Research in Social Stratification and Mobility 87 (2023) 100827 Contents lists available at ScienceDirect Research in Social Stratification and Mobility journal homepage: www.elsevier.com/locate/rssm Using the decomposition of the mutual information index as an alternative approach to the study of educational inequality trends in Spain☆ Ildefonso Marqués-Perales a, Sandra Fachelli b, *, Pedro López-Roldán c a Department of Sociology, Facultad de Ciencias de la Educación, Universidad de Sevilla, Calle Pirotecnia s/n, 41013 Sevilla, Spain b Departament of Sociology, Facultad de Ciencias Sociales, Universidad Pablo de Olavide, Ctra de Utrera Km 1, 41013 Sevilla, Spain c Departament of Sociology, Facultat de Ciències Polítiques i de Sociologia, Universidad Autónoma de Barcelona, 08193 Bellaterra, Spain A R T I C L E I N F O A B S T R A C T Keywords: In this study, we investigate the relationship between social origin and educational attainment in Spain from Entropy 1946 to 1989, testing the theory of persistent inequalities and using two different methodological approaches: Marginal and internal changes the margin-free versus the margin-dependent traditions. More specifically, we will compare the results of the Mutual information index traditional log-linear measures used most frequently in this field with those of the mutual information index, a Educational inequalities Occupational upgrading measure that considers the marginal change in both educational expansion and occupational upgrading. Moreover, we apply the decomposition of this index, as proposed by Jann & Seiler, which allows us to simul­ taneously observe and distinguish between the internal component (margin-free: not accounting for changes in marginal distributions such as odds ratios) and the marginal component (non-margin free: including and measuring marginal effect). Previous research using margin-free models reveals educational inequalities in Spain to be constant, whereas our results show a marked decline in educational inequalities when the weights obtained by the marginal distribution are taken into account. We argue that this decomposition may enrich our view of educational inequalities without disregarding traditional margin-free measures. 1. Introduction Author). The reason for this is that, as the totality of social classes in­ crease their educational level, the differentials between them do not There is no doubt that, throughout the 20th century, the expansion of decrease substantially. In other words, educational inequalities between education has been one of the greatest historical improvements experi­ social classes are maintained because the more advantaged social class enced by advanced societies. Although the majority of the world’s increases its educational attainment rates as the less advantaged social population was illiterate at the beginning of the century, this figure has classes improve, thereby retaining existing gaps between classes (Ayalon now been reduced to approximately one in ten (Roser & Ortiz-Ospina, & Shavit, 2004). This process, in which the social class distances are­ 2016). The cost of covering the basic necessities of existence has replicated over time, was called the ‘translation of structure’ by Bour­ fallen, and inclusive education policies have been implemented, both of dieu and Passeron (1970). which have reduced the economic burden on families; thus, we would Some scholars consider that the invariance in educational inequality expect the expansion of education to also translate into a reduction of rates is in part an artefact of a property called ‘margin insensitivity’ of inequalities between classes. However, it has been widely reported that measures such as the unidiff coefficient in multiplicative log-linear in many societies, there has not been a significant reduction in educa­ models (Combessie, 1984; Marks, 2004; Hellevik, 2007). This tradi­ tional inequalities (Shavit & Blossfeld, 1993; Whelan et al., 2001; tional perspective of margin-free measures used in studies of social ☆ This article was prepared with the support of the Ministerio de Ciencia e Innovación and Agencia Estatal de Investigación (AEI/10.13039/501100011033/) within an R&D Coordinated Project with references PID2019-106548GB-C21 and PID2019-106548GA-C22, “Dynamics of social mobility in Spain” (DINAMOS, htt ps://pagines.uab.cat/dinamos). Earlier versions of the results of this work were presented at the Research Committee 28 on Social Stratification and Mobility in the context of the Fourth ISA Forum of Sociology, Porto Alegre, in February 2021 and RC28 Spring Meeting, LSE, London, on 21-23 April 2022. We acknowledge the help and teachings received from Jann Ben and Simon Seiler for the drafting of this paper. "Funding for open access publishing: Universidad Pablo de Olavide” * Correspondence to: Departament of Sociology, Facultad de Ciencias Sociales, Universidad Pablo de Olavide, Ctra de Utrera Km 1, Edificio 11, 4º 14, 41013 Sevilla, Spain. E-mail addresses: imarques@us.es (I. Marqués-Perales), sfachelli@upo.es (S. Fachelli), pedro.lopez.roldan@uab.es (P. López-Roldán). https://doi.org/10.1016/j.rssm.2023.100827 Received 24 August 2021; Received in revised form 7 July 2023; Accepted 9 July 2023 Available online 12 July 2023 0276-5624/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by- nc-nd/4.0/). I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 mobility can be complemented by a method that takes into account the present the conclusions. margins of the contingency table and, therefore, the importance of the categories of the variables analysed and their variation over time (or 2. Theoretical and empirical review across societies). In this study, we will focus on this alternative interpretation. In this section, we discuss the main theoretical approaches in which According to the margin-insensitivity property, by calculating odds the analyses of educational inequalities are framed. Additionally, we ratios with the cells of the joint distribution of a contingency table, we present some of the main empirical evidence supporting each theory, can free ourselves from the influence of marginal distributions and thus especially for the Spanish case. construct a net measure. By seeking a genuinely abstract measure that During the second half of the 20th century, two opposing theories of reflects equal opportunities, free-margin models assign equal impor­ inequality of educational opportunities were proposed: social repro­ tance to all barriers between classes, without regard to the different duction theory and modernisation theory. Social reproduction theory is compositions of the groups involved. In studying the relationship be­ based on the differential forms of socialisation of the social classes. tween social origin and education, we assume that odds ratios are According to this approach, families socialise their children into calculated independently of the distribution of marginals, so that any different subcultures by promoting different values towards school changes over time are weighted equally, i.e. barriers to access to edu­ (Bourdieu & Passeron, 1970). Families who come from the middle cation do not take into account the changes in the prevalence of social classes promote socially legitimised knowledge that is highly valued in classes or educational levels. In some cases, these changes involve a the school environment. Their education follows what has been called ‘a reduction in certain occupations or educational levels such as agricul­ concerted cultivation’ (Lareau, 2000) consisting of a virtuous learning ture or illiteracy or an increase in the service class in a significant way. circle operating between families and schools. It should be highlighted that margin-free measures were an un­ The first international study to report persistent rates of educational precedented step forward in the studies of social stratification in general inequality was initiated by Shavit and Blossfeld (1993). Of the 13 and educational inequalities in particular. With the development of the countries analysed in their study, only two, the Netherlands and Swe­ log-linear and log-multiplicative models (Goodman, 1978; Hauser, den, showed a decrease in class differentials in terms of education. 