T Lemattre
email: lemattre@lurpa.ens-cachan.fr

B Denis
email: denis@lurpa.ens-cachan.fr

J-M Faure
email: faure@lurpa.ens-cachan.fr

Lurpa

J-F Pétin

Using a meta-model to build operational architectures of automation systems for critical processes

This paper addresses the design of the operational architecture of a critical system control. This design results from the allocation of control functions onto physical devices by taking into account numerous constraints such as capability, safety, time performance or reliability constraints. This paper focuses on the two first ones, capability and safety constraints by proposing a method based on reachability analysis in a network of communicating automata. The link with complementary studies about time performance or reliability constraints is established using an UML architecture meta-model that captures and shares information about control architectures. Automata models and their parameters used for reachability analysis are derived from this meta-model and, conversely, generated operational architectures give rise to a set of metamodel instances that can be used as input for complementary evaluation.

Introduction

The architecture of an automation system can be characterized by three complementary views [START_REF] Pétin | An Architecture Description Language for In-Vehicle Embedded System Development[END_REF]:

• the functional architecture, consisting of interconnected control functions, which expresses the users' needs; each function operates a transformation from inputs (from the process or from other functions) to outputs (to the process or to other functions); the functions are leave functions which are considered as indivisible; moreover, in the case of critical systems, functions are characterized by a safety-level that represents how critic is the impact of their failure on material or human being;

• the physical architecture, consisting of controllers (PLCs or real-time industrial computers), which provides hardware resources for executing the control functions and generating signals towards the process according to the information gathered on this latter, and of one or several communication networks allowing data exchanges between the controllers and between the controllers and the monitoring/supervision system; in the case of critical systems, the devices involved in the physical architecture are characterized by an integrity factor, that represents the reliability of the device, thanks for example to internal hardware redundancies;

• the operational architecture, built by allocating the leave functions of the functional architecture onto the control devices involved in the physical architecture.

The design of operational architecture must be done by taking into account various and numerous constraints or criteria such as:

• capabilities constraints that are related to the physical capabilities of the controllers (such as energy, memory, numbers of inputs/outputs, CPU charge, etc.);

• safety constraints that are related to the deterministic allocation rules answering safety purposes; for example, the function safety level must be compliant with the controller integrity factor in which it is allocated (such as highest critical functions should be implemented in most reliable controllers); enabling or disabling the combination of different functions within a single controller (due to functional redundancies, commissioning or testing constraints, etc.) is another example of safety constraint;

• time performance constraints, for example the time needed by the control functions to react to stimuli occurrence from the process; obviously, the quality of service of the communication networks that link the controllers would have a crucial impact on time constraint analysis as well as the functional chain distribution among several controllers;

• reliability constraints, that are related to both controller and functional reliability and may impact the design of operational architecture in terms of functional and physical redundancies.

It clearly appears that the problem of designing an operational architecture that fulfil all these constraints cannot be addressed by a single technique or model: the two first constraints can be addressed by using deterministic and untimed model for constraint solving while the two last ones require respectively using temporal and stochastic models. In the current industrial practice, design of operational architecture is a tedious and time-consuming activity, based on the expertise of designers and then performed in a non-automated way. Note that actuators and sensors location cannot provide help since the system architecture is basically centralized (one or several computing centres) for safety constraints (protected area against fire, external attacks, etc.). This paper aims to automate part of this activity and focuses on the two first constraints related to capability and safety issues:

• generation an operational architecture that fulfil capability and safety constraints is seen as a constraint solving problem and is addressed using an original approach based on reachability analysis on a discrete state space using communicating automata;

• a meta-model that centralizes all useful information regarding the architecture is sketched out to share a common and consensual information about the operational architecture between the various design activities; the proposed meta-model is built to integrate capabilities and safety constraints the paper is focusing on, as a basis to be further enriched by time performance and reliability constraints.

In section 2, the architecture meta-model is given using UML class diagram to represent the objects of functional and physical architectures while Object Constraint Language (OCL) expressions [START_REF] Richters | On Formalizing the UML Object Constraint Language OCL[END_REF] allows the description of capabilities and safety constraints to be applied for operational architecture design. Illustrative examples of capability and safety constraints are provided by OCL constraints respectively related to the number of inputs/outputs and to the compliance between controller integrity factor and the function safety level. Then, an automatic method to allocate all leave functions to controllers, while respecting the capabilities and safety constraints identified in the architecture meta-model, is proposed in section 3. This method relies on reachability analysis on a network of communicating automata whose transition guards implements OCL constraints; conversely, the result of the first operational architecture design must be translated into meta-model instance diagrams to be used for further engineering activities (temporal performance evaluation, reliability analysis, etc.). Two case studies illustrate this way of building operational architecture in section 4 and prospects for further work are sketched in Section 5.

Architectures meta-model

The aim of the architecture meta-model is to capture useful information about functional, physical and operational architecture in order to facilitate the analysis of the control architecture with regards to several constraints or criteria and using various modelling and analysis techniques. Main benefit rely in the ability for various engineering activity (architecture sizing according to capabilities and safety constraints, architecture time performance analysis, architecture reliability analysis, etc.) to share common concepts (functions, controllers, inputs/outputs etc) and their attributes using UML class diagrams and to formalise the various constraints to be satisfied using Object Constraint Language (OCL) expressions. This Model Driven Engineering way of thinking, based on the use of UML/SysML for system specification before using dedicated models such as state-based formalisms for behavioural design, has been promoted in automation engineering by projects such as Oooneida [START_REF] Auinger | Requirements and solutions to Software encapsulation and engineering in next generation manufacturing systems: OOONEIDA approach[END_REF] or Corfu [START_REF] Tranoris | A tool supported engineering process for developing control applications[END_REF] or by UML profile for Process Automation [START_REF] Katzke | UML-PA as an Engineering Model for Distributed Process Automation[END_REF].

The paper focuses on the automatic design of the operational architecture based on capability constraint related to the number of I/O and safety constraint related to the relationship between function safety level and controller integrity factor. This first attempt is currently being extended using OCL to capture additional capability or safety constraints within the context of the industrial project that supports this work.

Functional architecture

A functional architecture (Figure 1) is a set of interconnected functions which receives and sends data from/to the process. Hence, the part of the meta-model (Figure 3) which describes this architecture contains two main classes: Function and Data.

• A function f i ∈ F is defined as a 9-tuple (SL i , NILp i , NIAp i , NOLp i , NOAp i , NIL f i , NIA f i , NOL f i , NOA f i ) with:
-SL i ∈ Safety Level ; the lower this level is, the more critical the function. In the sequel of this paper, the functions are ranked in four levels; hence, the set Safety Level is Safety Level = {1, 2, 3, 4}; -NILp i , NIAp i ∈ N: numbers of respectively logic and analogic input data received from the controlled process; -NOLp i , NOAp i ∈ N: numbers of respectively logic and analogic output data sent to the controlled process;

-NIL f i , NIA f i ∈ N: numbers of respectively logic and analogic input data coming from other functions;

-NOL f i , NOA f i ∈ N: numbers of respectively logic and analogic output data going to other functions.

• Data are typed; they are either logic or analogic. Moreover, they are ranked in three disconnected sets:Input data (received from the process),Output data (sent to the process) and Inter-functions data.

It matters to underline that any data issued from a function is sent either to the process or to another function, as the three sets are disconnected. Moreover, self-loops are forbidden (a same data cannot act both as input and output for a same function) and the flow is directed; inter-functions data must then satisfy the following relation: -CF ∈ Criticality Factor of the controller, the lower the criticality factor is, the more dependable the controller must be; in the sequel of this paper, it will be assumed that this factor can take 3 values: Criticality Factor = {1, 2, 3};

context function INV : f 1 <> f 2 (1)
-LImax, AImax ∈ N maximum numbers of respectively logic and analogic input interfaces of the controller;

-LOmax, AOmax ∈ N maximum numbers of respectively logic and analogic output interfaces of the controller.

• A network flow models communication between two controllers. It is then assumed that relation (2) holds, i.e. that the source and the target of a flow are different.

context controller

INV : c1 <> c2 (2

Operational architecture

The operational architecture is built by allocating the leave functions of the functional architecture onto the controllers of the physical architecture. Following assumptions will be taken:

1. a controller can host between 1 to n functions; 2. a function must be allocated to one and only one controller.

These assumptions are included in the meta-model in the form of the relation between the classes Function and Controller. If F is the set of functions f i , with i ∈ N * and C the set of controllers c j , with j ∈ N * , the allocation of f i to the controller c j will be denoted by c j ← f i .

Moreover, each allocation has to satisfy capabilities and safety constraints which are formalized thanks to OCL expressions below.

Capabilities constraints

Capability constraints can be modelled using OCL expressions in terms of inequations where controller attributes are involved in. In our study, capability constraints (and controller attributes) are limited to the number of I/O interfaces. Roughly speaking, these constraints mean that the sums of inputs/outputs from/to the process of the functions that are allocated to a given controller must not exceed the numbers of input/output interfaces of this controller.

