Social Inequality in Dropout from Higher Education in Germany. Towards Combining the Student Integration Model and Rational Choice Theory

Students from a lower socioeconomic background have a higher risk of dropping out of higher education. The underlying mechanisms of this association between socioeconomic background and higher education dropout are not well understood. Previous research in higher education has followed Tinto’s model of academic and social integration to explain dropout but has largely neglected social inequality therein. In contrast, social stratification research draws on rational choice theory to explain social inequality in educational attainment but has rarely been applied to explain dropout from higher education. In our paper, we combine these two strands of research. Utilizing data from the National Educational Panel Study (NEPS), we draw on a largescale, representative sample of students in Germany to quantify the relative contribution of each theoretical approach for explaining social inequality in dropout from higher education. Binary logistic regression models reveal that both students’ integration and costs-benefit considerations are associated with their dropout risk net of each other. While academic and social integration appears to better predict dropout, rational choice theory accounts for a larger proportion of social inequality therein. We conclude that combining Tinto’s model and rational choice theory provides a more comprehensive perspective of dropouts from higher education and social inequality therein.

among students who enroll in higher education, those with less-educated parents and working-class families are more likely to drop out before obtaining a degree. This finding holds for the US (Chen, 2012;Ishitani, 2006) as well as for European countries (Larsen et al., 2013), such as Italy (Argentin & Triventi, 2011;Contini et al., 2017), the UK (Smith & Naylor, 2001;Vignoles & Powdthavee, 2009), and Germany (Heublein et al., 2017;Müller & Schneider, 2013). While numerous studies describe social inequality in higher education, the underlying mechanisms that drive the association between SEB and higher education dropout are not well understood.
Previous research on dropout from higher education has traditionally followed Tinto's (1975Tinto's ( , 1987 seminal work on the so-called student integration model (henceforth: SIM). From this perspective, students' academic and social integration into the higher education system are the pivotal predictors of degree completion; students' SEB is only of marginal interest. Despite having dominated theoretical discussions for decades (Kuh et al., 2006), empirical support for the SIM, while vast in number (for a review, see Braxton et al., 1997), remains mostly limited to the US context. Whether the model proves fruitful for explaining dropout beyond the US higher education system remains an open question. Moreover, only few studies have applied the SIM to explain social inequality (cf. Soria et al., 2013); whether differences in academic and social integration account for social inequality in higher education dropout is not known.
In contrast, social stratification researchers have established rational choice theory (henceforth: RCT) as the predominant micro foundation for explaining social inequality in educational attainment (Boudon, 1974;Breen & Goldthorpe, 1997;Erikson & Jonsson, 1996). Differentiating between performance-based (primary effects) and decision-based (secondary effects) factors, this theory assumes that social inequality in educational attainment arises from the expected costs, benefits, and success probability that vary systematically with students' SEB. While RCT has received considerable empirical support in various countries (Daniel & Watermann, 2018;Gabay-Egozi et al., 2010;Need & De Jong, 2001), it has rarely been applied to explain dropout from higher education.
In our paper, we combine these two predominant theories on higher education dropout and on social stratification. Conceptually, we treat the SIM and RCT as complementary rather than competing theoretical approaches. We argue that combining the SIM and RCT provides an integrating perspective for identifying the mechanisms underlying both dropout from higher education and the social disparities therein.
More specifically, we ask whether the SIM and RCT explain social inequality in dropout from higher education in Germany. We choose Germany as an example case because the association between SEB and educational attainment is comparatively strong (Stocké et al., 2011). At the same time, the German higher education system provides an interesting contrast to those of other countries with high levels of social inequality, such as the UK and especially the US (Allmendinger, 1989;Jackson, 2013). There are striking differences regarding access to and tuition for higher education and differences in everyday campus life that will likely affect social stratification, students' integration, and perceived costs and benefits from higher education. Moreover, unique survey data from Germany suits our research questions particularly well by combining a nationwide representative sample of students with questions specifically designed to measure students' integration (Dahm et al., 2016) and the perceived costs, benefits, and success probability associated with higher education (Stocké et al., 2011).
The contribution of our paper is threefold. First, we add to the limited number of empirical applications of the SIM outside of the US higher education context. Second, we are among the first to assess whether the SIM explains social inequality in higher education dropout. Third, our paper contributes to the scarce literature that applies RCT to explaining dropout from higher education.