1978; Goldthorpe, 1980), a new methodology was instituted, making it A detailed explanation of the evolution of educational inequalities possible to address important theoretical questions. To find an ideal was made by Raftery and Hout (1993) in a study performed some years measure of equality of opportunity, the log-multiplicative layer effect or earlier. On the one hand, the theory of maximally maintained inequality unidiff model (Xie, 1992; Erikson & Goldthorpe, 1993) created an ag­ postulates that class differentials remain unchanged over time because, gregation technique that takes an unweighted average across all bar­ as the working classes saturate the basic educational levels, the middle riers. Because of its parsimony and predictive ability, this model was set classes occupy the next-highest level. On the other hand, the theory up as a reference model. Being more precise, "the overall size of the log effectively maintained inequality (Lucas, 2001) highlights that these odds ratios shifts as we move from table to table means that each log changes are not only quantitative but also qualitative in nature. Differ­ odds ratio in the same table is scaled up or down by the same factor" ences in education do not follow a vertical, but rather, a horizontal, (Breen, 2020: 23).1 pattern of inequality that is manifested in the type and not simply in the This article aims to test the theory of persistent inequalities using a amount of education received. new methodological strategy that highlights the importance of marginal The research by Shavit and Blossfeld (1993) was refuted by Breen changes in social background and educational level, without dispensing et al. (2009), who found evidence of decreasing educational inequalities with the usual procedures used in studies of educational inequality. We in eight European countries due to improvements in nutrition and propose to study the role played by social origin in educational expan­ health, as well as educational reforms that reduced the effort required by sion using the mutual information index (Theil, 1967, 1972; Seiler & parents to send their children to school. Jann, 2019a, 2019b), which is traditionally used in studies regarding The modernisation theory or the liberal theory of industrialism school segregation. This index allows us to observe changes over time in predicts that industrialisation affects the structure and process of strat­ the relationship between social origin (O) and educational attainment ification (Treiman, 1970). In this sense, the theory holds in which all (E). However, to do this, inequalities of educational opportunity (IEO) industrialised societies substitute ascription by achievement, as tech­ are divided into two components: (1) the effects arising from marginal nological development and modernisation demand the prevalence of changes, which are expressed in the marginal distributions and (2) ef­ merit-based selection and a reduction in the association between origin fects arising from internal changes, which are expressed in the joint and education. However, this theory has been challenged over the last distribution. Since a society can be better defined by some barriers 20 years, to the point where some authors have stated that ‘modern­ rather than others, in some cases, treating each barrier according to its isation theory it is wrong’ (Hout & DiPrete, 2006:8). relevance may be informative. Alongside the two main theories, one of which postulates the con­ We will use this decomposition approach with data from Spain, a stancy of educational inequalities and the other that sustains the exis­ case of special interest because of two phenomena. First, Spain is one of tence of their reduction, we can also identify a third position that the countries where educational expansion has been the most intense considers that the decline in inequalities is limited to the golden years of (Barro & Lee, 2013), and second, many of the studies that have been capitalism (Barone, 2019; Barone, 2020). These were years of intense carried out found evidence of persistent educational inequalities. industrialisation, full employment, intensification in the urbanisation This article is organised as follows. In Section 2, we present the main process and educational expansion at the low educational levels. Barone theories on educational inequalities and comment on the empirical & Ruggera (2021: 21) report that ‘this finding suggests the relevance of studies. In Section 3, we present some of the advantages introduced in structural changes, possibly operating through a cost-equalising mech­ the calculations, encompassing both internal and marginal changes. In anism’, which led to a reduction of the educational inequalities and Section 4, we present and justify the hypotheses of our work. In Section stopped since the 1970s. Similarly, Shavit (2014) pointed out that the 5, we present the methodology of analysis of the mutual information process of educational equalisation was limited to the low educational index. In Section 6, we present the data sources and variables used. In levels. Consequently, the association marking the decline in educational Section 7, we present the results obtained. Finally, in Section 8, we inequalities is weakened for the youngest cohorts. During the last few decades, the Spanish scenario has been charac­ terised by strong economic and occupational modernisation and a major 1 Later, based on the same methodological framework, Cox et al. (2009) expansion of education. Thus, the case of Spain is a stimulating oppor­ developed an average global log–odds ratio that does not assume this property. tunity to explore these dynamics. The most complete bibliographical 2 I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 review of IEO carried out to date for Spain (Fernández-Mellizo, 2014, groups). This means that educational inequalities will generally change 2022) shows a lack of agreement among those researchers dedicated to when the distribution of respondents according to social origin or the study of educational inequalities. Methodological heterogeneity has educational attainment increases. Given educational expansion and the led to contradictory conclusions, although most studies point towards father’s occupational upgrading, taking into account marginal distri­ stability. Among the studies showing persistent inequalities is the work bution may be highly relevant. by Calero and Bonal (1999), which analyses the 1981 and 1991 popu­ Is a measure that considers only internal changes enough in certain lation censuses using descriptive statistics and that of Martínez-García circumstances? There is no problem when marginals do not contribute (2002), which uses the 1991 Sociodemographic survey and applies substantially to the results; however, in the case under study, can one linear regression and conditional logit. Among the studies showing de­ abstract from educational expansion and the father’s occupational creases are those of Carabaña (1993) using the 1970 and 1981 censuses upgrading? And does the size weight of barriers between groups not and applying descriptive