The numbers of interfaces are positive integers:

context controller INV : LImax > 0 (3)
INV : AImax > 0 (4)
INV : LOmax > 0 (5)

INV : AOmax > 0 (6)
The numbers of inputs/outputs from/to the process of a given function are defined by the number of existing relationship between this functions and its associated I/O; this information can be built from the instance diagram that defines a particular functional architecture using the OCL size() function:

context function DEF : /NILp = self.In-Log-p -> size() (7)
DEF : /NIAp = self.In-Ana-p -> size() (8)
DEF : /NOLp = self.Out-Log-p -> size() (9) DEF : /NOAp = self.Out-Ana-p -> size() (10)
Then, the capabilities constraints state that the sum of I/O functions allocated to a given controller must be lower or equal than the controller I/O capability; it is modelled using OCL sum() function as follows:

context controller INV : f unction./NIL-> sum() ≤ LImax (11) INV : f unction./NIA-> sum() ≤ AImax (12) INV : f unction./NOL-> sum() ≤ LOmax (13) INV : f unction./NOA-> sum() ≤ AOmax (14)

Safety constraints

One of the main safety constraints is related to the compliance between function safety level and controller integrity factor in which the function is allocated. More precisely, the most critical functions (SL = 1) must be allocated only to controllers whose criticality factor equals 1, the medium-critical controllers (CF = 2) can accept functions whose safety level equals 2 or 3 and the less critical controllers can accept functions whose safety level equals 3 or 4. Given the instances diagrams of the functional and physical architectures, these constraints are then expressed as follows: 3 Automatic construction of operational architecture

context
For a given functional architecture, it is possible to build the corresponding instances diagram from the metamodel of Figure 3; this diagram contains obviously as many instances of the class function as there are 'leave' functions in the functional architecture and is one of the inputs of the method to construct automatically operational architectures; L will denote the number of 'leave' functions.

The other input is a parametric instances diagram of physical architecture where the number of controllers, noted M, and the criticality factor of each controller are parameters. This approach allows several operational architectures to be defined from the same functional architecture while satisfying all capability and safety constraint; the choice of a solution among this set is under the responsibility of the automation system designer and can be represented as particular relationships between function and controller instances of the meta-model.

Principles

The aim of the method to construct automatically operational architectures is then to allocate a set of functions to a set of controllers, while respecting the capabilities and safety constraints (Figure 4).

As the capabilities constraints consider only the numbers of inputs/outputs from/to the process, the interfunctions data are no more useful data when allocating the functions. Hence, a function f i becomes merely a 5tuple f i = (SL i , NILp i , NIAp i , NOLp i , NOAp i ). Each controller c j is still represented by a 5-tuple, but the initial value of CF, when no function is allocated to this controller, will be 0. The criticality factor will be defined during functions allocation while respecting the safety constraints. The problem of allocation can then be seen as a bin packing problem than can be solved by optimisation methods such as linear programming or heuristic even if suffering from the exponential explosion problem. Approaches proposed by [START_REF] Behrmann | Production Scheduling by Reachability Analysis -A Case Study[END_REF] and [START_REF] Subbiah | Short-Term Scheduling of Multi-Product Batch Plants with Sequence-Dependent Changeovers Using Timed Automata Models[END_REF] showed the effectiveness of the timed automata and reachability analysis to address scheduling problems thanks to a modular, parametric and instanciable modelling. The method proposed in this paper takes advantage of these approaches but differs from them because time is not considered in an allocation problem. Based on communicating automata, the automatic allocation of functions is based on two principles:

• modelling the allocation problem as a set of competing call-response mechanisms between models, in the form of communicating automata, of allocation requests and of requests acceptances;

• investigating whether the execution of this set, which is a network of communicating automata, can lead to a state reachable from the initial state where all functions are allocated.

Toy example

Figure 6 represents a functional architecture which comprises five functions f 1 , f 2 , f 3 , f 4 , f 5 , whose safety level is in {1, 2} ⊂ SL, and which are defined as follows:

f 1 = (2, 5, 4, 1, 3); f 2 = (1, 5, 6, 2, 4); f 3 = (2, 1, 3, 6, 4); f 4 = (1, 6, 8, 5, 2); f 5 = (1, 3, 4, 5, 2).
These functions have to be allocated on a set of M=3 controllers c 1 , c 2 , c 3 , whose capabilities are the same:

∀ j ∈ {1, 2, 3}, LImax = AImax = LOmax = AOmax = 10
One possible allocation of these functions to the three controllers is described in Figure 5. This solution was obtained by first allocating the function f 1 to the controller c 1 , thus fixing the value of its criticality factor to CF 1 = 2. The function f 2 was then allocated to the controller c 2 , thus fixing the value of its criticality factor to CF 2 = 1. Then the function f 3 was allocated to the controller c 1 because its safety level is consistent with CF 1 = 2 and the sums of the numbers of inputs/outputs of the two functions do not exceed the capabilities of the controller. The function f 4 was then allocated to the controller c 3 because the sums of logic and analogic inputs of functions f 2 and f 4 are beyond the capabilities of the controller c 2 . The function f 5 was finally allocated to controller c 2 , because the remaining capabilities of the controller c 3 were too small for f 5 be allocated to this controller.

Once all functions are allocated, the list of functions which are allocated to each controller is known. These lists are instances of the relation between the two classes controller and function in the meta model (Figure 3).

Generic models of allocation request and of requests acceptance

As mentioned in 3.1, allocation of functions is modelled in this approach as a set of competing call-response mechanisms between models of allocation requests and of requests acceptances. Figures 6 and7 present respectively the generic model of an allocation request sent by a function and of the acceptance of requests by a controller; these models are denoted δ and α.

Definition of the formalism used

The formalism used is a network of automata communicating through shared variables and synchronized by transition labels. Every transition of these automata may comprise a synchronization label, a guard (condition transition which must be true to fire the transition) and variables (numbers of input/output interfaces used, criticality factor) updates. The following conventions are used in these models:

• the initial locations are indicated by a source arc;

• the marked locations are indicated by two concentric circles;

• the location names are in bold;

• the label names are in italics and followed by an ! (resp. ?) for emission (resp. reception) labels;

• the variables updates are underlined;

• the guards (transition conditions) are in normal characters.

allocation request model

The initial location of the model is 'Function not allocated'. Only one transition, which corresponds to the emission of an allocation request can be fired from this location. Once this request has been emitted, the model waits (in the location 'allocation Possible?') for the response from an acceptance model, which can be:

• Refusal, then the model returns to the initial location;

• Ok, then the model evolves to the location "Function allocated" which is a terminal marked location.

Refusal?

Request! Ok?

Function not allocated

Function allocated Allocation possible ?

Figure 6. Generic model of allocation request (δ )

Requests acceptance model

Four transitions of this model include guards which are obtained from the OCL expressions which define the capabilities and safety constraints. The guard Violation of one of the allocation constraints, for instance, mean that at least one of the constraints is not satisfied if the requesting function is allocated to the considered controller. Some notations must be introduced to ease the description of this model.

• Let: F j = { f i ∈ F|c j ← f i } be the set of functions f i which are allocated to c j ;

• Let: I j = {i ∈ {1, ..., L}|c j ← f i } be the set of index of functions f i which are allocated to c j .

Then the notations ∑

i∈I j NILp i ; ∑ i∈I j NIAp i ; ∑ i∈I j NOLp i ; ∑ i∈I j NOAp i
represent respectively the sums of logic and analogic inputs, logic and analogic outputs from/to the process of the functions which are already allocated to controller c j at a given moment. The initial values of these sums, at the beginning of the allocation, are obviously equal to 0. From the initial location, which is also marked, this model can evolve to the location 'Checking constraints' only upon reception of an allocation request. The three transitions which can be fired from this latter location correspond to:

• the violation of at least one of the allocation constraints, and the label Refusal is then emitted;

• the acceptance of a request whereas no other function has been previously allocated to the controller (guard 'First allocation' true). The variables

∑ i∈I j NILp i ; ∑ i∈I j NIAp i ; ∑ i∈I j NOLp i ; ∑ i∈I j NOAp i , are
then updated and the criticality factor CF j is set to the value of the safety level of the function;

• the acceptance of a request whereas at least one other function has been previously allocated (guard 'Additional allocation' true). Only the variables

∑ i∈I j NILp i ; ∑ i∈I j NIAp i ; ∑ i∈I j NOLp i ; ∑ i∈I j
NOAp i , are updated. The criticality factor CF j remains unchanged.

In the latter two cases, the label Ok is emitted. From the location 'Analysis of the state of the controller', two evolutions are possible which correspond to:

• the fact that all capabilities of the controller have been reached :

∑ i∈I j NILp i = LImax j ; ∑ i∈I j NIAp i = AImax j ; ∑ i∈I j NOLp i = LOmax j ; ∑ i∈I j NOAp i = AOmax j ;
(guard 'no other possible allocation' true), then the model evolves to the marked location 'Controller saturated';

• the fact that at least one capability of the controller is not reached (guard 'other possible allocation' true), then the model evolves to the initial location 'Controller waiting'.

Instantiated model

This model is a network of communicating automata NA = δ 1 δ 2 ... δ L α 1 α 2 ... α M that includes:

• as many instances (δ 1 , δ 2 , ..., δ L ) of the model in Figure 6 as there are functions,

• M instances (α 1 , α 2 , ..., α M ) of the model in Figure 7; the choice of the parameter M is let to the designer. An obvious value to obtain a solution is to set M = L; in that case indeed, functions allocation is always possible if no function owns a number of inputs or outputs larger than the corresponding capability of controllers. From the observation of the solution yielded with M = L, a more compact solution, with a smaller number of controllers, can be obtained as it will be shown in section 4.

A synchronous evolution of two automata is possible only if these two automata emit and receive one of the following label pairs:

• Request! and Request?;

• Ok! and Ok?;

• Refusal! and Refusal?.

To avoid inconsistencies such as the fact that an instance α i emits a reply to an instance δ j which is not the one having emitted the allocation request, the call-response mechanism must be designed as a critical section protected by a semaphore. The achievement of this critical section depends on the implementation and will not be discussed further in this paper.

Definition of the reachability property searched

All the functions are allocated when the marked location is reached in all the instances of δ . In this case, the active location of the instances of α may be the terminal location or the initial location, which are both marked. Hence, the reachability property to check can be informally stated as follows: From the initial state, is it possible to reach a state of the network of automata such that the active location is a marked location in all the automata of the network?