The Student Integration Model
The vast majority of sociological studies on higher education dropout approach the subject from the perspective of Tinto's (1975Tinto's ( , 1987 SIM and its core concepts academic and social integration (Braxton et al., 1997). This model distinguishes between an academic and a social domain within higher education institutions and views students' dropout as resulting from failure to integrate into either of these two domains.
Integration into the academic domain comprises structural and normative components (Tinto, 1975, p. 104). In addition to meeting formal performance requirements, such as maintaining a certain GPA and passing exams, students have to adjust to the norms and achievement orientations commonly held by higher education institutions. A perceived mismatch between achievement expectations and actual academic performance leads students to consider alternatives to higher education and drop out. Likewise, students who do not share the achievement orientations held by their institution are less likely to engage in their studies and more likely to drop out (Tinto, 1987, pp. 53-56).
To integrate into the social domain of higher education institutions, students must establish and cultivate contacts with faculty and fellow students (Tinto, 1975, pp. 106-107). Besides providing academic support, interactions with faculty stimulate students' intellectual development and increase institutional commitment. Conversely, students who feel unfairly treated or unaccepted by faculty are less likely to commit to their studies and more likely to drop out (Tinto, 1987, pp. 116-117). Likewise, establishing relationships with fellow students assures friendship support, which leads to collective affiliation and decreases the likelihood of dropping out (Tinto, 1975, p. 107).
Regarding social inequality in dropout, the SIM posits that students from a lower SEB face difficulties integrating into the academic and the social domain of higher education (Tinto, 1987, pp. 70-72). On average, these students enter higher education with lower levels of academic performance (Terenzini et al., 1996), which then translate into lower levels of academic integration. Moreover, students from a lower SEB hold norms that are incompatible with those held by their more privileged peers, making it more difficult to establish social relationships (Tinto, 1987, pp. 95-99). According to the SIM, social inequality in dropout from higher education arises from differences in the levels of academic and social integration.
In contrast to the core predictions of the SIM, very few studies have focused on its hypotheses regarding social inequality. While some studies suggest that social and academic integration mediate the association between SEB and dropout (Pascarella & Chapman, 1983), other studies find that parental education predicts students' dropout risk even at comparable levels of academic integration (Donovan, 1984). Two recent studies find lower levels of academic and social integration among working-class students (Soria et al., 2013) and first-generation students (Soria & Stebleton, 2012). However, neither study empirically tests whether these differences in academic and social integration account for social inequality in dropout.
At its core, the SIM posits students' academic and social integration as the decisive factors influencing their dropout decision. Despite the relevance of students' integration, researchers have criticized this perspective for largely disregarding economic factors, such as financial considerations, for dropping out (Breier, 2010;Cabrera et al., 1992). 1 Several studies find that financial aid reduces dropout rates (Chen & St. John, 2011;Melguizo et al., 2011) most notably for students from a lower SEB (Alon, 2011;Arendt, 2013). Therefore, we argue that a theoretical model that aims not only at explaining higher education dropout but also social inequality therein should include economic factors. More specifically, we suggest complementing the SIM with RCT, which explains dropout as resulting from cost-benefit considerations. Boudon (1974) suggests RCT as a general theoretical framework for explaining social inequality in educational attainment. This approach distinguishes between so-called primary and secondary effects. Primary effects refer to, on average, lower academic abilities of students from a lower SEB that stem from differences in cultural resources among social classes (Bourdieu & Passeron, 1990;Erikson & Jonsson, 1996). Secondary effects are class-specific educational choices, conditional on academic ability (Jackson et al., 2007), which result from evaluating the costs, benefits, and success probabilities of educational pathways (Breen & Goldthorpe, 1997). Taking primary effects as given, RCT aims at explaining educational choices and social disparities therein as resulting from cost-benefit considerations (Breen & Goldthorpe, 1997, p. 277).