statistics; Ballarino et al. (2009) using the 1991 generate sociologically important effects? sociodemographic survey and applying cumulative logit and Di Paolo If the aim is to analyse barriers by social class, i.e. to analyse the (2012) using the 2005 Statistics on Income and Living Conditions survey mobility opportunities of some classes versus others, the margin-free and applying ordinal probit analysis. Finally, studies that show increases measurement models are certainly fully satisfactory. However, they and decreases are those of Torres-Mora and Peruga (1997) using the are not satisfactory if we want to have a global idea of the degree of 1991 Sociodemographic survey and applying logit models and Carabaña social openness, combining marginal and internal changes in a society in (1999) using the 1991 Sociodemographic survey and applying linear space (in regions or countries), or over time (across cohorts, ages or regression. periods). Among other things, it is not very realistic to reach a conclu­ The results on IEO have also been investigated within the field of sion about the degree of openness or closure of a society without social mobility studies or, more specifically, the analysis of the origin considering the weight of the barriers experienced by each of the groups (O)–education (E)–destination (D) (OED) social mobility triangle. As that form it. According to Seiler and Jann : 7) (2019a), we have gone too Breen and Müller :7) (2020) point out, ‘(social fluidity between origins far with this type of analysis, as is illustrated by the case of the city-state and destinations depends on an indirect relationship between O and D of Singapore. via E… and a direct relationship between O and D…)’. From this ‘…when studying industrialization or modernization processes, the perspective, most analyses of Spain in this context report constancy in diminishing weight of the farming classes is of special importance, as the relationship origin-education (OE) (Author; Gil, Bernardi & Luijkx, it is a defining (or at least a characteristic) feature of these processes 2020; and Author). Additionally, all of them use the uniform difference (Treiman, 1970; Kuznets, 1955). Marginalization of agriculture model (unidiff), which assumes a common pattern of parent–child could mean that the size of the farming class approaches zero, for log-odds ratios that vary according to a specific scaling factor for each example because of the complete urbanization of an area; Singapore cohort studied. (Fields 1994) could serve as an almost perfect real world example. This extreme case is helpful for illustrating why ignoring changes in 3. Margin-free and margin-dependent measurement class distribution can produce misleading results when analyzing the changing effects of social origin’. In the analysis of the relationship between variables in a contingency table, we can apply two types of statistical techniques: those that ac­ This is a clear example in which the analysis technique based on log- count for the link by neutralising the effect of the margins of the table linear models and the calculation of odds ratios such as unidiff can oc­ (margin-free techniques) or those that take margins into account casionally lead us to incomplete results. Let us take the example of (margin-dependent techniques). We report on some of the benefits of Spanish society, which has undergone rapid major class transformations incorporating variations in the marginal distribution in the calculations as an exemplar of late industrialisation. The unidiff coefficients may lead along with internal adjustments, in particular, for educational in­ us to partial conclusions if we do not take into account that agricultural equalities. We describe how to enrich our view of this phenomenon, workers, landowners and day labourers have declined sharply in num­ usually focused only on internal structures, by incorporating informa­ ber and other classes (professionals and service workers) have increased tion from the margins. We explain how on some occasions, this may offer in number. In other words, since log-linear and log-multiplicative us valuable information about educational inequality trends. models make their calculations using a sort of aggregation technique In the search for a relative measurement free from the enforced for odds ratios, exchanges with different prevalence levels are consid­ movements in a table, inherent for instance in the calculations of ab­ ered equally important. It is not always the best strategy to give the same solute mobility, two types of measurements have been created. On the weight to cells that contain hardly any units as to cells that are heavily one hand, the margin-free measurement specifies the net relative change populated. The marginal distribution must be considered because it af­ beyond any marginal transformation. To check the probabilities of ac­ fects the importance of the dependencies between origin and destination cess to a given social class from different positions, we have no option owing to the proportion of society affected by these associations (Seiler but to abstract from changes in composition (Vallet, 2007). Thus, the & Jann, 2019a). odds ratio statistic is used as a building block. It is understood to be a net Taking a different approach, Hellevik (2007) considers that the measure devoid of any morphological influence. Since it is free from the persistence in educational inequality rates derives from the fact that influence of marginal distribution, it is a net measure of the impact of they are analysed with measures of association rather than with mea­ social origin on the final educational destination, and it can be consid­ sures of inequality. If income inequality measures were used to analyse ered a ‘pure’ or abstract measure of educational inequality. Thus, two educational inequality, a substantial decrease in inequality by social tables that are radically different in their marginal distribution can be origin would be seen, at least as far as access to university is concerned. identical in their odds ratios (Vallet, 2007). Moreover, if we multiply all Moreover, the same author stresses that the entrenched belief in net cells in a given row or column by a constant, the odds ratios will not be changes unaffected by marginal changes has been a major obstacle to altered. That is, the analyses of the tables will not be affected by different reaching an agreement on the relationship between social class and marginal distributions’ sample sizes of the different layers (cohort or education. country) ‘since the odds ratios remain the same’ (Wong, 2010:6). The paradigm based on log-linear models has almost certainly been On the other hand, the margin-dependent measurement takes into the most significant advance in the study of social mobility and educa­ account the changes in margins. For example, the index of dissimilarity tional inequalities. However, its use in isolation and societies in trans­ is margin-dependent in terms of the units under study (e.g. neighbour­ formation generates, in some specific cases, partial visions of reality that hoods or schools) but not in terms of the groups (e.g. racial/income should be corrected to reflect the social changes that represent such an 3 I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 outstanding improvement in having increasingly educated societies or the changes in the strength of association between the social origin (O) higher occupational levels. and education (E) across cohorts (C) and which are evaluated by the Bayesian information criterion (BIC) statistic. The two models are 4. Hypotheses expressed as follows: logFijkl = µ + λOi + λEj + λCk + λSl + λOCS ECS OE ikl + λjkl + λij To further elucidate the educational inequalities in Spain, we pro­ pose two complementary hypotheses that lead us to expect two different results in the relationship between social origin and educational level logFijkl = µ + λOi + λEj + λCk + λSl + λOCS ECS ikl + λjkl + βk ϕij reached by offspring. The first equation assumes a log-linear model formed by the main H1. : Following the margin-free tradition, we expect to find educa­ effects and second-order interactions, which shows the constancy of the tional inequalities unchanged in Spain. relationship between social origin and education over time. The second replaces the term λOE ij by a constant second-order interaction βkΦij, which When we consider the relationship between social origin and represents the multiplication of the OE interaction by a scaling factor educational attainment without taking into account the influence of the that expresses, given the same general pattern of behaviour, the level of margins, we can expect educational inequalities to remain constant. This attenuation or accentuation of inequality of social origin over time. is a consistent finding in previous studies in Spain using this type of We use the mutual information index to take into account the in­ methodology. Researchers following the margin-free tradition have ternal and marginal effects of educational inequality simultaneously. found that educational inequality rates remained unchanged and The logic of the index goes back to the approaches of information theory concluded that the improvement of economic conditions and educa­ (Shannon, 1948), introduced in the social sciences by Theil (1967, tional opportunities did not have a strong impact on educational in­ 1972), in which the concept of entropy indicates the degree of uncer­ equalities. Indeed, educational improvements of the working classes tainty of a particular piece of information and responds to the criterion (both skilled and unskilled) and the agricultural classes (both land­ that the more information one has about an event, the less information owners and day labourers) were not important enough to compensate one gains when observing it. Entropy can be used to measure the amount for the gap in educational inequality in comparison with the upper of information shared; hence, the higher the entropy is, the lower the classes. amount of information is. Thus, we contrast the difference between the H2. : We expect to find a decrease in educational inequalities in Spain information gained a posteriori by introducing a certain measure, X, when the barriers have been weighted for their relevance. regarding an a priori unconditional initial situation, Y, which lacks that information. What we gain or learn about the initial situation is a If we do not supplant the margin-free tradition with the marginal measure of the importance of X in knowing Y that allows us to analyse tradition, certain changes that should be considered to be tending to­ the relationship between the two variables. wards fluidity under a conception of an open society tend rather towards Following the example of Seiler and Jann (2019a; 2019b), this can be constancy or rigidity, especially when the tables we are analysing are achieved by comparing two distributions: the first corresponds to how very different in their composition. However, on some occasions, mea­ much information we have about an event Y (for example, the educa­ sures that do not account for proportional differences give us incomplete tional level achieved by a group of people), i.e. its unconditional dis­ information. If we rely on margin-free measures, the movement of small tribution, obtained a priori; the second, obtained a posteriori, farmers and agricultural workers to educational levels beyond primary corresponds to how much information is obtained if the conditional education will not be calibrated according to the demographic impor­ distribution of Y over another variable X is known. In our case, this tance that urbanisation and educational expansion had (thus, two co­ would mean comparing the unconditional distribution of education (E) horts with the same odds ratios will contribute equally to educational with its distribution conditional on social origin (O) (Fig. 1). If the dif­ inequality, regardless of the numerical magnitude of the two phenom­ ference is substantial, it may be observed that the social class of origin ena). When there are major changes in the social structure of origin and exerts a subsequent strong influence on educational attainment. the educational pyramid, for example, when farmers and agricultural The entropy of the distribution of variable E (the education achieved) workers are fewer in number and people with only non-compulsory ∑J ( ) is defined a priori as follows: E(PE ) = − j=1 p Ej .ln(p(Ej )). educational levels decrease in number, it may be beneficial to give Here, p(Ej ) is the unconditional probability of reaching the education different weights and propose a complementary hypothesis, as above. To specify this hypothesis, we need to think that younger cohorts will have greater fluidity if we take into account the prevalence of barriers Education and not just how pronounced they are. In other terms, their marginal Total OE ISCED ISCED ISCED ISCED ISCED structure will be more beneficial reducing social origin impact. 5-6 3-4 2 1 0 I 5. Methodology of analysis II Marginal distribution Oi The methodological strategy we follow to account for our hypotheses IIIa Origin Class establishes a two-pronged approach. Our objective is to determine whether an analysis based on the mutual information index (hereinafter IVab Internal distribution referred to as the M-index) leads to conclusions different from an anal­ IVc ysis based on the unidiff model, i.e. whether taking into account mar­ V+VI p (E | Oi ) ginal distributions matters. Furthermore, using counterfactual decomposition, we will discover the contribution of both internal and IIIb marginal changes. We will then present the structure of education by VIIa cohort and by social class and gender and changes in the parents’ VIIb occupational structure. For relative changes within the margin-free tradition, we apply log- Total Marginal distribution E j n linear and log-multiplicative models to perform a time-varying analysis of the relationship between social origin and education. Thus, we obtain Fig. 1. : Marginal and internal distributions of the contingency table between the constant fluidity first and then the unidiff models, which estimate the origin class (O) and education (E). 