Nathalie Picard
email: nathalie.picard@u-cergy.fr

André De Palma
email: andre.depalma@ens-cachan.fr

Sophie Dantan
email: sophie.dantan@u-cergy.fr

Intra-household discrete choice models of mode choice and residential location

Keywords: unitary models, collective models, mode choice, residential location, Pareto optimality

We review the literature on household decision-making in economics as well as in transportation and urban economics. This literature starts with Gary Becker in Economics, 30 years ago, and has recently been introduced to study specific questions related to space. We consider two examples: residential location and mode choice for dual earner households. In these two examples, the decision is the outcome of a bargaining process within the household. We show that the results may differ substantially when compared with those of the standard approach.

Introduction

Transportation and urban economics are mainly based on individual decision making processes: residential and job location, mode choice, departure time, or activity patterns. Aggregation of preferences and choices, although crucial, has received very little attention in the profession. The main tool used in transportation and urban economics, discrete choice models, ignores this multi-dimensional facet of decision making. One way to understand this paper is to see it as a first step in the modeling of decision processes within the family, for question raised in the transportation and urban economics context.

One ultimate scope of the research in transport and urban economics is to shed some light to help the public (and occasionally the private) bodies to design better policies. Domains of application are numerous and include infrastructure choices, subsidization, taxes, prices, etc. It is obvious for economists that policies affect individual and therefore aggregate behaviors. Poor policies may have negative consequences, or they may just never become effective due to strong political or grass root opposition. Toll policies provide an obvious example of such opposition, since they are often not accepted by the general public. Different stakeholders are opposed to potential inequalities and lack of fairness potentially induced by such policies. Opposition is often expressed in the media as: "x% of users are opposed", or less often as: "y% of low income people are opposed". All researchers know that tolls may have regressive effects, in the sense that high income individuals may be better-off, while low income individuals may be worse off. However, literature neglects one major point: policies may also generate inequalities within the same family. Household accessibility can be improved in standard models, but it is important to also understand how changes in accessibility affect the different members of the household. A given policy may, for example, improve accessibility for men, and decrease accessibility for women (while the average accessibility is improved). The reason may be schematically explained as follows.

Consider a dual earner household owing a single car used by the woman, who works relatively close by. Assume also that some roads (not used by the woman) are now tolled. One wishes to assess the impact of those tolls. Congestion will presumably be reduced on the tolled roads, but may increase on alternatives roads, especially on the road used by the woman. If the man decides to use the car on the tolled road (now more attractive), he may be better off (since travel time is reduced), while the woman may be worse off for two different reasons: no more access to the car, and less income. How the disposable income is spent and who is using the car are decisions depending on the bargaining process within the family. There is sometimes a "dictator" who decides for everything, but most often, the decisions are the outcome of a bargaining process involving multiple decision makers. The computation of who is winning and who is losing cannot be made without a subtle description of the intrahousehold decision process. We refer the reader to the previous papers published by the authors and coauthors of this current paper (see [START_REF] De Palma | Discrete Choice Decision-Making with Multiple Decision Makers within the Household[END_REF], [START_REF] Picard | Intra-household Decision models of Residential and Job Location, in Integrated transport and land use modeling for sustainable cities[END_REF], [START_REF] Inoa | Effect of an accessibility measure in a model for choice of residential location, workplace, and type of employment[END_REF], [START_REF] De Palma | Individual and couple decision behavior under risk: evidence on the dynamics of power balance[END_REF]).

The objective in this paper is to make a first step to bridge the gap between two strands of literature which developed independently up to now. The first one covers location choice models that consider more than one household member or worker, but consider, implicitly or explicitly, a single decision maker. The second one focuses on the decision process in other contexts, when several decision makers with diverging preferences and constraints have to coordinate for some decisions. These two strands of literature differ not only with respect to their topics, but also with respect to the nature (discrete versus continuous) of the decisions analyzed. As a consequence, these two strands of literature used and developed totally different techniques. This distinction, at the core of new theoretical and empirical developments, is of the scope of this paper. More details on these two literature strands are provided in [START_REF] De Palma | Discrete Choice Decision-Making with Multiple Decision Makers within the Household[END_REF] and in [START_REF] Picard | Intra-household Decision models of Residential and Job Location, in Integrated transport and land use modeling for sustainable cities[END_REF]. Our ultimate objective is to make a step forwards in the analysis of joint mode choice and residential location choice in a dual-earner household, and measure the empirical implications of the bargaining process in some specific examples.

The paper is organized as follows. Section 2 reviews the first strand of literature, on household models not interested in the bargaining process itself, starting with the definition and limits of "unitary models", followed by specific applications in transport and residential location contexts. Section 3 reviews the second strand of literature, which focuses on the within-family interactions and bargaining process, usually in domains hardly related to transport or location. Section 4 then proposes a few applications of household decisions related to transport or location involving several decision-makers, highlighting the importance of the bargaining process. Section 5 discusses insights for future work and concludes.

2 Household models neglecting the bargaining process

Unitary models

In the economic literature, models considering a single decision maker are usually referred to as unitary models. This literature has been developed in economics, since the early work of Gary Becker ([5], [START_REF] Becker | A theory of marriage: Part i[END_REF], [START_REF] Becker | A theory of marriage: Part ii[END_REF], [START_REF] Becker | A theory of social interactions[END_REF], [START_REF] Becker | A Treatise on the Family[END_REF]), who was awarded the Nobel Prize for such seminal contribution.

Many researchers, especially in family economics, tend to criticize this assumption, questioning the hypothesis that the behavior of multi-person households can be described by a single decision-maker model satisfying the basic preference axioms (such as transitivity or continuity, for example), or the income pooling property (no matter who earns income in the family, it is spent the same way). Incidentally, we know, since the Arrow impossibility theorem, the difficulty to aggregate preference within small groups.

Different research streams concerned with household decisions have developed independently in different disciplines such as in labor supply, transportation, time and task allocation, or residential location and job choices. In these unitary models, the household is treated as a single decision-making unit; the interactions within the household are not explicitly modeled and the decision-making process outcome is, implicitly or explicitly, considered as resulting from a representative individual (as if the household were a black box which needs not be opened).

More recently, in the urban and transport literature, researchers have started to study the family determinants of choices affecting the family, even though they did not yet study the within-family bargaining process. In the current models, household interactions are either introduced through explanatory variables defined at the household level, or simply disregarded (in models of activity-travel demand, for example, see [START_REF] Srinivasan | Modeling household interactions in daily in-home and out-of-home maintenance activity participation[END_REF]). Examples of household-level explanatory variables are number of household members, of active members, of children as well as household income, and other household dummy variables (e.g., household head occupational status, tenure status, age, etc. See [START_REF] Townsend | The effects of household characteristics on the multi-day time allocations and travel/activity patterns of households and their members[END_REF] or [START_REF] Golob | A model of activity participation and travel interactions between household heads[END_REF], inter alia).

Transport models

There is still a long way to achieve the goal of providing a theoretical and empirical framework to model and apply the framework of the economics of the family. Family decisions have been neglected too long in transportation. Two special issues ( [START_REF] Bhat | Modeling intra-household interactions and group decision-making[END_REF], [START_REF] Timmermans | Modeling household activity travel behavior: Examples of state of the art modeling approaches and research agenda[END_REF]) provide the most notable exceptions, although they are not interested in analyzing or modeling the bargaining process itself. Transportation should be very keen to have access to this type of models, since their applications are numerous. Let mention, for example, choices related to residential location, workplace, car ownership, children's school, mode, departure time, activity patterns and the like. This review is completed with the study of different types of accessibility measures including recent work on time-geography measures of accessibility. We also refer the reader to another extensive review on past research on household (unitary) models in the transportation literature ([32]).

Residential location

The residential and job location choice literature is dominated by models considering a single decision-maker in each household. In such early models, the analysis of joint decisions (such as residential location choices, involving monetary cost, but also emotional and time dimensions) is usually simplified by assuming away the within-household bargaining process. De facto, it is assumed that there is a single decision maker, who can be seen as a more or less benevolent dictator, to use the terminology introduced more than 30 years ago, by Gary Becker, the father of Economics of the Family.

In the context of residential location, it is highly probable that some intra-household bargaining process plays an important role in dual-earner households. Intra-household bargaining process refers to the negotiation process within the household with respect to joint decisions, such as residential location, use of the car (when there is a single car), or choice of a holiday location. In particular, in the context of residential location, each worker in the household has his own preferences and constraints depending on his own workplace, generally different from the workplaces of the other workers in the same household.

Recently, economists and transport researchers have worked on explaining the effect of intra-household variables on the choice of residential location and workplace ( [START_REF] Waddell | Accessibility and residential location: The interaction of workplace, housing tenure, residential mobility and location choices[END_REF], [START_REF] Abraham | Specification and estimation of nested logit model of home, workplaces, and commuter mode choices by multiple-worker households[END_REF], [START_REF] Sermons | Representing the differences between female and male commute behavior in residential location choice models[END_REF]). Their analysis addressed the multiple-member household decisions, but not the multiple decision-maker problem, which involves a within-household bargaining process. Overall, intra-household considerations remain limited in this literature; the theoretical and empirical contributions on the effect of intra-household variables on the residential location and workplace choices are scarce. In particular, there are no published contributions on withinhousehold bargaining process in location decisions (with the exception of the ongoing research [START_REF] Chiappori | Couple residential location and spouses workplaces[END_REF] sum-up in the Section 4.4.

Household models recognizing the bargaining process

Past and current research in family economics

By contrast to the models reviewed in Section 2, collective models recognize that household decisions result from a bargaining process involving several decision-makers with individual specific bargaining power. These non-unitary models developed in the economics of the family literature do not merely extend existing unitary model, and discrete choice models. They introduce new concepts, which are specific to within family interactions: negotiation, altruism, repeated interaction and Pareto optimality. See [START_REF] Vermeulen | Collective household models: Principles and main results[END_REF] and [START_REF] Vermeulen | Where does the unitary model go wrong? Simulating tax reforms by means of unitary and collective labour supply models[END_REF]) for reviews of economic literature on unitary and collective household models. More precisely, when there is more than one decision-maker in the household, a complex bargaining process usually takes place, and the resulting decisions would appear irrational if they were analyzed using a single decision-maker model. Since the 80's, economic literature has developed models analyzing the within-family bargaining process in other contexts such as labor supply or consumption. For example, [START_REF] Lundberg | Separate spheres bargaining and the marriage market[END_REF] developed a specific bargaining process in the presence of multiple interrelated decisions made by multiple decision-makers, whereas [START_REF] Chiappori | Rational household labor supply[END_REF] developed the so-called collective model. Collective models take account of the fact that multi-person households face joint decisions involving more or less altruistic members with specific preferences and constraints. In collective models, decisions are assumed Paretooptimal in the sense that the solution of the bargaining process is on the Pareto frontier. By definition, when a decision is Pareto optimal, it is not possible to increase the well-being of one household member without decreasing the well-being of at least another member. In other words, spouses jointly make decisions leading to a Pareto-optimal outcome, if it is not possible to make one spouse better-off without making the other spouse worse-off. Paretooptimality can be bested empirically, and is usually not rejected. By contrast, the assumptions and/or conclusions of unitary models are usually rejected in multi-person households.