Rational Choice Theory
The costs associated with higher education include direct costs, such as tuition fees; indirect costs, such as books and travel; as well as opportunity costs, such as forgone labor market earnings until degree completion. Because students from working-class families have fewer economic resources than students from a more advantageous SEB, the former tend to perceive objectively equal costs as more subjectively burdensome (Stocké, 2007). Consequentially, compared with students from a more advantageous SEB, working-class students are more sensitive to information about the actual costs of attending university (Barone et al., 2017;Mangan et al., 2010) and perceive higher education as more costintensive (Becker & Hecken, 2009;Daniel & Watermann, 2018;Schindler & Lörz, 2012).
The benefits from higher education operate through a mechanism that Breen and Goldthorpe (1997, pp. 283-285) call relative risk aversion. Trying to avoid intergenerational downward mobility, students seek to maintain their parents' socioeconomic status. Students from the service class, therefore, aspire to higher occupational status than their working-class peers, increasing the former's motivation to graduate. Despite ongoing debates about the assumption of equal concern for status maintenance across social classes (Barone et al., 2018;Tutić, 2017), studies have shown that the motive of status maintenance partly explains social inequality in educational choices (Need & De Jong, 2001;Van de Werfhorst et al., 2007).
Finally, the probability of success rests on students' beliefs about their academic abilities (Breen & Goldthorpe, 1997, pp. 285-286). Because of primary effects, students from a lower SEB enter higher education with, on average, fewer academic abilities (Terenzini et al., 1996). Therefore, they will estimate for themselves a lower success probability. Moreover, even at comparable levels of academic abilities, students from less educated parents might lack the tacit knowledge to navigate the higher education system that their more privileged peers possess (Erikson & Jonsson, 1996, p. 22;Schindler & Reimer, 2011, p. 265).

Towards Combining the Student Integration Model and Rational Choice Theory
While the SIM specifically aims at explaining dropout from higher education, RCT aims at an explanation of educational choices in general and social inequality therein. Both theoretical approaches emphasize certain aspects while disregarding others. Emphasizing students' experiences in the academic and social domain of higher education, the SIM tends to disregard external influences in general and cost-benefit considerations in particular (Cabrera et al., 1992). However, such influences might be crucial for explaining social inequality in dropout. RCT, on the other hand, emphasizes cost-benefit considerations but disregards student's sense of belonging to the higher education institution (Tinto, 1986, p. 363), which reduces the risk of dropping out.
We suggest combining the SIM and RCT, arguing that the two theoretical approaches compensate for each other's potential blind spots. While Tinto (1975, p. 98) explicitly frames students' academic and social integration as cost-benefit considerations, he focuses on intrinsic motivation, defining costs in terms of dissatisfaction with academic failure and benefits in terms of satisfying friendships that students experience during their studies. In contrast, RCT views higher education as a means to an end and focuses on extrinsic motivation, such as monetary costs until degree completion and the likelihood of status maintenance, both of which relate to future events. We argue that students are likely to be both intrinsically and extrinsically motivated by immediately perceived rewards from integrating into and the anticipated costs and benefits from completing higher education. Therefore, combining the SIM and RCT provides a more comprehensive theoretical approach.
Others have preceded us in combining the SIM with RCT. Using longitudinal data on a sample of first-year students in the Netherlands, Beekhoven et al. (2002) test the SIM and RCT against a model comprising variables from both theoretical approaches. They find that the combined model best predicts academic progress, measured in terms of completed credits after five years. Unlike Beekhoven et al. (2002), we are less interested in academic progress but more in students' risk of dropping out of higher education and especially in explaining the social inequality therein.