4 I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 level j based on a priori information without considering the social class 1) The effect arising from internal changes, which are expressed in the of origin. joint or conditional distribution and the OE association intensity and A posteriori, the entropy of the variable E given the information of the patterns taken changes over time. That is, how pronounced the the variable O (the social class of origin) is defined as follows: class barriers are, as the odds ratio also measured. ) 2) The effect of marginal changes, which are expressed in marginal ∑ I ∑J ( ) E PE|O = − p(Oi )⋅ ( ( ( p Ej |Oi )⋅ ln p Ej |Oi ) distribution: education (E) and class origin (O) over time in our case. i=1 j=1 That is, how prevalent the barriers are: the effect of educational expansion and changes in class structure. Here, p(Ej |Oi ) is the probability of reaching the education level j conditioned to a posteriori knowledge of the social class of origin i. The M-index between the two cohorts (c1 and c2, or countries, etc.) The mutual information index, M, is obtained as the difference be­ is the result of these two contributions: tween the two above expressions, as follows: ( ( )) Mc1− Mc2 = ΔM (Δmarginals, Δinternal) = Marginal Contribution + Internal ( ) ∑ I ∑ J ( ) p Ej |Oi Contribution M = E(PE ) − E PE|o = p Oi , Ej ln ( ) i=1 j=1 p Ej ( ) When we compare the M-index by cohort, the part of the difference Here, p Oi , Ej refers to the joint probability of class origin and that is due to differences in the marginals can be obtained by a coun­ educational destination. terfactual decomposition that maintains the internal association (a The M-index is a measure of association that accounts for both in­ contribution that is margin-free, like unidiff), but the marginal distri­ ternal association and marginal distribution. Obtaining internal barriers butions change. From a technical point of view, the separation of the two in the OE relationship through ‘margin-free’ measures (odds ratios) is effects is performed by a counterfactual decomposition, i.e. by calcu­ not the best way, because OE combinations only concern pairs of cate­ lating the M-index based on simulated data in which the associations are gories/cells: it is fine for one-to-one comparisons but not for compari­ maintained but the marginal distributions are swapped. The method sons between countries or cohorts as a whole. Each OE combination, used in this article is based on the composition of Deming and Stephen when relative changes are measured excluding marginal changes, is not (1940), which is the one used in Stata (Seiler & Jann, 2019a; Jann & appropriate for showing existing dependencies between variables, and Seiler, 2019). This raking procedure consists of an iterative proportional to generalise relative changes, it is necessary to consider different fitting. weights of marginal distribution ( how many people the relative change affects: a relative social change, or a barrier in a specific cell, which 6. Data sources and variables affects a small proportion of individuals is not the same as a relative change that affects a large part of the population), because changes in To test the proposed hypotheses, we pooled cross-sectional data from marginal distributions impact OE associations (Seiler & Jann, 2019a: 8). three nationally representative surveys for Spain: the living conditions The M-index can be expressed as a simple average of the components survey (EU-SILC/ECV) conducted in 2005 (n = 30,375), 2011 specific to each observed case c, considering the following two generic (n = 29,210) and 2019 (n = 30,375) (INE, 2008, 2013 and 2020). These variables x and y: surveys are conducted by the Spanish National Statistics Institute as a ( ) joint project designed at the European level for studying income and 1∑ n p(yc |xc ) living conditions and poverty and they add in those years a specific E(Mc ) = M = ln n c=1 p(yc ) module with the possibility of measuring the intergenerational trans­ This can be generalised to a multidimensional set of variables X (e.g. mission of poverty, which includes the parents’ data.4 for social origin) and another set of additional control variables V as We chose the sample of the special survey module ‘Intergenerational follows: transmission of poverty’. To have a population with complete higher ( ) ( ) education, we considered people aged 30 years or more. The 2011 M ∗ = E PY|V – E PY|V,X survey changed the reference population that was asked about their parents’ occupation by lowering the age limit from 65 to 59 years. To Thus, the M-index can be compared across categories of third vari­ homogenise the three samples, we took this age as the top limit. With ables such as countries or, in our case, cohorts. these filters, the final sample size was 43,669 cases: 14,166 in 2005, The a priori model would be p(yc|vc) and the a posteriori one p(yc|vc, 13,701 in 2011 and 15,802 in 2019. xc). The estimation process involves 1) estimating a multinomial logit of The variables of education and social class were considered y over v and obtaining the predicted probabilities p(yc|vc), 2) estimating following a standard of social stratification studies. Education was coded a multinomial logit of y over v and x and obtaining the predicted according to the International Standard Classification of Education probabilities p(yc|vc,xc) and 3) calculating for each observed case c the (ISCED) 1997 into five categories based on the more detailed original estimates.2. Spanish classifications. Educational level was classified in the following ( ) p (yc |vc , xc ) ̂ categories: ISCED 0, no education or up to pre-school; ISCED 1, primary; m ̂ c = ln p (yc |vc ) ̂ ISCED 2, lower secondary; ISCED 3–4, upper secondary and post- secondary non-tertiary; and ISCED 5 A and 6, tertiary. An important aspect of this measure is that it can be decomposed into The social class of parents was constructed for the respondents two effects. Comparing cohorts over time, the difference in the mutual following the classification of Erikson et al. (1979) and the operation­ information index between two cohorts (c1 and c2) is the result of two alisation proposed by Ganzeboom and Treiman (1996). We used the contributions.3. Erikson, Goldthorpe & Portocarero scheme in eight categories: I (service class: higher-grade professionals, administrators and officials; managers in large industrial establishments and large proprietors); II (service class: lower-grade professionals, administrators and officials; 2 We use the Stata command ‘mindex’ by Jann and Seiler (2019) for our analyses. In Stata, the command is ‘mindex education i.father_class, over (cohort)’. 3 4 In Stata, the command is ’mindex education i.father_class, over(cohorte­ A sensitivity test was applied to check that the period effect did not alter the ducation) refgroup(1) decompose vce(boostrap, reps(100))’ results. See Appendix B, Table B1. 5 I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 higher-grade technicians and managers in small industrial establish­ ments); IIIa (higher-grade routine non-manual workers); IIIb (low­ er-grade routine non-manual workers); IVab (small employers and self-employed); V + VI (skilled manual workers); VIIa (unskilled manual workers) and VIIb + IVc (semi- and unskilled workers in agri­ culture and farmers and smallholders). For the class of origin, we considered the dominance criterion to take into account both parents (Erikson, 1984) by attributing to family origin the highest occupation of the father or mother when the respondent was between 12 and 16 years old. We constructed four birth cohorts distributing the sample between 1946 and 1989, with the following division: 1946–1956, 1957–1967, 1968–1977 and 1978–1989. The range of ages included in each cohort by the survey is presented in Table 1. 