Within-college human capital and racial ethnic differences in academic performance
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- Bu səhifədəki naviqasiya:
- WITHIN-COLLEGE HUMAN CAPITAL AND RACIAL ETHNIC DIFFERENCES IN ACADEMIC PERFORMANCE
- WITHIN-COLLEGE CAPITAL
- Academic and Intellectual Skills
- Global and Specific Self-Esteem, Academic Self-Confidence and Identity
- Academic Effort and Engagement
- DESIGN, MEASURES AND METHODS Study Design
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WITHIN-COLLEGE HUMAN CAPITAL AND RACIAL ETHNIC DIFFERENCES IN ACADEMIC PERFORMANCE* Kenneth I. Spenner Duke University Sarah Mustillo Purdue University Nathan D. Martin Duke University April 2, 2008 Word Count: 8,978 Running Head: Within-College Human Capital and Academic Performance *Direct correspondence to Kenneth Spenner, Department of Sociology, Duke University, Durham, North Carolina, 27708; email: kspen@soc.duke.edu. The authors gratefully acknowledge the support for this research provided by grants from the Andrew W. Mellon Foundation and Duke University. The authors bear sole responsibility for the contents of the paper. Ken Spenner is Professor of Sociology and Psychology, and Director of the Markets & Management Studies Program at Duke University. Over the years his research has focused on careers, work and personality, technology and market transitions in Eastern Europe. In recent years his research centers various aspects of undergraduate education at an elite university, using the Campus Life and Learning data.
ABSTRACT
WITHIN-COLLEGE HUMAN CAPITAL AND RACIAL ETHNIC DIFFERENCES IN ACADEMIC PERFORMANCE Gaps in academic performance among racial and ethnic groups persist at all levels of education in the United States. Bowen and Bok’s (1998) seminal study refocused attention on such gaps in higher education. They found the differences are large (for example, 30 percentile points between black and white college students in the College and Beyond sample), enduring over recent decades, and perhaps even somewhat larger in the most selective institutions. Most studies that attempt to explain the gap in terms of human capital factors focus on pre-college differences in human capital and generally find that pre-college factors account for up to about one-half of the black-white gap (Bowen and Bok 1998; Massey et al. 2003; see Jencks and Phillips [1998] for review). Below we present new evidence that the academic achievement gap among racial and ethnic groups at an elite university declines substantially over the course of the college career even after controlling for pre-college factors. This raises the important question: what accounts for the decline? The research that attempts to understand the gap contains several lacunae. First, the typical study design is cross-sectional or a limited panel, and includes controls for pre-college socioeconomic and family background factors, and perhaps single-point in time measures of within-college capital such as test scores or major choice. Studied outcomes include verbal, quantitative and subject matter competence, cognitive skills and growth, grades, persistence in school, and graduation rates. Fine-grained studies of within-college capital acquisition are in more limited supply (Pascarella and Terenzini [2005] provide comprehensive review). For example, how do effort, time use and academic engagement differ over the course of college among white, black, Asian and Latino students? Do racial and ethnic differentials in academic self-concept and identities help us understand trajectories of academic achievement? Second, the major methodological challenge to such designs is unobserved heterogeneity. How do we know the design measured all of the capital factors that might explain the gap? Left out unmeasured factors will bias estimates of the gap (Allison 1994; Halaby 2004). A few studies in economics attempt to control for unobserved heterogeneity but few to none of the studies in education and sociology using college samples attempt such controls, for example with fixed-effects, panel designs (Pascarella and Terenzini 2005: 67). Our study design offers a contribution in attempting to control for unobserved heterogeneity in a multi-point panel design taken over the college years, with fixed and random effects statistical models. Our research design features a prospective panel study of two entering cohorts at an elite Southern university. A probability sample of entering students were surveyed in the summer before starting college and then in the spring semester of the first, second and fourth college years. As a dependent variable for academic achievement we use semester grade point average (GPA) in the spring semester of each survey year as taken from Registrar’s records. WITHIN-COLLEGE CAPITAL Human capital generally refers to the knowledge, skills, health and values that people possess (Becker 1975). Human capital is distinct from the financial capital and physical assets that people hold. We also hold it conceptually distinct from (but correlated with) cultural (Bourdieu 1986) and social capital (Coleman 1988), which we have or will address with our panel study elsewhere (XXXXX 2005; XXXXX 2008). For this paper, our interest is in forms of human capital that are changeable during college and that might be germane to academic performance. Our reading of the literature identifies three sub-types of human capital within college populations: (1) academic and intellectual skills; (2) global and specific (academic) self-esteem, academic self-confidence, and identity; and (3) academic effort and engagement. This conceptualization is not exhaustive but rather is aimed at capturing the major variations, and ones that are susceptible to reasonable measurement in a panel design. Academic and Intellectual Skills Pascarella and Terenzini (2005: 65-75) review a large body of research that shows an increase by a quarter to nearly a full standard deviation over the college career in a set of subject-matter and general academic competencies and skills. These skills include: understanding fundamental concepts or theories, applying knowledge and concepts, analyzing ideas and arguments, synthesizing and integrating information, oral expression and writing. The measures in various studies include both standardized and self-report measures of skills (Maes et al. 1996; Thorndike and Andrieu-Parker 1992). These skills are related to earned grades. The few studies that examine racial and ethnic differences in changes in these skills are inconsistent. For example, Myerson and colleagues (1998) find that black students experience larger gains in the Armed Forces Qualification Test (a composite of verbal and quantitative scores) compared with a cohort of white students in the National Longitudinal Study of Youth, after controlling for SES and age. In contrast, Flowers (2002) reports that black students experienced significantly smaller gains on standardized tests compared with white students, using data from the National Study of Student Learning and data from 56 institutions in the College Basic Academic Subject Examination, and a more extensive set of controls including demographics, pre-college test scores, indicators of academic effort, and social involvement in college life.