Hypotheses for the German Higher Education Context
The German higher education system differs from the higher education systems of other European countries and even more so from the US. The differences will likely affect the extent of social inequalities and the associations of students' academic performance, integration, and cost-benefit considerations with the risk of dropping out. Although we cannot empirically test such differences between higher educational systems, they are informative for deriving our hypotheses.
Both the SIM and RCT assume that students from a lower SEB have, on average, lower levels of academic performance. Unlike the US, Germany has a highly stratified school system (Allmendinger, 1989;Neugebauer et al., 2013) that allocates students to different secondary school types according to academic performance very early, typically after fourth grade (Pietsch & Stubbe, 2007). 2 Only one secondary school type, the "Gymnasium", then provides direct access to higher education. 3 Germany's early tracking system leads to a comparatively homogeneous student population eligible to entering higher education (Mare, 1980). Still, social disparities in academic performance persist even during the transition into tertiary education (Schindler & Reimer, 2011). 4 Therefore, despite early selection processes, we expect that students from a lower SEB have, on average, lower levels of academic performance (H1a).
From the perspective of the SIM, academic performance in higher education reflects students' formal academic integration, while RCT views academic performance as stemming from primary effects that partly determine the success probability. We prefer to think of academic performance as an additional predictor not belonging exclusively to either theoretical approach. In line with previous research in the US (Mayhew et al., 2016, pp. 556-557), Europe (Larsen et al., 2013), and Germany (Neumeyer & Alesi, 2018), we expect a negative association between academic performance and dropout risk (H1b). Following our argument that students from lower SEB enroll with lower levels of academic performance, we further expect differences in academic performance partly explain social inequalities in dropout (H1c).
According to the SIM, students from a lower SEB integrate less well into the academic and social domains of higher education. Unlike in the US or the UK, only very few students live on campus in Germany. Moreover, students from lower SEB are more likely to live with their parents rather than share an apartment with fellow students (Middendorff et al., 2013, p. 415). Thus, students from lower SEB have fewer opportunities of interacting and integrating with fellow students. Also, the student-faculty ratio in Germany is typically worse than in the US (Hüther & Krücken, 2018, p. 61), making interactions with faculty more difficult. Before this background, we expect that students from lower SEB are less integrated, especially with their fellow students (H2a).
Despite likely differences between Germany and other countries in the degree of students' integration, the association between integration and dropout risk should generally be comparable across countries. Following the SIM, we expect insufficient academic and social integration increases the risk of dropping out (H2b). Combining the lower levels of integration among students from lower SEB and the association of integration with dropout risk, we further expect social disparities in academic and social integration to partly explain social inequalities in dropout (H2c).
According to RCT, students from a lower SEB perceive higher costs, lower benefits, and a lower probability of completing higher education (H3a). Compared with the US and many European countries, especially the UK, tuition for higher education is very low in Germany (OECD, 2021). Until 2006, there were no tuition fees and only nominal administrative fees. Moreover, depending on their parents' income, students are eligible for state subsidies. However, between 2006 and 2014 (when our data was collected), some federal states temporarily charged tuition fees of up to 500 Euro (approximately 725 US $) per semester (Kroth, 2015, pp. 59ff.). Empirical evidence suggests that financial considerations related to tuition fees are more relevant for students from lower SEB (Kroth, 2015). These findings are in line with the assumptions of RCT that students from lower SEB perceive objectively equal costs as subjectively more burdensome. While students from lower SEB perceive higher education as more costly, they expect few additional benefits. Unlike in the US, where high school graduates either enroll in higher education or enter the labor market directly, Germany has a standardized vocational education and training program (VET). Especially for students from lower SEB, the VET provides access to qualified positions in the labor market that will likely maintain their parents' socioeconomic status (Becker & Hecken, 2009). At the same time, unlike in the US, the higher education experience does not seem to generate additional labor market returns for dropouts in Germany (Schnepf, 2017).
Overall, our arguments suggest that the cost-benefit considerations are likely to vary with higher education systems. Nevertheless, following RCT, higher perceived costs, lower benefits, and a lower success probability should generally increase the risk of dropping out (H3b). Furthermore, accounting for differences in cost-benefit considerations should partly explain social inequalities in the risk of dropping out (H3c).

Data
We base our empirical analyses on data from the National Educational Panel Study (NEPS): Starting Cohort First-Year Students . The initial one-stage stratified clustered sample comprises 17,910 first-year students at German universities and universities of applied sciences (Zinn et al., 2017), covering the most relevant higher education institutions and the vast majority of higher education students. 5 Students entered higher education in the winter term of 2010/11 and have been followed up each semester. Interviews were conducted alternating between computer-assisted telephone and computerassisted web interviews. Participation rates varied between 60% and just above 70% over the 14 available panel waves (Zinn et al., 2020).
In two respects, the NEPS dataset suits our research questions particularly well. First, the prospective research design enables us to measure all explanatory variables of interest 5 There are two major types of higher education institutions in Germany. Universities traditionally have a more research-oriented academic focus, while universities of applied sciences have a more vocational orientation. Admission to the latter is somewhat less restrictive as it requires one year less of secondary education ("Fachabitur"). Also, universities of applied sciences do not offer all fields of study (e.g., medicine, law) (Hüther & Krücken, 2018). However, traditional differences between higher education institutions have continuously been decreasing (Enders, 2016), and students have always been allowed to change between universities and universities of applied sciences (KMK, 2003). The NEPS sample only excludes administrative colleges, which are run by the federal states and prepare students for working in public administration as civil servants. Less than 1% of the student population in Germany attends such institutions. before observing students' academic outcomes, reducing potential bias due to retrospective self-reports (Schwarz, 2007). Second, the NEPS questionnaires include a series of items that were specifically designed to measure both students' academic and social integration (Dahm et al., 2016) and the decision parameters of RCT (Stocké et al., 2011). Table 1 provides an overview of our variables, which we briefly describe here. Our focal predictor is students' SEB, which we measure in terms of parental education and occupation. We collapse the levels 0 to 4 of the ISCED-97 classification (OECD, 1999) into one category representing non-tertiary education and contrast these first-generation students with those whose parents' have attained tertiary education (ISCED-97 levels 5 or 6). We measure occupation using the EGP class scheme (Erikson et al., 1979). 6 Here, we differentiate between the working class (IIIb, VI, and VII) and the service and mixed classes (I, II, IIIa, IV, and V). We use the higher level of education and occupational class among both parents, respectively.

