model: Nextgeneration statistical tools for analyzing interdependencies across the life course[END_REF]. Without separating the within-person and between-person effects, results of the autoregressive, standard cross-lagged panel model analyses demonstrated a reciprocal association between empathy and prosocial behaviour over time, suggested that there were significant positive cross-lagged effects of empathy on prosocial behaviour and vice-versa. This suggests that adolescents who report higher empathy relative to their peers are also expected to show higher prosocial behaviour over time and adolescents who report higher prosocial behaviour relative to their peers are also expected to be more empathic over time. Overall, the associations at the between-person level were in accordance with previous studies (e.g., [START_REF] Van Der Graaff | Prosocial behavior in adolescence: Gender differences in development and links with DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 35 empathy[END_REF]. However, to the best of our knowledge, studies clarifying whether the identified reciprocal associations were due to between-person differences or withinperson fluctuations is lacking. Our results suggest that when these longitudinal associations were tested at the within-adolescent level, while controlling for stable differences and associations at the between-adolescent level, the results showed that at the within-person level, adolescents' empathy was related to within-person changes in prosocial behaviour over time but prosocial behaviour was not related to within-person changes in empathy, after controlling for stable between-person differences. In addition, these processes were found to be similar for males and females. The comparison of the results of the two types of models highlights the necessity of going beyond rank-order DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 19 changes in empathy and prosocial behaviour in understanding the 'true' nature of their associations over time.

Consistent with the hypothesis, these results showed that adolescents' empathy was related to later changes in their prosocial behaviour. This is in line with empirical studies from sociocognitive developmental approaches and social neuroscience theories, which supports the idea that empathy drives prosocial behaviour [START_REF] Batson | These things called empathy: Eight related but distinct phenomena[END_REF][START_REF] Benita | Integrative emotion regulation predicts adolescents' prosocial behavior through the mediation of empathy[END_REF][START_REF] Davis | Empathy as a driver of prosocial behaviour: Highly conserved neurobehavioural mechanisms across species[END_REF][START_REF] Eisenberg | Prosocial development[END_REF]. It further confirms previous research showing that individuals who are less empathic toward others are less inclined to behave prosocially in a variety of different scenarios [START_REF] Batson | Empathy-induced altruistic motivation[END_REF][START_REF] Batson | An additional antecedent of empathic concern: Valuing the welfare of the person in need[END_REF][START_REF] Davis | Empathy as a driver of prosocial behaviour: Highly conserved neurobehavioural mechanisms across species[END_REF][START_REF] Hein | I feel how you feel but not always: the empathic brain and its modulation[END_REF][START_REF] Masten | An fMRI investigation of empathy for 'social pain' and subsequent prosocial behavior[END_REF][START_REF] Morelli | Empathy and well-being correlate with centrality in different social networks[END_REF].

The current research extends these studies by demonstrating the effects of empathy on changes in prosocial behaviour at the within-person level. We found that early adolescents' empathy was related to within-adolescent changes in prosocial behaviour over time, that is, adolescents' changes in empathy, compared to their own empathy, were positively related to change in their prosocial behaviour. It suggests that the propensity to empathise with others might be one of the mechanisms underlying development of prosocial behaviour.

In addition, the results of RI-CLPM showed that within-adolescent change in empathy and prosocial behaviour were correlated. This means that within-adolescent changes in empathy were associated with within-adolescent changes in prosocial behaviour. Thus, if adolescents' empathy change, for instance, empathy increases, this should be related to adolescents' prosocial behaviour. The positive results on the within-DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 20 adolescents level indicate that, during early adolescence, fluctuations in adolescents' empathy were related to adolescents' prosocial behaviour change over time.

An important aspect to keep in mind is that the reported within-person effects of empathy on prosocial behaviour reflect averaged within-person effects. There is considerable within-adolescent heterogeneity in the processes that link empathy to prosocial behaviour, and future studies may explain this heterogeneity in within-person effects.