7. Results Fig. 2. : Educational attainment in average years of schooling between 1950 7.1. Descriptive results: education and social origin over time and 2010 for Spain, Germany, the Netherlands, United States, the United Kingdom and Sweden. There is no doubt that educational expansion in Spain has been one Source: Based on Barro and Lee (2013) with the average years of schooling of the population aged 16–64. See: http://www.barrolee.com/. of the most profound in Europe. In a very short period, Spain went from having a large illiterate population to being fully comparable to the most highly educated of societies. Fig. 2 illustrates the comparison of the (Treiman & Yip, 1989; Breen, 2019). trends in educational expansion in European countries. We compare Finally, as we move cohort by cohort, we note a decline in the lowest Spain with countries that had an earlier and more intense period of educational levels in the manual classes and an expansion of the middle industrialisation (the United States, Sweden, the Netherlands, the levels. Moreover, as this progress is taking place, the service and United Kingdom and Germany). We see that in the early 20th century, administrative classes are increasing their presence in university Spain began to distance itself from these countries as a result of a education. convulsive period that involved the loss of the colonies, military coups, In short, with a greater lag than in other societies, educational the civil war and the post-war period (Martorell & Juliá, 2012). Indeed, expansion has followed the same pattern, in which the lower classes while all these industrialised countries had shown a sharp rise in completed basic education and the higher social classes improved their educational attainment by 1950, Spain had barely improved its position positions in university education. in comparison with the beginning of the century. However, since the beginning of the 21st century, this gap is narrowing and Spain is 7.2. The margin-free tradition: what it can tell us approaching the position of most of the more developed countries. This trend can be observed in more detail if, instead of looking at Next, we will present the analysis of relative changes under the average education per year, we analyse educational attainment by the margin-free tradition using a margin-insensitive measure through log- cohorts involved in our study. As can be seen in Fig. 3, the shape of our linear models. We compare the constant fluidity model, which postu­ first cohort (1946–1956) resembles a submarine with a broad base with lates stability in relative mobility rates, with the uniform difference basic education that progressively narrows over time. In the beginning, model, which allows for a cohort variation. Table 2 presents the results 44% of the Spanish population was in ISCED levels 0 and 1. However, for obtained for the evolution of inequality of educational opportunities for the last of these cohorts (1978–1989), the shape practically reversed, both sexes in Spain for those born between 1946 and 1989. As can be although with a greater proportion of people in the middle levels, seen, following the BIC statistic, the unidiff model does not improve the reaching almost 85% in levels 3–6 in the youngest cohort.5 constant model, suggesting that for both sexes, the inequality of In Fig. 4, we present the evolution over time in the association be­ educational opportunities remained constant over the years analysed. tween educational attainment and social origin.6 The lowest educational Although the last cohort showed some increased rigidity, this is not level is associated with the agricultural and working classes of origin and statistically significant. Even if our selection criterion was the chi- the highest with the service and non-manual routine classes of origin square, we would have to accept that the change did not lead to a reduction in educational inequalities but to an increase in rigidity. The following figure shows the unidiff coefficients in men and Table 1 women.. Survey, cohort and age of people of the sample. Included only people from 30 to 59 7.3. The margin-dependent tradition: What it can tell us Cohort Survey Fig. 6 presents the overall result of the mutual information index.7 Its 2005 2011 2019 interpretation is as follows: given an initial value for the first cohort, a 1946 1956 49–59 55–59 Empty reduction in the value in the next cohort implies a decrease in inequality 1957 1967 38–48 44–54 52–59 of educational opportunities, while an increase in the value implies an 1968 1977 30–37 34–43 42–51 1978 1989 Empty 30–33 30–41 increase in inequality. As can be seen, the results for men and women do not differ too much. Both experienced a decline in educational in­ Source: The authors based this information on EU-SILC-2005, EU-SILC-2011 and equalities from the first to the fourth cohort. This decrease has halted in EU-SILC-2019 the last cohort. In other words, by weighting the barriers between social origin and 5 See accurate details in Table A1a and A1b of Appendix A. 6 7 See details in Tables A2 and A3 of Appendix A. See details in Table A4 of Appendix A. 6 I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 1946-1956 1957-1967 1968-1977 1978-1989 ISCED5-6 ISCED3-4 ISCED2 ISCED 0-1 Fig. 3. : Evolution of educational attainment by cohort: Total population. Source: Based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019. Fig. 4. : Evolution of educational attainment according to social origin by cohort: Total population. Source: The authors based on EU-SILC-2005 EU-SILC-2011 and EU-SILC-2019. Table 2 Relative analysis of inequality of educational opportunities: Log-linear models of constancy and uniform variation across cohorts. Model CSO CSE L2 BIC DI DF OE Men (n = 21261) Constant 429,55 -2639,55 4,49 308 Unidiff 416,67 -2622,53 4,32 305 Cohort 1946–1956 1957–1967 1968–1977 1978–1989 Coefficient 1,00 0,97 1,03 1,19 Standard error 0,06 0,06 0,08 (0,85–1,08) (0,91–1,14) (1,07–1,30) Model CSO CSE L2 BIC DI DF OE Women (n = 22485) Constant 475,35 -2610,98 4,74 308 Fig. 5. : Relative analysis of inequality of educational opportunities: Unidiff Unidiff 467,30 -2588,97 4,72 305 coefficients. Cohort 1946–1956 1957–1967 1968–1977 1978–1989 Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019. Coefficient 1,00 1,13 1,02 1,14 Standard error 0,06 0,06 0,08 7.4. The decomposition (1,07–1,19) (0,96–1,08) (1,06–1,22) Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019 An analysis of how far the marginal changes go and how far the in­ Note. L2: Likelihood ratio; BIC: Bayesian information criterion; DI: Dissimilarity ternal changes go would help us to resolve important questions in the index; DF: Degree of Freedom field of educational inequalities over time. By decomposing the mutual information index, we can determine the contribution of the marginal educational attainment according to their relevance, we obtain a changes from one cohort to another, distinguishing it from internal different picture of educational inequality trends. Therefore, taking into changes. In other words, as the mutual information index is based on a account the marginal changes, according to the mutual information weighted average taking into consideration the marginal distribution, index, we can conclude that Spain experienced a substantial reduction in we now decompose it to separate marginal from internal changes, so educational inequalities over a long period (1946–1977). In men, the that this internal measure has an unweighted average. important reduction appears between the first and second cohorts and in Fig. 7 shows the decomposition of the mutual information index women a decrease in all cohorts except the last one is observed. How­ between the marginal and the internal effects, first for men and then for ever, the 1980s witnessed the end of this reduction in school in­ women.8 We can see that the evolution of the internal component is equalities. To interpret these results more accurately, the overall index needs to be decomposed. 8 See details in Table A5 of Appendix A. 7 I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 both the educational expansion and the occupational upgrading of their parents. If these changes are taken into account to weigh the average, considering the composition of groups, educational inequalities show a decline. This has been possible even though the internal differences remain constant (or even increase for men in the last cohort). Thus, education inequalities have declined as a consequence of changes in the structure of Spanish society, which have rendered some barriers less important despite their persistence. 