Variables
We measure academic performance using both the grade point average (GPA) of students' higher education entrance certificate and the GPA during their studies, typically reported after the first year. Because register data is not available, we rely on students' selfreports. We reverse the coding of the German GPA so that higher values indicate better academic performance.
We measure students' academic integration and social integration using a total of 13 items (see Dahm et al., 2016). Two subscales constitute our measure of students' academic integration: Achievement orientation (Cronbach's alpha = .74) captures students' compliance with higher education performance standards while fulfillment of achievement expectations (Cronbach's alpha = .78) measure students' satisfaction with their academic performance. Two different subscales measure students' social integration: Interactions with faculty (Cronbach's alpha = .79) capture how students assess their relationship to faculty members while interactions with fellow students (Cronbach's alpha = .84) reflect students' assessment of their relations to fellow students. All scales are composed of Likert-type items, which we recode so that higher values reflect a higher degree of integration.
All parameters of RCT are measured with single Likert-type items (also, see Stocké et al., 2011). The item that measures direct and indirect costs of higher education asks students how difficult it is for their family to pay for commuting, books, and tuition fees. A second item measures opportunity costs in terms of students' perceived financial pressure resulting from forgone labor market earnings until graduation. In both items, higher values indicate higher costs. To capture the concept of relative risk aversion, we follow Barone et al., (2018, p. 557) and measure students' occupational aspirations in terms of ISEI-08 (Ganzeboom & Treiman, 1996) scores. The success probability is measured by students' perceived likelihood of obtaining a degree.
Our primary outcome is a binary indicator variable that takes the value of 1 if a student has dropped out of higher education before obtaining a degree and 0 if they have .90 GPA (× − 1) 1.0-1.5 "very good" 1.6-2.5 "good" 2.6-3.5 "satisfactory" 3.6-4.0 "sufficient"  .20 Other .08  Bildungsberichterstattung, 2020, p. 195), we exclude all students (about 2%) who have not obtained a degree and are still studying by the time of the last interview. In all analyses, we control for the type of entrance certificate, the type of higher education institution, the field of studies, and the pursued degree. We also control for students' migration background, gender, and age at enrollment.

Analytic Sample and Multiple Imputation of Missing Values
From the initial sample of 17,910 students, we exclude 581 (3.2%) students, who were already enrolled in higher education before the winter-term of 2010/11, a further 284 (1.6%) students who were already 35 years or older when they first enrolled in higher education, and 83 students (< 1%) with missing values on basic variables such as their field of studies. These exclusions leave us with 16,962 students, half of which have missing values in less than 3 out of the 22 variables that we use in the analyses (see Fig. 2 and Table 5, appendix). The proportion of missing values across variables ranges from less than 5% for parental education, GPA of the entrance certificate, and most measures of RCT, to slightly over 50% for GPA during studies (see Table 6, appendix). There are no missing values in the basic control variables.
Assuming that variables are missing at random, we address missing values by multiple imputations via chained equations (Rubin, 1987;Van Buuren, 2012). In the imputation models, we include all variables of the substantive analyses and, if available, repeated measures. In order to improve the imputations of missing values on the outcome, we additionally include a measure of students' intentions of dropping out of higher education. Considering missing values due to non-participation, we further include the total number of completed interviews by each student. Finally, we account for the sampling design by including weights for strata and probability weights for participation in the first panel wave (Rubin, 1996). We use Stata 16 (StataCorp., 2019) to create a total of 100 complete datasets, ensuring that the number of imputations is larger than the fraction of missing information (White et al., 2011). Following Von Hippel (2007), we restrict our analyses to 11,057 cases that do not have missing values on the outcome. 8

Analytic Approach
In a first step, we assess social inequality in GPA, academic and social integration, and the costs, benefits, and success probability (H1a, H2a, and H3a). We standardize all variables to zero mean and unit variance. In a second step, we use binary logistic regression models to predict students' dropout risk from their SEB, GPA, and the predictors of the SIM and RCT (H1b, H2b, and H3b). 9 Last, we assess the extent to which the theoretical approaches mediate the association between SEB and the dropout risk (H1c, H2c, H3c) using the nonlinear decomposition method suggested by Karlson et al. (2012).