Implementation with a formal verification tool

The techniques of formal verification by model checking [START_REF] Bérard | Systems and Software Verification[END_REF] aim to prove that a model satisfies or does not satisfy a formal property, which may be a reachability property. It is hence natural to consider the implementation of the method proposed by using such a technique. This requires first to formally state the property searched, given in textual way in the previous section. Using the quantifiers of the CTL temporal logic, this property, noted P, can be written 'P: EF Full allocation' Full allocation designating the state of the network such that the active location is a marked location for all automata. This property is verified if there exists at least one trace from the initial state of the network which reaches the state Full allocation.

Case studies 4.1 Choice of the formal verification tool

Several model-checking-based formal verification tools, such as NuSMV, SPIN, UPPAAL, may be considered for achieving the reachability analysis on which the search for an allocation solution relies. The UPPAAL tool [START_REF] Behrmann | UPPAAL implementation secrets[END_REF] was selected for these studies because it has a very ergonomic graphical interface and can provide an execution trace even in case of positive proof. It is important to note that only these features have motivated this choice; the ability of this tool to check properties on timed models does not constitute a selection criterion, as the communicating automata considered in this work are not timed. In the following case studies, the UPPAAL parameters have been set so that reachability analysis be done depth first to fasten analysis.

First case study

This case aims to illustrate the approach. The functional architecture consists of twenty functions which are defined in Table 1, and the controllers features are as follows: ∀ j ∈ {1, ..., M}, LImax j = AImax j = LOmax j = AOmax j = 32 A first reachability analysis with an initial number of controllers M = 20 provides a solution in which only N = 5 controllers are hosting at least one function (15 controllers are not used). By performing a new analysis with an initial number of controllers M = 4, a more compact solution can be found, in which 4 controllers are actually used. It is not possible to further reduce the number of controllers, as an analysis with M = 3 does not provide any solution. The final allocation solution for N = 4 controllers is detailed in Table 2.

Second case study

To assess scalability of the proposal, a study based on 200 functions was subsequently undertaken. These 200 functions are all different from each other, and their characteristics are randomly distributed as follows:∀i ∈ {0, ..., L}, SL i ∈ {1, 2, 3, 4}, {NILp i ∈ {0, ..., 9}, NIAp i ∈ The controllers used are all the same and their characteristics are: ∀ j ∈ {1, ..., M}, LImax j = AImax j = LOmax j = AOmax j = 32. A first reachability analysis was conducted, providing an allocation solution in which some controllers are not hosting any function; the number of really useful controllers is then N = 37. The analysis performed with M = 36 also provides a solution. The number of controllers required to achieve the operational architecture cannot however be reduced further, as the analysis performed with M = 35 controllers does not provide any solution. The allocation solution hence uses at least N = 36 controllers. Table 3 shows the durations of the different reachability analyses conducted in this case study. These values show that the approach proposed is quite feasible in the industrial context of operational architectures design.

Conclusion and Perspectives

The contribution of this paper is twofold. First an architectures meta-model that contains all information on the architectures of automation systems for critical processes has been proposed; OCL expressions have been introduced to express the generic constraints on the operational architecture. Second, a method to construct automatically an operational architecture from instances diagrams and the OCL expressions has been presented; this proposal is based on a novel modelling of the allocation of functions and reachability analysis of a network of communicating automata.

Ongoing works are aiming at introducing new capabilities and safety constraints and to propose not only one allocation solution but a set of solutions in the automatic construction of an operational architecture. Comparison to other methods, like integer linear programming, is also planned on the basis of several case studies.

These results will allow in a near future to focus on validation of operational architectures, e.g. by time performances assessment; this will imply to extend the archi-tectures meta-model so as to introduce time features of the classes.

Figure 1 . Example of functional architecture 2 . 2

 122 Figure 1. Example of functional architecture

Figure 2 .

 2 Figure 2. Example of physical architecture

Figure 3 .

 3 Figure 3. Architectures meta-model

  controller INV : self.CF = 1 implies function.SL = 1 (15) INV : self.CF = 2 implies(function.SL = 2 or function.SL = 3) (16) INV : self.CF = 3 implies(function.SL = 3 or function.SL = 4) (17)

Figure 4 .

 4 Figure 4. Aim of the constructionThe problem of allocation can then be seen as a bin packing problem than can be solved by optimisation methods such as linear programming or heuristic even if suffering from the exponential explosion problem. Approaches proposed by[START_REF] Behrmann | Production Scheduling by Reachability Analysis -A Case Study[END_REF] and[START_REF] Subbiah | Short-Term Scheduling of Multi-Product Batch Plants with Sequence-Dependent Changeovers Using Timed Automata Models[END_REF] showed the effectiveness of the timed automata and reachability analysis to address scheduling problems thanks to a modular, parametric and instanciable modelling. The method proposed in this paper takes advantage of these approaches but differs from them because time is not considered in an allocation problem. Based on communicating automata, the automatic allocation of functions is based on two principles:

2 Figure 5 .

 25 Figure 5. Example of allocation of 5 functions on 3 controllers

Figure 7 .

 7 Figure 7. Generic model (α) of requests acceptance

Table 1 . Functions description

 1

	Functions SL i NILp i NIAp i NOLp i NOAp i
	1	1	3	5	2	4
	2	1	4	6	1	3
	3	2	3	7	2	1
	4	2	3	4	4	3
	5	2	7	5	6	2
	6	2	7	2	8	6
	7	1	4	5	2	4
	8	2	3	6	4	5
	9	1	3	7	4	2
	10	1	7	2	6	4
	11	3	3	5	2	4
	12	4	4	6	1	3
	13	3	3	7	2	1
	14	4	3	4	4	3
	15	3	7	5	6	2
	16	2	7	2	8	6
	17	4	4	5	2	4
	18	3	3	6	4	5
	19	4	3	7	4	2
	20	1	7	2	6	4

Table 2 . Controllers features and functions distribution for the set of functions of Table 1

 2 { f 11 , f 14 , f 15 , f 17 , f 18 , f 19 } { f 1 , f 2 , f 7 , f 9 , f 10 , f 20 } 28 27 21 21{0, ..., 9}, NOLp i ∈ {0, ..., 9}, NOAp i } ∈ {0, ..., 9}.

	c j CF j	Fj	∑ i∈Ij	NILp i	∑ i∈Ij	NIAp i	∑ i∈Ij	NOLp i	i∈Ij ∑	NOAp i
	c 1 3 c 2 2 c 3 3	{ f 12 , f 13 } { f 3 , f 4 , f 5 , f 6 , f 8 , f 16 }		7 30 23		13 26 32		3 32 22		4 23 20
	c 4 1

Table 3 . Duration of the reachability analy- sis with L=200

 3

	Initial numbers of controllers	200	37	36
	Duration	110 s 46 s 50 s


Julie Vallée
email: julie.vallee@parisgeo.cnrs.fr

Emmanuelle Cadot

Christelle Roustit

Isabelle Parizot

Pierre Chauvin

The role of daily mobility in mental health inequalities: the interactive influence of activity space and neighbourhood of residence on depression

Keywords: France, Paris, activity space, daily mobility, residential and non residential neighbourhood, neighbourhood exposure, cross-level interaction, mental health, depression

come

The role of daily mobility in mental health inequalities:

the interactive influence of activity space and neighbourhood of residence on depression.

Introduction

Neighbourhoods of residence have recently emerged in social epidemiology and public health literature as a relevant dimension to be taken into account in studies of health inequalities [START_REF] Diez-Roux | Investigating neighborhood and area effects on health[END_REF][START_REF] Kawachi | Neighborhodds and Health New[END_REF]. Health geography literature has also greatly contributed to the understanding of the central concepts of "place" and "space" in health studies [START_REF] Curtis | Health and Inequality: Geographical Perspectives[END_REF][START_REF] Gatrell | Geographies of Health[END_REF][START_REF] Jones | Health, Disease and Society: An Introduction to Medical Geography[END_REF].