Collective models and other within-households bargaining models aim at answering the theoretical and empirical criticisms addressed to unitary models of family decision-making. They were developed in two major directions. "Strategic" models rely directly on the theory of non-cooperative games (see, e.g. [START_REF] Ashworth | Household models[END_REF] or [START_REF] Leuthold | An empirical study of female income transfers and the work decision of the poor[END_REF]), while "collective" models ( [START_REF] Chiappori | Rational household labor supply[END_REF], [START_REF] Chiappori | Collective labor supply and welfare[END_REF]) rely on the basic assumption that the household decision process leads to Pareto-efficient allocations. The bargaining process may be either explicit (as in [START_REF] Mcelroy | Nash-bargained decisions: Toward a generalization of the theory of demand[END_REF], or in [START_REF] Lundberg | Separate spheres bargaining and the marriage market[END_REF]), or non-specified (as in ( [START_REF] Chiappori | Rational household labor supply[END_REF], [START_REF] Chiappori | Collective labor supply and welfare[END_REF]). In the latter case, the bargaining process is very general and not restrictive, only assuming Pareto-optimality. Pareto-optimality hypothesis seems natural for analyzing household decisions since family members, who interact over a long period, are probably able to find mechanisms leading to efficient decisions.

As shown in [START_REF] Chiappori | Collective labor supply and welfare[END_REF], collective models can be used to study the welfare level of each household member, and therefore to analyze and measure in a consistent way the redistributive effects of any economic policy, not only at the household level but also at the individual level. More specifically, he showed that, under some rather plausible conditions, individual utility functions can be recovered from household behavior (and disentangled from bargaining power effects, whereas bargaining effects induce a bias in the measurement of preference parameters in unitary models.

In addition, a few contributions use experimental economics methodology to compare the decisions taken by the husband alone, the wife alone, and then by the spouses together (see [START_REF] Bateman | An experiment on risky choice amongst households[END_REF], [START_REF] Beharry-Borg | An analytical framework for joint vs separate decisions by couples in choice experiments: The case of coastal water quality in tobago[END_REF] or [START_REF] De Palma | Individual and couple decision behavior under risk: evidence on the dynamics of power balance[END_REF]). However, these papers do not explicitly analyze and model the decision mechanism within the couple.

Application to residential location

Beyond the simple differentiation of location choice by socio-demographic characteristics, recent models dealing with family location decisions allow for the identification of differences between females and males, and between multiple-worker households and singleworker households. In the context of Paris Area, we analyze in this paper differences between spouses' commuting times and we show that spouses' disparities in commuting is a key element in the intra-household decision process. The single-worker household approach might leave aside important intra-household considerations that influence commuting time and accessibility to jobs. As a consequence, the one-worker oriented location choice models may lead to misleading conclusions for dual-earner households. More precisely, we argue that the framework introduced in [START_REF] Inoa | Effect of an accessibility measure in a model for choice of residential location, workplace, and type of employment[END_REF], could also be adapted to analyze residential and job location choices in a two-worker household. By accounting for two accessibility variables (one for each spouse), their three-level nested Logit model could be used to study the interdependency of residential location and workplace, while accounting for variation of preferences for job types across individuals. This could be performed both in single-and in two-worker households.

Modeling intra-household decisions in specific examples

An important issue in modeling residential and job location is the interdependence of these two decisions, both at the individual level and at the household level. It can be argued that the choice of residential location is made conditional on the workplace, or vice versa ( [START_REF] Waddell | Accessibility and residential location: The interaction of workplace, housing tenure, residential mobility and location choices[END_REF]). We can also argue that the choice of residential location is made in different time frames according to the life cycle of individuals. What households then consider in their joint location decisions, with a varying degree of importance along their life cycle, are the chances of each active member in the household to access a good job in each alternative location, which determines the accessibility to jobs of each household member, from any given residential location.

We present here an innovative perspective on the interdependence between residential location and workplace of spouses, taking into account the process of negotiation within the household, the travel time of each spouse, and the job accessibility for each spouse. For this we models of residential location to try to give clear answers to various theoretical and empirical issues concerning: the residential location conditional on workplaces and the joint choice of residential location and workplace of each spouse.

The work has allowed expanding and developing models that correct the bias in the individual value of time of spouses, separately measuring the influence of explanatory variables on the decision-making process within the household and the value of time of spouses by taking into account the bargaining power to analyze the effect of the commuting time of spouses to the household residential location choice. For more details on the modeling framework of the results developed below see [START_REF] Chiappori | Couple residential location and spouses workplaces[END_REF].

Mode choice and spouses travel times

The differences between male and female observed travel times depend (1) on spouses endogenous joint mode choices, (2) on the distribution of male and female jobs over the region (marginal distribution of the destination of morning commuting trips), (3) on the distribution of dwellings over the region (marginal distribution of the origin of morning commuting trips), as well as (4) on the household joint residential and job location choices. The link between the marginal distributions of origins (home) and destinations (work place) and their joint distribution is partially explained by the spouses bargaining powers for location choice in the household. If the women's bargaining power is larger than the men's one, then the joint distribution of household residential location and spouses' job locations results in a reduction in woman's travel time compared to man's travel time.

In this section, we make a first step at disentangling the effects of these different sources, focusing on the relative reduction in actual women's and men's travel times (compared to all potential travel times) resulting from their respective bargaining powers.

Using data form the 1999 Census in Paris Region, we analyze the distributions of spouses' potential and actual travel times, and interpret their differences in the light of respective bargaining powers. Paris Region is composed by 8 "départements'': Paris, 3 départements around Paris ("inner ring'"), and départements farther away ("outer ring'"). The region contains 10,724,748 inhabitants for a total of 4,510,369 households. Household location and workplace are observed only in a 5% sample. About half of the individuals in this 5% sample work, which represents 242,516 workers, among which 239,499 work in Paris area. We finally restrict our sample to couples in which both spouses live and work in the Paris Region, which leads to 60,798 dual earners households (the difference corresponds to singles and one worker families).

Commuting network distances and travel times in chosen alternatives are computed using individual information on actual residential location and workplace, and reported in Table 1, in our sample of 60,798 dual-earner households. These distances and travel times correspond to the actual, or observed commuting trips. The upper (resp. medium) part of the table reports distance and travel time information for women (resp. men), and the lower part of the table reports the average differences and tests the significance of these differences. In the chosen alternatives, women's commuting distances and travel times are shorter than men's. The average difference is 2.27 km and 5.67 minutes through the public transportation network, and 1.85 km and 2.23 minutes through the private car network. The standard errors are those of the average difference between female and male actual commuting distances and travel times. The t-Stats are the ratios of the Mean to Std Err columns. Under the null hypothesis that the male and female average distances and travel times are equal, this statistic follows a Student distribution. For the 4 variables considered, the absolute values of the t-Stats are very large, and the null hypothesis is clearly rejected: actual commuting distances and travel times are significantly lower for women than for men. This difference may come from two different phenomena: (1) the marginal distributions of dwellings (irrespectively of the male and female workplaces) is such that dwellings are on average closer to female jobs than to men jobs; (2) the distribution of dwellings conditional on female and male workplaces is such that, for a specific household, the dwelling is closer to the woman's workplace than to the man's workplace.

In order to quantify the first phenomenon, we randomly select 9 hypothetical residential locations for each household, independently from the husband and wife workplaces. To do so, we consider a uniform distribution over the 5 million dwellings of the region. Equivalently, we perform importance sampling of communes, which weights proportional to the number of dwelling in this commune (see [START_REF] Antoniou | Methodological contributions to econometrics, in Integrated transport and land use modeling for sustainable cities[END_REF] for more details on importance sampling). Following this strategy, we build a sample of dwelling locations independent from both spouses' workplaces, and the distribution of these hypothetical dwellings is statistically identical to the marginal distribution of dwellings in the region.

The resulting distances and travel times are reported in Table 2, which is built on a sample of 60,798*9=547,182 hypothetical (unchosen) alternatives for residential location. The structure of Table 2 is the same as the one of Table 1, except that the figures are now those of hypothetical commuting trips. In the hypothetical case in which households would randomly choose a residential location independent from husband's and wife's workplaces, the average commuting distances and travel times would be approximately multiplied by 2. Stated differently, the fact that households (partially) adjust their residential location choice to the spouses' workplaces divides by 2 the spouses commuting distances and travel times, compared to the hypothetical case in which they would randomly pick one of the existing dwellings.

More interestingly, in this hypothetical situation, women's commuting distances and travel times would be slightly larger than men's travel times. This means that, on average, dwellings are marginally closer to male jobs than to female jobs.

The fact that actual commuting distances and travel times are shorter for wives than for men according to Table 1 cannot come from the marginal distributions of dwellings and jobs because the relationship is reversed in Table 2, but it comes from the conditional distribution of dwellings given spouses workplaces. Stated differently, the endogenous household residential location favors wives rather than husbands, which suggests a larger bargaining power for women than men in the residential location choice.

This result is also consistent with [START_REF] Abraham | Specification and estimation of nested logit model of home, workplaces, and commuter mode choices by multiple-worker households[END_REF], who explained that the probability of moving is more strongly related to commuting distance for women than for men, which results in shorter commuting distance for women after a relocation.

Note also that the endogenous residential location reduces more actual travel time of women compared to men by public transit than by private car. Given that women have a stronger tendency than men to commute by public transit, this result suggests that households locate so as to favor woman's commuting by transit.

The fact that both dwellings and jobs are rather located in the center of the region divides by about 2 the distances and by about 2.5 the travel times in comparison with the hypothetical situation in which dwellings and jobs would be uniformly and independently distributed among the communes. In this even more hypothetical case, the average distance between dwellings and jobs would be 66 km both through public and private networks, and it would represent an average travel time of 215 min by public transportation and 80 minutes by car.

