Bénédicte Apouey
email: benedicte.apouey@psemail.eu

M Albouy

M Fartaoui

G Linger

A Messanvi

H Villemeur

Xuan

Conditions of Existence and Subjective Perceptions of Retirement: Quantitative Evidence from France

Keywords: Retirement, Career's end, Ageing, Perceptions, Social conditions, Satisfaction, France Retirement, Career's end, Ageing, Perceptions, Social conditions, Satisfaction, France

This article explores subjective perceptions of retirement in France, using original quantitative data on the customers of a not-for-profit insurance company. The sample contains individuals aged 40-84, who are either in the labour force (N=923) or retired (N=705). Perceptions of retirement are measured using closed questions on views of the retirement transition (these views can be positive, negative, or neutral) and definitions of retirement (retirement can be interpreted as a period of freedom, boredom, greater risk of precariousness, etc.). Using a number of different social indicators, we examine whether differences in social conditions translate into heterogeneous perceptions. We also investigate whether social differences in perceptions fade away with increasing age. Both working-age individuals and retirees generally have a positive view of the retirement transition and often define retirement as a period of freedom. Perceptions of retirement are shaped by social conditions: a higher level of education and income, greater wealth, better health, and stronger social involvement go hand in hand with rosier perceptions. Moreover, we uncover a strengthening of this social gradient with increasing age. Finally, perceptions are positively correlated with satisfaction in various domains, for retirees.

Introduction

Population ageing is one of the most significant ongoing social transformations: the proportion of older persons in the population is rapidly growing in many countries. In France, while individuals aged 60+ represented 20.4% of the population in 2000, the share reached 23.9% in 2013. i Predictions show that in 2030 the proportion of people over 65 years of age will be for the first time greater than that of individuals under 20. ii In Western societies, the transition from work to retirement is experienced by most people and the retirement stage is longer than ever before.

While retirement was historically synonymous with old age, this is no longer the case. How do workers and retirees perceive retirement --defined as the transition out of work and the life phase following this transition --nowadays? On the one hand, the perception of retirement could be mainly negative. Indeed, in our work-oriented societies, retirement could be associated with a lack of purpose, a sense of loss, and boredom [START_REF] Arbuz | Le Départ à la Retraite : Perceptions et Accompagnement[END_REF]. iii In addition, retirement may go hand in hand with a decrease in income and even with income precariousness. Moreover, for individuals for whom the workplace is an important context to socialize, retirement may represent a high risk of social exclusion. On the other hand, retirement may also be perceived in a more positive way, as a time of autonomy and freedom [START_REF] Gestin | Temps, Espaces et Corps à la Retraite : Des Paradoxes à Penser[END_REF][START_REF] Repetti | Since I Retired I Can Take Things as They Come. For Example, the Laundry': Gender, Class and Freedom in Retirement in Switzerland[END_REF]. As an example, for some people, retirement could represent an opportunity to spend time on things they never had time for before [START_REF] Arbuz | Le Départ à la Retraite : Perceptions et Accompagnement[END_REF]. According to the recent literature, respondents generally point out that retirement represents freedom [START_REF] Quéniart | Older Women and Their Representations of Old Age: A Qualitative Analysis[END_REF][START_REF] Roland-Lévy | Social Representations of Retirement in France: A Descriptive Study[END_REF].

In societies stratified by multiple factors, the perception of retirement is likely to be heterogeneous rather than homogeneous. Since social conditions shape individual opportunities and constraints, they are probably important structuring principles behind perceptions. According to Pierre Bourdieu's constructivist structuralist approach, differences in social conditions of existence (due to social positions) lead to differences in perceptions, appreciations, and practices [START_REF] Bourdieu | Choses Dites[END_REF].

Drawing on this theory, a qualitative study by [START_REF] Laberge | Les Conceptions Du « Bien-Vieillir » d'aînées de Milieux Favorisés et Défavorisés[END_REF] finds differences in personal notions of "ageing well" between low and high socioeconomic status (SES) women in Canada.

Gerontology researchers have argued that understanding subjective perceptions of retirement (and more generally of ageing and old age) as well as satisfaction with retirement is all the more important as they relate to future wellbeing and health. For example, poor self-perceptions of ageing iv (i.e. dissatisfaction with one' own ageing) at baseline increase the risk of mortality in late life [START_REF] Sargent-Cox | Longitudinal Change of Self-Perceptions of Aging and Mortality[END_REF]. Negative self-perceptions of ageing predict deterioration in general, physical, and mental health [START_REF] Wurm | Optimism Buffers the Detrimental Effect of Negative Self-Perceptions of Ageing on Physical and Mental Health[END_REF]. Moreover, negative attitudes to ageing as regards to "physical and psychological loss" strengthen the risk of becoming physically frail or pre-frail [START_REF] Gale | Attitudes to Ageing and Change in Frailty Status: The English Longitudinal Study of Ageing[END_REF].

In this article, we explore subjective perceptions of retirement in France, using recent data.

Quantitative research on the relationship between social conditions and perceptions of retirement is limited for this country, as far as we know. For instance, research often collects information on these perceptions among retirees, but not younger adults. Moreover, there has been no systematic study on the link between social conditions and perceptions. Finally, whether the role of social conditions is weaker or stronger for older individuals, compared to younger ones, remains an open question. We try to fill these gaps in the literature by analysing quantitative data from an original survey, the Harmonie Mutuelle Ageing Well ("Bien Vieillir Harmonie Mutuelle" --BVHM) survey, carried out in 2016. This survey provides information on the customers of Harmonie Mutuelle (HM) which is the largest not-for-profit insurance company in France. The sample consists of 1,628 middle-aged and older adults aged 40 to 84. The data thus contain younger adults, "young old" individuals, and "old old" persons (but not "oldest old" adults). We capture subjective perceptions of retirement by asking respondents closed questions on their views of the retirement transition and on their definitions of retirement: respondents report whether they have a positive, negative, or neutral view of the retirement transition, and whether they define retirement as a time of freedom, a well-deserved rest, or a period with a greater risk of financial precariousness, among other things.

Using this information, we first describe perceptions of retirement, on average, in our sample.

Rather than focusing on retirees only, we explore these perceptions for economically active individuals and retirees separately and examine whether perceptions are the same or not in these two population groups. Our analysis thus complements previous work by [START_REF] Mutran | Factors That Influence Attitudes toward Retirement[END_REF] who compare attitudes toward retirement before and after retirement, using US longitudinal data. In our cross-sectional data, differences in perceptions between the two population groups could be explained by an age and/or a cohort effect. An age effect means that perceptions would change as individuals get older. Regarding the cohort effect, groups of respondents may have been exposed to different labour markets, due to the rise of employment precarity and unemployment in recent decades in France, which could translate into different perceptions. Moreover, in a context of frequent legislation changes regarding pension age and benefits in the country, active individuals may feel less secure (regarding retirement) than already retired individuals, which could lead to less positive perceptions of retirement in the active population.

We then investigate the role of the conditions of existence in the perception of retirement, by taking advantage of respondents' diverse backgrounds. We specifically focus on the influence of SES, health status, and social activities. SES is measured using information on education (which is a proxy for cultural capital) and income and wealth (which capture economic capital). Health and social activities are measured by indices of general, physical, and mental health, and involvement in community volunteering and in associations (which are related to social capital). Importantly, we ask whether the role of social conditions weaken or strengthen with increasing age. Exploiting original information on perceptions, as well as data on a number of social indicators, we thus try to illustrate the pervasiveness of social inequalities during the ageing process. By doing so, we supplement the qualitative study of [START_REF] Laberge | Les Conceptions Du « Bien-Vieillir » d'aînées de Milieux Favorisés et Défavorisés[END_REF] for Canada.

Finally, we check that the subjective perception of retirement correlates with mental wellbeing for retirees. We here analyse data on individual satisfaction in the finances, leisure, social life, health, and housing domains. These satisfaction domains are widely used in the literature in economics of happiness and wellbeing [START_REF] Van Praag | Happiness Quantified: A Satisfaction Calculus Approach[END_REF].

Background

To understand the ageing and retirement processes, theoretical insights from the model of the institutionalisation of the life course may be useful. In this model, the life course is structured around work and divided in three periods (which are keyed to chronological age): preparation, "activity," and retirement [START_REF] Kohli | The World We Forgot: A Historical Review of the Life Course[END_REF]. Pension systems played a major role in this institutionalisation. Some claim that a de-standardisation of life course transitions has been taking place since the 1970s: for instance, the age boundary of retirement may have become less clear.

However, there is disagreement over the extent of this de-institutionalisation process: for example, while [START_REF] Kohli | The Institutionalization of the Life Course: Looking Back to Look Ahead[END_REF] argues that de-standardisation is limited, [START_REF] Guillemard | Les Défis du Vieillissement[END_REF] highlights that the process is important and goes hand in hand with insecurity and uncertainty in biographical trajectories.

In France, the retirement age has risen over the last decades. Indeed, while the country has a long tradition of social insurance, a number of pension reforms has been implemented (in 1993, 2003, 2010, and 2014) to provide financial incentives for older workers to remain longer in the labour market. As a result of reforms, while the labour force participation of older workers was declining in the 1970s and 1980s, it has been steadily increasing since the mid-2000s, and the retirement age has risen. In 2017, people retired when they were 62.5 years old on average (including early retirement). v These recurring reforms have generated a feeling of uncertainty regarding retirement age and pension benefits, in particular among working-age individuals, which may partly lead to differences in retirement perceptions between working-age persons and retirees. Importantly, most people disagree with the increase in the retirement age in France. Indeed, the normative perception that retirement is a social right obtained by older cohorts of workers is widely shared, and many people wish this right was not progressively endangered. A survey on retirement intentions carried out in France in 2004-2005 vi shows that most respondents (aged 54-59) think that 60 is the ideal age to retire [START_REF] Rapoport | Les Intentions de Départ à la Retraite des Salariés du Privé Agés de 54 à 59 Ans[END_REF]. Education and income level, job satisfaction, as well as health status, are positively correlated with preferred retirement age. [START_REF] Aouici | Le Passage à la Retraite en Période de Réformes: Vécu et Expériences[END_REF] mainly exploits qualitative data (collected in 2010) from the same survey and compares anticipated and actual retirement ages. She highlights the roles of laws, family environment, and working conditions in the gap between these ages, in particular.

Our article is closely related to the interdisciplinary literature on subjective meanings and perceptions of retirement, in France and other countries. These perceptions may have changed over the past decades, due to labour market changes, the deterioration of working conditions [START_REF] Askenazy | Les Désordres du Travail : Enquête sur le Nouveau Productivisme[END_REF], the rise of "successful ageing" [START_REF] Rowe | Successful Aging: The MacArthur Foundation Study[END_REF], retirement income growth, and life expectancy increase, among other things.