With regard to mental health, studies have revealed the effect of neighbourhood structural features (such as the socioeconomic composition and the built and services environment) and neighbourhood social processes (such as disorder, social cohesion and perceived violence) on depression, even after adjustment for individual factors in multilevel models [START_REF] Echeverria | Associations of neighborhood problems and neighborhood social cohesion with mental health and health behaviors: the Multi-Ethnic Study of Atherosclerosis[END_REF][START_REF] Evans | The built environment and mental health[END_REF][START_REF] Galea | Urban built environment and depression: a multilevel analysis[END_REF][START_REF] Kim | Blues from the neighborhood? Neighborhood characteristics and depression[END_REF][START_REF] Mair | Are neighbourhood characteristics associated with depressive symptoms? A review of evidence[END_REF][START_REF] Ostir | Neighbourhood composition and depressive symptoms among older Mexican Americans[END_REF][START_REF] Ross | Neighborhood Disadvantage and Adult Depression[END_REF]. Some authors have also underscored the importance of not assuming that the effects of the residential context on health operate identically for every resident [START_REF] Macintyre | Neighborhoods and Health: an overview[END_REF][START_REF] Stafford | Gender differences in the associations between health and neighbourhood environment[END_REF]. Focusing on depression, some studies have shown that the strength of the residential context effect varies according to gender [START_REF] Berke | Protective association between neighborhood walkability and depression in older men[END_REF][START_REF] Fitzpatrick | Depressive symptomatology, exposure to violence, and the role of social capital among African American adolescents[END_REF][START_REF] Gutman | Continuities in depression from adolescence to young adulthood: contrasting ecological influences[END_REF], racial/ethnic group [START_REF] Gary | Neighborhood characteristics and mental health among African Americans and whites living in a racially integrated urban community[END_REF][START_REF] Henderson | Neighbourhood characteristics, individual level socioeconomic factors, and depressive symptoms in young adults: the CARDIA study[END_REF][START_REF] Ostir | Neighbourhood composition and depressive symptoms among older Mexican Americans[END_REF] and socioeconomic status [START_REF] Weich | Income inequality and the prevalence of common mental disorders in Britain[END_REF][START_REF] Weich | Contextual risk factors for the common mental disorders in Britain: a multilevel investigation of the effects of place[END_REF]. Daily mobility has sometimes been mentioned to explain why a stronger association between neighbourhood characteristics and mental health was observed amongst certain population subgroups. For instance, observing that women were more vulnerable to the characteristics of their neighbourhood of residence, Stafford et al. postulate that this may be due to the fact that women spend more of their time in their neighbourhood than men (2005). The authors of one of the few studies that consider the nonresidential neighbourhood environment in addition to the characteristics of the residential neighbourhood stress the need to include nonresidential neighbourhood exposure in order to accurately measure the association between the residential neighbourhood and self-rated health [START_REF] Inagami | Non-residential neighborhood exposures suppress neighborhood effects on self-rated health[END_REF]. It could be interesting not to limit placebased health research to analyses based on residential location and to take into consideration the possible mediating role of attributes in extra-local places [START_REF] Chaix | Neighbourhoods in eco-epidemiologic research: delimiting personal exposure areas. A response to Riva, Gauvin, Apparicio and Brodeur[END_REF][START_REF] Cummins | Commentary: investigating neighbourhood effects on health-avoiding the 'local trap[END_REF][START_REF] Matthews | Spatial Polygamy and the Heterogeneity of Place: Studying People and Place via Egocentric Methods[END_REF][START_REF] Rainham | Conceptualizing the healthscape: Contributions of time geography, location technologies and spatial ecology to place and health research[END_REF]. In this paper the idea is then to examine how the spatial extent of daily mobility -within or outside the neighbourhood of residence -can modify an individual"s vulnerability to residential neighbourhood characteristics. It is reasonable to assume that the residential environment may exert a stronger influence on the mental health of people spending most of their time in their local area than on the mental health of people whose daily travels go beyond residential neighbourhood boundaries.

Apart from the hypothesis that daily mobility could modify the strength of association of attributes of residential neighbourhood with depression, we can assume that daily mobility might be directly associated with depression. Two opposite hypotheses can be formulated a priori about the relationship between daily mobility and depression: (1) spatially limited daily mobility reflects spatial and social confinement and can be seen as a behaviour correlated with a higher risk of being depressed or, conversely, (2) spatially limited daily mobility is linked both to a deep attachment to the neighbourhood of residence and to neighbourhood well-being and can be seen as a behaviour correlated with a lower risk of being depressed. The complexity of the relationship between the spatial extent of daily mobility and well-being was pointed out in an empirical study of 40 adults with serious mental illness neighbourhoods [START_REF] Townley | Understanding the experience of place: expanding methods to conceptualize and measure community integration of persons with serious mental illness[END_REF]. The authors found that those whose daily activities remained close to home tended to have lower life satisfaction but a stronger sense of community in their neighbourhood. For our part, we sought to investigate, in the general population, whether spatially limited daily mobility might be associated with a higher risk of being depressed or, instead, with a lower risk of being depressed. These two seemingly incongruent hypotheses regarding the potential influence of neighbourhood characteristics on depression could both be shown to be correct, depending on the characteristics of the neighbourhood under study. Indeed, in deprived neighbourhoods, spatially limited daily mobility could be associated with a higher risk of being depressed because it could indicate strong exposure to unpleasant residential circumstances, e.g., violence, incivilities, exterior noise, and a lack of key services [START_REF] Curry | Pathways to Depression: The Impact of Neighborhood Violent Crime on Inner-City Residents in Baltimore, Maryland, USA[END_REF][START_REF] Leslie | Are perceptions of the local environment related to neighbourhood satisfaction and mental health in adults?[END_REF][START_REF] Ross | Neighborhood disadvantage, disorder, and health[END_REF]. On the contrary, in advantaged neighbourhoods, spatially limited daily mobility could be associated with a lower risk of being depressed precisely because it could indicate a deeper attachment to pleasant circumstances. However, to the best of our knowledge, neither the association between daily mobility and depression nor the interactions between daily mobility and neighbourhood deprivation on depression have yet been investigated.

In this paper, neighbourhood deprivation was defined not only through subjective neighbourhood assessments (first individual, then aggregated to neighbourhood level) but also through census based information. As a reciprocal relationship between mental state and area perception probably exists and may lead to a reporting bias [START_REF] Mair | Are neighbourhood characteristics associated with depressive symptoms? A review of evidence[END_REF], it was indeed interesting to measure neighbourhood deprivation independently of the perceptions of sample respondents using census as a source of information [START_REF] Fagg | Neighbourhood perceptions among inner-city adolescents: Relationships with their individual characteristics and with independently assessed neighbourhood conditions[END_REF]. The idea is to examine whether relationship between depression and neighbourhood deprivation (defined alternately from individual neighbourhood assessments, collective neighbourhood assessments or neighbourhood census social composition) yield concordant results.

To study the role of daily mobility in depression, we propose here to use the concept of "activity space", which can be defined as the space within which people move about or travel in the course of their daily activities. Various measures of activity space have been used in time geography to identify social differences in people"s access to opportunities [START_REF] Golledge | Spatial Behaviour -A geographic perspective[END_REF], in particular, to health-care facilities [START_REF] Arcury | The Effects of Geography and Spatial Behavior on Health Care Utilization among the Residents of a Rural Region[END_REF][START_REF] Nemet | Distance and health care utilization among the rural elderly[END_REF][START_REF] Sherman | A suite of methods for representing activity space in a healthcare accessibility study[END_REF]. Very recently, the concept of activity space also contributed to measuring community integration [START_REF] Townley | Understanding the experience of place: expanding methods to conceptualize and measure community integration of persons with serious mental illness[END_REF] or exposure to the food environment [START_REF] Kestens | Using experienced activity spaces to measure foodscape exposure[END_REF]. In this research, we use a simplified measure of activity space: the measure of the concentration of daily activities in the perceived neighbourhood. In a previous paper on cervical screening behaviour among women in the Paris metropolitan area, we provided strong arguments in favour of the association between this measure of activity space and participation in health-care activities. We showed that women who reported concentrating their daily activities within their neighbourhood had a statistically greater likelihood of delayed cervical screening. We also pointed out that the strength of association of attributes of residential neighbourhood with participation in cervical screening was significantly higher among women whose activity space was limited to their neighbourhood of residence [START_REF] Vallée | The combined effects of activity space and neighbourhood of residence on participation in preventive health-care activities: The case of cervical screening in the Paris metropolitan area (France)[END_REF].

The present paper deals with individual and neighbourhood deprivation features associated with depression in the Paris metropolitan area in 2005, with special attention to the combined association of activity space and neighbourhood characteristics, after controlling for individual-level confounders.

Our objectives were to determine (1) whether the association of residential neighbourhood deprivation with depression was greater among people who reported concentrating their daily activities within their neighbourhood of residence, and (2) whether the relationship of activity space with depression varied according to neighbourhood deprivation.

Materials and methods

Study sample

The SIRS (French acronym for Health, Inequalities and Social Ruptures) survey was conducted between September and December 2005 among a representative sample of the adult French-speaking population in the Paris metropolitan area (Paris and its suburbs, a region with a population of 6.5 million). This survey constituted the first wave of a socioepidemiological population-based cohort study, which is a collaborative research project between the French National Institute for Health and Medical Research (INSERM) and the National Centre for Scientific Research (CNRS). This cohort study was approved by France"s privacy and personal data protection authority (Commission Nationale de l"Informatique et des Libertés [CNIL]).

In the present paper, data collected in 2005 were examined cross-sectionally. The SIRS survey employed a stratified, multistage cluster sampling procedure. The primary sampling units were census blocks called "IRISs" ("IRIS" is a French acronym for blocks for incorporating statistical information). They constitute the smallest census unit areas in France (with about 2,000 inhabitants each in the Paris metropolitan area) whose aggregate data can be used on a routine basis. In all, 50 census blocks were randomly selected (overrepresenting the poorer neighbourhoods) from the 2,595 eligible census blocks in Paris and its suburbs. Subsequently, within each selected census block, households were randomly chosen from a complete list of dwellings in order to include at least 60 households in each surveyed census block. Lastly, one adult was randomly selected from each household by the birthday method. A questionnaire containing numerous social and health-related questions was administered face-to-face during home visits. Further details on the SIRS sampling methodology were published previously (Chauvin & Parizot, 2009;[START_REF] Renahy | Health information seeking on the Internet: a double divide? Results from a representative survey in the Paris metropolitan area, France, 2005-2006[END_REF][START_REF] Roustit | Exposure to interparental violence and psychosocial maladjustment in the adult life course: advocacy for early prevention[END_REF].

In 2005, 29% of the people contacted declined to answer, and 5% were excluded because they did not speak French (3%) or were too sick to answer our questions (2%) [START_REF] Renahy | Health information seeking on the Internet: a double divide? Results from a representative survey in the Paris metropolitan area, France, 2005-2006[END_REF]. The final sample consisted of 3,023 persons with a mean of 60.5 participants per census block (range: 60 to 65).

The mean age of the SIRS population was 47 years (range: 18 to 97). The mean monthly household income was 1,734 € per consumption unit (range: 50 to 10,000 € per CU). The 3,023 SIRS respondents were predominantly female (61%), French (86%), with a postsecondary education (45%) and in the workforce (55%). Table 1 presents respondents characteristics in the full sample.