The geographical distribution of actual commuting time for women and men by public transportation and private care are depicted in Figure 1 and Figure 2, respectively, by gender. Actual commuting times in public transportation and private car are increasing functions of the distance to the center of the region, in which most of the jobs are concentrated. The difference in travel times between the outer ring and the central part of the region (Paris and inner ring) is more pronounced for public transportation than for private car, which reflects the fact that the public transportation network is concentrated in the center of the region and is poorly adapted to trips from the outer ring to the outer ring. The geographical distribution of relative (woman-to-man) travel times depicted in Figure 3 shows that women work closer to their dwelling than men in the outer ring, whereas, for households living in Paris or in the inner ring, the husband's workplace is closer to the dwelling than the wife's workplace. This is consistent with the fact that men's jobs are concentrated in the core of the region, whereas female jobs are more uniformly spread over the region.

Car ownership and mode choice

Using the same sample as in subsection 4.1, we estimate a multinomial Logit model of residential location conditional on husband and wife workplaces, as a function of local dwelling price (per square meter, in log) and spouses' actual travel time, by mode and gender. Table 3 shows results of two different models. In the first model, the Value of Time (coefficient of the travel time variable) only depends on mode and gender. In the second model, the travel time variable is crossed not only with mode and gender, as in the first model, but also with the number of cars owned by the household.

The Price has the usual negative sign and is highly significant in both models. According to the first model, household location is slightly more sensitive to woman's than man's travel time by private car (|-0.0212| is significantly larger than |-0.0183|), but it is twice more sensitive to woman's than man's travel time by transit. The second model helps understanding such differences.

Travel time by private car (both for husband and wife) has no influence on household location for households without any car, which is totally consistent. Households do not care how much time husband or wife would spend commuting by car, when they have no access to any car.

For each household, 9 unchosen alternatives are generated using importance sampling. The weights are proportional to the number of dwellings in the commune and travel times are computed using the dynamic transport network model METROPOLIS ( [START_REF] Palma | Real cases applications of the fully dynamic metropolis tool-box: an advocacy for large-scale mesoscopic transportation systems[END_REF]). Alternatives correspond to the 1300 communes of the Paris Region. The city of Paris accounts for 20 arrondissements that correspond to 20 communes. The associated coefficients are the only ones which are not significant.

The difference between the coefficients of travel time by private car for the wife in households with 0 or 1 car is not statistically significant. For the husband, it is the difference between 1 and 2 cars which is not significant. This means that the husband's commuting time by private car becomes relevant in residential location choice as soon as there is at least one car in the household. By contrast, the wife's commuting time by car becomes really relevant only when there are at least 2 cars in the household. This gives a strong indication that the husband has the priority to use the car to commute when there is competition between spouses for the use of the unique car.

The influence of travel time by public transportation on residential location is decreasing with the number of cars in the household, both for the wife and for the husband. This influence nearly disappears for men when there are two cars in the household, suggesting that the husband usually commutes by car and does not care about transit travel times when there are 2 cars in the household. These results are consistent with the fact that, on the one hand, public transportation is a substitute for private car when household members can easily reach a station, which is usually the case in Paris and inner ring, whereas it is a complement when individuals have to drive to the station in order to use public transportation. The fact that husband's transit commuting time plays virtually no role in residential location of two-car households, whereas the wife's transit commuting time still matters suggests that, when there are two cars in the household, men will anyway entirely commute by car, whereas wives may go to the station by car.

Independent, unitary and collective mode choice of spouses

In order to analyze in more detail mode choice within couples and evaluate the importance of competition for using the car for commuting trips, we compare three models. In the first model, mode choice of the two spouses is estimated independently, as is done in a (too large) majority of articles interested in mode choice. The estimation results are reported in Table 4. In the right-hand side of the table, the model is estimated separately and independently for households owning only one car and for households owning at least two cars (independent binary Logit model). The households owning no car are not considered in the sample since the only mode available for the members of these households is public transport, so there is no mode choice decision in these households. The "all" columns on the left correspond to a model imposing equal preferences whatever the number of cars available to the household.

The explanatory power of the models is measured by the pseudo-R² reported in the bottom row. This explanatory power is very low in the 1-car case because this simplistic model assuming that spouses' mode choices are independent misses the most important determinant of mode choice, which is competition for the unique car. The models presented in Table 5 will address this problem. Mode choice is explained by travel time. We assume that the value of time (VOT) depends on gender, mode, age (expressed in number of decades after 40), occupation (reference=blue collar) and nationality (reference=French). The coefficients are negative since the utility (and not the costs) are estimated. The coefficients estimated are proportional to the marginal disutility of commuting time, that is to the opposite of the VOT. Note that the coefficients estimated do not measure directly the value of time, but are only proportional to it, since the coefficients in a Logit model are defined up to a multiplicative constant.

The model imposing that preferences do not depend on the number of cars available (columns "all") shows that woman's VOT is 70% larger than man's VOT, and that this ratio is the same by public transport and by car. In the 1-car sample, this ratio is the same by car, but VOT by public transport is 90% larger than man's VOT by public transport. By contrast, in the 2-cars sample, VOT by public transport is only 14% larger for women than for men whereas, VOT by car is 30% larger for women than for men. Such differences can hardly be explained by something else than by an endogenous selection bias related to the number of cars.

In addition to the value of time, we consider a fixed utility (for a null commuting time) for travelling by car rather than by public transportation. The negative value of this coefficient in the 1-car sample could suggest that women dislike driving, but the fact that this coefficient is positive in other models suggests that the negative coefficient in the 1-car sample rather results from the competition for the unique car. Consistently with the results in Subsection 4.2, when there is only one car in the household, this unique car is generally used by the man rather than by the woman.

For the woman, the fixed utility of commuting by car depends on the number of children, by age group, in order to reflect the fact than women often drop children to child daycare or to school while they go to work in the morning, or pick them when they come back in the evening. The fact that these coefficients are positive and generally increasing with children's age is consistent with the fact that child daycare facilities are usually very close to households, whereas the distance to households tends to increase when going from preschool to primary school and secondary school.

Table 5 presents a more realistic model in which spouses jointly choose their mode. We develop a multinomial Logit model estimated by maximum likelihood. We consider a bargaining model in which the household maximizes the weighted average utility of the woman and of the man. The weight of woman's utility corresponds to her bargaining power, whereas the weight of man's utility is normalized to 1 minus woman's weight. The model could be further enriched by considering that the bargaining power depends on each spouse's characteristics such as age, nationality or education The models presented in Section 2 consider a weighted average of commuting times and interpret them as a weighted average of individual utilities. However, in such models, the weights associated to spouses' commuting times in the household utility function merge the effect of individual-specific preferences (VOT) and of bargaining powers. By contrast, our structural model explicitly takes into account the bargaining process and disentangles individual preferences (both VOT and fixed cost of commuting by a given mode) from bargaining power.

Our structural joint model recognizes that the choice set of the couple depends on the number of cars available, and that there may be competition for the use of the car when the household has only one car.

Three alternatives are available whatever the (positive) number of cars, and perfectly observed:

 Both spouses use public transport (reference case, with the utility of both spouses normalized to 0  The woman commutes by car and the man by public transport  The man commutes by car and the woman by public transport The fourth case in which both spouses commute by car corresponds to 3 sub-cases involving different travel times:  Drive alone: Each spouse uses his/her own car: this alternative is available only in households with at least 2 cars  (CP,CD) = Car Passenger for the woman, Car Driver for the man: The man drops the woman in his way to his own job. This implies a detour for the man, that is a trip longer than his trip in the drive alone case  (CD,CP) = Car Driver for the woman, Car Passenger for the man: The woman drops the man in her way to her own job. This implies a detour for the woman, that is a trip longer than her trip in the drive alone case The data do not say which of these three sub-cases is selected by the household when both spouses answer that they commute by car, but the model computes different travel times (and therefore different utility levels) in each of these three sub-cases. Note: For woman's bargaining power, significance is for the difference with equal weights (0.5).

This structural model gives estimates of the spouses' values of time which are more consistent than those resulting from Table 4. The bias resulting from the endogenous selection of the 1-car and 2-car samples is at least partly corrected; the differences across samples in the ratios of woman-to-Man VOT are considerably reduced and become non-significant. Woman's VOT is now estimated to be 40% larger than man's VOT, which is more plausible than the 70% estimated by independent models for spouses' mode choice. Recall that, in a Logit model, coefficients are estimated only up to a multiplicative constant, and this constant differs across models which coefficients are reported in Table 4 and Table 5.

As a result, coefficients in Table 4 and Table 5 cannot be compared directly, but ratios of coefficients can be compared across these tables.

The estimation of the bargaining power is slightly in favor of the woman, who has 51.8% of the bargaining power, which leaves 48.2% for the man. However, the difference is significant only at the 10% level. The advantage of the woman is more important (54.9% compared to 45.1% for the man) and more significant (5% level) in the 1-car sample.

The negative coefficient of constant of the drive alone alternative for the woman in the 1car sample confirms that, when there is only one car in the household, the man has a strong priority to use it.

Our structural model also allows to estimate the fixed benefit (controlling for modespecific travel times) of commuting by car rather than by public transport. This fixed benefit can be identified separately for each spouse when they drove alone, but only the average benefit for the household can be identified when they drive together. Controlling for the number of cars, the weighted average benefit for the household when both spouses commute together in the same car is larger when the man drives (line "Intercept (CP,CD)") than when each spouse drives his/her own car (line "Weighted average intercept"), but it is lower when the woman drives (line "Intercept (CD,CP)"). This means that the household is globally happier when the man drives the woman than when each one drives alone, and the household is even less happy when the woman drives the man. This reflects the fact that the man prefers driving than being driven, and the woman prefers being driven than driving the man, but it is not possible to identify each effect separately. The comparison of the constants in the "All" columns could mean that the spouses dislike commuting together in the same car (since the weighted average fixed benefit of commuting by car separately is larger than the weighted average fixed benefit of commuting together, whoever drives). We rather believe that this comes from the endogenous selection bias related to the number of cars.