Contrary to our expectations, the pattern of longitudinal associations between prosocial behaviour and empathy did not differ between early adolescents' males and females in the standard cross-lagged-panel models nor in the random-intercept crosslagged panel models. The main focus of the present study, however, was on the longitudinal associations among the study variables. The issue of gender differences in the interplay among empathy and prosocial behaviour maybe more complex, since other studies found mixed results. The study by [START_REF] Van Der Graaff | Prosocial behavior in adolescence: Gender differences in development and links with DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 35 empathy[END_REF] examined the interplay among empathy concern, perspective taking and prosocial behaviour and only found a moderation of adolescents' gender in the association between prosocial behaviour and empathy concern. Specifically, they found this association to be limited to females only. This finding can be interpreted in the context of gender role expectations, in contrast to perspective taking, males are more likely to report lower scores of empathic concern and be less affected than females that may be influenced to a larger extent by environmental variables, such as the parental and peer environment.

The results for the standard CLPM used for comparison purposes that aggregates between-and-within-person variances revealed that empathy and prosocial behaviour were positively related over time, with the effect of empathy on prosocial behaviour DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 21 being stronger than the reciprocal effect of prosocial behaviour on empathy. In contrast, the RI-CLPM that allows to examine how empathy and prosocial behaviour influence each other at the within-adolescent level while controlling for between-adolescents differences, showed that only adolescents' empathy was related to within-adolescents changes in their prosocial behaviour over time. Thus, once we accounted for adolescents' trait-like stability in each of these constructs via the inclusion of the random intercepts, these cross-lagged effects of prosocial behaviour were no longer significant.

Our results evidence that at the between-person level there was a reciprocal pattern of associations between empathy and prosocial behaviour over time. These associations between empathy and prosocial behaviour over time might be accounted for by stable personal characteristics (e.g., personality traits), considering that the between-person effects often reflect the more trait-like effects, whereas the withinperson effects often reflect the more state-like effects [START_REF] Zhou | A four-wave, cross-lagged model of problematic internet use and mental health among Chinese college students: Disaggregation of within-person and between-person effects[END_REF]. Future studies are needed to identify the specific factors that may account for the current identified between-person associations between empathy and prosocial behaviour.

In addition, the within-person-effects address the extent to which an individual's behaviour at any time point deviates from their typical average and the extent to which those deviations are related within an individual. We found positive associations between empathy and prosocial behaviour within each time point and across time, indicating that empathy was related to later prosocial behaviour rather than the reverse.

The within-person findings suggest that adolescents' higher level of empathy at one grade level was linked to their relatively higher level of prosocial behaviour at the next grade level. Importantly, such findings were based on three annual waves of data across DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 22 the entire early adolescence period and the patterns of within-person effects between empathy and prosocial behaviour were quite consistent across time lags. However, the optimal time lag is not yet clear and this issue requires more systematic attention in future research. Although some psychological attributes may change in the short-term, such that for example daily experiences lead to daily changes in behaviour, other psychological attributes might only change on the longer term after repeating experiences that accumulate and lead to changes in attitudes or general tendencies. The selection of the time lag for this study was based on the common lag chosen by studies related to the constructs under investigation, that allows us to observe and capture most substantial changes. The measures of prosocial behaviour and empathy used in the current study assess empathic and prosocial tendencies rather than very specific prosocial or empathic behaviours.

Limitations, Strengths and Future Directions

The results should be interpreted considering some limitations. First, despite the longitudinal design and the insights gained of the direction of effects, no causal conclusion can be drawn considering that it is impossible to exclude the possibility that particular associations are due to variables that were not measured in the study design.