Variables

Depression

Depression was investigated by the Mini-International Neuropsychiatric Interview (M.I.N.I.). This is a short, structured diagnostic interview designed in such a manner as to permit administration by nonspecialist interviewers. The depression index was determined by a 10-item questionnaire for measuring the occurrence of major depressive disorders during the previous two weeks. Using the validated and usual cut-off score (four positive answers out of ten questions), a binary variable indicating the absence or presence of depression was created. The internal and external validity of this binary variable had been demonstrated in the French population [START_REF] Lecrubier | The Mini International Neuropsychiatric Interview (MINI). A short diagnostic structured interview: reliability and validity according to the CIDI[END_REF][START_REF] Sheehan | The Mini-International Neuropsychiatric Interview (M.I.N.I.): the development and validation of a structured diagnostic psychiatric interview for DSM-IV and ICD-10[END_REF].

Activity space

In this paper, activity space was assessed from the respondents" statements about the location of their regular domestic and social activities. In the SIRS survey, people were asked where they usually 1) go food shopping; 2) use services (bank, post office); 3) go for a walk; 4) meet friends; and 5) go to a restaurant or café. For each of these five activities, three answers were proposed: 1) mainly within their residential neighbourhood; 2) mainly outside their residential neighbourhood; and 3) both within and outside their residential neighbourhood. The neighbourhood of residence was not defined, and its boundaries were left to the individual"s own assessment and perception -what we call here the "perceived neighbourhood".

A measure of activity space was then created: activities said to be performed mainly within the neighbourhood were assigned a value of 100%, while those performed "both within and outside the neighbourhood" or "mainly outside neighbourhood" were assigned a value of 50% or 0%, respectively.

Upon adding these values and dividing the sum by the total number of reported activities, we obtained an individual score measuring the concentration of daily activities in the perceived neighbourhood.

The respondents were then ranked on the basis of this score, which ranged from 0 (for people who reported doing every proposed activity mainly outside their neighbourhood of residence) to 1 (for people who reported doing every proposed activity mainly within their neighbourhood of residence).

Participants who did not provide answers for any of the five proposed activities (n=12) were excluded from the analysis. This score was then used as a proxy of personal exposure to the neighbourhood of residence.

To analyze cross-level interaction more easily, we also decided to isolate people who reported doing the vast majority of their daily activities within their perceived neighbourhood of residence. We grouped together people with a score greater than or equal to 0.8. They were those who reported that they did either (a) every activity within their perceived neighbourhood of residence, (b) one or two activities both within and outside their perceived neighbourhood of residence, or (c) only one activity mainly outside their perceived neighbourhood of residence. Finally, 520 of the 3,011 respondents (17.5%) were then considered as having an activity space significantly limited to their perceived neighbourhood of residence, while 2,491 (82.5%) were considered as having an activity space larger than their perceived neighbourhood. In a previous published research which used data from the same survey, we showed that being a foreigner, having a low level of education, living in a low-income household, being unemployed, retired or at home, having lived in the neighbourhood for more than 20 years, being physically limited, living in a neighbourhood with a high shop density and with a high mean income increase significantly the likelihood to have an activity space limited to the neighbourhood of residence [START_REF] Vallée | The combined effects of activity space and neighbourhood of residence on participation in preventive health-care activities: The case of cervical screening in the Paris metropolitan area (France)[END_REF].

Other variables

In addition to sex, age, nationality, level of education, occupational status and employment status, we considered whether the person was living in a couple relationship. Functional limitation was investigated as well, by the global activity limitation indicator from the Minimum European Health Module [START_REF] Cox | The reliability of the Minimum European Health Module[END_REF]. The respondents were asked whether they perceived themselves as having a

severe limitation of at least six months" duration in performing activities people usually engage in.

Neighbourhood of residence

Neighbourhood deprivation

As stated in the introduction, we considered neighbourhood deprivation in three different manners: (1) from individual assessment, (2) from collective assessment and (3) from census based measures of social composition.

(1) Individual neighbourhood assessments were obtained by asking the participants to rate ten neighbourhood characteristics using a 4-point Likert scale. The neighbourhood attributes were the quality of the public transportation, the quality of the schools, the presence of shops, cohesion between the residents, the condition of the roads and buildings, the local authorities" efforts involving the neighbourhood, the level of unemployment, the neighbourhood"s remoteness, its condition compared to that of other urban neighbourhoods, and its quality as a place to raise children. The responses to these items were summed to create a total index score and divided into two roughly equal groups. In all, 1,298 of the 3,023 participants were classified as having a positive assessment of their neighbourhood of residence, 1,725 a negative assessment.

(2) We also aggregated the neighbourhood assessments by all the participants living in the same census block to create a contextual variable. Neighbourhoods in which a majority of participants had a positive individual assessment of their neighbourhood were categorized as neighbourhoods with a positive collective assessment (n=21), while those in which a majority of participants had a negative individual assessment were categorized as neighbourhoods with a negative collective assessment (n=29).

(3) Neighbourhood census social composition was based on individual socio-occupational data (occupational status and current employment status) from the 1999 census information [START_REF] Préteceille | La division sociale de l'espace francilien[END_REF]. These census socio-occupational data were then aggregated by census blocks (i.e. IRIS units).

The idea was here to characterize neighbourhood deprivation independently of the perceptions of our sample respondents. The 50 census blocks which were selected in our health survey were categorized, either as working-class neighbourhoods (n=20) or as upper-or middle-class neighbourhoods (n=30).

Neighbourhood location

Finally, we accounted for location in the Paris metropolitan area. The surveyed census blocks were classified either as being in central Paris (inner-city areas) or outside central Paris (suburban areas), with 13 and 37 census blocks, respectively, in each of these two groups.

Neighbourhood spatial delimitations

In this paper, we used two different spatial delimitations of neighbourhood of residence.

-One of them was based on the participants" self-defined areas. It is the one they referred to when they were asked about their daily activities and to give their neighbourhood assessments.

-The other was based on the census blocks (i.e. IRIS units). It was used to characterize neighbourhood social composition (as working-class neighbourhoods or as upper-or middleclass neighbourhoods) and location in the Paris metropolitan area.

Statistical methods

All the proportions presented in this article were weighted to account for the complex sample design (notably, the design effect associated with cluster sampling and the overrepresentation of poorer neighbourhoods) and for the poststratification adjustment for age and sex according to the general population 1999 census data. Significant differences in weighted proportions were measured by the Pearson chi-squared test.

Statistical associations between the individual and neighbourhood characteristics and depression were examined using multi-level logistic regression models of 3,011 individuals (i.e. respondents for whom activity space data were available) at level 1 nested within 50 surveyed census blocks (i.e. IRIS units) at level 2. In these analyses the mean number of respondents per census block was 60.2 (range: 58 to 65). Multilevel logistic regression models were fitted using the xtmelogit command in Stata10 software.

In each regression model, we introduced sex, age, nationality, level of education, occupational status, employment status, couple relationship status, functional limitation, and urban location to control for residual confounding and entered, alternately, the following measures of neighbourhood deprivation:

individual neighbourhood assessment (Model 1), collective neighbourhood assessment (Model 2) and neighbourhood census social composition (Model 3). We then calculated the cross-level interaction terms between the measure of activity space and each of these three measures of neighbourhood deprivation. We examined whether the effects of activity space on depression differed across neighbourhood deprivation and, conversely, whether the effects of neighbourhood deprivation on depression differed across activity space. A p-value < 0.05 was considered significant for all the statistical analyses presented.

Results

Individual characteristics and depression

Of the overall population of 3,023 persons, 11.6% were depressed (Table 2). In bivariate analysis (Table 2), we observed that the proportion of depressed people was statistically higher for women (15.4%) than for men (7.4%; p<0.01); for foreigners (15.5%) than for people of French nationality (11%; p=0.05); for people with a low level of education (15.9%) than for those with high level of education (8.0%; p<0.01); for lower white-collar workers (18.9%) than for upper-white collar workers (5.4%; p<0.01); for people who were unemployed (13.6%), retired or at home (14.6%) than for those who were working or studying (10.1%; p=0.02); for those who were not in a couple relationship (15%) than those who were (9.9%; p<0.01); and for people with a severe functional limitation (33.5%) than for those without a severe limitation (10.3%; p<0.01). In multivariate analysis, we observed that women, people without a postsecondary education, those with a low occupational status, blue-collar workers, lower white-collar workers, tradespeople and salespeople, those who were retired or at home, those who were not in a couple relationship and those with a severe functional limitation had a significantly higher risk of depression. In multivariate analysis, nationality became non-statistically associated with depression.

As regards the relationship between age and depression, we observed that the proportion of depressed people was slightly higher -though the difference was not statistically significant -among the respondents aged 60 years or older (12.9%) than in those under the age of 30 (10.5%). In multivariate analysis, after adjusting for certain age-related variables, such as employment status and functional limitation, the trend reversed, in that the under-60s had a significantly higher risk of being depressed than those aged 60 years or older.

Neighbourhood characteristics and depression

In bivariate analysis (Table 2), we observed that the proportion of depressed people was statistically higher:

among those who had given (individually) a negative assessment of their neighbourhood (14.4%) compared to those who had given a positive assessment (8.9%; p <0.01);

among those living in neighbourhoods where the collective neighbourhood assessment was mainly negative (13.5%) compared to those living in neighbourhoods where the collective neighbourhood assessment was mainly positive (9.9%; p=0.03);

and, lastly, among those living in working-class neighbourhoods (17.2%) compared to those living in upper-or middle-class neighbourhoods (9.6%; p<0.01).

In multivariate analysis (Table 2), we found that the respondents who had given a negative assessment of their neighbourhood had a significantly higher likelihood of being depressed (OR=1.57; 95% CI=1.24-1.99. See Model 1 in Table 2). However, those living in neighbourhoods where the neighbourhood assessment was mainly negative did not have a significantly higher likelihood of being depressed (OR=1.21; 95% CI=0.93-1.59; See Model 2 in Table 2). Lastly, those living in workingclass neighbourhoods had a significantly higher likelihood of being depressed (OR=1.57; 95% CI=1.22-2.02; See Model 3 in Table 2), even after adjustment for individual characteristics.