More precisely, employing survey data collected in 1955 in the Paris area, an article published in the late 1950s showed that more than 40% of retirees reported that they were not satisfied [START_REF] Treanton | Les Réactions à la Retraite. Une Étude Psycho-Sociologique[END_REF] cited in [START_REF] Caradec | Retraite 'à la Carte' et 'Libre Choix' Individuel. Réflexion sur les Transformations Contemporaines de la Retraite[END_REF]). Moreover, using data from a survey carried out in 1961, a report published in the early 1960s concluded that almost one third of individuals thought about their future retirement with apprehension [START_REF] Laroque | Politique de La Vieillesse[END_REF]. [START_REF] Caradec | L'épreuve de la Retraite. Transformations Sociétales, Expériences Individuelles[END_REF] argues that until the 1970s, retirement was often considered as a period of rest, but that in the 1970s it started being seen as a time of leisure, and that nowadays it is often interpreted as a period of useful activities, in the omnipresent context of "active" and "successful ageing" ideals.

Accordingly, the recent literature shows that the meaning of retirement is mainly positive and that retirement is often a synonym for freedom. For instance, using data from the Survey of Health, Ageing and Retirement in Europe (SHARE), [START_REF] Blanchet | Aspiration à la Retraite, Santé et Satisfaction au Travail : Une Comparaison Européenne[END_REF] find that 58% of French respondents aged 50-64 would like to transition to retirement as soon as possible. In addition, [START_REF] Crenner | Etre Retraité : Quelle Identité Après le Travail ?[END_REF] exploits quantitative data on retirees from the French "Histoire de vie" survey and highlights that transitioning to retirement is a positive experience, for most people. Two thirds of people report that they transitioned to retirement at the right time in their life. This perception depends on the type of occupation and on demographic characteristics, but not on health status, in particular. For most people, the transition to retirement takes place during a good period in their life. Moreover, in their study on 1,686 recently retired people from six European countries, [START_REF] Fouquereau | Perceptions of and Satisfaction with Retirement: A Comparison of Six European Union Countries[END_REF] show that satisfaction with retirement is high in France. The major determinants of satisfaction with retirement are health, resources, and anticipated satisfaction. In particular, French (and British) retirees are more likely to report that regaining freedom and control, experiencing a lower level of stress and responsibilities, and engaging in social activities significantly contribute to satisfaction, compared to participants from other countries. On a related matter, [START_REF] Roland-Lévy | Social Representations of Retirement in France: A Descriptive Study[END_REF] investigate social representations of retirement among students (N=100), people in the workforce (N=200), and recent retirees (N=50) in France. Social representations are assessed using a free association task (for the word "retirement") and a characterizing task. The overall representation of retirement is positive in the three population groups. However, representations are different between groups: while for students the representation is centered around "old age" (which is negatively perceived) and "time to rest" (positive in this study), working adults perceive retirement as a period of "rest," "freedom," and "free time" (positive), and the representation of retirees highlights "freedom," "less stress," "inactivity" (positive), and "end of work" (neutral). The characterizing task also shows that representations of students and adults are different. Note that positive perceptions of retirement do not imply that negative stereotypes have totally disappeared in France. Indeed, [START_REF] Macia | Perception of Age Stereotypes and Self-Perception of Aging: A Comparison of French and Moroccan Populations[END_REF] study stereotypes on older adults in a sample of older French (N=260) (and Moroccan) adults.

Stereotypes are measured by asking individuals whether society perceive older adults in a positive, negative, or neutral way. Results show that age stereotypes are widespread in France: 40.4% of respondents think that they are negatively perceived, but only 20.2% think that they are positively perceived.

We take inspiration from these studies to create our measures of perceptions (see their presentation below). Consistent with these articles, our findings show that perceptions of retirement are mainly positive in our data. However, in contrast with these studies, we systematically explore heterogeneity in perceptions according to a number of social indicators and investigate whether a social convergence process takes place for older individuals.

Another strand of the literature for France provides a synthetic and general view of retirement, by presenting typologies of retirement practices [START_REF] Caradec | L'épreuve de la Retraite. Transformations Sociétales, Expériences Individuelles[END_REF]. In particular, using quantitative data, [START_REF] Guillemard | La Retraite[END_REF] developed a typology containing five types of practices and argued that the most common type is "retirement-withdrawal" ("retraite-retrait" in French), i.e. "social death," in which practices are limited to functional activities (such as bathing, sleeping, etc.). She also showed the roles of income, social relationships, health status, and education in individual practices of retirement. However, since the publication of this book, the relative importance of each type of retirement practice has changed.

Even if our paper focuses on France, it is also connected to the substantial literature on the subjective perception of retirement and ageing in other countries. For example, using quantitative longitudinal data on workers aged 58-64 from the Raleigh-Durham-Chapel Hill metropolitan area in the US in 1992 and 1994, [START_REF] Mutran | Factors That Influence Attitudes toward Retirement[END_REF] show that individuals are more positive about retirement after they become retired. Authors identify some factors influencing attitudes toward retirement, among workers and retirees. For workers in preretirement (in 1992), these factors include social background, individual identity, self-esteem, and depression. Moreover, using indepths interviews with 14 women before and after the retirement transition in the US, [START_REF] August | Women's Retirement Meanings: Context, Changes, and Organizational Lessons[END_REF] highlights several meanings associated with retirement: a new period of life, a final phase, a period of losses, and a pinnacle of accomplishment or relief. Moreover, in their qualitative study of 25 interviews based on anchored theorisation, [START_REF] Quéniart | Older Women and Their Representations of Old Age: A Qualitative Analysis[END_REF] explore representations of old age among women aged 65+ in Québec. Women reject the preconception on "older or elderly women" and conceptions of old age are often positive. Ageing means accepting the passage of time (i.e. physical changes) and having time for oneself and for others. For women aged 65-74, retirement offers freedom: it is a period of time when people can focus on their own interests. For women over 75, old age means the opportunity to care for loved ones. In addition, [START_REF] Van Dyk | The Many Meanings of 'Active Ageing'. Confronting Public Discourse with Older People's Stories[END_REF] conduct 55 qualitative interviews with retirees aged 60-72 in Germany and show that the meaning of retirement is very diverse for older people and that "late freedom" is highly valued. [START_REF] Repetti | Since I Retired I Can Take Things as They Come. For Example, the Laundry': Gender, Class and Freedom in Retirement in Switzerland[END_REF] conduct a qualitative analysis with 15 Swiss retirees and show that both men and women describe retirement as a time of freedom. However, using a feminist gerontological approach, they find that the meaning of freedom differs between genders and reflects the gender division of labour. Finally, the study by [START_REF] Laberge | Les Conceptions Du « Bien-Vieillir » d'aînées de Milieux Favorisés et Défavorisés[END_REF] is also interesting to us because it emphasizes heterogeneity in individual notions of "ageing well" according to socioeconomic background. Building on Bourdieu's sociological theory [START_REF] Bourdieu | La Distinction. Critique Sociale Du Jugement[END_REF][START_REF] Bourdieu | Choses Dites[END_REF] and using qualitative data on 36 women aged 65 to 75 in Canada, the authors show that high SES women think that ageing well means accepting declining health and maintaining a positive attitude, whereas low SES women define ageing well as having their needs met. For both low-and high-SES women, enjoying the simple pleasures of life is an important component of ageing well. vii In a nutshell, recent studies often show that retirement is perceived in a positive way as a time of freedom. Quantitative research uncovering social differences in perceptions is limited for France. In what follows, we build on existing evidence to broaden knowledge for this country.

Data and method

Survey

Data come from the BVHM survey, which covers customers of HM and was carried out in 2016 [START_REF] Apouey | Présentation de l'Enquête Bien vieillir Harmonie Mutuelle[END_REF]. First, 10,721 customers aged 40-84 were contacted by phone by a polling institute on behalf of HM and were asked whether they agreed to participate in the survey. 4,811 individuals answered in the positive and were sent a paper questionnaire. Finally, 1,769 individuals filled a questionnaire and sent it back to the polling institute. The total completion rate is rather small (16.50%=100*(1,769/10,721)%), which is partly due to the length of the questionnaire. The sample was constructed using the quota method and the final sample is representative of the customers of HM along three criteria (gender, age group, and type of insurance coverage i.e. individual versus group insurance coverage). Importantly, the sample excludes the oldest old.

Data are unique in containing information on perceptions of retirement, SES, health status, social activities, and demographic characteristics. They have been used to study preparation for old age, service needs among the elderly, and perceptions of ageing well (Apouey, 2018a(Apouey, , 2018b, forthcoming), forthcoming).

In this study, we focus on individuals who are either in the economically active population or retired. We drop the small number of individuals who are not in the labour force but not retired either. The analysis sample contains 923 active individuals and 705 retirees.

Subjective perceptions of retirement

Subjective perceptions of retirement are captured by questions on the view of the retirement transition on the one hand, and on the definition of retirement on the other hand. These questions on perceptions were designed by the author, in partnership with "Chaire Transitions Démographiques Transitions Economiques" and HM.

First, the view of the retirement transition is measured using the following question: "Is your view about the transition to retirement positive or negative?" with the following possible answers: "Positive," "Negative," and "Neither positive nor negative." viii We code our variable in the following way: 1=Negative, 2=Neither positive nor negative, and 3=Positive. Note that this question is different from a question capturing stereotypes, which would focus on perceptions in society. For instance, in his study on age stereotypes, [START_REF] Macia | Perception of Age Stereotypes and Self-Perception of Aging: A Comparison of French and Moroccan Populations[END_REF] employ the following question: "In your opinion, how does society perceive older adults? Rather positively, rather negatively, in a neutral way." Second, we measure definitions of retirement thanks to the following question: "In the following list of propositions, which one gives the best definition of retirement, in your opinion? (1) A welldeserved rest; (2) A period of time during which one can be bored, feel useless, or feel empty; (3)

The joy of no longer hearing of work; (4) A greater risk of financial precariousness; and (5) A feeling of freedom, the feeling to start a new life." ix Respondents select one answer. The corresponding variable is thus a nominal variable with five categories. While "freedom" is clearly a positive perception whereas "boredom" and "precariousness" are negative, it is unclear to us whether "rest" is positive or negative. The importance of the response items has been highlighted in previous research on ageing. For instance, [START_REF] Caradec | L'épreuve de la Retraite. Transformations Sociétales, Expériences Individuelles[END_REF] argues that the social meaning of retirement was a time of "rest" in France until the 1970s, and that this meaning is now less widespread, even if it has not completely disappeared. Moreover, in their study on social representations in France, Roland-Lévy and Berjot (2009) find that retirement means "time to rest," "rest," "freedom," "inactivity," and "end of work" for a significant share of French respondents, while [START_REF] Van Dyk | The Many Meanings of 'Active Ageing'. Confronting Public Discourse with Older People's Stories[END_REF] highlight the importance of "late freedom" for older individuals in Germany.