8. Conclusions The results of our study lead to a revision of the usual conclusions shown in the analyses of the inequality of educational opportunities in the case of Spain. Concerning previous theories, it could be noted that, with the application of the mutual information index measure, the evi­ Fig. 6. : The mutual information index by sex and cohort. dence collected mainly called into question the theory of persistent in­ Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019. equalities, although this idea would have to be nuanced due to the slowing of this trend in the last cohort. These are the results when we similar to the evolution of the unidiff parameters for both men and consider the prevalence of educational barriers that different classes women, although for men, in the last cohort, a statistically significant have to navigate. If we do not consider this prevalence, which would be behaviour towards greater rigidity is observed. another correct way to measure the relative change, the evidence points Conversely, the marginal component shows a decrease in in­ towards constancy. In our view, this index, therefore, enriches the equalities for both men and women over time. analysis of the IEO by offering us a new perspective. In short, the results show that the reduction in educational in­ In Europe, educational inequalities declined considerably after the equalities has been driven by marginal changes, while internal changes post-war period during the Trente Glorieuses (Breen et al., 2009), but remain more or less constant, except in the last cohort where is reached a point at which they stopped weakening (Barone & Ruggera, increased. 2018). The only difference with regard to the trends followed by other Now, it is much easier to interpret our results. Those who were born European countries is that in Spain, the reduction of educational in­ between 1946 and 1989 experienced a great transformation in terms of equalities occurred later, after the post-war period, at a time when Fig. 7. : Decomposition of the mutual information index separating the effect of the relative changes (Internal) from the marginals (Marginal) by sex and cohort. Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019. 8 I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 economic improvements and educational reforms were introduced occupational upgrading. When these changes are considered, weighting (Breen, 2020; Authors). the average and taking into account the composition of groups, educa­ In this study, we have tried to go a step further by introducing a tional inequalities have declined substantially. This has been possible single measure that would allow us to account for both internal and even while the internal differences have remained constant (although in marginal changes. This measure is based on information theory and the men there was a rise in the last cohort). notion of entropy. Based on the mutual information index to explain The mutual information index moderates its downward trend due to educational inequalities over time, we have considered the joint and the performance of the last cohort. When we compare tables that differ decomposable measurement of the effects of change over time of in­ greatly in their marginal composition, using the mutual information ternal and marginal relationship patterns. We have done this by index enriches our methodological perspective and leads us to elaborate applying the most extensive scheme employed to date with data for alternative theoretical conclusions. In this sense, the index allows us to Spain. Instead of the usual six or seven social classes of the Erikson, ask questions that we could not otherwise ask with margin-free table Goldthorpe and Portocarero model, we used a scheme of eight social comparisons. Thus, educational inequalities have declined because the classes. structure of Spanish society has changed in such a way that strong We think that the decomposition of the mutual information index barriers have become less important (though they are still strong). (Seiler & Jann, 2019a, 2019b) may help us enrich the study of educa­ There is an interesting field to explore and assess the potential of this tional inequalities, thus broadening the perspective on this topic. At the measure. By applying it to other case studies to obtain a general over­ theoretical level, we have tried to explain why considering the weight of view comparing different countries, it will be particularly interesting to barriers via marginal changes may offer an alternative interpretation of know how it works in peripheral countries where the processes of educational inequality trends. We have, thus, emphasised two processes: industrialisation and structural change have been more delayed than in (a) occupational upgrading (of parents) and (b) educational expansion the central countries. In this sense, we believe that this measure can (of respondents). The former could have contributed to an overall open up a vast field of analysis and debate that will complement the improvement in economic conditions leading to nutritional improve­ comparative studies that have traditionally been carried out in this field ment and the release of family expenditure; moreover, the latter was of international research throughout the different generations of social fostered by the creation and implementation of a less selective national mobility studies. system (Roser & Ortiz-Ospina, 2016). Our main results confirm the two hypotheses proposed. Thus, following the margin-free tradition, we find educational inequalities Declaration of Competing Interest unchanged in Spain, or even more, the last cohort shows in men an in­ crease in rigidity. Regarding the second hypothesis, however, in Spain, it The authors declare that they have no known competing financial is observed that the cohorts born between 1946 and 1989 experienced interests or personal relationships that could have appeared to influence significant changes as a consequence of educational expansion and the work reported in this paper. Appendix A Table A1a Educational level by cohort: Total population. Four levels of education Educational level Cohort 1946–1956 1957–1967 1968–1977 1978–1989 Total ISCED0_1 43.7% 16.9% 9.6% 4.4% 16.9% ISCED2 20.5% 22.3% 17.5% 10.6% 18.8% ISCED3–4 16.5% 36.0% 40.1% 52.6% 36.7% ISCED5–6 19.3% 24.8% 32.8% 32.4% 27.6% Total 100.0% 100.0% 100.0% 100.0% 100.0% Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019. Table A1b Educational level by cohort: Total population. Five levels of education Educational level Cohort Total 1946–1956 1957–1967 1968–1977 1978–1989 ISCED0 3.70% 1.40% 1.00% 1.10% 1.60% ISCED1 40.10% 15.50% 8.60% 3.30% 15.30% ISCED2 20.50% 22.30% 17.50% 10.60% 18.80% ISCED3–4 16.50% 36.00% 40.10% 52.60% 36.70% ISCED5–6 19.30% 24.80% 32.80% 32.40% 27.60% Total 100.0% 100.0% 100.0% 100.0% 100.0% Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019. 