Furthermore, we did not observe a statistical association between depression and urban versus suburban location in bivariate analysis. However, in the last multilevel model (Model 3 in Table 2), we did observe that the respondents living in inner-city areas were statistically more depressed than those living in suburban areas (OR=1.40; 95% CI=1.05-1.86).

Spatial disparities in depression in the Paris metropolitan area

Depending on the surveyed census block, the crude proportion of people with depression varied from 3.1% to 31.6%, as shown in Figure 1. The crude between-area variation (VarU 0j ) was statistically significant, which indicates that there were spatial disparities in depression between the 50 census blocks surveyed in the Paris metropolitan area (Table 2). We observed that the between-area variation in depression remained statistically significant when only individual variables were introduced into the model (not shown) but became statistically nonsignificant when measures of neighbourhood deprivation were introduced into other models in addition to individual variables (Models 1, 2 and 3 in Table 2). This means that spatial disparities in depression between the 50 surveyed census blocks in the Paris metropolitan area were fully explained by selected individual and neighbourhood characteristics.

Activity space, neighbourhood characteristics and depression

We did not observe a statistical association between activity space and depression either in bivariate analysis or in multivariate analysis when individual and contextual characteristics were considered.

However, interaction between activity space and neighbourhood characteristics was found to be statistically significant regardless of the method for characterizing neighbourhood deprivation:

individual neighbourhood assessment (p<0.01; See Tables 3 and4); collective neighbourhood assessment (p=0.01; See Tables 5 and6) or neighbourhood census social composition (p<0.01; See Tables 7 and8). Relationship between activity space and depression according to neighbourhood deprivation Table 3 compares the point estimates of the odds ratios for the association between activity space and depression among the respondents who had individually given a negative or a positive assessment of their neighbourhood of residence. Those with an activity space limited to their perceived neighbourhood had a significantly higher risk of being depressed if they gave a negative assessment of their neighbourhood (OR=1.61; 95% CI=1.10-2.36) but a significantly lower risk of being depressed if they gave a positive assessment (OR=0.58; 95% CI=0.36-0.93).

Similarly, we observed (Table 5) that the respondents with an activity space limited to their perceived neighbourhood had a higher risk (but not statistically significant) of being depressed if they lived in neighbourhoods where the collective assessment was negative (OR=1.38; 95% CI=0.95-2.00) but a significantly lower risk of being depressed if they lived in neighbourhoods where the collective assessment was positive (OR=0.54; 95% CI=0.33-0.89). Lastly, it is seen in Table 7 that the participants with an activity space limited to their perceived neighbourhood had a significantly higher risk of being depressed (OR=1.62; 95% CI=1.06-2.49) if they lived in a working-class neighbourhood but a significantly lower risk of being depressed (OR=0.65; 95% CI=0.43-0.98) if they lived in an upper-or middle-class neighbourhood (as measured from census socio-occupational data).

Briefly, an activity space limited to the perceived neighbourhood may be seen as a protective factor with regard to depression for those living in advantaged neighbourhoods and as a risk factor with regard to depression for those living in deprived neighbourhoods -regardless of the method used for characterizing neighbourhood deprivation. Greater effect of neighbourhood characteristics on depression according to activity space Table 4 compares the point estimates of the odds ratios for individual neighbourhood assessments among the respondents with an activity space limited to their perceived neighbourhood and those with a larger activity space. Their mental health was statistically more affected by a negative individual neighbourhood assessment if they had reported an activity space limited to their neighbourhood (OR=3.29; 95% CI=1.91-5.67) compared to those who had reported an activity space extending beyond their perceived neighbourhood (OR=1.30; 95% CI=1.00-1.70).

Similarly, we observed (Table 6) that the odds ratios for the association between a negative collective neighbourhood assessment and depression were statistically greater in the participants with an activity space limited to their perceived neighbourhood (OR=2.30; 95% CI=1.27-4.19) compared to those whose activity space extended beyond their perceived neighbourhood (OR=1.03; 95% CI=0.77-1.40).

Lastly, Table 8 compares the point estimates of the odds ratios for the association between neighbourhood census social composition and depression among the respondents with an activity space limited to the perceived neighbourhood and those with a larger activity space. Their mental health was statistically more vulnerable to neighbourhood deprivation if they had an activity space limited to their perceived neighbourhood (OR=4.05; 95% CI=2.11-7.79) compared to those whose activity space extended beyond their perceived neighbourhood (OR=1.31; 95% CI=0.98-1.77).

Briefly, living in a deprived or negatively perceived neighbourhood had the most negative effect in terms of depression in the respondents with an activity space limited to their perceived neighbourhood of residence.

Sensitivity analysis

Individual and collective neighbourhood assessments were first continuously measured. Before categorizing these two neighbourhood variables, we checked in multilevel logistic regression models that they were linearly related (p=0.01) to depression. We decided to convert these two continuous variables to binary variables to more easily analyze the cross-level interactions according to these assessments (Table 3 andTable 5).

For the same reason, we decided to divide the respondents into two groups on the basis of their activity space score. We chose a cut-off of 0.8, which led to 17.5% of the respondents being classified as having an activity space limited to their neighbourhood of residence. If a cut-off of 0.7 had been used, it would have led to 29.3% of the respondents being classified as having a limited activity space. In this case, cross-level interactions would have been statistically significant for individual neighbourhood assessment (p<0.01) and neighbourhood census social composition (p=0.04), but not for collective neighbourhood assessment (p=0.18). If a cut-off of 0.6 had been used, it would have led to nearly half of the surveyed population (42.6%) being categorized as having a limited activity space, and the cross-level interactions would have been statistically significant for individual neighbourhood assessment (p<0.01), but not for neighbourhood census social composition (p=0.15) or collective neighbourhood assessment (p=0.32). Finally, when the score measuring the concentration of their daily activities in their neighbourhood was considered as a continuous variable in the models, the cross-level interactions were found to be statistically significant for individual neighbourhood assessment (0.03), but not for collective neighbourhood assessment (p=0.18) or neighbourhood census social composition (p=0.10). In conclusion, even if our models" ability to detect significant interactions was obviously dependent on the cut-off chosen for reasons of statistical power, we observed that (i) the point estimates of the odds ratios for neighbourhood deprivation were always higher among the respondents with a limited activity space than among those with a larger activity space, and that (ii) the point estimates of the odds ratios for a limited activity space were always greater than 1 in deprived neighbourhoods (which indicates a higher risk of being depressed) and less than 1 in more privileged neighbourhoods (which indicates a lower risk of being depressed).

Discussion

Greater sensitivity to residential characteristics in people with a limited activity space In this research, we observed greater vulnerability to neighbourhood deprivation in the respondents who concentrated their daily activities in their perceived neighbourhood of residence. This vulnerability could be due to the fact that a limited activity space may increase contact with the people, institutions, spatial structures and norms present in the neighbourhood of residence [START_REF] Hanson | Perspectives on the geographic stability and mobility of people in cities[END_REF]. On the other hand, a larger activity space may permit exposure to a greater variety of neighbourhoods, people and social norms. Urban geographers and environmental psychologists who have studied the role of daily mobility in neighbourhood attachment have described daily travels within the overall city as a way to escape the constraints of one"s residential neighbourhood [START_REF] Authier | Le quartier à l'épreuve des mobilités "métapolitaines[END_REF][START_REF] Gustafson | Mobility and Territorial Belonging[END_REF][START_REF] Ramadier | Mobilité quotidienne et attachement au quartier : une question de position?[END_REF]. This interpretation was still being developed by [START_REF] Inagami | Non-residential neighborhood exposures suppress neighborhood effects on self-rated health[END_REF], when they studied adult self-rated health and found that residential neighbourhood effects are suppressed by exposure to other environments.

Association between activity space and depression according to neighbourhood deprivation status. This research also revealed that an activity space limited to the neighbourhood of residence was protective with regard to depression for people living in advantaged neighbourhoods. We can postulate that spatial confinement may have indeed promoted individual well-being for people living in places where social cohesion and public and private facilities were assessed positively by the residents. In addition, we can also hypothesize that spatial confinement within advantaged neighbourhoods results from a choice rather than a constraint. Consequently, an activity space limited to one"s residential neighbourhood may be associated with well-being in privileged neighbourhoods because this spatial behaviour is voluntary.

On the other hand, spatially limited daily mobility appeared to be associated with more frequent depression in deprived areas. This inverse association could be a consequence of constrained spatial confinement within deprived neighbourhoods due to the material and physical difficulties or symbolic barriers to moving outside such neighbourhoods.

Lastly, the opposite relationship between activity space and depression according to neighbourhood deprivation status may explain why we did not observe a significant association between activity space and depression in the models with no interaction (e.g., OR=0.99; 95% CI=0.74-1.33; See Model 2 in Table 2). In these models, we did not, in fact, observe an association between activity space and depression precisely because this association varied in an opposite direction according to neighbourhood deprivation status.

Activity space, functional limitation and depression

Previous research has shown that functional limitation (or disability) was frequently related to greater depression [START_REF] Bierman | Timing, Social Support, and the Effects of Physical Limitations on Psychological Distress in Late Life[END_REF][START_REF] Gayman | Physical limitations and depressive symptoms: exploring the nature of the association[END_REF]. Some studies have suggested a reciprocal, potentially spiralling relationship between depression and functional limitation, notably in late life [START_REF] Bruce | Depression and Disability in Late Life: Directions for Future Research[END_REF][START_REF] Carriere | Modelling complex pathways between late-life depression and disability: evidence for mediating and moderating factors[END_REF]. In this research, we also observed that participants with severe functional limitation were significantly more depressed (Table 2). However, reducing the role of daily mobility in health to functional limitation alone would be restrictive. Indeed, we observedaccording to neighbourhood deprivation status -a relationship between activity space and health, even after adjusting for functional limitation. These results suggest that one should explore both functional limitation and activity space when examining in detail the role of daily mobility in depressive symptoms.