SES, health, and social activities

We measure SES using education, income, and wealth. Our SES variables thus capture both cultural capital (education) and economic capital (income and wealth). Education is coded in three categories: low education (less than "baccalauréat," which serves as the reference category), medium education ("baccalauréat" graduate), and high education (more than "baccalauréat").

Finally, we measure income and wealth using the logarithm of household income and wealth (plus one).

The data contain rich information on general, mental, and physical health. General health is captured by the well-known self-assessed health measure [START_REF] Idler | Self-Rated Health and Mortality: A Review of Twenty-Seven Community Studies[END_REF]. Individuals indicate whether their health is very poor, poor, fair, good, or very good. Because a very small number of respondents are in very poor health, we merge the very poor and poor responses into one category. Regarding mental health, we employ the Kessler Psychological Distress Scale (K6). This self-administered scale involves six questions to estimate anxiety and mood disorders. The K6 score is an abbreviated version of the K10 scale. Higher scores indicate better mental health. Finally, for physical health, we employ a dummy for whether the individual reports that she has high blood pressure and a dummy for whether she is overweight or obese. x

Satisfaction in different domains

We gather information on satisfaction in the finances, leisure, social life, health, and housing domains using the following questions: "On a scale from 0 to 10, please indicate your level of satisfaction with the financial situation of your household / leisure / your relationship with your family, friends, and neighbours / health / housing." These satisfaction questions are widely used in the economics and psychology literature [START_REF] Huppert | Measuring Well-Being across Europe: Description of the ESS Well-Being Module and Preliminary Findings[END_REF][START_REF] Van Praag | Happiness Quantified: A Satisfaction Calculus Approach[END_REF].

Control variables

We use two sets of control variables. "Controls 1" include gender, age, and marital status (i.e. a dummy capturing marital life), and "controls 2" include "controls 1" plus education, income, general health, volunteering, and involvement in an association.

When an explanatory variable (like income) has more than 75 observations with missing values, we create a dummy variable for whether information is missing and include it in our model.

Method

We estimate several regression models. First, we regress perceptions -i.e. view of the retirement transition and definition of retirement -on SES, health, and social activities, adjusting for demographic characteristics, separately for retirees and younger adults. For retirement view, which is an ordered variable, we use an ordered logit model, and for the retirement definition question, in which responses are unordered, we employ a multinomial logit model. For both explained variables, we report the average marginal effect for each category of response.

To investigate whether the effect of social conditions on perceptions changes as respondents get older, we re-estimate our model including an interaction term between social condition and age. We employ a linear model (ordinary least squares).

Last, we regress satisfaction in different domains on perceptions and control variables, for retirees, using a linear model.

Our study is descriptive and shows correlations between our variables of interest. For instance, the association between SES, health, social activities, and retirement perceptions cannot be interpreted as the causal effect of conditions of existence on perceptions. Indeed, some reverse causality may be at work, with perceptions influencing conditions of existence. Moreover, there may be hidden factors --such as genetic endowments, life events, and preferences --influencing both right-and left-hand side variables. Omission of these factors may create a spurious correlation between variables.

Results

Sample description

Table A in the Supplementary Material provides summary statistics for gender, age, marital status, and conditions of existence. There are 38% of males in the economically active population, versus 47% in the retiree sample. The average age is 51 years for active individuals and 70 years for retirees. While 77% of active persons have a marital life, this is the case of 72% of retirees. There is some evidence that retirees are less educated, have a lower income, but are wealthier than active individuals. In addition, retirees are less healthy (for all health variables except mental health) and more likely to be involved in community volunteering and in associations.

Descriptive statistics for the perception variables are presented in Table 1. In the full sample, that contains both individuals in the economically active population and retirees, more than half of the respondents have a positive view of the retirement transition (52%), and less than 10% have a negative view. Moreover, 38% of the sample has neither a positive nor a negative view. Answers to the question on the definition of retirement are consistent with this positive view: retirement is defined as a period of freedom by half of the sample. Moreover, 30% of the sample report that this is a well-deserved rest. The other three possible definitions are chosen by less than 10% of the sample each.

In the descriptive statistics, perceptions seem more positive among retirees than among active individuals. Indeed, 59% of retirees have a positive view, versus 47% of active individuals.

Similarly, retirement represents a period of freedom for 53% of retirees, versus 48% of active individuals; and a risk of precariousness for only 3% of retirees, but 7% of active individuals.

Finally, it is defined as a well-deserved rest for 32% of retirees, versus 28% of active individuals.

However, active individuals and retirees have different socio-demographic characteristics, as shown in Table A. In a regression model that controls for gender, age, marital status, and education, we find that retirees are significantly less likely to answer "precariousness" and more likely to answer "rest" than younger adults, all other things being equal (results available upon request). There is no significant difference between the two population groups for other perception responses. Note that the model presented in Table B in Supplementary Material, which distinguishes between the employed, the unemployed, and retirees, and includes a larger number of right-hand side variables, supports these findings.

< Insert Table 1 about here >

Using descriptive statistics, we also illustrate the association between better conditions of existence and more positive perceptions. Figure 1 presents the distribution of responses to the question on retirement definitions according to income level, general health, and involvement in an association.

Subfigures for the economically active population are presented on the left-hand side, while those for retirees are shown on the right-hand side. To generate the figure, we distinguish between individuals whose income is lower than median income and those whose income is greater than median income, individuals who report very poor or poor health and those who report fair, good, or very good health, and individuals who are involved in an association and those who are not.

In the active population, high-income individuals are more likely to define retirement as a period of freedom and less likely to define it as a period of potential precariousness. Indeed, while only 41% of low-income individuals answer "freedom," this is the case for 52% of high-income individuals.

Similarly, active individuals who are in better health or who are involved in an association are more likely to mention freedom and less likely to mention the risk of precariousness.

Findings for retirees are in line with those for younger adults. However, for retirees, the gap in the definition of retirement as a period of freedom between low-and high-income individuals and between unhealthy and healthy individuals is even greater than for workers. Indeed, only 41% of low-SES individuals answer "freedom," versus 62% of high-SES respondents. Similarly, 44% of unhealthy individuals answer "freedom," versus 63% of healthy respondents. We also find a sizeable difference in the "rest" answer between low-and high-income individuals and between unhealthy and healthy respondents. For instance, while 41% of low-income individuals answer "rest," this is the case for only 26% of high-income persons. Finally, like in the active population, being involved in an association increases the chances of defining retirement as a period of freedom. Results on social differences in views of the retirement transition are consistent with results on definitions of retirement (results available upon request).

< Insert Figure 1 about here > Regression results: social differences in the active population Multivariate regression analysis indicates that higher SES, better health, and more social activities are associated with improved perceptions of retirement, all other things being equal. This result is robust to the use of different measures for SES, health, and social activities and it holds both in the economically active population and for retirees.

More precisely, in Table 2, we report marginal effects from an ordered logit (in columns (1) to (3)) and a multinomial logit (in columns ( 4) to ( 8)) model for perceptions of retirement, for the economically active population. For instance, Panel A shows the results of regressions of perceptions of retirement on education (and control variables). While there is no significant association between education and the view of the retirement transition, high education decreases the likelihood of defining retirement as a well-deserved rest and as the joy of not hearing of work, and increases the chance of defining retirement as a period of freedom. As an example, compared with a low level of education, a high level of education increases the probability of answering "freedom" by 18 percentage points.

Income and wealth also play a role: while a higher income level decreases the probability of a negative view and increases that of a positive view, greater income and wealth levels decrease the chance of defining retirement as a period or precariousness and as a well-deserved rest, and increase that of defining it as a period of freedom.

With increasingly more problematic mental health, there are more negative views of the retirement transition. Moreover, (good) mental health is positively associated with a positive view of the transition and is negatively associated with the probability of mentioning "precariousness." Mental and physical health is negatively associated with answering "rest." For instance, the likelihood that an individual with normal blood pressure answers "rest" is less than that of an individual with high blood pressure by 12 percentage points. Similarly, being overweight or obese increases the probability of answering "rest" by 9 percentage points. On a related matter, better general and mental health decreases the chance of defining retirement as a period of boredom. In addition, general and mental health is positively correlated with the probability of mentioning "freedom." For example, the probability that a respondent in very good health answers "freedom" is greater than that of an individual in fair health by 13 percentage points.

Finally, social activities are also associated with positive perceptions. For example, compared with individuals who do not volunteer, individuals who volunteer are more likely (by 10 percentage points) to have a positive view of the retirement transition; and respondents who are involved in an association are more likely (by 14 percentage points) to define retirement as a time of freedom.

< Insert Table 2 about here >

Regression results: social differences for retirees

Like in the economically active population, SES, health, and social activities are positively correlated with positive perceptions, for retirees.

In this population group, none of the SES variables is significantly associated with views of retirement (Table 3). However, all three SES measures are negatively correlated with the definitions emphasizing "rest" and "precariousness" and positively correlated with the definition highlighting "freedom." Compared with a low level of education, a high level of education increases the probability of answering "freedom" by 28 percentage points, and decreases the likelihood of answering "rest" and "precariousness" by 25 and 4 percentage points.

General, mental, and physical health is negatively correlated with a negative view of retirement (and positively associated with a positive view). Moreover, better health decreases the likelihood of defining retirement as a well-deserved rest, a period of boredom, the joy of not hearing of work, and a risk of precariousness, and increases the chance of defining retirement as a time of freedom. Very good health decreases the probability of answering "rest" and "the joy of not hearing of work" by 17 and 6 percentage points, while increasing the probability of mentioning "freedom" by 26 percentage points (compared to fair health).

Social activities also go hand in hand with more positive views and definitions. For example, volunteering increases the probability of answering "freedom" by 13 percentage points.

< Insert Table 3 about here >

Regression results: full models

In Tables 2 and3, we explored the roles of SES, health, and social activities separately. We now estimate a full model that includes several indicators of social conditions at the same time (education, income, general health, volunteering, and association). Results (presented in Table B in Supplementary Material) indicate that each family of social condition variables has an independent effect on perceptions. More precisely, even when we control for health and social activities, a higher income goes hand in hand with rosier definitions. Similarly, in this full model, better health still decreases the likelihood of negative perceptions while strengthening that of positive ones.