Table 6 shows that the structural model better predicts who will commute by car, especially when we impose the natural condition that preferences do not change when a second car is bought by the household (left-hand side of the table). The first lines suggests that, all together, the "Individual model", which assumes that spouses choose independently their mode, predicts the probability that both spouses commute by car slightly better than the structural model (45.47% versus 43.92%, whereas the actual probability is 49.54%). However, the subsequent lines show that the individual model strongly overestimates (41.24% instead of 21.82%) the probability that both spouses commute by car in the 1-car sample, and strongly underestimates it in the 2-cars sample (49.27% instead of 74.50%). The predictions of the structural model are far closer to reality. This means that the natural assumption that preferences do not change when a second car is bought by the household is strongly rejected when combined with the less realistic but too common assumption that spouses choose individually and independently their mode. But the natural assumption combined with our structural model is approximately consistent with the observations. The right-hand side of the table shows that, when we relax the assumption that preferences are the same in the 1-car and 2-cars samples, the predictions of the individual model are only marginally farther from reality than the predictions of the structural model. This implies that modeling errors would lead to the wrong conclusion that preferences change dramatically when the households buys a second car. By contrast, our structural model correctly predicts joint mode choice even under the quite realistic assumption that preferences are not affected by car ownership.

Finally our structural model leads to plausible predictions of the probabilities that spouses commute in separate cars or together, and of the probabilities that the man drops the woman, and that the woman drops the man. Recall that the information about who is driving is not available in the database. In our approach, this information can be computed exploiting the structure of the model and the observed influence on the probability that both spouses commute by car of the extra time implied by the detour (when the man drops the woman or when the woman drops the man).

Couple residential location conditional on spouses workplaces

We now discuss the results obtained by [START_REF] Chiappori | Couple residential location and spouses workplaces[END_REF] in a structural collective model assuming that spouses first choose their workplaces (anticipating the effect it will have on residential choice) and then the household collectively chooses the residential location conditional on each spouse's workplace. Like in Section 4.1, residential location then depends on the spouses' actual commuting times between the current workplace and the potential residential locations contemplated. However, by contrast with the reduced-form model estimated in Section 4.1, we analyze here a structural model explicitly taking into account both spouses individual preferences and respective bargaining powers rather than mixing them in a household utility function which may or may not be consistent with rationality. To the best of our knowledge, this the only contribution to the literature that takes into account the role of the within-family decision process and spouses bargaining powers in a residential location choice, and more generally in the measurement of the spouses values of time. The other contributions such as [START_REF] Abraham | Specification and estimation of nested logit model of home, workplaces, and commuter mode choices by multiple-worker households[END_REF] or [START_REF] Beharry-Borg | An analytical framework for joint vs separate decisions by couples in choice experiments: The case of coastal water quality in tobago[END_REF] do consider the influence of individual characteristics on residential location choices, but the associated coefficients mix the influence of individual preferences and bargaining powers.

The objective here is to analyze the within-family decision process involving bargaining between members with diverging preferences, objectives and constraints. Spouses may or may not have diverging preferences concerning local amenities, and these preferences probably differ from the preferences of singles. The same individual will tend to enjoy bars and discos when he is single, and open spaces when he is married with children. Given this change in preferences of the same individual when he marries, it is not possible to disentangle respective bargaining powers and spouses' preferences for local amenities, and we will not try to do so. Instead, we consider a joint preference of the household for local amenities.

On the opposite, it is obvious that the husband preference for his own commuting time (let's call it Value of Time, although it is more complex when utility is not linear in travel time because then VOT is given by the local derivative of utility with respect to commuting time, and it varies with commuting time) is different from the wife's preference for the husband commuting time. The influence of the husband's commuting time on household residential location mixes (1) the role of the husband's VOT, which a priori depends only on the husband's individual characteristics and (2) the role of respective spouses bargaining powers, which a priori depends both on the husband and wife individual characteristics. Similarly, the influence of the wife's commuting time on household residential location mixes the role of the wife's VOT and of the respective spouses bargaining powers. As a result, neglecting spouses' respective bargaining powers leads to biased estimates of the values of time of the household members.

In [START_REF] Chiappori | Couple residential location and spouses workplaces[END_REF], we develop a method to disentangle the bargaining power and the spouses VOTs, and to measure separately the influence of explanatory variables on the bargaining power and the value of time. We apply this method to the same data as in subsections 4.1 and 4.2. Each spouse's bargaining power is normalized to 1/2 in the reference case (the two spouses are French and 20 years old), and the husband and wife bargaining powers always sum to 1, so that bargaining powers can be interpreted as percentages. Any increase in the woman's bargaining power corresponds to a decrease of the same percentage for the husband's bargaining power. The econometric results show that the woman's age and man's nationality play a crucial role in determining bargaining power. The magnitude of the effects depends on the covariates considered, but some general patterns emerge. Each spouse bargaining power increases with age, but the increase is faster for the wife than for the man. Consider two equally old men. The wife of the first man is 10 years older than the wife of the second. Our estimates show that, in this case, the bargaining power of the first wife is 4.28% larger than that of the second woman. Consider now a third couple, in which the wife has the same age as the wife in the first couple, but the husband is 10 years older than the husband in the first couple. Then, the bargaining power of the husband in this third couple is 0.78% larger than that of the first husband. Consider now the first couple 10 years later. Each spouse is 10 years older and, as a result, the wife's bargaining power has increased more than the man's. As a consequence, the wife's bargaining power tends to increase over time (by 4.28%-0.78%= 3.5% each 10 years).

The bargaining power of the husband is reduced by about 4.5% when he is foreign, whereas the wife's bargaining power does not significantly depend on her nationality.

The econometric results of this model also show a large bias in the measurement of spouses' VOT when the bargaining power is not taken into account in a residential location model. For example, there is a 20% underestimation of the value of time for a 40-year-old French man, which becomes a 18% overestimation for a 40-year-old French man. For the wife, the VOT is always overestimated when the bargaining power is omitted, and the bias is 15% for a 40-year-old French woman and 10% for a 20-year-old French woman. There is an additional bias upwards in the measurement of the VOT of foreign men, when the bargaining power is omitted.

The commuting cost estimated in the model is a concave function of commuting time for each spouse. The value of time for a man (respectively woman) of 20 years old is about €11 (respectively €8) per hour at the origin (i.e. when the travel time tends to 0). The value of time is a decreasing function of age.

Accessibility in a residential location, workplace, and job type model in one-and two-worker households

We now discuss the extension to dual-earner households of the single or one-worker household structural model developed by [START_REF] Inoa | Effect of an accessibility measure in a model for choice of residential location, workplace, and type of employment[END_REF]. By contrast to Subections 4.2 and 4.4, this model assumes that residential location is chosen before individual workplaces. This model analyzes a three-level nested Logit model of residential location, workplace, and job type choice. Residential location is the upper-level choice, workplace the middle-level choice, and job type is chosen at the lower level.

The decision tree is fully consistent and rational: on one hand, workplace choice is anticipated at the upper-level residential location choice; on the other hand, the middle-level workplace choice is made conditional on the residential location actually chosen. This model allows defining and computing an individual-specific measure of attractiveness of geographical units, which takes into account the local distribution of job types, and individual preferences for job types. This measure of attractiveness of geographical units is fully consistent with the model, from the theoretical and econometric points of view. It corresponds to a log-sum term going up the decision tree from the lower level to the middle level. Going one step up in the decision tree, we obtain an individual-specific measure of accessibility to jobs, which takes into account the heterogeneity in VOT and preferences for different job types captured in the measure of attractiveness of potential workplaces.

The model is estimated on single or one-worker households, using the same data as in subsections 4.1 to 4.4. At the lower level of the decision tree, preferences for job types depend on gender, education and number of children (for women only). The same variables also explain observed heterogeneity in the Values of Time at the middle level of the decision tree. At the upper level, observed heterogeneity in preferences for local amenities is even richer, with the introduction of profession of household dead, and household composition and income, crossed with the local distribution of the same characteristics, in order to reflect the tendency of households to locate close to similar households.

Conclusion

We have discussed in this article different models describing the choice of mode, residential location and workplace of the various members of a family, mainly the husband and wife, but more sophisticated models should also take into account the children. The key concept introduced in this paper is the negotiation within the couple, that is to say, the decision-making process related to joint choices, such as the residential location, which is usually, but not always, unique in couples.

The tools we have discussed in this paper are based on the work developed during recent decades in the field of economics of the family. They will revolutionize discrete choice models used in transport and urban economics, in the sense that the decisions are no more described as the outcome of a unique decision maker. Instead, in collective discrete choice models, the decisions depend on preferences and constraints of two individuals tied by economic, but also emotional ties. Discrete choice theory still has to be adapted to this new way of approaching the individual decisions. Current literature in transportation [START_REF] Abraham | Specification and estimation of nested logit model of home, workplaces, and commuter mode choices by multiple-worker households[END_REF] has developed a few intra-family decision models. However these models ignore the bargaining process within the family, as studied by family economists ( [START_REF] Chiappori | Rational household labor supply[END_REF], [START_REF] Chiappori | Collective labor supply and welfare[END_REF]).

An objective of the research is to provide new tools for researchers in urban and transportation economics, as well as in engineering. In particular, we plan to provide collective choice models to the integrated transport and land use tool, UrbanSim. This article begins a search path that may be still very long and hopefully fruitful.

Figure 1 .

 1 Figure 1. Actual commuting times by public transportation for women (left-hand side) and for men (right-hand side)

Figure 2 .Figure 3 .