9 I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 Table A2 Origin by cohort: Total population. Origin 1946–1956 1957–1967 1968–1977 1978–1989 Total I 5.3% 6.0% 7.3% 9.4% 6.8% II 5.6% 6.1% 8.3% 9.2% 7.2% IVab 13.9% 15.8% 18.1% 18.9% 16.7% IIIa 3.1% 3.5% 4.7% 5.4% 4.1% IIIb 5.9% 7.0% 9.2% 12.2% 8.3% V + VI 17.0% 18.0% 18.9% 17.6% 18.0% VIIa 19.0% 21.4% 19.6% 17.2% 19.8% VIIb + IVc 30.3% 22.2% 14.1% 10.2% 19.1% Total 100.0% 100.0% 100.0% 100.0% 100.0% Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019. Table A3 Evolution of educational attainment according to social origin by cohort:. Total population Education ISCED_0 ISCED_1 ISCED2 ISCED3_4 ISCED5_6 Total Origin 1946–1956 I 1.9% 7.2% 12.2% 25,00% 53.6% 100.0% II 0.8% 16.7% 12.2% 23.2% 47.1% 100.0% IVab 2.4% 28.8% 18.2% 22.5% 28,00% 100.0% IIIa 1.4% 17.1% 15.2% 24.3% 41.9% 100.0% IIIb 1.5% 32.3% 20.2% 21.5% 24.4% 100.0% V + VI 1.8% 41.9% 27.4% 15.7% 13.2% 100.0% VIIa 3.4% 41.9% 22.9% 17.8% 14,00% 100.0% VIIb + IVc 6.3% 55.5% 21.6% 8.6% 8,00% 100.0% Origin 1957–1967 I 0.1% 1.4% 6,00% 29.4% 63.1% 100.0% II 0.5% 4,00% 10,00% 33.7% 51.8% 100.0% IVab 0.9% 9.7% 17.9% 40.5% 31.1% 100.0% IIIa 0,00% 5,00% 12.3% 38.3% 44.3% 100.0% IIIb 0.8% 10.4% 18.9% 44.6% 25.4% 100.0% V + VI 0.9% 17.3% 25.7% 37.9% 18.2% 100.0% VIIa 1.5% 16.9% 26.3% 36.8% 18.6% 100.0% VIIb + IVc 2.4% 26.2% 28.9% 30,00% 12.5% 100.0% Origin 1968–1977 I 0.1% 1,00% 4.1% 28.7% 66,00% 100.0% II 0.5% 3.4% 7.5% 30.9% 57.7% 100.0% IVab 0.7% 6.9% 13.9% 43,00% 35.5% 100.0% IIIa 0.3% 2.8% 5.4% 38.4% 53.1% 100.0% IIIb 0.6% 5.9% 16.9% 45.4% 31.1% 100.0% V + VI 0.8% 9.2% 20.1% 44.8% 25.2% 100.0% VIIa 1.1% 10.7% 22.3% 42.2% 23.7% 100.0% VIIb + IVc 1.9% 16,00% 28.2% 36,00% 17.8% 100.0% Origin 1978–1989 I 0.3% 0.5% 2.7% 27.7% 68.7% 100.0% II 0.3% 0.5% 4.2% 39.7% 55.2% 100.0% IVab 0.9% 3.5% 8.4% 51.6% 35.5% 100.0% IIIa 0.6% 0.3% 5.6% 52.5% 40.9% 100.0% IIIb 0.5% 1.8% 7.9% 63.1% 26.6% 100.0% V + VI 0.5% 3.7% 13.2% 59.7% 22.8% 100.0% VIIa 1,00% 2.8% 14,00% 58.5% 23.6% 100.0% VIIb + IVc 3.6% 7.6% 18.5% 55.5% 14.8% 100.0% Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019. Table A4 The mutual information index by sex and cohort. Men Cohort Coef. Std. Err. (*) 1946–1956 0.1006795 0.0074055 1957–1967 0.0704876 0.0039967 1968–1977 0.0678970 0.0042533 1978–1989 0.0737856 0.0067501 Women Cohort Coef. Std. Err. (*) 1946–1956 0.0896361 0.0067642 1957–1967 0.0802154 0.004146 1968–1977 0.0618521 0.0037563 1978–1989 0.0664117 0.0062509 10 I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 (*) Obtained by bootstrap (100 replications). Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU- SILC-2019 Table A5 Decomposition of the mutual information index separating the effect of the relative changes (Internal) from the marginals (Marginal) by sex and cohort. Men Internal Cohort Coef. Std. Err. (*) z P>z [95% Conf. Interval] 1946–1956 0 (omitted) 0 0 1957–1967 -0.0033498 0.0077342 -0.43 0.665 -0.0185086 0.0118090 1968–1977 -0.0045350 0.0082928 -0.55 0.584 -0.0207886 0.0117186 1978–1989 0.0343542 0.0120964 2.84 0.005 0.0106457 0.0580627 Marginal Cohort Coef. Std. Err. (*) z P>z [95% Conf. Interval] 1946–1956 0 (omitted) 0 0 1957–1967 -0.0268422 0.0032122 -8.36 0.000 -0.033138 -0.0205465 1968–1977 -0.0282475 0.0035520 -7.95 0.000 -0.0352093 -0.0212857 1978–1989 -0.0612481 0.0074652 -8.2 0.000 -0.0758796 -0.0466166 Women Internal Cohort Coef. Std. Err. (*) z P>z [95% Conf. Interval] 1946–1956 0 (omitted) 0 0 1957–1967 0.0106605 0.0079259 1.35 0.179 -0.0048741 0.0261950 1968–1977 0.0058826 0.0076767 0.77 0.444 -0.0091636 0.0209287 1978–1989 0.0200311 0.0133232 1.50 0.133 -0.0060819 0.0461441 Marginal Cohort Coef. Std. Err. (*) z P>z [95% Conf. Interval] 1946–1956 0 (omitted) 0 0 1957–1967 -0.0200811 0.0037185 -5.40 0.000 -0.0273693 -0.0127929 1968–1977 -0.0336666 0.0047081 -7.15 0.000 -0.0428943 -0.0244388 1978–1989 -0.0432554 0.0102121 -4.24 0.000 -0.0632708 -0.0232401 (*) Obtained by bootstrap (100 replications). Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019 Appendix B The homogeneity test in Table B1 analyses how the surveys employed have affected the different cohorts for men and women. We present a double log-multiplicative model made up of the following terms CSO CSE, βcOE and βsOE. The last interaction (βsOE) has been modelled according to the linear constraints that permit us to evaluate the survey changes on cohorts. With linear restrictions, we intend to force the coefficients of the models to comply with a specific relationship, in this case, an equality restriction. We want to unify those periods that do not introduce a change in the cohorts. In this way, we obtain a parsimonious model that helps us to detect which period may have a different effect per cohort. The first model allows only cohort variations. Conversely, the latter allows both cohort and linear variations for all surveys. The intermediate models allow variation by cohort but constrain some surveys over others; they are reflected within each parenthesis. Therefore, we chose in each case the models with the most negative BIC. Table B1 Homogeneity test models. Men Models L2 BIC DI DF 1. CSO CSO OExC OExS (No Survey Variation) 371.79 -2378.44 4.03 276 2. CSO CSO OExC OExS (Variation 1 vs. 2 3) 372.18 -2378.05 3.98 276 3. CSO CSO OExC OExS (Variation 1 2 vs. 3) 271.79 -2478.48 3.09 276 Unidiff parameters (Cohort) 1.00 1.19 (0.06) 1.15 (0.06) 1.10 (0.94) Standard errors 1.00 0.63 (0.33) Unidiff parameters (Survey) Standard errors 4. CSO CSO OExC OExS (Variation 2 vs. 1 3) 353.02 -2397.21 3.87 276 5. CSO CSO OExC OExS (Complete Variation) 309.04 -2440.19 3.50 276 Women Models L2 BIC DI DF 1. CSO CSO OExC OExS (No Survey Variation) 404.14 -2361.06 4.36 276 2. CSO CSO OExC OExS (Variation 1 vs. 2 3) 368.12 -2397.62 3.99 276 3. CSO CSE OExC OExS (Variation 1 2 vs. 3) 307.60 -2458.08 3.57 276 Unidiff parameters (Cohort) 1.00 1.02 (0.07) 0.95 (0.07) 0.92 (0.14) Standard errors 1.00 1.28 (0.12) Unidiff parameters (Survey) Standard errors (continued on next page) 11 I. Marqués-Perales et al. Research in Social Stratification and Mobility 87 (2023) 100827 Table B1 (continued ) Women Models L2 BIC DI DF 4. CSO CSO OExC OExS (Variation 2 vs. 1 3) 389.58 -2376.10 4.12 276 5. CSO CSO OExC OExS (Complete Variation) 328.66 -2437.01 3.75 276 Source: The authors based on EU-SILC-2005, EU-SILC-2011 and EU-SILC-2019 Note. L2: Likelihood ratio (or deviance); BIC: Bayesian information criterion; DI: Dissimilarity index; DF: Degree of Freedom. C: Cohort, S: Survey, O: Origin, E: Education In regard to men and women, the best fit corresponds to model 3, which allows only for the temporal change in the last of the survey. In men, unidiff parameters increased, while in women, these parameters decreased slightly. This means that if there is a survey effect, it only affects the last survey. Taking into account the unidiff parameters of the surveys, we do not see a change in the cohort’s parameters, which continue to show invariance. Given the size of errors in the partitioning of cohorts and surveys, we should be cautious about these differences. However, it is reasonable to posit that changes could occur after Spain’s 2008 financial crisis. We know that in terms of survey enrolment rates, the crisis brought a marked increase (Martínez-García & Molina, 2019). Future research may explore this area in further detail. Unsurprisingly, model 3 fits better by adding such a survey change, given the wide range that we cover in terms of cohorts (from 1946 to 1989) since we are comparing individuals who were born in extremely different periods. It would then be necessary to check whether, after the crisis, a growing number of people of service class origins, aged 30 or more, enrolled in higher education. However, considering this change, in any case, we do not detect a variation that implies a substantial reduction in educational inequalities. Assuming a hypothetical survey effect (and we should be cautious about such an assumption), the analysis reveals a high level of invariance or constancy in the effect of social origin on education. References Erikson, R., & Goldthorpe, J. H. (1993). The Constant Flux: A Study of Class Mobility in Industrial Societies. New York: Oxford University Press. Erikson, R., Goldthorpe, J. H., & Portocarero, L. (1979). Intergenerational class mobility Ayalon, H., & Shavit, Y. 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