Social activities also play an independent beneficial role.

In this full model, the role of education is complex: on the one hand, high education is positively correlated with a negative view of the retirement transition (and negatively correlated with a positive view), but it also increases the likelihood of defining retirement as a time of freedom.

Although this is not our prime concern, the analysis enables us to know whether gender, age, and marital status are important structuring principles behind perceptions or not. First, we generally do not find any significant difference between females and males. However, this probably does not imply that there is no gender discrepancy in perceptions. Indeed, previous research shows that ageing is a gendered process [START_REF] Bozon | L'Expérience du Vieillissement Autour de la Soixantaine en France[END_REF][START_REF] Foster | Gender and Active Ageing in Europe[END_REF], due to differences in work trajectory [START_REF] Bué | Temps Partiel des Femmes : Entre Choix et Contraintes[END_REF], domestic and caring activities [START_REF] Craig | Parenthood, Gender and Work-Family Time in the United States, Australia, Italy, France, and Denmark[END_REF]Haberken et al., 2015), and health status and life expectancy [START_REF] Cambois | Gender Career Divide and Women's Disavantage in Depressive Symptoms and Physical Limitations in France[END_REF][START_REF] Van Oyen | Gender Difference in Healthy Life Years Within the EU: An Exploration of the "Health-Survival" Paradox[END_REF] between females and males, in particular. Moreover, the literature also finds that men and women have different retirement expectations and describe retirement in different ways [START_REF] Onyx | Retirement expectations: gender differences and partner effects in an Australian employer-funded sample[END_REF][START_REF] Repetti | Since I Retired I Can Take Things as They Come. For Example, the Laundry': Gender, Class and Freedom in Retirement in Switzerland[END_REF]. In our study, it may simply be the case that our variables are not precise enough to capture gender differences. Using an in-depth qualitative method may thus be useful to better assess the influence of gender on perceptions.

Second, age is positively associated with positive views of the retirement transition (and negatively correlated with negative views). Moreover, the answer "freedom" becomes more common with increasing age. These changes with age could mean that individuals embrace new social norms as they get older, although our data do not enable to test this hypothesis.

Finally, not having a partner has a detrimental impact on perceptions, since it decreases (resp. increases) the likelihood of positive (resp. negative) views and increases the likelihood of defining retirement as a period of potential precariousness. This results is unsurprising, given the positive influence of having a spouse/partner while adjusting to retirement and re-organizing weekly activities [START_REF] Caradec | L'épreuve de la Retraite. Transformations Sociétales, Expériences Individuelles[END_REF]. Does the effect of social conditions on perceptions weaken with age? To investigate this point, we re-estimate our full model including interaction terms between social condition and age (Table C in Supplementary Material). There is some evidence that a divergence process is taking place, meaning that the relationship between social conditions and perceptions strengthens with age. For instance, the correlation between education or income and the likelihood of responding "rest" (is negative starting age 41 for education and 54 for income and) decreases with age, while the association between income and the likelihood of responding "freedom" (is positive starting age 42 and) increases with age. xi In other words, a higher income level is associated with a smaller probability of responding "rest" and this negative income impact is even more negative for older individuals. Symmetrically, a higher income level increases the probability of mentioning "freedom" and this effect is greater for older individuals. In contrast, the association between social activities (i.e. volunteering and association) and perceptions remains stable with age.

Rosier perceptions of retirement correlate with higher satisfaction for retirees

Finally, we check that rosier retirement perceptions are associated with higher levels of wellbeing for retirees. Specifically, we regress the satisfaction variables on the view of the retirement transition and on the definition of retirement (see Table D in Supplementary Material). We alternatively include "controls 1" and "controls 2" to check the robustness of findings. Compared with a view of retirement transition that is neither negative nor positive, a positive view is correlated with greater satisfaction in all domains, for the two sets of control variables. Findings on the definition of retirement support these results. People who define retirement as a period of precariousness report lower satisfaction in the finances, leisure, and health domains (when either "controls 1" or "2" are included) and in the housing domain (when "controls 1" are accounted for).

In contrast, individuals who define retirement as a period of freedom report higher levels of wellbeing in all domains, when "controls 1" are included, and in the social life and housing domains, when "controls 2" are included. Surprisingly, we also find that a negative view of retirement, as well as a definition of retirement as a period of boredom, is positively associated with satisfaction with financial situation. However, our results generally show that perceptions are closely related to individual wellbeing, with the expected signs.

Discussion

Employing quantitative data on a diverse sample of French respondents, we show that perceptions are undoubtedly positive: 52% of respondents have a positive view of the retirement transition (while only 10% have a negative view), and the most common definition of retirement is a period of freedom, both among younger adults and retirees. Our finding is thus in stark contrast with the negative stereotype associated with age: for our respondents, retirement does not mean social exclusion or "social death" [START_REF] Guillemard | De la Retraite Mort Sociale à la Retraite Solidaire[END_REF]. Our result is consistent with recent qualitative findings for Switzerland, that show that retirees describe retirement as a time of freedom [START_REF] Repetti | Since I Retired I Can Take Things as They Come. For Example, the Laundry': Gender, Class and Freedom in Retirement in Switzerland[END_REF]. It also corroborates earlier results on the positive social representations of retirement in France [START_REF] Roland-Lévy | Social Representations of Retirement in France: A Descriptive Study[END_REF] and old age in Canada [START_REF] Quéniart | Older Women and Their Representations of Old Age: A Qualitative Analysis[END_REF]. These findings on the positive image of retirement are not surprising given the power of our society's norm of "successful ageing" and the growing culture of personal selffulfilment (retirement is then perceived as a period during which one may realise her potential).

In the context of longer working lives in France (related to public policies, including pension reforms), these positive views of retirement may also mean that retirement is more rewarding than working life for some respondents. Previous research for France has highlighted that some older people are exposed to poor working conditions during the last years of their working lives, due to the general deterioration of working conditions since the 1980s [START_REF] Khireddine-Medouni | Augmentation des Taux de Prévalence de la Souffrance Psychique en Lien avec le Travail chez les Salariés Actifs en France entre 2007 et 2012, à Partir du Programme MCP[END_REF][START_REF] Pailhé | Working Conditions: How are Older Workers Protected in France?[END_REF] and to the negative perception of older workers in the workplace. For instance, some older workers are affected by negative relationships with colleagues and get boring tasks [START_REF] Aouici | Le Passage à la Retraite en Période de Réformes: Vécu et Expériences[END_REF]. In addition, some older workers are excluded from the labour market because of forced (early) retirement or unemployment [START_REF] Salembier | Fins de Carrière Autour des Années 2000 : Une Hausse des Situations de Chômage à l'Approche des 60 ans[END_REF]. These conditions and constraints generate stress, job dissatisfaction, and a feeling of being a burden in the workplace. This may partly explain the positive perception of retirement, by contrast with work (or unemployment).

Perceptions of retirement are different between retirees and younger adults in France: retirees are less likely to define retirement as a risk of financial precariousness, but more likely to define it as a period of rest (all other things being equal). Note that [START_REF] Mutran | Factors That Influence Attitudes toward Retirement[END_REF] also find differences in attitudes toward retirement, using US longitudinal data: in their study, attitudes are more positive among respondents who retire than among those who continue to work. In our cross-sectional data, the difference between retirees and younger adults may be interpreted as an age and/or a cohort effect. First, it could be due to a life stage effect: people would have different views about retirement after experiencing this new stage of life. Second, the difference between workers and retirees may also reflect differences in life trajectories, anticipations, or socializations between cohorts. In the context of recurring pension reforms (to improve sustainability) in France, younger cohorts may anticipate that their retirement income will decrease [START_REF] Caradec | L'épreuve de la Retraite. Transformations Sociétales, Expériences Individuelles[END_REF], which may explain why they put more emphasis on the risk of precariousness than retirees (cohort effect). Younger cohorts may also anticipate higher risks of financial precariousness during retirement because they experience higher degrees of employment insecurity during their working life, which goes hand in hand with lower pension payments (cohort effect). Understanding the gap in perceptions between the two population groups is an important avenue for future research. Do subjective perceptions of retirement depend on social and material circumstances? We quantitatively examine this point using a large number of social condition indicators. Indeed, social conditions (high SES, good health, and social involvement) may provide resources that help individuals cope with retirement and ageing. These resources may enable individuals to prepare (financially) for retirement (Apouey, 2018b), to reorganize their life after retirement, and to develop some (non-work) aspects of their identity. Our analysis confirms that perceptions markedly vary according to social position. Importantly, these social differences are found for both active individuals and retirees. Our result seems robust, since social heterogeneity emerges in descriptive statistics, but remains statistically significant in regression models, all other things being equal. For instance, household income, which is a proxy for economic hardship, is negatively associated with defining retirement as a period of potential precariousness. Similarly, superior economic and cultural capital allows respondents more often to perceive retirement as a period of freedom. Moreover, while these social differences are observed for a variety of social condition indicators (SES, health, and social capital measures), each social condition variable plays an independent role, which points to the cumulative effects of the different forms of capitals (see the full model). The effects we uncover are large: for instance, in our full model, high education increases the probability of perceiving retirement as a period of freedom by 12 percentage points (compared to low education).

While "rest" was the main social meaning of retirement until the 1970s [START_REF] Caradec | L'épreuve de la Retraite. Transformations Sociétales, Expériences Individuelles[END_REF], retirement still means "rest" for a high share of respondents (approximately 30%) in our very recent data.

Among retirees and individually active individuals, the probability of answering "rest" is significantly correlated with low SES and poor health. We hypothesize that this response is linked to poor working conditions or precarity on the labour market.

More generally, our results on the role of social conditions are consistent with previous evidence on differences in personal definitions of "ageing well" in France (Apouey, forthcoming) and Canada [START_REF] Laberge | Les Conceptions Du « Bien-Vieillir » d'aînées de Milieux Favorisés et Défavorisés[END_REF]. It is entirely possible that social differences in perceptions of retirement and ageing echo social differences in perceptions of working life. However, longitudinal data that would follow individuals before and after the retirement transition would be necessary to test this hypothesis. In any case, using original data on perceptions, our work emphasizes the persistence of social inequalities during the retirement period. We also provide evidence of divergence in perceptions of retirement between individuals from different social backgrounds with increasing age. Again, this result could be due to an age or a cohort effect. In the case of an age effect, our result may mean that the adverse effect of poor social conditions on perceptions accumulate as individuals get older.