 23 Figure 2. Actual commuting times by private car for women (left-hand side) and for men (right-hand side)

  002 *** *(occup="professional") -1.588 *** -1.180 *** -1.470 *** -0.417 -1.368 *** -1.181 *** *(occup="manager") -1.195 *** -1.673 *** -1.268 *** -0.848 * -0.479 * -1.116 *** *(occup="self-employed") -3.643 *** -5.545 *** -1.695 ** -3.732 *** -4.530 *** -5.903 *** *foreign 2.571 *** 1.310 *** 3.241 *** 0.774 * 1.871

Table 1 . Actual commuting distances and travel times

 1

		Women
	Variable	Mean Std Err Min	Max
	Network distance (public, km)	13.151 13.973	0.25	134.73
	Network distance (private, km)	10.710 11.716	0.25	107.32
	Travel time (public, minute)	42.574 34.936	1.00	384.56
	Travel Time (private, minute)	15.200 14.248	0.76	151.51
		Men
		Mean Std Err Min	Max
	Network distance (public, km)	15.417 15.047	0.25	134.73
	Network distance (private, km)	12.557 12.620	0.25	109.12
	Travel time (public, minute)	48.242 36.732	1.00	384.56
	Travel Time (private, minute)	17.431 15.199	0.76	149.36
	Difference (Women-Men)
		Mean Std Err t-Stat p-value
	Network distance (public, km)	-2.266 0.083	-27.21 <0.01%
	Network distance (private, km)	-1.847 0.070	-26.45 <0.01%
	Travel time (public, minute)	-5.668 0.206	-27.57 <0.01%
	Travel time (private, minute)	-2.231 0.084	-26.41 <0.01%

Table 2 . Hypothetical commuting distances and travel times using the marginal distributions of dwellings

 2

		Women
	Variable	Mean Std Err Min	Max
	Network distance (public, km)	31.712 21.177	0.33	182.33
	Network distance (private, km)	26.995 19.399	0.33	155.90
	Travel time (public, minute)	86.200 50.193	1.32	564.52
	Travel Time (private, minute)	35.020 20.778	1.00	233.01
		Men
		Mean Std Err Min	Max
	Network distance (public, km)	31.154 20.813	0.35	182.33
	Network distance (private, km)	26.464 19.020	0.35	155.90
	Travel time (public, minute)	84.844 48.704	1.40	564.52
	Travel Time (private, minute)	34.469 20.355	1.05	233.01
	Difference (Women-Men)
		Mean Std Err t-Stat p-value
	Network distance (public, km)	0,558	0,0401	13,90 <0.01%
	Network distance (private, km)	0,531	0,0367	14,46 <0.01%
	Travel time (public, minute)	1,356	0,0945	14,34 <0.01%
	Travel time (private, minute)	0,551	0,0393	14,01 <0.01%

Table 3 . Effect of spouses' commuting times on household location

 3

	Parameter	Coeff.	Std Err t Stat	Pr>|t|
	Model 1: Travel time by mode and gender
	Log(Price)	-3.7977 0.0314	-120.9	<.0001
	Travel time (public transportation) Wife	-0.0325 0.000467 -69.59	<.0001
	Travel time (public transportation) Husband	-0.0172 0.00047 -36.58	<.0001
	Travel time (private car) Wife	-0.0212 0.000979 -21.68	<.0001
	Travel time (private car) Husband	-0.0183 0.000968 -18.87	<.0001
	Model 2: Travel times by sex, mode, and number of cars
	Log(Price)	-4.0008 0.0323	-123.87 <.0001
	Travel time (public transportation) Wife, 0 car	-0.0578 0.00226	-25.57 <.0001
	Travel time (public transportation) Wife, 1 car	-0.0394 0.000693 -56.9	<.0001
	Travel time (public transportation) Wife, 2 cars	-0.0233 0.000645 -36.07 <.0001
	Travel time (public transportation) Husband, 0 car -0.0414 0.00232	-17.84 <.0001
	Travel time (public transportation) Husband, 1 car -0.0232 0.000697 -33.31 <.0001
	Travel time (public transportation) Husband, 2 cars -0.0094 0.000651 -14.46 <.0001
	Travel time (private car), Wife, 0 car	-0.0059 0.004761 -1.23	0.2191
	Travel time (private car), Wife, 1 car	-0.0114 0.001442 -7.89	<.0001
	Travel time (private car), Wife, 2 cars	-0.0319 0.01391	-22.97 <.0001
	Travel time (private car), Husband, 0 car	-0.0075 0.00474	-1.58 <.0001
	Travel time (private car), Husband, 1 car	-0.0169 0.001425 -11.89	<.0001
	Travel time (private car), Husband, 2 cars	-0.0191 0.00137 -13.95	<.0001

Table 4 : Independent mode choice

 4

		All		At least 2 cars	1 car
		Woman Man Woman Man Woman Man
	Intercept (private car)	0.118 *** 0.774 *** 1.723 *** 2.086 *** -0.816 *** 0.163 ***
	*# children aged 0-3 0.041		0.098 *		0.082 **
	*# children aged 4-6 0.080 ***		0.120 **		0.071 *
	*# children aged 7-11 0.171 ***		0.085 **		0.143 ***
	*# children aged 12-16 0.212 ***		0.178 ***		0.175 ***
	Travel time by public tr. -2.101 *** -1.226 ** -2.114 *** -1.859 *** -1.985 *** -1.043 ***
	Woman-to-Man VOT (PT)	1.713		1.137		1.903
	Travel time by car	-3.469 *** -2.076 *** -4.248 *** -3.261 *** -3.218 *** -1.899 ***
	Woman-to-Man VOT (car)	1.671		1.303		1.695
	*(age-40)/10 -0.138 *** 0.047	0.387 *** 0.491 *** 0.045	0.140 ***
	*(occup="employee") 0.320 *** -0.674 *** 0.502 ** 0.159	0.090	-0.993 ***
	*(occup="professional") -0.839 *** -0.721 *** -0.660 *** -0.226	-0.564 *** -0.671 ***
	*(occup="manager") -0.583 *** -0.930 *** -0.488 *** -0.459 ** 0.085	-0.411 ***
	*(occup="self-employed") -2.254 *** -3.554 *** -1.176 *** -2.307 *** -2.867 *** -3.750 ***
	*foreign 1.391 *** 0.661 *** 1.596 *** 0.506 ** 0.956 *** 0.369 ***
	# observations	34,915 34,915 18,377 18,377 16,538 16,538
	log-likelihood	-20,180 -18,345	-6,882	-5,986	-9,793 -10,439
	pseudo-R²	0.1661	0.242 0.4597 0.5301 0.1457 0.0894

*: Slightly significant (10% level). **: Significant (5% level). ***: Highly significant (1% level).

Table 5 . Spouses' joint mode choice in a structural bargaining model

 5

		All	At least 2 cars	1 car
		Woman Man Woman Man Woman Man
	Intercept (CP,CD)	0.419 ***	2.852 ***	-0.810 ***
	Intercept (CD,CP)	0.251 ***	2.373 ***	-0.934 ***
	Intercept (drive alone)	0.689 *** 2.026 *** 2.195 *** 3.195 *** -1.788 *** 0.144 *
	Weighted average intercept	1.333	2.692	-0.917
	*# children aged 0-3 0.094		0.237 **	0.131 *
	*# children aged 4-6 0.138 **		0.269 **	0.105
	*# children aged 7-11 0.263 ***		0.171 **	0.258 ***
	*# children aged 12-16 0.346 ***		0.353 ***	0.305 ***
	Travel time by public tr.	-3.945 *** -2.738 *** -4.160 *** -3.658 *** -3.653 *** -2.463 ***
	Woman-to-Man VOT (PT)	1.441	1.137	1.483
	Travel time by car	-6.393 *** -4.459 *** -7.869 *** -6.208 *** -5.379 *** -3.945 ***
	Woman-to-Man VOT (car)	1.434	1.267	1.363
	*(age-40)/10 0.104	0.379 *** 0.848 *** 1.014 *** 0.277 *** 0.481 ***
	*(occup="worker") 0.654 *** -1.229 *** 1.047 *** 0.401	0.318	-2.

Table 6 . Probability that both spouses commute by car, by model Preferences independent of the number of cars Preferences dependent on the number of cars VA, VA VP, VD VD, VP V,V VA, VA VP, VD VD,VP V,V All households (34,915 couples)

 6

	Observed from Census		49.54		49.54		49.54	49.54
	Individual model		45.47		45.47		48.59	48.59
	Structural model 23.99 12.31	7.62 43.92 22.08	17.79 9.23 49.10
		Households with 1 car (16,538 couples)
	Observed from Census	-	21.82	21.82	-	21.82	21.82
	Individual model	-	41.24	41.24	-	21.84	21.84
	Structural model	-	15.66	9.95 25.60	-	13.09 8.74 21.82
		Households with 2 cars (18,377 couples)
	Observed from Census		74.50		74.50		74.50	74.50
	Individual model		49.27		49.27		72.66	72.66
	Structural model 45.57	9.29	5.53 60.40 41.95	22.03 9.68 73.65


Gaël Chételat
email: chetelat@cyceron.fr

Marine Fouquet

Keywords: Apolipoprotein E4, structural MRI, FDG-PET, amyloid PET imaging, preclinical stage, early diagnosis, atrophy apolipoprotéine E4, IRM structurale, FDG-TEP, imagerie amyloïde, stade préclinique, diagnostic précoce, atrophie

Introduction. The E4 allele of the apolipoprotein E (APOE4) is the major known genetic risk factor for AD, with a dramatic increase in the risk of developing AD as the number of APOE4 alleles increases from 0 to 2. For this reason, asymptomatic APOE4 carriers as a group offer a great opportunity to seek for the presence of early biomarkers for AD. The present article reviews neuroimaging studies on APOE4 carriers, focusing on cognitively normal individuals and on the main neuroimaging biomarkers for AD, i.e. atrophy with structural MRI, hypometabolism with FDG-PET, and amyloid deposition with amyloid-PET imaging.

State of art. There is a great number of studies on the effect of APOE4 on brain structures, and they tend to show significant atrophy in APOE4 carriers compared to non carriers especially in regions susceptible to AD pathology such as the hippocampus. However, results are rather discrepant which suggests that the effect of APOE4 on brain structure is subtle. As for FDG-PET metabolism, the few studies show decreased metabolism, again especially in AD-sensitive regions such as posterior associative parietal areas, with a dose-dependent effect (i.e. worsening as the number of APOE4 alleles increases). Finally, there is a unanimous and major effect of APOE4 on amyloid deposition with an increase in Aβ load as the number of APOE4 alleles increases and a decrease in the age of predicted amyloid-positivity in APOE4 carriers. This graded effect of APOE4 on atrophy, hypometabolism, and amyloid deposition is consistent with multimodal neuroimaging studies suggestive of a predominant effect of APOE4 on amyloid rather than tau-related injury and on brain metabolism rather than brain structure. Neuroimaging studies also suggest that APOE4 effects may be mediated by both Aβ-dependent and Aβ-independent pathological processes. This contradicts the view that Aβ pathology is a necessary upstream event to neuronal injury in AD.

Perspectives and conclusion. Future studies would tell whether the mechanisms and sequences evidenced in carriers is comparable to those found in non carriers, but it is likely that APOE4 not only influences the risk for AD, but also modulates the physiopathological cascade. Altogether, APOE4 carriers offer a great opportunity to investigate brain changes in the asymptomatic stages of AD and to further our understanding of the physiopathology of the disease, although precaution is needed for interpretation in AD at large.