Social Disparities in the Theoretical Predictors
To test whether our predictors vary with students' SEB, we estimate a series of linear regression models predicting students' GPA, their academic and social integration, and the perceived costs, benefits (status maintenance), and success probability from parents' educational level and class position. We choose tertiary education (ISCED levels 5 and 6) and the composite of the service and mixed classes (EGP I, II, IIIa, IV, and V) as the reference categories, which we compare to non-tertiary educated (ISCED level 0 to 4) and working class (EGP IIIb, VI, and VII) students. Table 2 shows the social disparities in GPA of both students' entrance certificate and during their studies. The results for the GPA of the entrance certificate confirm our expectations (H1a): First-generation students enter tertiary education with, on average, one fifth Others have suggested to model similar data within the framework of event history analysis (Willett & Singer, 1991). This method centers on the question when an event occurs and models the time to that event.

Social Disparities in Academic Performance
We are, however, neither interested in the duration of studies nor in the timing of dropout (or graduation) but in the final academic outcome. Therefore, we follow previous research on social stratification in higher education (e.g., Barone et al., 2018;Daniel & Watermann, 2018;Schindler & Lörz, 2012) in applying (binary) logistic regression models. standard deviation lower grades (b = − 0.20; p < .001). These disadvantages are less pronounced with respect to parents' social class (b = − 0.11; p < .01). Turning to the first GPA during studies, we find that the differences according to parents' education are much lower (b = − 0.08; p < .01) whereas the differences according to social class are similar to those of the entrance certificate (b = -0.10; p < .05).

Social Disparities in Academic and Social Integration
The left panel of Fig. 1 depicts linear regression coefficients along with their 95% confidence intervals that indicate differences in students' academic and social integration according to their parents' educational level and class position (H2a). 10 The overall picture suggests relatively small differences in students' integration according to their SEB. Regarding academic integration, there is no evidence for social disparities in students' achievement orientations with respect to their parents' education or class position (b = 0.01; p > .05). We find slightly lower levels of achievement expectations among first-generation students (b = − 0.03; p > .05) and working-class students (b = − 0.05; p > .05) but neither difference is statistically significant (at the conventional 5% level), and we deem these differences substantively negligible. Turning to social integration, the picture for interactions with faculty basically resembles our findings for academic integration: There is no evidence for social disparities according to parents' education (b = − 0.03; p > .05) or class position (b = -0.01; p > .05). We do, however, find that first-generation students (b = − 0.10; p < .001) and working-class students (b = − 0.11; p < .05) appear to struggle to establish friendships during higher education. Overall, except for interaction with peers, these results do not confirm our hypotheses regarding social disparities in the predictors of the SIM.

Social Disparities in Costs, Benefits, and Success Probability
The right panel of Fig. 1 depicts social disparities in the perceived costs, benefits, and success probability (H2b). Compared to the results for the SIM (left panel), the overall picture suggests comparatively large differences according to students' SEB. Both first-generation students (b = 0.27; p < .001) and working-class students (b = 0.28; p < .001) perceive higher education as financially more burdensome than the respective reference groups. The pattern is similar for opportunity costs, although the respective differences according to parents education (b = 0.10; p < .001) and social class position (b = 0.15; p < .001) are only about half the size.
Regarding status maintenance, there is no strong evidence for differences in occupational aspirations according to parents' educational level (b = 0.04; p > .05) or social class (b = 0.02; p > .05). Considering that we are looking at a highly selective sample, we might expect that all higher education students aspire to similar occupations that typically require a higher education certificate.
Turning to the success probability, we find that first-generation students (b = − 0.13; p < .001) and working-class students (b = − 0.15; p < .001) perceive a lower likelihood of completing a degree. Overall, these results confirm our hypotheses regarding social disparities in the parameters of RCT.