While our analysis emphasizes social differences, some common features also emerge between social classes. In particular, in both low-and high-income groups, the most common definition of the retirement is a period of freedom (see the descriptive statistics).

Finally, we check that positive perceptions of retirement are associated with greater wellbeing (i.e. satisfaction in several life domains) for retirees. These associations remain significant (but are attenuated) when we adjust for covariates. This result speaks to previous findings on the link between age stereotypes and satisfaction in Germany [START_REF] Kornadt | Contexts of Aging: Assessing Evaluative Age Stereotypes in Different Life Domains[END_REF], for instance.

One important limitation of our study is that our sample is not representative of the entire French population, but of the customers of a large insurance company. In particular, compared with the French population of the same age group, the average individual in our sample is more likely to be married and to be a home-owner. For this reason, our results should be interpreted with some caution and further research based on a nationally representative is needed. Another limitation is that respondents may not understand questions on retirement perception and response categories in the exact same way. For example, in the response category "a feeling of freedom," the meaning of the word "freedom" could be different for low-and high-income individuals. In addition, our questionnaire lists a relatively small number of definitions of retirement, and it is thus possible that some important definitions are omitted: for instance, retirement may represent an opportunity to care for loved ones. To address this point, future quantitative research should include a broader set of definitions. Moreover, while we focus on a number of SES, health, and social activities measures, other variables, such as former occupation (for retirees) and the spouse's occupation, may also shape perceptions. Finally, our results cannot be interpreted as causal relationships, as discussed above.

In closing, the strengths of our study include the description of perceptions of retirement in France, the study of heterogeneity, and the use of a relatively large sample of respondents. Our findings point to the importance of social conditions in shaping perceptions of retirement. We also find that perceptions are associated with satisfaction in different domains. Future research should try to uncover the mechanisms behind these associations, in the context of successful ageing ideals. Notes: Differences in perceptions between the economically active population and retirees are statistically significant at conventional levels (except for "A period of time during which one can be bored, feel useless, or feel empty" for which the difference is not significant).

Tables and figures

Notes

i See https://www.insee.fr/fr/statistiques/1906664?sommaire=1906743#titre-bloc-3.

ii See: Libault D. (2019). Rapport de la concertation Grand âge et autonomie. Link: https://solidaritessante.gouv.fr/affaires-sociales/personnes-agees/concertation-grand-age-et-autonomie/article/rapport-de-la-concertationgrand-age-et-autonomie.

iii In this context, a large literature in Economics studies the effect of retirement on health and wellbeing.

iv measured with the Attitudes Toward Own Ageing subscale from Lawton's (1975) Philadelphia Geriatric Centre Morale Scale.

v See https://www.statistiques-recherches.cnav.fr/age-de-depart-a-la-retraite.html. vi The sample includes employed individuals from the general retirement scheme ("salariés du régime genéral").

vii Our article also relates to the international economics literature on the causal impact of retirement on mental health and wellbeing. This series of articles employ specific methodologies, such as the instrumental variable technique, to quantify the causal effect. Findings are somewhat contradictory: while several articles document a negative effect of retirement [START_REF] Dave | The Effects of Retirement on Physical and Mental Health Outcomes[END_REF], other articles find a positive impact [START_REF] Apouey | Retirement and Unexpected Health Shocks[END_REF]Zhu 2016). Nishimura, Oikawa, andMotegi (2018) argue that differences in results are often due to differences in the choice of the analysis method.

viii In French, the question is the following: "Avez-vous une vision positive ou négative du passage à la retraite ? x This dummy is constructed using self-reported height and weight.

xi More precisely, the effect of our income variable (log (income + 1)) on "rest" decreases from -0.005 (=0.319-0.006*54) at age 54 to -0.185 at age 84. The effect of the income measure on "freedom" increases from 0.003 at age 42 to 0.297 at age 84.

  (1) Positive ; (2) Négative ; (3) Ni positive ni négative." ix In French: "Parmi les propositions suivantes, quelle est celle qui définit le mieux, pour vous, la retraite ? (1) Un repos bien mérité ; (2) Une période où l'on peut parfois s'ennuyer, se sentir inutile, ou avoir un sentiment de vide ; (3) Le plaisir de ne plus entendre parler du travail ; (4) Le risque de tomber dans la précarité ; (5) Un sentiment de liberté, de renaissance, d'être au début d'une deuxième vie."

Table 1

 1

	: Descriptive statistics
			Proportions (%)
		Full	Economically	Retirees
		sample	active
			population
	View of the retirement transition
	Negative	9.6	12.7	5.5
	Neither negative nor positive	38.3	40.8	35.2
	Positive	52.1	46.5	59.3
	Total	100	100	100
	Definition of retirement
	A well-deserved rest	29.9	28.1	32.2
	A period of time during which one can be bored,	6.6	7.2	5.9
	feel useless, or feel empty
	The joy of no longer hearing of work	8.3	10.0	6.2
	A higher risk of financial precariousness	5.0	6.5	3.0
	A feeling of freedom, the feeling to start a new life 50.2	48.2	52.7
	Total	100	100	100

Table 2 :

 2 Correlates of perception of retirement, in the economically active population

		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
		View of the retirement transition		Definition of retirement
		Negative	Neither	Positive	Rest	Boredom	Not	Precarious	Freedom
			negative				hearing	ness
			nor				of
			positive				work
	Socioeconomic status
	Panel A. Education
	Low edu	Ref	Ref	Ref	Ref	Ref	Ref	Ref	Ref
	Medium edu	-0.021	-0.027	0.049	-0.057	0.012	-0.054**	0.012	0.087*
		(0.018)	(0.025)	(0.043)	(0.042)	(0.024)	(0.026)	(0.023)	(0.045)
	High edu	-0.008	-0.010	0.018	-0.139***	0.014	-0.047**	-0.008	0.181***
		(0.016)	(0.019)	(0.036)	(0.033)	(0.020)	(0.023)	(0.018)	(0.037)
	Panel B. Income
	Ln(income+1)	-0.019** -0.022** 0.041** -0.063*** -0.007	-0.019 -0.028*** 0.118***
		(0.009)	(0.010)	(0.019)	(0.020)	(0.012)	(0.013)	(0.007)	(0.026)
	Panel C. Wealth
	Ln(wealth+1)	-0.004	-0.005	0.009	-0.020**	0.002	-0.010* -0.014*** 0.043***
		(0.005)	(0.005)	(0.010)	(0.009)	(0.006)	(0.006)	(0.005)	(0.010)
	Health
	Panel D. General health
	Very poor or poor	0.002	0.002	-0.004	-0.031	-0.022	-0.000	0.077*	-0.024
		(0.031)	(0.030)	(0.062)	(0.060)	(0.034)	(0.042)	(0.041)	(0.065)
	Fair	Ref	Ref	Ref	Ref	Ref	Ref	Ref	Ref
	Good	-0.024	-0.028	0.052	-0.058*	-0.021	-0.025	-0.010	0.114***
		(0.017)	(0.019)	(0.035)	(0.035)	(0.021)	(0.024)	(0.018)	(0.038)
	Very good	-0.029	-0.035	0.065	-0.062	-0.043*	-0.011	-0.012	0.129**
		(0.021)	(0.026)	(0.047)	(0.045)	(0.024)	(0.031)	(0.024)	(0.050)
	Panel E. Mental health
	Good mental health (K6 scale) -0.005*** -0.006*** 0.012*** -0.007** -0.003*	-0.001 -0.005*** 0.016***
		(0.002)	(0.002)	(0.004)	(0.003)	(0.002)	(0.002)	(0.002)	(0.004)
	Panel F. High blood pressure
	High blood pressure	-0.021	-0.029	0.050	0.118*	-0.043*	-0.008	-0.045**	-0.023
		(0.025)	(0.039)	(0.065)	(0.067)	(0.024)	(0.040)	(0.020)	(0.068)
	Panel G. Overweight or obese
	Overweight or obese	-0.003	-0.004	0.007	0.091***	-0.004	-0.018	-0.026	-0.042
		(0.014)	(0.018)	(0.031)	(0.030)	(0.017)	(0.020)	(0.016)	(0.034)
	Social activities
	Panel H. Social activities
	Volunteering	-0.046** -0.056** 0.102**	0.090**	-0.014	0.004	-0.086**	0.006
		(0.020)	(0.024)	(0.043)	(0.040)	(0.025)	(0.028)	(0.043)	(0.049)
	Association	-0.012	-0.015	0.027	-0.081**	0.014	0.012	-0.080** 0.135***
		(0.016)	(0.020)	(0.036)	(0.036)	(0.019)	(0.023)	(0.032)	(0.039)
	Controls 1	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
	Observations	923	923	923	919	919	919	919	919
	Notes: Each column and panel corresponds to a specific regression. Columns (1) to (3) contain marginal effect from a
	ordered logit model. Columns (4) to (8) report marginal effects from a multinomial logit model. "Controls 1" are
	included, i.e. gender, age, and marital status. Standard errors in parentheses.
	*** p<0.01, ** p<0.05, * p<0.1.

Table 3 :

 3 Correlates of perception of retirement, for retirees On the x axis, "-" denotes low income or poor health, "+" denotes high income or good health, "No" means that the individual is not involved in an association, and "Yes" means that she is. Each column and panel corresponds to a specific regression. We use linear probability models. "Controls 2" (i.e. gender, age, marital status, education, income, self-assessed health, volunteering, and association) and labour market status are included. Standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1.