Résumé :

Introduction. L'allèle E4 de l'apolipoprotéine E (APOE4) est le plus important facteur de risque génétique pour la maladie d'Alzheimer (MA), avec une augmentation dramatique de ce risque lorsque le nombre d'allèles APOE4 augmente. Les porteurs asymptomatiques de l'allèle APOE4 représentent donc une population particulièrement intéressante pour évaluer, à l'échelle du groupe, la présence de biomarqueurs précoces de la MA. La présente revue, qui correspond à une conférence donnée dans le cadre des Journées Internationales de la Société Française de Neurologie, rapporte les résultats des études d'imagerie chez les porteurs APOE4 en se focalisant uniquement sur les sujets asymptomatiques et en considérant essentiellement les techniques d'imagerie les plus reconnues : l'IRM structurale pour étudier l'atrophie, la TEP-FDG pour le métabolisme et l'imagerie TEP des dépôts amyloïdes.

Etat des connaissances. De très nombreuses études ont évalué les effets de l'APOE4 sur la structure cérébrale. Les résultats sont très divergents, même s'ils suggèrent dans l'ensemble un effet subtile dans le sens d'une diminution de volume dans les régions les plus sensibles à la MA telles que l'hippocampe. Les rares études en TEP-FDG indiquent une baisse du métabolisme là encore dans les régions altérées précocement dans la MA comme le cortex associatif pariétal postérieur, avec un effet dose-dépendant (i.e. fonction du nombre d'allèles APOE4). Enfin, l'effet sur les dépôts amyloïdes est unanime et massif, indiquant une augmentation dose-dépendante chez les porteurs APOE4, et une diminution de l'âge prédit de positivité pour l'amyloïde chez les porteurs. Cet effet graduel de l'APOE4 sur l'atrophie, l'hypométabolisme, puis les dépôts amyloïdes est retrouvé dans les études multimodales qui suggèrent un effet prédominant de l'APOE4 sur les processus amyloïde-dépendants plutôt que tau-dépendants, et sur le métabolisme plutôt que sur la structure. Ces études multimodales, y compris celles réalisées au moyen d'autres techniques d'imagerie, suggèrent également que l'effet

Introduction

The apolipoprotein E (ApoE) is a cholesterol carrier glycoprotein that supports lipid transport and injury repair in the brain. The human apolipoprotein E (APOE) gene exists as three polymorphic alleles: ε2, ε3 and ε4 which have a worldwide frequency of 8%, 78% and 14%, respectively [START_REF] Schipper | Apolipoprotein E: implications for AD neurobiology, epidemiology and risk assessment[END_REF]. The 4 allele of APOE (APOE4) is the major known genetic risk factor for AD. It has a dosedependent effect on the risk and age at onset of Alzheimer's disease (AD): as the number of alleles increases from 0 to 2, the risk of developing AD increases from 20 to 90% and the mean age at onset decreases from 84 to 68 years [START_REF] Corder | Gene dose of apolipoprotein E type 4 allele and the risk of Alzheimer's disease in late onset families[END_REF]. More recent estimations reported a lifetime risk of AD of 50 and 60% for APOE4/4 females and males versus 23% and 30% for APOE4/3 males and females, respectively [START_REF] Genin | APOE and Alzheimer disease: a major gene with semi-dominant inheritance[END_REF]. A meta-analysis of clinical and autopsy-based studies reported odds ratios for AD of 2.6, 3.2, and 14.9 for APOE2/4, 3/4, and 4/4, respectively [START_REF] Farrer | Effects of age, sex, and ethnicity on the association between apolipoprotein E genotype and Alzheimer disease. A metaanalysis. APOE and Alzheimer Disease Meta Analysis Consortium[END_REF]. By contrast, the APOE2 allele has a protective effect, with odds ratios of 0.6 for APOE2/3 or 2/2. It is important to note however that APOE4 is neither necessary nor sufficient to develop AD: there are non carriers amongst AD cases and APOE4 carriers that will not develop AD (Saunders et al., 1993).

Because APOE4 significantly increases the risk of having AD, and consequently because the proportion of individuals who will develop AD is greater amongst APOE4 carriers than amongst non carriers, cognitively normal APOE4 carriers allow to assess the preclinical / asymptomatic stage of the disease. More specifically, by comparing neuroimaging data between carriers and non carriers, one might be able to detect potential early neuroimaging biomarkers of AD that would be present in asymptomatic stages. The present review is derived from a talk given at the French Society of Neurology aiming at highlighting what APOE4 carriers tell us about neuroimaging biomarkers for AD in the preclinical stage. Therefore, only studies on cognitively normal individuals will be included, focusing on the main neuroimaging biomarkers for AD considered in the revised criteria for preclinical AD [START_REF] Dubois | Revising the definition of Alzheimer's disease: a new lexicon[END_REF][START_REF] Sperling | Toward defining the preclinical stages of Alzheimer's disease: recommendations from the National Institute on Aging-Alzheimer's Association workgroups on diagnostic guidelines for Alzheimer's disease[END_REF], i.e. atrophy with structural MRI, hypometabolism with FDG-PET, and β-amyloid deposition with amyloid-PET imaging, with only brief mention to other imaging modalities.

APOE4 and structural MRI / atrophy

Studies assessing the effect of APOE4 on brain structure using structural MRI are numerous but lead to diverging findings. Thus, several studies have reported a significant atrophy (or cortical thinning) in APOE4 carriers compared to non carriers, most often in brain regions most susceptible to AD, i.e. medial temporal structures (hippocampus, entorhinal cortex, etc;[START_REF] Plassman | Apolipoprotein E epsilon 4 allele and hippocampal volume in twins with normal cognition[END_REF][START_REF] Tohgi | Reduced size of right hippocampus in 39-to 80-year-old normal subjects carrying the apolipoprotein E epsilon4 allele[END_REF][START_REF] Lemaître | No epsilon4 gene dose effect on hippocampal atrophy in a large MRI database of healthy elderly subjects[END_REF][START_REF] Lind | Reduced hippocampal volume in non-demented carriers of the apolipoprotein E epsilon4: relation to chronological age and recognition memory[END_REF][START_REF] Wishart | Regional brain atrophy in cognitively intact adults with a single APOE epsilon4 allele[END_REF][START_REF] Burggren | Reduced cortical thickness in hippocampal subregions among cognitively normal apolipoprotein E e4 carriers[END_REF][START_REF] Honea | Impact of APOE on the healthy aging brain: a voxel-based MRI and DTI study[END_REF][START_REF] Suthana | Reduced hippocampal CA2, CA3, and dentate gyrus activity in asymptomatic people at genetic risk for Alzheimer's disease[END_REF][START_REF] O'dwyer | Reduced hippocampal volume in healthy young ApoE4 carriers: an MRI study[END_REF]. Other regions are also reported such as the lateral temporal and prefrontal cortex [START_REF] Wishart | Regional brain atrophy in cognitively intact adults with a single APOE epsilon4 allele[END_REF] or parietal areas [START_REF] Honea | Impact of APOE on the healthy aging brain: a voxel-based MRI and DTI study[END_REF]. Yet, there have also been numerous studies reporting no significant differences between APOE4 carriers and non carriers in terms of brain volume or cortical thickness [START_REF] Soininen | Decreased hippocampal volume asymmetry on MRIs in nondemented elderly subjects carrying the apolipoprotein E epsilon 4 allele[END_REF][START_REF] Schmidt | Apolipoprotein E e4 allele in the normal elderly: neuropsychologic and brain MRI correlates[END_REF][START_REF] Reiman | Hippocampal volumes in cognitively normal persons at genetic risk for Alzheimer's disease[END_REF][START_REF] Jack Cr | Hippocampal atrophy and apolipoprotein E genotype are independently associated with Alzheimer's disease[END_REF][START_REF] Cohen | Effect of apolipoprotein E genotype on hippocampal volume loss in aging healthy women[END_REF][START_REF] Han | Verbal paired-associate learning by APOE genotype in non-demented older adults: fMRI evidence of a right hemispheric compensatory response[END_REF][START_REF] Schuff | MRI of hippocampal volume loss in early Alzheimer's disease in relation to ApoE genotype and biomarkers[END_REF][START_REF] Filippini | Distinct patterns of brain activity in young carriers of the APOE-epsilon4 allele[END_REF][START_REF] Filippini | Differential effects of the APOE genotype on brain function across the lifespan[END_REF][START_REF] Dennis | Temporal lobe functional activity and connectivity in young adult APOE varepsilon4 carriers[END_REF][START_REF] Kukolja | Medial temporal lobe dysfunction during encoding and retrieval of episodic memory in non-demented APOE epsilon4 carriers[END_REF]Liu et al., 2010a;[START_REF] Westlye | Increased hippocampal default mode synchronization during rest in middle-aged and elderly APOE ε4 carriers: relationships with memory performance[END_REF][START_REF] Bunce | APOE genotype and entorhinal cortex volume in non-demented community-dwelling adults in midlife and early old age[END_REF][START_REF] Protas | Posterior cingulate glucose metabolism, hippocampal glucose metabolism, and hippocampal volume in cognitively normal, late-middle-aged persons at 3 levels of genetic risk for Alzheimer disease[END_REF][START_REF] Hostage | Alzheimer's Disease Neuroimaging Initiative. Dissecting the gene dose-effects of the APOE ε4 and ε2 alleles on hippocampal volumes in aging and Alzheimer's disease[END_REF] or even showing greater volume [START_REF] Honea | Impact of APOE on the healthy aging brain: a voxel-based MRI and DTI study[END_REF] or cortical thickness [START_REF] Espeseth | Accelerated age-related cortical thinning in healthy carriers of apolipoprotein E epsilon 4[END_REF] in APOE4 carriers compared to non carriers. Some studies suggest that the effect of APOE4 is more marked in young that in elderly individuals [START_REF] Lind | Reduced hippocampal volume in non-demented carriers of the apolipoprotein E epsilon4: relation to chronological age and recognition memory[END_REF][START_REF] Wishart | Regional brain atrophy in cognitively intact adults with a single APOE epsilon4 allele[END_REF], but the reverse has also been reported [START_REF] Mueller | Selective effect of Apo e4 on CA3 and dentate in normal aging and Alzheimer's disease using high resolution MRI at 4 T[END_REF][START_REF] Mueller | Selective effect of age, Apo e4, and Alzheimer's disease on hippocampal subfields[END_REF].