Methodological bias

The frequency with which each activity was engaged in was not considered in the computation of activity space score. It could nonetheless affect the quality of the activity space score, particularly for discriminating social activities. Eating out at a restaurant weekly as opposed to once a year could reflect a different way to move about within a city. If frequency data were available -which is not the case in our survey database-it could be interesting to weight the activity space score with frequency data.

The main limitation of the simplified measure of activity space proposed in this paper stems from the impossibility of distinguishing between the actual spatial extent of daily mobility and the perceived neighbourhood delimitation [START_REF] Vallée | The combined effects of activity space and neighbourhood of residence on participation in preventive health-care activities: The case of cervical screening in the Paris metropolitan area (France)[END_REF]. Our measure of activity space was not defined on the basis of the precise location of the daily activities, but was instead directly linked to the respondents" neighbourhood representation, since they were asked to place their activities within or outside what they considered their neighbourhood of residence. So, when studying in the same analysis both measure of activity space based on the individual perceived neighbourhood delimitation and the measure of neighbourhood deprivation based on the census neighbourhood delimitation (See Model 3 in Table 2), a potential spatial discrepancy can appear and lead to a misinterpretation of personal exposure to the neighbourhood for some individuals [START_REF] Vallée | The combined effects of activity space and neighbourhood of residence on participation in preventive health-care activities: The case of cervical screening in the Paris metropolitan area (France)[END_REF]. Depending on the individual, the spatial delimitation of the perceived neighbourhood may be the same, smaller or larger than the census unit. Unfortunately, data collected in 2005 did not enable us to study precisely individual perceived neighbourhood boundaries. To address this potential spatial discrepancy it was therefore interesting to take into account neighbourhood individual assessments which were logically based on individual perceived neighbourhood boundaries. When considering both the model of activity space, based on the respondents" statement, and the measure of neighbourhood deprivation, based on respondents" assessments (See Model 1 in Table 2), the potential spatial discrepancy should disappear.

However, we must also consider that an individual"s neighbourhood delimitation may vary according to the question. A "one-size-fits-all" definition of the neighbourhood may be too simplistic [START_REF] Stafford | Small area inequalities in health: are we underestimating them?[END_REF]. For example, a person may delimit his or her neighbourhood as a small building block when reporting the location of his or her domestic and social activities but delimit his or her neighbourhood as a larger space when evaluating the quality of his or her residential environment.

Reporting bias is another important methodological bias, as discussed in the literature investigating neighbourhood effects on depression [START_REF] Mair | Are neighbourhood characteristics associated with depressive symptoms? A review of evidence[END_REF]. Reporting bias exists if people who are depressed are more likely to give a more negative assessment of their neighbourhood because of their depression, even if, objectively, the neighbourhood conditions are actually good. To address this potential reporting bias, we used census information to characterize neighbourhood deprivation independently of the perceptions of our sample respondents.

In short, we suggest that reporting bias might be less problematic when the neighbourhood characteristics of interest were measured from information collected independently of the sample respondents (i.e. census information). Conversely, we suggest that potential spatial discrepancy might be less problematic when the measures of neighbourhood deprivation were derived from individual or collective participant perceptions. Models incorporating either individual neighbourhood assessments (Model 1), collective neighbourhood assessments (Model 2) or neighbourhood census social composition (Model 3) yielded concordant results. Regardless of the method for characterizing neighbourhood deprivation, similar significant relationships between depression, activity space and neighbourhood deprivation were then observed.

Reverse causation and causality

Cross sectional analyses such as those presented here cannot be discussed in terms of causality. Specifically, it is not known (i) if a limited activity space leads to depression in deprived areas and to well-being in privileged areas, or (ii) if depression leads individuals to concentrate their daily activities within their neighbourhood if they live in a deprived area and to extend their activity space beyond their neighbourhood if they live in privileged areas. Even if it appears implausible to assume that depression leads people in privileged areas to extend their activity space, it would be interesting to analyse longitudinal data collected in 2005 and 2009 among the same population (within the framework of the SIRS cohort) in order to overcome causality interpretation problems, which affect all cross-sectional studies.

Longitudinal analysis could also be useful in limiting reverse causation bias associated with residential trajectories. This reverse causation bias would arise if depressed people were particularly inclined to move into or to remain within deprived neighbourhoods and mentally healthy people were particularly inclined to move into or to remain in advantaged neighbourhoods [START_REF] Curtis | Socio-Geographic Mobility and Health Status: a longitudinal analysis using the National Population Health Survey of Canada[END_REF][START_REF] Deverteuil | Mental health and the city: intra-urban mobility among individuals with Schizophrenia[END_REF]. In such case, exposure to neighbourhood characteristics would be a consequence, not a cause of, depression. In cross-sectional studies, residential mobility could thus lead to a misinterpretation of the relationship between depression and exposure to neighbourhood characteristics.

Model adjustment

As in other research studying the association between neighbourhood characteristics and depressive symptoms, individual-level confounders, such as age, gender, couple relationship status, education, occupational status and employment status, were included in our models to control for residual confounding. However, there is no consensus as to what the key confounders are [START_REF] Mair | Are neighbourhood characteristics associated with depressive symptoms? A review of evidence[END_REF].

With regard to respondent's 'origin' we used nationality data to distinguish between French people and foreigners. Statistical data on 'race' or 'ethnicity' cannot, by law, be collected or used in research in France. In the final models, we preferred to include occupational status rather than household income, even if people from poor households were found to be more inclined to concentrate their daily activities within their perceived neighbourhood [START_REF] Vallée | The combined effects of activity space and neighbourhood of residence on participation in preventive health-care activities: The case of cervical screening in the Paris metropolitan area (France)[END_REF] and to be significantly more depressed (models not shown). However, household income and occupational status were too closely correlated to be integrated simultaneously into the same regression models. To be consistent with the choice of socio-occupational data which were used to measure neighbourhood census social composition at the IRIS level, we opted for individual occupational status.

We also accounted for functional limitation because we wanted to be able to study activity space after adjusting for disability. Finally, we included in our models a variable describing urban neighbourhood location because it can be postulated that central urban residence may be associated with a higher risk of depression [START_REF] Ross | Neighborhood Disadvantage and Adult Depression[END_REF]. We observed a significant difference in this regard in only one of the three models (Model 3 in Table 2). Actually, we kept urban location in the regression models to take into account differences in the size of perceived neighbourhood according to urban and suburban location, which were previously described in the Paris metropolitan area (Humain-Lamoure, 2010).

Further implications

Investigating activity space promises a deeper understanding of the mechanisms involved in depression. As for mental health policy, this study suggests that more attention should be paid to people who are spatially confined within a deprived neighbourhood, considering that such confinement may explain, at least in part, the social inequalities in depression prevalence observed in most cities in developed countries. From a medical perspective, since it is known that the accuracy of depression recognition by non-psychiatric physicians is generally low, particularly in underprivileged populations [START_REF] Cepoiu | Recognition of depression by non-psychiatric physicians--a systematic literature review and meta-analysis[END_REF], it could be useful to inform primary care physicians that not only social isolation but also constrained spatial confinement may bring about conditions conducive to depression. As for city planning policy, this research underscores the importance of enabling people to overcome any material or physical difficulties so that they can move about outside their neighbourhood of residence. Special efforts in public transportation for deprived populations and confined neighbourhoods could then help improve individual well-being.

Since the sample we studied was from the largest metropolitan area in France, where the population is more educated and has higher incomes on average (but also shows the deepest social disparities) and where there is a highly developed public transportation system and numerous (but unevenly distributed) services and recreation resources, our results cannot be extrapolated to smaller urban settings or to rural areas, even if it can be assumed that there could be a similar association between spatial confinement and depression in such places. On the other hand, it would be interesting to replicate such an approach in other major urban settings, particularly in other world cities that share with Paris similar urban and social patterns.

Conclusion

The social epidemiology, health geography and public health literature studying the relationship between neighbourhood and mental health has seldom accounted for the role of activity space. This paper points out that it may be useful to examine how activity space -representing an individual's experience of place and degree of mobility -modifies in a significant way the strength of neighbourhood effects on depression and how neighbourhood characteristics reverse the association between activity space and depression. Data on activity space as measured by the concentration of daily activities in the perceived neighbourhood are easy to collect in a large sample and permit a deeper understanding of the mechanisms linking the neighbourhood of residence and mental health.

Considering only the effect of the neighbourhood of residence on health and excluding non-residential exposure would lead to a "local trap" [START_REF] Cummins | Commentary: investigating neighbourhood effects on health-avoiding the 'local trap[END_REF] and consequently to an exposure misclassification [START_REF] Basta | Neighborhoods, daily activities, and measuring health risks experienced in urban environments[END_REF]. We therefore suggest taking activity space into account more systematically when studying the neighbourhood determinants of health outcomes.

Funding/Support

The SIRS survey was supported by the French National Institute for Health and Medical Research (INSERM), the Institute for Public Health Research (IReSP), the Directorate-General of Health (DGS), the Interministerial Delegation for Urban Affairs (DIV), the European Social Fund, the Regional Council of Île-de-France, and the City of Paris. Support for Julie Vallée"s postdoctoral research was provided by the Fondation de France.

Figure 1 .

 1 Figure 1. Spatial disparities in depression in the Paris metropolitan area in 2005.

Table 1 .

 1 Selected characteristics of SIRS respondents in 2005

	Total sample
	(n=3,023)

Table 2 .