		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
		View of the retirement transition		Definition of retirement
		Negative Neither	Positive	Rest	Boredom	Not	Precarious	Freedom
			negative				hearing	ness
			nor				of
			positive				work
	Socioeconomic status
	Panel A. Education
	Low edu	Ref	Ref	Ref	Ref	Ref	Ref	Ref	Ref
	Medium edu	-0.006	-0.025	0.032	-0.198***	0.021	0.002	-0.009	0.184***
		(0.010)	(0.040)	(0.050)	(0.043)	(0.026)	(0.027)	(0.018)	(0.050)
	High edu	0.004	0.013	-0.016 -0.254***	0.042	-0.030	-0.037*** 0.278***
		(0.011)	(0.039)	(0.050)	(0.040)	(0.028)	(0.020)	(0.009)	(0.047)
	Panel B. Income
	Ln(income+1)	-0.002	-0.008	0.011	-0.203***	0.017	-0.044*** -0.024*** 0.255***
		(0.007)	(0.027)	(0.034)	(0.030)	(0.020)	(0.014)	(0.009)	(0.035)
	Panel C. Wealth
	Ln(wealth+1)	-0.001	-0.003	0.003	-0.055***	0.000	-0.008	-0.010** 0.072***
		(0.003)	(0.011)	(0.014)	(0.012)	(0.007)	(0.006)	(0.004)	(0.013)
	Health
	Panel D. General health
	Very poor or poor	0.016	0.051	-0.067	-0.029	-0.001	0.015	0.044	-0.030
		(0.015)	(0.046)	(0.061)	(0.059)	(0.033)	(0.036)	(0.027)	(0.062)
	Fair	Ref	Ref	Ref	Ref	Ref	Ref	Ref	Ref
	Good	-0.007	-0.029	0.036	-0.119***	-0.026	-0.003	0.006	0.142***
		(0.008)	(0.033)	(0.041)	(0.040)	(0.020)	(0.021)	(0.013)	(0.042)
	Very good	-0.024** -0.112** 0.136** -0.165***	-0.033	-0.056***	-0.006	0.260***
		(0.010)	(0.052)	(0.061)	(0.058)	(0.028)	(0.020)	(0.017)	(0.062)
	Panel E. Mental health
	Good mental health (K6 scale) -0.003** -0.011*** 0.014***	-0.008	-0.007*** -0.005** -0.005*** 0.024***
		(0.001)	(0.004)	(0.005)	(0.005)	(0.002)	(0.002)	(0.002)	(0.005)
	Panel F. High blood pressure
	High blood pressure	-0.004	-0.016	0.020	0.121**	-0.015	0.066*	0.001	-0.172***
		(0.011)	(0.041)	(0.051)	(0.052)	(0.023)	(0.034)	(0.017)	(0.052)
	Panel G. Overweight or obese
	Overweight or obese	0.019** 0.075** -0.094** 0.075**	0.007	0.034*	0.003	-0.118***
		(0.008)	(0.030)	(0.037)	(0.036)	(0.018)	(0.018)	(0.013)	(0.039)
	Social activities
	Panel H. Social activities
	Volunteering	-0.020** -0.072** 0.092**	-0.030	0.005	-0.075**	-0.027	0.127***
		(0.010)	(0.034)	(0.044)	(0.042)	(0.021)	(0.032)	(0.022)	(0.045)
	Association	-0.018** -0.065** 0.083**	-0.007	-0.033	-0.001	-0.022	0.063
		(0.009)	(0.030)	(0.038)	(0.037)	(0.021)	(0.020)	(0.017)	(0.040)
	Controls 1	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
	Observations	705	705	705	704	704	704	704	704
	Notes: Each column and panel corresponds to a specific regression. Columns (1) to (3) contain marginal effect from an
	ordered logit model. Columns (4) to (8) report marginal effects from a multinomial logit model. "Controls 1" are
	included, i.e. gender, age, and marital status. Standard errors in parentheses.
	*** p<0.01, ** p<0.05, * p<0.1.

Definition of retirement. Controls 2 included

  Each column and panel corresponds to a different regression. We estimate OLS models. "Controls 1" (i.e. gender, age, and marital status) are included in Panels A and C. "Controls 2" (i.e. gender, age, marital status, education, income, general health, volunteering, and association) are included in Panels B and D. Standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1.

	Rest	Ref	Ref	Ref	Ref	Ref
	Boredom	0.23	-0.27	-0.28	-0.34	0.11
		(0.34)	(0.40)	(0.30)	(0.25)	(0.31)
	Not hearing of work	-0.42	-0.43	-0.02	-0.75***	-0.14
		(0.34)	(0.38)	(0.29)	(0.25)	(0.30)
	Precariousness	-0.85*	-1.69***	0.46	-0.93***	-0.28
		(0.48)	(0.55)	(0.42)	(0.35)	(0.43)
	Freedom	0.18	0.33	0.39**	-0.03	0.32*
		(0.18)	(0.21)	(0.16)	(0.13)	(0.16)
	Controls 2	Yes	Yes	Yes	Yes	Yes
	Observations	643	631	660	649	653
	Notes:

, including adherence to the legal requirement in France. The survey obtained ethics approval from the national ethics committee, i.e. "Commission Nationale de l'Informatique et des Libertés" (CNIL). The empirical work presented in the paper does not require any ethical review.

Statement of funding:

The author was supported by "Chaire Transitions Démographiques, Transitions Economiques (TDTE)" hosted by "Fondation du Risque." The author prepared the questionnaire of the survey, in partnership with "Chaire TDTE," HM, and "FYM Etudes." HM funded data collection, which was implemented by "FYM Etudes." Data were provided by HM. HM, "Chaire TDTE," and "FYM Etudes" had no role in the design, execution, analysis and interpretation of data, or writing of the study.

Rest


Guillaume Desagulier
email: gdesagulier@univ-paris8.fr

introduction terminology commonalities differences exploring not predicting further reading References Multifactorial Exploratory Approaches fundamentals

commonalities differences exploring not predicting further reading References an example from home

Bresnan et al. (2007)

The dative alternation in English is influenced by several factors, such as:

• the meaning of the verb • multiple correspondence analysis (MCA)

• principal component analysis (PCA)

• exploratory factor analysis (EFA)

the fifth one is more recent

  When a linguistic phenomenon is influenced by several factors at the same time, its analysis is multifactorial. Experience tells us that, arguably, just about anything in language is multifactorial.

	outline outline outline outline outline	introduction introduction introduction introduction introduction	terminology terminology terminology terminology terminology	commonalities commonalities commonalities commonalities commonalities	differences differences differences differences differences	exploring not predicting exploring not predicting exploring not predicting exploring not predicting exploring not predicting	further reading further reading further reading further reading further reading	References References References References References
	looking for patterns in the data exploring a data set generating hypotheses the empirical cycle multifactorial, multivariate, and multidimensional
				previous works	ask a theory-informed question	theory
						champions_league
						europa_league
					formulate hypotheses	non_relegable relegable
					operationalize hypotheses
			update hypotheses		gather corpus data	non_relegable inform back theory
				quantify your observations	test hypotheses	relegable use statistics
					interpret the results
				no	are the significant? results	yes
	Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches				Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier

Once corpus linguists have collected sizeable amounts of observations, they look for patterns in the data. When the data set is too large, it becomes impossible to summarize the table with the naked eye and summary statistics are needed. This is where exploratory data analysis steps in.

Exploring a data set means separating meaningful trends from the noise (i.e. "random" distributions) 1

1 Even though "language is never, ever, ever random" (Kilgarriff 2005)

In theory, exploratory data analysis is used to generate hypotheses because the linguist does not yet have any assumption as to what kinds of trends should appear in the data. In practice, however, linguists collect observations in the light of specific variables precisely because they expect that the latter influence the distribution of the former.

Table 1 :

 1 Once operationalized by the linguist, these multiple factors are captured by means of several independent variables. When observations of the linguistic phenomenon are captured by several variables, the analysis is multivariate.We are now entering the world of data tables from a maths/stats viewpoint! The analysis becomes (multi)dimensional when the complex table is decomposed into meaningful dimensions. a word-word cooccurrence matrix different quite thing writing natural place sort become men . . .

	• the length/animacy/definiteness/pronominality/accessibility of the
	recipient/theme
	• the realization of the recipient
	• etc.
	Multifactorial Exploratory Approaches	Guillaume Desagulier

Table 2 :

 2 a word-vector matrix of some adjectives from the BNC (snapshot) the tables are complex objects of which we want to get a synthetic view.• we do it by finding clusters in the data. This is no trivial task because we need to make sure the clusters are valid. • the reward is a graphic representation of the data set in terms of neat clusters.

	adjectives	V1	V2	V3	V4	V5	V6	. . .
	gloomy	-0.36405	-0.44487	-0.33327	-0.16695	-0.52404	0.31066	. . .
	sacred	0.60337	-0.20526	-0.042822 -0.33008	-0.68957	0.26654	. . .
	jaundiced	-0.32168	-0.58319	-0.34614	-0.12474	0.10368	0.1733	. . .
	loud	0.24615	-0.24904	-0.18212	-0.14834	-0.06532	-0.3393	. . .
	memorable 0.30206	0.20307	0.062304	0.66816	0.048326	0.034361	. . .
	justified	-0.080959 -0.23694	-0.43372	-0.31442	-0.31528	0.0057226 . . .
	scant	-0.14467	-0.29329	0.10832	-0.11123	-0.57925	-0.27022	. . .
	continuous -0.15253	-0.082764 -0.40871	-0.53719	0.0822	-0.31482	. . .
	imposing	0.32043	0.155	-0.10547	-0.23157	-0.35657	-0.097553 . . .
	weighty	0.085281	0.015087	0.58454	0.0094917 -0.082617 0.36811	. . .
	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .
	Multifactorial Exploratory Approaches					Guillaume Desagulier

the first four are based on eigenvalue decomposition

• correspondence analysis (CA)

•

  • like a complex object in real life, a data table has to be rotated so as to be observed from an optimal angle • although the dimensions of a data table are eventually projected in a two-dimensional plane, they are not spatial dimensions • if the table has K columns, the data points are initially positioned in a space R of K dimensions to allow for easier interpretation, dimensionality-reduction methods decompose the cloud into a smaller number of meaningful planes. • the methods covered in this course summarize the table by measuring how much variance there is and decomposing the variance into proportions. • these proportions are eigenvalues in CA, MCA, and PCA. They are loadings in EFA (and a special kind of PCA not covered in this course). 2 all four methods offer graphs that facilitate the interpretation of the results • although convenient, these graphs do not replace a careful interpretation of the numerical results. CA takes as input a contingency table, i.e. a table that cross-classifies observations on a number of categorical variables • entries in each cell are integers, namely the number of times that observations (in the rows) are seen in the context of the variables (in the columns).

	outline outline outline outline outline outline outline outline	introduction introduction introduction introduction introduction introduction introduction introduction	terminology terminology terminology terminology terminology terminology terminology terminology	commonalities commonalities commonalities commonalities commonalities commonalities commonalities commonalities	differences differences differences differences differences differences differences differences	exploring not predicting exploring not predicting exploring not predicting exploring not predicting exploring not predicting exploring not predicting exploring not predicting exploring not predicting	further reading further reading further reading further reading further reading further reading further reading further reading	References References References References References References References References
	you do not compile a data set randomly dimensions dimensionality reduction graphic summary differences CA CA
		Table 3 is an example of a contingency table. It displays the frequency
		counts of four types of nouns (rows) across three corpus files from the
		BNC-XML (columns).
		The main difference between these methods pertain mainly to the kind of
		data that one works with
	Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches				Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier

• t-SNE

what does exploratory mean?