Increased rate of atrophy has been shown in APOE4 carriers compared to non carriers in some longitudinal studies, especially in medial temporal structures [START_REF] Cohen | Effect of apolipoprotein E genotype on hippocampal volume loss in aging healthy women[END_REF][START_REF] Chen | Correlations between apolipoprotein E epsilon4 gene dose and whole brain atrophy rates[END_REF][START_REF] Morra | Automated mapping of hippocampal atrophy in 1-year repeat MRI data from 490 subjects with Alzheimer's disease, mild cognitive impairment, and elderly controls[END_REF]Donix et al., 2010b;[START_REF] Hua | Sex and age differences in atrophic rates: an ADNI study with n=1368 MRI scans[END_REF][START_REF] Risacher | Longitudinal MRI atrophy biomarkers: relationship to conversion in the ADNI cohort[END_REF][START_REF] Chiang | Impact of apolipoprotein E4-cerebrospinal fluid β-amyloid interaction on hippocampal volume loss over 1 year in mild cognitive impairment[END_REF][START_REF] Lu | Apolipoprotein E genotype is associated with temporal and hippocampal atrophy rates in healthy elderly adults: a tensor-based morphometry study[END_REF].

Normal elderly APOE2 carriers were found to show greater cortical thickness in the temporal cortex compared to APOE3 [START_REF] Fan | Cortical thickness is associated with different apolipoprotein E genotypes in healthy elderly adults[END_REF][START_REF] Fennema-Notestine | Presence of ApoE ε4 allele associated with thinner frontal cortex in middle age[END_REF] and in the dorsolateral prefrontal cortex compared to APOE4 [START_REF] Fan | Cortical thickness is associated with different apolipoprotein E genotypes in healthy elderly adults[END_REF], as well as slower rates of hippocampal atrophy compared to APOE3 homozygotes [START_REF] Chiang | Impact of apolipoprotein E4-cerebrospinal fluid β-amyloid interaction on hippocampal volume loss over 1 year in mild cognitive impairment[END_REF] or reduced ventricular expansion [START_REF] Hua | Tensor-based morphometry as a neuroimaging biomarker for Alzheimer's disease: an MRI study of 676 AD, MCI, and normal subjects[END_REF]). Yet, again, contradictory findings have been reported, including a lack of difference between APOE2 and APOE 3 in hippocampal volume [START_REF] Chiang | Hippocampal atrophy rates and CSF biomarkers in elderly APOE2 normal subjects[END_REF] and cortical thickness (Liu et al., 2010b), or even decreased hippocampal and amygdala volumes in E2 equivalent to that found in E4 carriers (den [START_REF] Heijer | Hippocampal, amygdalar, and global brain atrophy in different apolipoprotein E genotypes[END_REF], consistent with findings of increased AD neuropathology in APOE2 carriers [START_REF] Berlau | APOE epsilon2 is associated with intact cognition but increased Alzheimer pathology in the oldest old[END_REF]. Whether these changes represent a genetically-determined feature or is the result of pathological processes is still under debate. The fact that such differences between APOE4 carriers and non carriers have been evidenced in children [START_REF] Shaw | Cortical morphology in children and adolescents with different apolipoprotein E gene polymorphisms: an observational study[END_REF] and even in neonates [START_REF] Knickmeyer | Common Variants in Psychiatric Risk Genes Predict Brain Structure at Birth[END_REF], tends to argue for at least a part of genetic determination. On the other hand, longitudinal studies showed increased rates of atrophy (see above), suggesting that APOE4 also has a pathologic effect that tend to exacerbate with time.

Altogether, these findings suggest that the APOE4 allele may induce changes in the structure of the brain, more particularly in regions susceptible to AD pathology, but that this effect is rather subtle, which would account for the divergence of findings.

APOE4 and FDG-PET / hypometabolism

FDG-PET studies investigating the effects of APOE4 are less numerous than those on brain structure but overall the conclusions are similar: in general, studies report a decrease of metabolism in APOE4 carriers compared to non carriers and this decrease mainly concerns AD-sensitive brain areas. The findings are less discrepant than for brain structuremaybe because the effects are more marked on metabolism than on brain structure (see below "multimodality"), or because there are much less PET studies and negative or unexpected results would tend not to be published (see Methodological highlights below).

More specifically, decreased metabolism in APOE4 carriers has first been reported by [START_REF] Small | Apolipoprotein E type 4 allele and cerebral glucose metabolism in relatives at risk for familial Alzheimer disease[END_REF] in elderly with memory complaints and at-least two relatives with AD and by [START_REF] Reiman | Preclinical evidence of Alzheimer's disease in persons homozygous for the epsilon 4 allele for apolipoprotein E[END_REF] in truly cognitively normal elderly. It was found in AD-sensitive regions, i.e. predominantly in posterior cingulate, parietal and temporal areas but also in the prefrontal cortex. Interestingly, the same findings were obtained in young (20-39 years old) carriers (Reiman et al.,, 2004). Moreover a genedose effect was reported in each of these brain regions in late-middle-age individuals, with greater decrease in individuals with two APOE4 alleles compared to those with only one (Reiman et al., 2005). Accelerated rates of decline in resting-state cerebral blood flow in frontal, parietal and temporal cortices have also been reported in one previous study using H 2 O-PET [START_REF] Thambisetty | APOE epsilon4 genotype and longitudinal changes in cerebral blood flow in normal aging[END_REF].

Findings obtained with perfusion MRI (an MRI method that measures cerebral blood flow) were however discrepant compared to those with FDG-PET. Thus, one study reported increased perfusion in middle-age APOE4 carriers compared to non carriers [START_REF] Fleisher | Cerebral perfusion and oxygenation differences in Alzheimer's disease risk[END_REF], and another laboratory found cingulate cortex dysfunction in elderly but not young carriers [START_REF] Filippini | Distinct patterns of brain activity in young carriers of the APOE-epsilon4 allele[END_REF][START_REF] Filippini | Differential effects of the APOE genotype on brain function across the lifespan[END_REF].

Further studies are needed to directly compare the effects of APOE4 using both techniques on the same sample.

APOE4 and amyloid deposition / amyloid PET imaging

The effect of APOE on amyloid deposition as assessed with recently developed amyloid-imaging (i.e.

PET associated with PIB, florbetapir, or other amyloid ligands) is clearly more unanimous and marked. Thus, APOE4 has been consistently shown in many studies to be associated with a significant increase in Aβ deposition or a greater proportion of amyloid-positive individuals in normal elderly [START_REF] Reiman | Fibrillar amyloid-beta burden in cognitively normal people at 3 levels of genetic risk for Alzheimer's disease[END_REF][START_REF] Hinrichs | Cortical binding of pittsburgh compound B, an endophenotype for genetic studies of Alzheimer's disease[END_REF][START_REF] Morris | APOE predicts amyloidbeta but not tau Alzheimer pathology in cognitively normal aging[END_REF][START_REF] Rowe | Amyloid imaging results from the Australian Imaging, Biomarkers and Lifestyle (AIBL) study of aging[END_REF][START_REF] Villemagne | Longitudinal Assessment of Aβ and Cognition in Aging and Alzheimer Disease[END_REF][START_REF] Jagust | Alzheimer's Disease Neuroimaging Initiative. Apolipoprotein E, not fibrillar β-amyloid, reduces cerebral glucose metabolism in normal aging[END_REF][START_REF] Kantarci | APOE modifies the association between Aβ load and cognition in cognitively normal older adults[END_REF][START_REF] Mathis | In vivo assessment of amyloid-β deposition in nondemented very elderly subjects[END_REF][START_REF] Mielke | Indicators of amyloid burden in a population-based study of cognitively normal elderly[END_REF][START_REF] Fleisher | Apolipoprotein E ε4 and age effects on florbetapir positron emission tomography in healthy aging and Alzheimer disease[END_REF]. Thus, across studies in cognitively normal elderly, 35 to 49% of APOE4 carriers are 11 C-PIBpositive while they are 9 to 37% amongst non carriers (see Table 1). Similar findings were obtained using FDDNP ligand binding to both NFT and plaques, with increased binding in asymptomatic elderly APOE4 carriers in the frontal cortex [START_REF] Small | Influence of cognitive status, age, and APOE-4 genetic risk on brain FDDNP positron-emission tomography imaging in persons without dementia[END_REF].

The effect of APOE on amyloid deposition was found to be dose-dependent, such that PiB increased progressively with each additional E4 allele [START_REF] Reiman | Fibrillar amyloid-beta burden in cognitively normal people at 3 levels of genetic risk for Alzheimer's disease[END_REF][START_REF] Morris | APOE predicts amyloidbeta but not tau Alzheimer pathology in cognitively normal aging[END_REF]. This effect is also supposed to be region-specific, i.e. to be more pronounced in some brain regions than in others, but the regions showing a predominant effect differed amongst studies (e.g. the frontal cortex in [START_REF] Reiman | Fibrillar amyloid-beta burden in cognitively normal people at 3 levels of genetic risk for Alzheimer's disease[END_REF]versus posterior regions in Fleisher et al., 2013). Moreover, the presence of the APOE4 allele increases the prevalence of conversion from amyloid-negative to amyloid-positive [START_REF] Vlassenko | Amyloid-beta plaque growth in cognitively normal adults: longitudinal [11C]Pittsburgh compound B data[END_REF] and decreases the age of predicted amyloid-positivity of about 20 years, i.e. from 76 to 56 years in non carriers versus carriers respectively [START_REF] Fleisher | Apolipoprotein E ε4 and age effects on florbetapir positron emission tomography in healthy aging and Alzheimer disease[END_REF]. APOE4 has also been shown to modify the relationships between amyloid deposition and cognitive function, with increased Aβ load being associated with decreased performances especially in APOE4 carriers [START_REF] Kantarci | APOE modifies the association between Aβ load and cognition in cognitively normal older adults[END_REF].

The effect of APOE4 on amyloid imaging thus appears to be marked, unquestionable and consistent amongst study. APOE4 has even been reported to be the best predictor of the presence of amyloid in the brain of healthy elderly amongst age, sex, APOE genotype, family history, or cognitive performance. Yet, it is interesting to note that one study reported that PiB binding has a high heritability, and 74% of the heritable component cannot be explained by APOE4 genotype [START_REF] Hinrichs | Cortical binding of pittsburgh compound B, an endophenotype for genetic studies of Alzheimer's disease[END_REF].

Methodological highlights

Results are partly discrepant, especially regarding the effect of APOE4 on brain structure.