 2 Individual and neighbourhood risk factors for depression in the population in the Paris metropolitan area(2005)

			Bivariate		Multilevel logistic regression
	Depression		(n=3,023) 1		(n=3,011)
		% 2	Total	p-value 3	Model 1	Model 2	Model 3
	Total	11.6 (3,023)	-		OR (95% CI)
	Individual variables (level 1):
	Sex
	Male Female	7.4 15.4 (1,843) (1,180)	<0.01	Ref. 2.06 (1.58-2.69)*	Ref. 2.08 (1.59-2.71)*	Ref. 2.07 (1.59-2.71)*
	Age
	18-29 years	10.5 (524)		1.90 (1.17-3.08)*	1.87 (1.15-3.04)*	1.80 (1.11-2.92)*
	30-44 years 45-59 years	11.0 (956) 12.4 (797)	>0.05	1.88 (1.23-2.88)* 1.98 (1.35-2.90)*	1.87 (1.22-2.86)* 1.98 (1.35-2.89)*	1.80 (1.18-2.75)* 1.92 (1.31-2.80)*
	≥ 60 years	12.9 (746)		Ref.	Ref.	Ref.
	Nationality
	French Foreign	11.0 (2,588) 15.5 (435)	0.05	Ref. 1.16 (0.85-1.58)	Ref. 1.12 (0.82-1.57)	Ref. 1.11 (0.81-1.51)
	Level of education
	Postsecondary	8.0	(1,348)		Ref.	Ref.	Ref.
	Secondary school None or primary school	15.4 (1,283) 15.9 (392)	<0.01	1.45 (1.08-1.94)* 1.43 (0.94-2.18)	1.45 (1.08-1.94)* 1.43 (0.94-2.18)	1.39 (1.04-1.86)* 1.34 (0.88-2.04)
	Occupational status (current or last)
	Never worked	10.8 (254)		1.19 (0.68-2.08)	1.19 (0.68-2.07)	1.14 (0.65-1.99)
	Blue collar	11.7 (415)		2.26 (1.39-3.68)*	2.21 (1.35-3.59)*	2.06 (1.26-3.35)*
	Lower white-collar Tradespeople, salespeople	18.9 (888) 12.5 (121)	<0.01	2.21 (1.46-3.33)* 2.31 (1.23-4.35)*	2.20 (1.46-3.32)* 2.24 (1.19-4.21)*	2.08 (1.38-3.14)* 2.24 (1.19-4.20)*
	Intermediary occupation	11.2 (668)		1.49 (0.99-2.23)	1.45 (0.97-2.18)	1.40 (0.93-2.10)
	Upper white-collar	5.4	(677)		Ref.	Ref.	Ref.
	Current employment status
	Working or studying	10.1 (1,815)		Ref.	Ref.	Ref.
	Unemployed	13.6 (272)	0.02	1.29 (0.90-1.85)	1.28 (0.89-1.83)	1.28 (0.90-1.848)
	Retired/At home	14.6 (936)		1.48 (1.06-2.08)*	1.49 (1.06-2.10)*	1.47 (1.04-2.06)*
	Living in a couple relationship
	Yes No	9.9 15.0 (1,663) (1,360)	<0.01	Ref. 1.63 (1.30-2.04)*	Ref. 1.65 (1.32-2.06)*	Ref. 1.64 (1.32-2.05)*
	Severe functional limitation
	No Yes	10.3 (2,805) 33.5 (218)	<0.01	Ref. 3.17 (2.25-4.45)*	Ref. 3.19 (2.27-4.48)*	Ref. 3.19 (2.28-4.47)*
	Activity space
	Larger than perceived neighbourhood Limited to perceived neighbourhood	11.1 (2,491) 13.2 (520)	>0.05	Ref. 1.05 (0.78-1.40)	Ref. 0.99 (0.74-1.33)	Ref. 0.99 (0.75-1.32)
	Individual neighbourhood assessment
	Positive	8.9	(1,298)	<0.01	Ref.	-	-
	Negative	14.4 (1,725)		1.57 (1.24-1.99)*	-	-
	Contextual variables (level 2):
	Collective neighbourhood assessment
	Positive Negative	9.9 13.5 (1,746) (1,277)	0.03	--	Ref. 1.21 (0.93-1.59)	--
	Neighbourhood census social composition
	Upper-or middle-class neighbourhood Working-class neighbourhood	9.6 17.2 (1,202) (1,821)	<0.01	--	--	Ref. 1.57 (1.22-2.02)*
	Location
	Suburban areas	11.6 (2,234)	>0.05	Ref.	Ref.	Ref.
	Paris	11.7 (789)		1.30 (0.97-1.76)	1.24 (0.91-1.68)	1.40 (1.05-1.86)*
	Between-area variation:	Crude Model:		Model 1:	Model 2:	Model 3:
	VarU 0j (95% CI)	0.146 (0.067-0.314)		0.044 (0.008-0.251)	0.055 (0.013-0.238)	0.018 (0.000-0.693)

Table 3 .

 3 Association between activity space and depression as determined from the multilevel logistic regression model for two subpopulations, by type of individual neighbourhood assessment

	Depression 1	People with: A positive assessment of their (perceived) neighbourhood	A negative assessment of their (perceived) neighbourhood	Entire population n=3,011	Interaction Activity space x individual
		n=1,291	n=1,720		neighbourhood
		Odds Ratio (95% CI)		assessment
	Activity space
	Larger than the perceived neighbourhood Ref. Limited to the perceived neighbourhood 0.58 (0.36-0.93)*	Ref. 1.61 (1.10-2.36)*	Ref. 0.97 (0.72-1.29)	p<0.01
	1 After adjustment for sex, age, nationality, level of education, occupational status, employment status, couple
	relationship status, functional limitation and urban location
	* p <0.05

Table 4 .

 4 Association between individual neighbourhood assessment and depression as determined from the multilevel logistic regression model for two subpopulations, by type of activity space

		People with an activity space:		Interaction
	Depression 1	Larger than the perceived neighbourhood	Limited to the perceived neighbourhood	Entire population n=3,011	Activity space x individual
		n=2,491	n=520		neighbourhood
		Odds Ratio (95% CI)		assessment
	Individual neighbourhood assessment
	Positive Negative	Ref. 1.30 (1.00-1.70)*	Ref. 3.29 (1.91-5.67)*	Ref. 1.55 (1.22-1.96)*	p<0.01
	1 After adjustment for sex, age, nationality, level of education, occupational status, employment status, couple
	relationship status, functional limitation and urban location
	* p <0.05

Table 5 .

 5 Association between activity space and depression as determined from the multilevel logistic regression model for two subpopulations, by type of collective neighbourhood assessment

		People living in neighbourhoods:		Interaction
		Whose residents gave a	Whose residents	Entire population
	Depression 1	mainly positive assessment	gave a mainly negative assessment	n=3,011	Activity space x collective
		n=1,269	n=1,742		neighbourhood
		Odds Ratio (95% CI)		assessment
	Activity space
	Larger than the perceived neighbourhood Limited to the perceived neighbourhood	Ref. 0.54 (0.33-0.89)*	Ref. 1.38 (0.95-2.00)	Ref. 0.97 (0.72-1.29)	p=0.01
	1 After adjustment for sex, age, nationality, level of education, occupational status, employment status, couple
	relationship status, functional limitation and urban location
	* p <0.05

Table 6 .

 6 Association between collective neighbourhood assessment and depression as determined from the multilevel logistic regression model for two subpopulations, by type of activity space

		People with an activity space:		Interaction
	Depression 1	Larger than the perceived neighbourhood	Limited to the perceived neighbourhood	Entire population n=3,011	Activity space x collective
		n=2,491	n=520		neighbourhood
		Odds Ratio (95% CI)		assessment
	Collective neighbourhood assessment
	Positive Negative	Ref. 1.03 (0.77-1.40)	Ref. 2.30 (1.27-4.19)*	Ref. 1.22 (0.93-1.60)	p=0.01
	1 After adjustment for sex, age, nationality, level of education, occupational status, employment status, couple
	relationship status, functional limitation and urban location
	* p <0.05

Table 7 .

 7 Association between activity space and depression as determined from the multilevel logistic regression model for two subpopulations, by type of neighbourhood census social composition

		People living in:
	Depression 1	Upper-or middle-class neighbourhoods n=1,815	Working-class neighbourhoods n=1,196	Entire population n=3,011	Interaction Activity space x neighbourhood
		Odds Ratio (95% CI)		social composition
	Activity space
	Larger than the perceived neighbourhood Limited to the perceived neighbourhood	Ref. 0.65 (0.43-0.98)*	Ref. 1.62 (1.06-2.49)*	Ref. 0.97 (0.72-1.29)	p<0.01
	1 After adjustment for sex, age, nationality, level of education, occupational status, employment status, couple
	relationship status, functional limitation and urban location
	* p <0.05

Table 8 .

 8 Association between neighbourhood census social composition and depression as determined from the multilevel logistic regression model for two subpopulations, by type of activity space

		People with an activity space:
	Depression 1	Larger than the perceived neighbourhood n=2,491	Limited to the perceived neighbourhood n=520 Odds Ratio (95% CI)	Entire population n=3,011	Interaction Activity space x neighbourhood social composition
	Neighbourhood census social
	composition
	Upper-or middle-class neighbourhood	Ref.	Ref.	Ref.	p<0.01
	Working-class neighbourhood	1.31 (0.98-1.77)	4.05 (2.11-7.79)*	1.57 (1.21-2.04)*
	1 After adjustment for sex, age, nationality, level of education, occupational status, employment status, couple
	relationship status, functional limitation and urban location
	* p <0.05

Except for the bivariate analysis between depression and activity space (n=3,011).

Taking into account the complex sample design.

Pearson (design-based). * p <0.05

Acknowledgements

The authors would like to thank three anonymous referees for their helpful comments.