(again)

the linguist makes no assumption as to what kinds of groupings are to be found in the data. In practice, however, you compile a table of data because you expect to find meaningful groupings. Therefore, if you find no meaningful grouping, this is because your rows and your columns are independent. Chances are that you might want to rethink the design of your study, especially your choice of explanatory variables.

goal

• we seek to explore a cloud of points from a data set in the form of a rows × columns table with as many dimensions as there are columns.

• 2 See baayen2008analyzing.

•

Table 3 :

 3 An example of a contingency table

		A1J.xml A1K.xml A1L.xml row totals
	NN0	136	14	8	158
	NN1	2236	354	263	2853
	NN2	952	87	139	1178
	NP0	723	117	71	911
	column totals	4047	572	481	5100
	Multifactorial Exploratory Approaches				Guillaume Desagulier

Table 4 :

 4 A sample input table for MCA(Desagulier 2017, p. 36)

	corpus file mode	genre	exact match	intensifier syntax	adjective
	KBF.xml	spoken conv	a quite ferocious mess	quite	preadjectival	ferocious
	AT1.xml	written biography	quite a flirty person	quite	predeterminer	flirty
	A7F.xml	written misc	a rather anonymous name	rather	preadjectival	anonymous
	ECD.xml	written commerce	a rather precarious foothold rather	preadjectival	precarious
	B2E.xml	written biography	quite a restless night	quite	predeterminer	restless
	AM4.xml	written misc	a rather different turn	rather	preadjectival	different
	F85.xml	spoken unclassified a rather younger age	rather	preadjectival	younger
	J3X.xml	spoken unclassified quite a long time	quite	predeterminer	long
	KBK.xml	spoken conv	quite a leading light	quite	predeterminer	leading
	Multifactorial Exploratory Approaches				Guillaume Desagulier

•

  PCA takes as input a table of data of i individuals or observations (rows) and j variables (columns) • the method handles continuous and nominal data • the continuous data may consist of means, reaction times, formant frequencies, etc. • the categorical/nominal data are used to tag the observations Table 6 is a table of 6 kinds of mean frequency counts further described by 3 kinds of nominal information.

	outline	introduction	terminology	commonalities	differences	exploring not predicting	further reading	References
	PCA
	Multifactorial Exploratory Approaches				Guillaume Desagulier

Table 5 :

 5 A sample data frame(Lacheret-Dujour et al. 2019) Like PCA, EFA takes as input a table of continuous data. However, it does not commonly accommodate nominal data. Typically, Table6minus the 3 columns of nominal data can serve as input for EFA.

	corpus sample fPauses fOverlaps fFiller fProm fPI	fPA	subgenre	interactivity	planning type
	D0001	0.26	0.12	0.14	1.79	0.28 1.54 argumentation	interactive	semi-spontaneous
	D0002	0.42	0.11	0.10	1.80	0.33 1.75 argumentation	interactive	semi-spontaneous
	D0003	0.35	0.10	0.03	1.93	0.34 1.76	description	semi-interactive	spontaneous
	D0004	0.28	0.11	0.12	2.29	0.30 1.79	description	interactive	semi-spontaneous
	D0005	0.29	0.07	0.23	1.91	0.22 1.69	description	semi-interactive	spontaneous
	D0006	0.47	0.05	0.26	1.86	0.44 1.94 argumentation	interactive	semi-spontaneous
	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .
	Multifactorial Exploratory Approaches							Guillaume Desagulier

Table 6 :

 6 A sample data frame(Lacheret-Dujour et al. 2019)

	corpus sample fPauses fOverlaps fFiller fProm fPI	fPA
	D0001	0.26	0.12	0.14	1.79	0.28 1.54
	D0002	0.42	0.11	0.10	1.80	0.33 1.75
	D0003	0.35	0.10	0.03	1.93	0.34 1.76
	D0004	0.28	0.11	0.12	2.29	0.30 1.79
	D0005	0.29	0.07	0.23	1.91	0.22 1.69
	D0006	0.47	0.05	0.26	1.86	0.44 1.94
	. . .	. . .	. . .	. . .	. . .	. . .	. . .


Guillaume Desagulier
email: gdesagulier@univ-paris8.fr

introduction principles case study Multifactorial Exploratory Approaches correspondence analysis

Keywords:

introduction principles case study

CA

 Benzécri (1984).

Because we are interested in determining whether two variables are interdependent, we formulate the hypotheses as follows: H 0 : the distributions of row variables and column variables are independent;

H 1 : the distributions of row variables and column variables are interdependent.

outline introduction principles case study the χ 2 test

One calculates the χ 2 value of a cell in the i th row and the j th column as follows:

χ 2 i,j = (O i,j -E i,j ) 2 E i,j (i)
where E i,j is the expected frequency for cell i, j and O i,j is the expected frequency for cell i, j. The χ 2 statistic of the whole table is the sum of the χ 2 values of all cells.

χ 2 = n i=1 (O -E ) 2 E (ii)
Multifactorial Exploratory Approaches Guillaume Desagulier outline introduction principles case study profile

• central to CA is the concept of profile

• to obtain the profile of a row, each cell is divided by its row total

Multifactorial Exploratory Approaches Guillaume Desagulier outline introduction principles case study

row profiles

Table 5 displays the row profiles of Table 3. The row profiles add up to 1.

column profiles

Likewise, one obtains the profile of a column by dividing each column frequency by the column total (Table 4). Again, the column profiles add up to 1.

symmetry

CA performs an analysis of rows and columns that is both simultaneous and symmetric.

complex prepositions

Leitner (1991) reports a study by [START_REF] Hirschmüller | The use of complex prepositions in Indian English in comparison with British and American English[END_REF] who compares the distribution of complex prepositions in three corpora of English: the Brown Corpus, the LOB Corpus, and the Kolhapur Corpus. The Brown Corpus is a corpus of American English (Francis and Kučera 1964). The LOB Corpus is the British counterpart to the Brown Corpus [START_REF] Leech | The LOB Corpus, POS-tagged version[END_REF][START_REF] Leech | The Kolhapur Corpus of Indian English: Intra-varietal description and/or intervarietal comparison[END_REF]. The Kolhapur Corpus is a corpus of Indian English [START_REF] Shastri | The Kolhapur Corpus[END_REF]).

Multifactorial Exploratory Approaches Guillaume Desagulier outline introduction principles case study

complex prepositions

Complex prepositions are multiword expressions (i.e. expressions that consist of several words): ahead of, along with, apart from, such as, thanks to, together with, on account of, on behalf of, or on top of.

Multifactorial Exploratory Approaches Guillaume Desagulier outline introduction principles case study complex prepositions

Hirschmüller observes a higher incidence of complex prepositions in the Kolhapur Corpus than in the other two corpora. He also observes that the most complex prepositions (i.e. prepositions that consist of three words and more) are over-represented in the corpus of Indian English.

Multifactorial Exploratory Approaches Guillaume Desagulier outline introduction principles case study

complex prepositions

We replicate Hirschmüller's study based on a two-fold assumption:

• complex prepositions are likely to be over-represented in the Kolhapur corpus; • within the corpus, complex prepositions are likely to be over-represented in the more formal text categories.

complex prepositions

The data set has been imported as a data frame. The table consists of 257 lines (one line per preposition type) and 19 columns (one column per variable). Each column stands for a context where the preposition is found. There are 3 kinds of columns:

• the first 3 columns correspond to the three corpora.

• the next 15 columns correspond to the text categories.

• The 19th column specifies the word length of the prepositions. This last column (prep.length) is loaded as a factor because it contains nominal data (for this reason, it is said to be qualitative).

Multifactorial Exploratory Approaches Guillaume Desagulier outline introduction principles case study complex prepositions

The first three columns are declared as active. Columns 4 to 18 are quantitative and declared as supplementary (col.sup=4:18). Column 19, which corresponds to the complexity of the preposition, is qualitative and therefore supplementary (quali.sup=19).

> ca.object <-CA(dfca, col.sup=4:18, quali.sup=19, graph=FALSE) "eigenvalues" 2 "$col" "results for the columns" 3 "$col$coord" "coord. for the columns" 4 "$col$cos2" "cos2 for the columns" 5 "$col$contrib" "contributions of the columns" 6 "$row" "results for the rows" 7 "$row$coord" "coord. for the rows" 8 "$row$cos2" "cos2 for the rows" 9 "$row$contrib" "contributions of the rows" 10 "$col.sup$coord" "coord. for supplementary columns" 11 "$col.sup$cos2" "cos2 for supplementary columns" 12 "$quali.sup$coord" "coord. for supplementary categorical var." 13 "$quali.sup$cos2" "cos2 for supplementary categorical var." 14 "$call" "summary called parameters" 15 "$call$marge.col" "weights of the columns" 16 "$call$marge.row" "weights of the rows"

complex prepositions

The standard graphic output of CA is a symmetric biplot in which both row variables and column variables are represented in the same space using their coordinates. In this case, only the distance between row points or the distance between column points can be interpreted accurately (Greenacre 2007, p. 72).

Only general observations can be made about the distance between row points and column points, when these points appear in the same part of the plot with respect to the center of the cloud of points (Husson, p.c.)

complex prepositions

Hirschmüller observed the following:

1 complex prepositions cluster in non-fictional texts, a preference that is amplified in the Kolhapur Corpus;

2 learned and bureaucratic writing shows a more pronounced pattern in the Kolhapur Corpus than in the British and American corpora.