Another caveat is that the data were based on self-reported paper and pencil measures, thus shared method and informant variance might inflate the identified effects, although utilising the cross-lagged approach somewhat reduced this concern by accounting for autoregressive effects and although the findings were generally consistent with prior research, future research will benefit from using multiple methods designs (e.g. observational) and multi-informant measures. Moreover, as in any longitudinal study, the period of the time lags between each wave warrant discussion. It is important to DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 23 clarify that the processes that may underlie the bidirectional within-adolescent associations between empathy and prosocial behaviour may be different on the shortterm and on the long-term, and inconsistency of estimates across studies could be partially explained by differences in the time lag (e.g., [START_REF] Kuiper | Drawing conclusions from cross-lagged relationships: Re-Considering the role of the Time-Interval[END_REF][START_REF] Orth | Testing prospective effects in longitudinal research: Comparing seven competing cross-lagged models[END_REF] Therefore, investigating processes at different time lags in panel designs it is important, considering that convergence, fit, consistence of estimates, and cross-lagged regression coefficients might vary, depending on the respective time lags between different sets of measurement occasions, and on whether the time lag between waves is appropriate for the substantive research question [START_REF] Dormann | Optimal time lags in panel studies[END_REF][START_REF] Orth | Testing prospective effects in longitudinal research: Comparing seven competing cross-lagged models[END_REF].

Consequently, this can lead to researchers drawing conflicting conclusions regarding the sign or dominance of relations. Future research should examine this issue in more detail, for example, by applying short-time optimal lag or continuous-time models [START_REF] Dormann | Optimal time lags in panel studies[END_REF][START_REF] Kuiper | Drawing conclusions from cross-lagged relationships: Re-Considering the role of the Time-Interval[END_REF]). In addition, longitudinal studies with longer follow-up than used in the present study are needed.

Notwithstanding these limitations, the current study has resulted in new insights and important avenues for future research into within-individual empathy and prosocial behaviour associations during early adolescence. It shows long-term associations of empathy on prosocial behaviour. So far, the empathy-prosocial behaviour links have been examined through models assessing aggregated between - and within person effects. To the best of our knowledge, this is the first study to have made use of RI-CLPM to disentangle and examine between-person differences and within-person processes. Indeed, by using this new technique, we were able to examine within-person changes and our results reveal that adolescents' empathy was related to within-person DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 24 changes in their prosocial behaviour over time and this finding could lend support to interventions that aim to improve adolescents' prosocial behaviour by targeting empathy. The identified within-person effects from empathy to increases in prosocial behaviour over time suggest that ameliorating adolescents' empathy may produce corresponding improvements in their prosocial behaviour. In addition, this study focused on empathy and prosocial behaviour in early adolescence where research is still scarce and which is an important developmental period for empathy and prosocial behaviour. This implication is unique given that prior research has primarily focused on between-person comparisons and although the between-person comparisons can demonstrate the link between empathy and prosocial behaviour across individuals, it provides little evidence showing that the changes in empathy within an individual may actually be associated with the corresponding changes in his or her prosocial behaviour [START_REF] Curran | The challenge of modeling co-developmental processes over time[END_REF] Future research should continue to move beyond a unidimensional approach of prosocial behaviour in an attempt to more accurately capturing the multidimensional nature of prosocial behaviour [START_REF] Carrizales | Brief report: How many dimensions in the prosocial behavior scale? Psychometric investigation in French-speaking adolescents[END_REF]. This could be reached by acknowledging the different types of prosocial behaviour related to different situational contexts [START_REF] Carlo | Are all prosocial behaviors equal? A socioecological developmental conception of prosocial behavior[END_REF], by how prosocial behaviour differs according to the target of the behaviour, such as family, friends, and strangers [START_REF] Padilla-Walker | Adolescents' prosocial behavior toward family, friends, and strangers: A DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 34 person-centered approach[END_REF]. Similarly, some studies on empathy that considered variables such as the group context, the personal motivations and the situational factors found that among stable social groups, patterns of empathic responding might be relatively consistent across time and context [START_REF] Gutsell | Intergroup differences in the sharing of emotive states: Neural evidence of an empathy gap[END_REF].