Methodological differences amongst study (e.g. acquisition parameters, quantification methods, etc) may explain part of this variability. Some of the conflicting results may also be attributable to the antagonist pleiotropy of APOE4 effects across different stages of the life span, such that APOE4 may be beneficial in earlier ages and may only confer risk of cognitive decline later in life [START_REF] Tuminello | The apolipoprotein e antagonistic pleiotropy hypothesis: review and recommendations[END_REF]. It is also possible that non-intuitive findings are not reported and confounding variables are rarely modeled [START_REF] Donix | Family history and APOE-4 genetic risk in Alzheimer's disease[END_REF]. Finally, most studies on APOE include individuals with a familial history of AD which may have blurred the findings as different genetic factors may interact. Thus, APOE genotype and family history risk were shown to have independent and/or additive contribution to structural changese.g. hippocampal or precuneus and frontal atrophy (Donix et al., 2010a;[START_REF] Honea | Reduced gray matter volume in normal adults with a maternal family history of Alzheimer disease[END_REF][START_REF] Schipper | Apolipoprotein E: implications for AD neurobiology, epidemiology and risk assessment[END_REF] or metabolism decrease [START_REF] Mosconi | Maternal family history of Alzheimer's disease predisposes to reduced brain glucose metabolism[END_REF]2009).

APOE4 effects on other neuroimaging modalities

While this review primarily focuses on the main recognized neuroimaging biomarkers for AD, namely structural MRI, FDG-PET and amyloid-PET, the effect of APOE4 has also been assessed using other neuroimaging modalities. In brief, the results are consistent with the idea that APOE4 cognitively normal elderly, as a group, do present some signs of AD pathology, e.g. decreased fractional anisotropy as measured with DTI in brain regions usually impaired in AD [START_REF] Honea | Impact of APOE on the healthy aging brain: a voxel-based MRI and DTI study[END_REF], decreased local and regional interconnectivity measured from graph theory applied to fiber tractography data [START_REF] Brown | Brain network local interconnectivity loss in aging APOE-4 allele carriers[END_REF], or disruption of functional connectivity measured with restingstate fMRI [START_REF] Sheline | APOE4 allele disrupts resting state fMRI connectivity in the absence of amyloid plaques or decreased CSF Aβ42[END_REF]. By contrast, a finding often described in fMRI studies of APOE4 carriers is increased activations (often, but not only, within the hippocampus) compared to non carriers in episodic memory tasks [START_REF] Trachtenberg | The effects of APOE on brain activity do not simply reflect the risk of Alzheimer's disease[END_REF]. For example, [START_REF] Bookheimer | Patterns of brain activation in people at risk for Alzheimer's disease[END_REF] reported an increase in both the magnitude and the extent of brain activation during memory-activation tasks in regions affected by Alzheimer's disease, including the left hippocampal, parietal, and prefrontal regions, in cognitive normal carriers of the APOE4 allele compared to non carriers. This increased activity is usually interpreted as compensatory recruitment to support memory performance, although works in mice suggest that it may rather reflect a detrimental function of hippocampal inhibitory function [START_REF] Gallagher | Episodic memory on the path to Alzheimer's disease[END_REF].

Mechanisms and multimodal studies

Despite the well-documented effect of APOE4, the underlying mechanisms are not completely clear.

Postmortem neurohistopathology studies suggest that APOE4 leads to increased production / aggregation of Aβ or decreased clearance. While the effect on amyloid plaque burden is well established, its effect on tau and tau-related mechanisms is much less clear [START_REF] Kim | The role of apolipoprotein E in Alzheimer's disease[END_REF]. Apart from an influence on β-amyloid pathology, APOE4 may decrease synaptic function, neurogenesis, glucose metabolism, vascular and mitochondrial functions and lipid and cholesterol metabolism, and increase neurotoxicity, aberrant brain activity, brain atrophy and tangle formation [START_REF] Liu | Apolipoprotein E and Alzheimer disease: risk, mechanisms and therapy[END_REF].

The review of the literacy from single-modality neuroimaging studies as described above reveals discrepant -and therefore probably subtle -effects of APOE4 on brain structure, less heterogeneous (but rare) findings for FDG-PET, and a clear effect on amyloid deposition, all mainly occurring in AD-sensitive areas. This graded effect of APOE4 on atrophy, hypometabolism, and amyloid deposition, illustrated in Figure 1, is consistent with several results from multimodal neuroimaging studies. It has been shown for example that the effect of APOE4 on Aβ deposition or CSF-Aβ (known to be highly correlated with cortical Aβ deposition) was greater than the effect of the clinical group, while there was a reverse effect for CSF-tau and MRI atrophy [START_REF] Vemuri | Effect of apolipoprotein E on biomarkers of amyloid load and neuronal pathology in Alzheimer disease[END_REF][START_REF] Murphy | Mapping the effects of ApoE4, age and cognitive status on 18F-florbetapir PET measured regional cortical patterns of beta-amyloid density and growth[END_REF]. [START_REF] Chen | Correlations between FDG PET glucose uptake-MRI gray matter volume scores and apolipoprotein E ε4 gene dose in cognitively normal adults: a cross-validation study using voxel-based multi-modal partial least squares[END_REF] reported that FDG-PET showed a stronger association to APOE4 compared to structural MRI, and concluded that PCC hypometabolism appears earlier than observable MRI-based structural abnormalities. Similarly, [START_REF] Protas | Posterior cingulate glucose metabolism, hippocampal glucose metabolism, and hippocampal volume in cognitively normal, late-middle-aged persons at 3 levels of genetic risk for Alzheimer disease[END_REF] found significant differences between APOE4 carriers and non carriers in their posterior cingulate glucose metabolism, but not in their clinical ratings, neuropsychological test scores, hippocampal volumes, or hippocampal glucose metabolism measurements, and they suggest that a reduction in posterior cingulate glucose metabolism precedes a reduction in hippocampal volume or metabolism in cognitively normal persons at increased genetic risk for AD.

In human, it is thought that there are both Aβ-dependant and Aβ-independent effects of APOE4 acting in concert to exacerbate the pathological and clinical phenotypes of AD [START_REF] Huang | Abeta-independent roles of apolipoprotein E4 in the pathogenesis of Alzheimer's disease[END_REF][START_REF] Liu | Apolipoprotein E and Alzheimer disease: risk, mechanisms and therapy[END_REF] see Figure 1). Multimodal studies confirm this view by showing that the effects of APOE4 on brain structure and function is at least partly independent from its effect on amyloid deposition. Thus, [START_REF] Desikan | Apolipoprotein E epsilon4 does not modulate amyloid-β-associated neurodegeneration in preclinical Alzheimer disease[END_REF] showed that the APOE4 allele was associated with Aβ deposition but, contrary to CSF-p-tau, didn't affect Aβ-associated neurodegeneration in clinically healthy individuals. As another example, comparing cognitively normal APOE4 carriers to non carriers, [START_REF] Sheline | APOE4 allele disrupts resting state fMRI connectivity in the absence of amyloid plaques or decreased CSF Aβ42[END_REF] showed decreased resting-state fMRI connectivity in the absence of amyloid plaques or CSF-Aβ abnormalities, suggesting that these changes antedate the pathological effects of fibrillar Aβ deposition. On the same line, [START_REF] Jagust | Alzheimer's Disease Neuroimaging Initiative. Apolipoprotein E, not fibrillar β-amyloid, reduces cerebral glucose metabolism in normal aging[END_REF] found an effect of APOE4 on both amyloid deposition and FDG-PET metabolism, but showed that the metabolism decrease was independent from the presence of Aβ, concluding that APOE genotype, and not aggregated fibrillar forms of Aβ, contributes to reduced glucose metabolism in aging. The detection of APOE4-related structural and functional changes in children [START_REF] Shaw | Cortical morphology in children and adolescents with different apolipoprotein E gene polymorphisms: an observational study[END_REF] or young adults [START_REF] Filippini | Distinct patterns of brain activity in young carriers of the APOE-epsilon4 allele[END_REF], i.e. well before the appearance of Aβ accumulation in the brain, is also suggestive of Aβ-independant effects of APOE4. In sum, it is thought that there are both Aβ-dependant and Aβ independent effects of APOE4 acting in concert to exacerbate the pathological and clinical phenotypes of AD [START_REF] Huang | Abeta-independent roles of apolipoprotein E4 in the pathogenesis of Alzheimer's disease[END_REF][START_REF] Huang | Alzheimer mechanisms and therapeutic strategies[END_REF].

Conclusion

Widely accepted neuroimaging biomarkers for AD (hippocampal atrophy, posterior parietal associative cortex hypometabolism and amyloid deposition) can be detected in the asymptomatic stage in APOE4 carriers. There is a graded effect of APOE4 across imaging modalities with a predominant effect on Aβ deposition and more subtle effects on brain structure (Figure 1). Moreover, APOE4 effects seem to be mediated both by Aβ-dependent and Aβ-independent pathological processes. Future studies would tell whether the mechanisms and sequences evidenced in carriers is comparable to those found in non carriers, but it is likely that APOE4 not only increases the risk for AD, but also modulates its physiopathological cascade. Altogether, APOE4 carriers offer a great opportunity to further our understanding of the physiopathology of the disease, although precaution is needed for interpretation in AD at large.

Figure 1 :

 1 Figure 1: Schematic representation illustrating the graded effect of APOE4 on amyloid deposition, metabolism and brain structure, showing a clear predominance of its effect on Aβ pathology (thick arrows). This figure also illustrates that ApoE4 operates through both Aβ-dependent and Aβindependent processes.

Table 1 :

 1 Prevalence of amyloid-positive individuals amongst cognitively normal elderly according to APOE4 status. % Aβ+: percentage of amyloid-positive individuals; *: mean and/or standard deviation not indicated.

	References	PET ligand	Dataset	Mean age	APOE4 non	APOE4
				(standard	carriers	carriers
				deviation)	n	% Aβ+	n	% Aβ+
	Rowe et al., 2010 PIB	AIBL	71.6 (7.4) yrs	101	21%	76	49%
	Morris et al.,	PIB	Washington	66.8 (10.7) yrs	158	9%	83	35%
	2010		University
	Jagust et al.,	Florbetapir	ADNI	77.2 (6.0) yrs	135	22%	40	50%
	2012
	Rodrigue et al.,	Florbetapir	Dallas	>60*	70	16%	17	35%
	2012		lifespan study	(60 to 89) yrs
	Mielke et al.,	PIB	Mayo clinic	78*	361	37%	122	64%
	2012			(75 to 82) yrs

Acknowledgments:

The authors thank Renaud Lajoie for providing Table 1.

Conflicts of interest: none to disclose.