The CA plot reflects these tendencies... Multifactorial Exploratory Approaches Guillaume Desagulier

Multifactorial Exploratory Approaches

  to determine whether rows and columns are independent, CA relies on the χ 2 test • it tests the significance of the overall deviation of the table from the independence model • the test computes the contribution of each cell to χ 2 and sums up all contributions to obtain the χ 2 statistic

	outline outline	introduction introduction	principles principles	case study case study
	the χ 2 test the χ 2 test
	Multifactorial Exploratory Approaches		Guillaume Desagulier

•

Table 1 :

 1 The row profiles of Table3

		A1J.xml A1K.xml A1L.xml row total
	NN0	0.8608	0.0886	0.0506	1
	NN1	0.7837	0.1241	0.0922	1
	NN2	0.8081	0.0739	0.1180	1
	NP0	0.7936	0.1284	0.0779	1
	column average	0.7935	0.1122	0.0943	1
	Multifactorial Exploratory Approaches				Guillaume Desagulier

Table 2 :

 2 The column profiles of Table3

		A1J.xml A1K.xml A1L.xml row average
	NN0	0.0336	0.0245	0.0166	0.0310
	NN1	0.5525	0.6189	0.5468	0.5594
	NN2	0.2352	0.1521	0.2890	0.2310
	NP0	0.1787	0.2045	0.1476	0.1786
	column total	1	1	1	1
	Multifactorial Exploratory Approaches				Guillaume Desagulier

Table 3 :

 3 An example of a contingency table

		A1J.xml A1K.xml A1L.xml row totals
	NN0	136	14	8	158
	NN1	2236	354	263	2853
	NN2	952	87	139	1178
	NP0	723	117	71	911
	column totals	4047	572	481	5100
	Multifactorial Exploratory Approaches				Guillaume Desagulier

Table 4 :

 4 The column profiles of Table3

		A1J.xml A1K.xml A1L.xml row average
	NN0	0.0336	0.0245	0.0166	0.0310
	NN1	0.5525	0.6189	0.5468	0.5594
	NN2	0.2352	0.1521	0.2890	0.2310
	NP0	0.1787	0.2045	0.1476	0.1786
	column total	1	1	1	1
	Multifactorial Exploratory Approaches				Guillaume Desagulier

Table 5 :

 5 The row profiles of Table32 is obtained by dividing the χ 2 statistic by the sample size• CA interprets inertia geometrically to assess how far row/column profiles are from their respective average profiles • the larger φ 2 , the more the data points are spread out on the map

		A1J.xml A1K.xml A1L.xml row total
	NN0	0.8608	0.0886	0.0506	1
	NN1	0.7837	0.1241	0.0922	1
	NN2	0.8081	0.0739	0.1180	1
	NP0	0.7936	0.1284	0.0779	1
	column average	0.7935	0.1122	0.0943	1
	Multifactorial Exploratory Approaches				Guillaume Desagulier

• distances between profiles are measured with inertia. It is with the total inertia of the table (φ 2 ) that CA measures how much variance there is • φ

• each column of the table contributes one dimension. The more columns in your table, the larger the number of dimensions • when there are many dimensions, summarizing the table becomes very difficult. To solve this problem, CA decomposes φ 2 along a few dimensions that concentrate as large a proportion of inertia as possible. These proportions of inertia are known as eigenvalues Two descriptors help interpret the dimensions:

• contribution: if a data point displays a minor contribution to a given dimension, its position with respect to this dimension must not be given too much relevance • quality of projection: used to select the dimension in which the individual or the variable is the most faithfully represented

These supplementary rows and/or columns help interpret the active rows and columns. As opposed to active elements, supplementary elements do not contribute to the construction of the dimensions.

  The eig object allows to see how many dimensions there are to inspect. Because the input table is simple and because the number of active variables is low, there are only two dimensions to inspect.

	outline outline	introduction introduction	principles principles	case study case study
	complex prepositions
	> ca.object$eig
	eigenvalue percentage of variance cumulative percentage of variance
	dim 1 0.020398336	82.34156	82.34156
	dim 2 0.004374495	17.65844	100.00000
	Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches	Guillaume Desagulier Guillaume Desagulier

Printed book


Guillaume Desagulier
email: gdesagulier@univ-paris8.fr

Multifactorial Exploratory Approaches multiple correspondence analysis

Keywords: Multifactorial Exploratory Approaches Guillaume fucked, fucker, fucking, gosh, and shit. Two Multifactorial Exploratory Approaches Guillaume

) provides an analysis of sex differences in the 10M-word spoken section of the British National Corpus (BNC). Schmid shows that women use certain swear-words more than men, although swear-words which tend to have a perceived 'strong' effect are more frequent in male speech. Schmid's study is based on two subcorpora, which are both sampled from the spoken section of the BNC. The subcorpora amount to 8,173,608 words.

nominal data

As pointed out yesterday, MCA takes as input a table of nominal data. • each variable does not break down into too many categories Otherwise, the contribution of each dimension to φ 2 is small, and a large number of dimensions must be inspected(which kind of defeats the purpose)

beware of inertia

There are no hard and fast rules for knowing when there are too many dimensions to inspect. However, when the eigenvalue that corresponds to a dimension is low, we know that the dimension is of little interest (the chances are that the data points will be close to the intersection of the axes in the summary plot).

how men and women swear in the BNC-XML

The contributions are not equally shared among men and women since for every 100 word spoken by women, 151 are spoken by men. To calculate the distinctive lexical preferences of men and women, while taking the lack of balance in the contributions into account, Schmid's measures rely on the difference coefficient.

• age (6 levels: Ag0, Ag1, Ag2, Ag3, Ag4, Ag5)

• social class (4 levels: AB, C1, C2, DE) how men and women swear in the BNC-XML

Age breaks down into 6 groups:

• Ag0: respondent age between 0 and 14;

• Ag1: respondent age between 15 and 24;

• Ag2: respondent age between 25 and 34;

• Ag3: respondent age between 35 and 44;

• Ag4: respondent age between 45 and 59;

• Ag5: respondent age is 60+.

Social classes are divided into 4 groups:

• AB: higher management: administrative or professional.

• C1: lower management: supervisory or clerical;

• C2: skilled manual;

• DE: semi-skilled or unskilled. > str(df) 'data.frame': 293289 obs. of 4 variables: $ word : Factor w/ 8 levels "bloody","damn",..: 2 2 7 7 7 2 7 2 7 7 ... $ gender : Factor w/ 2 levels "f","m": 2 2 2 2 2 2 2 2 2 2 ... $ age : Factor w/ 6 levels "Ag0","Ag1","Ag2",..: 6 6 6 6 6 6 6 6 6 6 ... $ soc_class: Factor w/ 4 levels "AB","C1","C2",..: 1 1 1 1 1 1 1 1 1 1 We convert df$word back to a factor. The number of levels has been reduced to five.

> df$word <-as.factor(df$word)

Multifactorial Exploratory Approaches Guillaume Desagulier outline introduction principles case study References

How men and women swear in the BNC-XML

As in CA, we can declare some variables as active and some other variables as supplementary/illustrative in MCA. We declare the variables corresponding to swear words and gender as active, and the variables age and social class as supplementary/illustrative. Running a MCA involves the following steps:

• determining how many dimensions there are to inspect;

• interpreting the MCA graph.

Multifactorial Exploratory Approaches Guillaume Desagulier outline introduction principles case study References

How men and women swear in the BNC-XML

We run the MCA with the MCA() function. We declare age and soc_class as supplementary (quali.sup=c(3,4)). We do not plot the graph yet (graph=FALSE).

> mca.object <-MCA(df, quali.sup=c(3,4), graph=FALSE)

Again, the eig object allows us to see how many dimensions there are to inspect.

> How men and women swear in the BNC-XML

The MCA map is plotted with the plot.MCA() function. Each category is the color of its variable (habillage="quali"). The title is removed (title="").

  most explicit swear words (f-words) cluster in the rightmost part of the plot. These are used mostly by men. Female speakers tend to prefer a softer swear word: bloody. upper part (gosh and shit) are used primarily by upper-class speakers. F-words, bloody, and damn are used by lower social categories. Age groups are positioned close to the intersection of the axes. This is a sign that the first two dimensions bring little or no information about them. upper-right corner)gosh and shit, used by male and female upper class speakers;cluster 2 (lower-left corner) bloody, used by female middle-class speakers; cluster 3 (lower-right corner) f-words and damn, used by male lower-class speakers.

•

  the male/female distinction disappears • a divide is observed between f-words and bloody (left), used mostly by younger and middle-aged speakers, and gosh and damn (right), used mostly by upper-class speakers from age groups 3 and 5. • the most striking feature is the outstanding position of shit in the upper-left corner.

Table 1 :

 1 A sample input table for MCA(Desagulier 2017, p. 36)

	corpus file mode	genre	exact match	intensifier syntax	adjective
	KBF.xml	spoken conv	a quite ferocious mess	quite	preadjectival	ferocious
	AT1.xml	written biography	quite a flirty person	quite	predeterminer	flirty
	A7F.xml	written misc	a rather anonymous name	rather	preadjectival	anonymous
	ECD.xml	written commerce	a rather precarious foothold rather	preadjectival	precarious
	B2E.xml	written biography	quite a restless night	quite	predeterminer	restless
	AM4.xml	written misc	a rather different turn	rather	preadjectival	different
	F85.xml	spoken unclassified a rather younger age	rather	preadjectival	younger
	J3X.xml	spoken unclassified quite a long time	quite	predeterminer	long
	KBK.xml	spoken conv	quite a leading light	quite	predeterminer	leading
	Multifactorial Exploratory Approaches				Guillaume Desagulier

For MCA to yield manageable results, it is best if

• the table is of reasonable size (not too many columns)

  With gsub(), we replace each word with the single tag f-words.

	outline	introduction		principles		case study	References
	how men and women swear in the BNC-XML
	We can group fuck, fucking, fucked, and fucker into a single factor:
	f-words. bloody	damn f-words	gosh	shit
	146203	32294	33184	60678	20930
								...
	> table(df$word)
	bloody	damn	fuck fucked fucker fucking	gosh	shit
	146203	32294	9219	11	467	23487	60678	20930
	Multifactorial Exploratory Approaches					Guillaume Desagulier

> df$word <-gsub("fuck|fucking|fucker|fucked", "f-words", df$word, ignore.case=TRUE) > table

(df$word)

  Ideally, we would want to see a sharp decrease after the first few dimensions, and we would want these first few dimensions to account for as much share of φ 2 as possible. Here, no sharp decrease is observed.

	outline outline outline	introduction introduction introduction	principles principles principles	case study case study case study	References References References
	How men and women swear in the BNC-XML How men and women swear in the BNC-XML
		0.5
		0.4
		0.3
		0.2
		0.1
	round(mca.object$eig, 2)
	dim 1	0.56 eigenvalue percentage of variance cumulative percentage of variance 22.47 22.47 dim 1 dim 2 dim 3 dim 4 dim 5 0.0
	dim 2	0.50	20.00	42.47
	dim 3	0.50	20.00	62.47
	dim 4	0.50	20.00	82.47
	dim 5	0.44	17.53	100.00
	Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches		Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier

The number of dimensions is rather large and the first two dimensions account for only 42.47% of φ 2 . To inspect a significant share of φ 2 , e.g. 80%, we would have to inspect at least 4 dimensions. This issue is common in MCA. The eigenvalues can be vizualized by means of a scree plot. It is obtained as follows.

> barplot(mca.object$eig[,1], + names.arg=paste("dim ", 1:nrow(mca.object$eig)), las=2)

Practical Handbook of Corpus Linguistics

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