Social Inequality in Dropout Risks
Next, we turn to our main research questions and test whether social inequality in the risk of dropping out can be attributed to differences in students' academic performance, their academic and social integration, as well as the expected costs, benefits, and success probability. We present results from binary logistic regression models choosing dropout as the reference outcome. 11 Table 3 shows the social inequality in dropout risk. In the baseline model (Table 3, Model 1), first-generation students (OR 1.53; p < .001) and working-class students (OR 1.74; p < .001) have a higher risk of dropping out, net of each other, than the respective reference groups.
Model 2 confirms that students' GPA in their entrance certificate (OR 0.48; p < .001) and their studies (OR 0.55; p < .001) is negatively associated with the risk of dropping out (H1b). Furthermore, accounting for differences in students' GPA reduces inequality with 11 We present results as odds ratios. We provide results as average marginal effects as robustness checks in Tables 11 and 12 in the appendix. We use the community-contributed software MIMRGNS (Klein, 2014) to estimate average marginal effects. respect to parental education (OR 1.26; p < .05) and social class (OR 1.51; p < .01), respectively (H1c).
In Model 3, we include measures of students' academic and social integration. As expected (H2b), we find that, net of each other, all four dimensions are negatively associated with the risk of dropping out. The association is strongest for achievement expectations (OR 0.60; p < .001). We also find that the inequality with respect to parents' education (OR 1.46; p < .001) and social class (OR 1.63; p < .01) slightly decrease when compared to Model 1, weakly supporting hypothesis H2c.
Model 4 includes students' perceived costs, benefits, and success probability, each of which is associated with dropout risks (H3b). Except for the success probability (OR Table 3 Social inequality in dropout Results from binary logistic regression models (Odds ratios). Odds ratios (exponentiated coefficients) from M = 100 imputed datasets. Raw coefficients corrected according to Karlson et al. (2012). Robust standard errors in parentheses. Ref.: reference category. GPA, SIM (student integration model), and RCT (rational choice theory) variables are standardized to zero mean and unit variance. Models include controls for type of entrance certificate, type of higher education institution, field of studies, pursued degree, migration background, gender, and age *p < .05, **p < .01, ***p < .001 0.57; p < .001), the respective associations are somewhat weaker than those for academic and social integration. Moreover contrary to our expectations, higher opportunity costs are associated with a slightly lower dropout risk (OR 0.91; p < .05). One possible ad-hoc explanation is that students might feel obliged to persist in order to justify high opportunity costs. Accounting for differences in the perceived costs, benefits, and success probability reduces the association of both parents' education (OR 1.34; p < .01) and class position (OR 1.52; p < .01) with the risk of dropping out (H3c). The respective reductions are comparable to those observed when accounting for students' GPA (Model 2). In Model 5, we include all theoretical predictors simultaneously. The overall picture still supports most of our hypotheses regarding the predictors' associations with dropout risk (H1b, H2b, and H3b). However, compared to the separate models, the odds ratios, especially those of the variables that capture students' academic performance in higher education such as GPA (OR 0.80; p > .05), achievement expectations (OR 0.72; p < .001), and success probability (OR 0.74; p < .001), are somewhat closer to 1. In contrast, the odds ratio of the GPA of the entrance certificate remains rather stable (OR 0.50; p < .001), suggesting that a substantial part of students' performance in higher education is driven by their academic performance at the end of secondary education.
In the full model (Model 5, Table 3), social inequality in dropout risk substantially decreases with respect to parents' education (OR 1.17; p > .05), which is no longer statistically significant different from 1, and social class position (OR 1.39; p < .05).
In a last step, we apply the nonlinear decomposition method suggested by Karlson et al. (2012) to the results of Model 5. 12 Table 4 disentangles the relative contribution of each theoretical approach for explaining social inequality in dropout risk. We find that social disparities in GPA account, on average, for about 27% of the social inequality in dropout. The SIM accounts for just above 9% while RCT accounts for roughly 16% of social inequality in dropout from higher education. Overall, controlling for social disparities in the theoretical predictors reduces about half of the social inequality in dropout risk.