Conclusions and Implications

DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 25

The present study offers relevant new insights for understanding the withinperson processes in the development of early adolescents' empathy and prosocial behaviour. This study represents the first examination disaggregating within-person and between-person effects in the association between empathy and prosocial behaviour.

Moreover, it expands and contributes to current knowledge examining bidirectional longitudinal associations between empathy and prosocial behaviour during early adolescence. Results suggest that the contribution of empathy to early adolescents' prosocial behaviour over time may be attributable to both the more stable, trait-like differences between individuals and the more state-like fluctuations within individuals.

These results have important theoretical and practical implications. The positive results on the within-person level call for further empirical research regarding the time scale at which developmental processes, such as the one studies here, take place. From a practical perspective, for example, prevention and intervention programmes interested in enhancing adolescents' prosocial behaviour should consider empathy as the main proximal target and a potential avenue of intervention strategies, as our results suggested that empathy is a driver of prosocial behaviour.

  = 7,561, p = .10, CFI= .99, RMSEA=.05, SRMR = .02. The CLPM results are depicted in Figure 1. Model comparisons indicated that the stability path, cross-lagged effects, and within-time correlations were time invariant and could be constrained over time DISENTANGLING BETWEEN-AND WITHIN-PERSON ASSOCIATIONS 14 without worsening the model fit. The constrained CLPM showed good fit: S-Bχ 2 (10) = 19.192, p <.05, CFI= .97; RMSEA=.05, SRMR = .07. The detailed parameters estimates are reported in Table

Figure 1

 1 Figure 1

Figure 2

 2 Figure 2

Table 1

 1 Descriptive statistics and Correlation coefficients between prosocial behaviours and empathyNote. ωs = omega coefficient. All correlations are significant at p < .001.

		Total	Males	Females	6th Grade	7th Grade
	Variable	(N=383)	(n=193)	(n=190)	(n=212)	(n=171)	ωs	1	2	3	4	5	6
		M (SD)	M (SD)	M (SD)	M (SD)	M (SD)
	1. Prosocial behaviours (Time 1)	3.75 (.63)	3.51 (.40)	3.99 (.29)	3.79 (.42)	3.74 (.36)	.75	-	.408	.310	.489	.301	.213
	2. Prosocial behaviours (Time 2)	3.71 (.68)	3.49 (.53)	3.93 (.29)	3.69 (.40)	3.72 (.55)	.77		-	.441	.445	.545	.367
	3. Prosocial behaviours (Time 3)	3.77 (.70)	3.55 (.46)	3.99 (.36)	3.83 (.39)	3.69 (.60)	.80			-	.243	.328	.509
	4. Empathy (Time 1)	3.78 (.59)	3.52 (.35)	4.05 (.23)	3.81 (.39)	3.80 (.31)	.67				-	.448	.273
	5. Empathy (Time 2)	3.70 (.62)	3.46 (.34)	3.93 (.33)	3.70 (.34)	3.68 (.44)	.69					-	.525
	6. Empathy (Time 3)	3.62 (.66)	3.38 (.34)	3.90 (.39)	3.69 (.43)	3.58 (.43)	.69						-

We excluded the following items: Item 5: I'm empathic with those who are in need, Item 8: I intensity feel what others feel, Item 12: I easily put myself in the shoes of those who are in discomfort, Item 16: I immediately sense my friends' discomfort even when it is not directly communicated to me.

Note. Cross-lagged panel model (Figure 1), Random Intercept Cross-Lagged Panel Model (Figure 2). CI = 95% Confidence Interval. Standardized effects (β) are indicators of effect size. N = 383, measurements = 3. The substantial interpretation of correlations, cross-lagged effects and correlated change is different in both models. In a standard cross-lagged panel model, parameters reflect how an individual's relative position compared to the rest of the samples are correlated or can be predicted, which aggregates within-and between person sources of variation. In a Random Intercept Cross-Lagged Panel Model, parameters reflect how within-person variations relative to their own scores are correlated or can be predicted.