Discussion
In our paper, we sought to move from describing social inequalities in higher education dropout to an explanatory model thereof. In this endeavor, we suggested combining the two predominant theories from higher education dropout and social stratification: Tinto's  (1975,1987) SIM and RCT (Boudon, 1974;Breen & Goldthorpe, 1997). Drawing on a nationwide, representative sample of higher education students in Germany, we tested these theories' predictions regarding dropout risks and each theory's contribution to explaining social inequality in dropout from higher education. Our results confirm the SIM's hypotheses that students' academic and social integration are associated with their dropout risk. These findings are in line with various studies from the US (Braxton et al., 1997) and add to the still limited number of empirical applications of the SIM outside of the US context (Chrysikos et al., 2017;Klein, 2019). Contrary to our expectations, we hardly find any social disparities regarding students' academic and social integration. One notable exception is that students from lower SEB struggle to establish friendships with their fellow students. Our results differ from those of others, who report social disparities in both academic and social integration (Soria & Stebleton, 2012;Soria et al., 2013). The diverging results might be due to differences in the educational systems between the US and Germany. The latter is highly selective regarding the transition into secondary education (Müller et al., 2017), which leads to a homogeneous student population in tertiary education in terms of achievement orientations and academic integration in general. Further research should investigate the effects of selectivity of educational systems on students' integration and higher education dropout.
Regarding RCT, our results confirm that the perceived costs, benefits, and success probability are associated with the dropout risk. These findings strengthen the argument that financial considerations affect students' dropout decisions (Cabrera et al., 1992) and provide an empirical underpinning of micro-level assumptions that underlie studies on the effects of financial aid on dropout (Chen & St. John, 2011;Melguizo et al., 2011). As expected, the parameters of RCT vary with students' SEB in that first-generation students and students from working-class families perceive higher costs, lower benefits, and a lower success probability.
Overall, our results suggest that both the SIM and RCT offer empirically valid explanations for students' dropout from higher education in Germany. The theoretical predictors are associated with the dropout net of each other, suggesting that the SIM and RCT are complementary. While the SIM appears to predict dropout better, RCT accounts for a higher proportion of social inequality. However, even our combined model cannot fully account for the disadvantages of working-class students, and we leave it to future research to further explore additional factors that explain the remaining social inequality in dropout.
While our paper extends previous research by offering a more comprehensive theoretical approach to explaining dropout from higher education and social inequality therein, we acknowledge certain limitations. First, our dataset suffers from panel attrition, due to which we do not observe the educational outcome of a substantial proportion of students (about 30%). While the respective complete case analyses will underestimate the overall dropout rate, the extent to which our results regarding social inequality in dropout are affected is less clear. Obviously, we cannot completely rule out bias due to unobserved confounders. However, drawing on a rich set of covariates, our imputation model recovers the officially reported dropout rate for Germany (Heublein et al., 2020) quite well. Moreover, observing several focal variables and strong predictors of degree completion, such as parental education and entrance certificate GPA for the vast majority (more than 95%) of the students, we have some confidence in our results.
Another limitation follows from our analytic approach. Focusing on the final academic outcome and disregarding the timing of dropout, we do not capture potential variations in the associations between social inequalities, students' integration, and cost-benefit considerations over the time spent in higher education. For example, one might speculate that students from lower SEB integrate more slowly into the higher education system, increasing their dropout risk primarily in earlier semesters. Given that the literature appears to be missing studies on students' integration in later semesters (Nora & Crisp, 2012, p. 243), we suggest them for future research, possibly within the framework of event history analysis (Willett & Singer, 1991).
On a similar note, while our theoretical perspective demonstrates how the SIM and RCT compensate for each other's blind spots, we have not addressed possible interrelations between the variables of both theories. For instance, Beekhoven et al. (2002) have suggested that students' integration might increase their perceived probability of success and vice-versa. Future research should investigate such interrelations and whether they account for the remaining social inequality in dropout from higher education.
With these limitations in mind, what can we learn from our results? First and foremost, combining the SIM and RCT leads to a more comprehensive understanding of both higher education dropout and social inequality therein. Students appear to be equally concerned with the intrinsic rewards from integrating into higher education communities and the expected costs and benefits from completing higher education. However, we also learn that social disparities in these factors do not fully account for the social inequality in dropout.
Finally, our results should be interpreted with the peculiarities of the German higher education context in mind. Although the SIM does not seem well suited for explaining social inequality in dropout from higher education in Germany, results might differ for higher education systems in other countries. For instance, in Germany, only a minority of the students live on campus (Middendorff et al., 2013). Because social integration appears to be more relevant for residential students (Pascarella & Chapman, 1983), the SIM might better explain social inequality in dropout in countries where more students are living on campus, such as the US. Likewise, compared to Germany, other European countries, such as the UK, the Netherlands, and Italy, charge higher tuition fees (OECD, 2021). If higher tuition fees increased social stratification during the transition into higher education (Kroth, 2015), we would expect financial considerations to be less relevant for the subsequent decision to drop out. Ultimately, the extent to which students' academic and social integration and the perceived costs, benefits, and success probability account for social inequality in dropout in different higher education systems remains an empirical question.   I, II

Table 11
Robustness check: Social inequality in dropout Results from binary logistic regression models (Average marginal effects). Average marginal effects from M = 100 imputed datasets. Coefficients corrected according to Karlson et al. (2012). Robust standard errors in parentheses. Ref.: reference category. GPA, SIM (student integration model), and RCT (rational choice theory) variables are standardized to zero mean and unit variance. Models include controls for type of entrance certificate, type of higher education institution, field of studies, pursued degree, migration background, gender, and age *p < .05, **p < .01, ***p < .001