Self-esteem (global) generally refers to an individual’s attitude toward the self as a totality (Rosenberg et al. 1995). Self-esteem has been linked to academic performance in a number of studies (Osborne 1995; Massey et al. 2003; Morgan and Mehta 2004), including racial and ethnic differences in such. Few studies address within-college differences, and most studies are interested in self-esteem as an outcome rather than a cause of academic performance, a possibility that we will examine. Rosenberg and colleagues (1995) suggest that specific self-esteem -- in this case academic self-concept -- is more closely related to behavioral outcomes including academic performance as measured by grades compared with global self-esteem. Data from the Youth in Transition Study support their argument for high school boys. We will examine variations in both global and specific (academic) self-esteem. Closely related to self-esteem, academic self-confidence refers to a person’s judgment about the likelihood that their actions will lead to a successful academic outcome. Psychological research has long held (Bandura 1982) that people who are confident in their ability are more likely to endure and persist in the face of difficulty, likely a resource in challenging academic settings. Identities refer to self-in-role meanings for role incumbents and group members (i.e., parent, fraternity member, good student) (see Burke [2004] for theoretical statement and review; Reitzes and Burke [1980] specifically address the college student identity). Identities have affective, motivational and behavioral implications. For the college student identity, one key dimension of role performance is academic achievement. The college student identity might vary over the college career as a function of earlier academic performances, one’s social networks, and intellectual climate in courses and social settings. We posit that closer identification with being a good student will yield stronger academic performance and improved academic performance over time. As a corollary, those with stronger identification with a good student role should invest more time and energy in academic matters and engagement in the academic enterprise in college, which introduces the third area of within-college capital.
Research in education, economics and sociology shows that college students vary in the levels of effort and academic engagement in college and these variations are positively associated with a range of outcomes, including test scores, subject matter competencies, and other measures of academic performance and skill acquisition (Astin 1993; Pascarella and Terenzini [2005:119-120] provide review). Our focus will be on several sub-dimensions of this broader concept. Effort attributions refer to the self-inferences that students make about the role of their own effort in academic tasks, for example the extent to which they worked hard in challenging academic courses. At the level of individual challenging courses, we find that effort attributions do predict course grade outcomes (XXXXX 2006). A number of studies address the role of time use, or hours spent studying, as an indicator of effort and academic engagement (Schuman et al. 1985; Rau and Durand 2000; Stinebrickner and Stinebrickner 2004). Most studies show small-to-modest positive effects of time spent studying and in related activities on indicators of academic performance, although there is disagreement about the size of effects and the optimal measures of time use (see exchange between Schuman [2001] and Rau [2001] and detailed methodological work by Juster and Stafford [1991]). In general, multiple and more detailed measures provide larger estimates of the effects of study time on outcomes (Stinebrickner and Stinebrickner 2004). We posit that time use varies over the college career and we investigate whether such variations are related to trajectories of academic performance. One weakness of time use measures is that they do not directly capture the efficiency with which students expend time, which might vary substantially over time and is a function of a myriad of factors. Our design includes some information on effective time management, which we bring in to adjust for this limitation. Finally, the larger education literature suggests multiple dimensions of academic engagement are positively associated with knowledge acquisition and academic performance (Pascarella and Terenzini 2005). This might include not only time allocation but engagement of faculty and fellow students to work on challenging material, use of tutoring and academic skills centers and advisors, particularly for students having academic difficulty, and engaging resources at the class level that might assist in challenging classes, such as teaching assistants, organized review sessions, and peer study groups. More generally, we posit that academic engagement varies over the college career, and perhaps by racial and ethnic group, and may be associated with trajectories of academic achievement. DESIGN, MEASURES AND METHODS Study Design The larger research design, The Campus Life and Learning Project (CLL), involves a multi-year, prospective panel study of two consecutive incoming cohorts of students admitted to Duke University and who accepted admission (incoming classes of 2001 and 2002; graduating classes of 2005 and 2006). Duke is a private research university located in Durham, North Carolina with a total undergraduate enrollment of about 6,000 students from the United States and a number of foreign countries. The sampling frame included undergraduate students who planned to enroll in the college of arts & sciences and in the engineering school. The design randomly selected about one-third of white students, about two-thirds of Asian students, all black and Latino students, and about one-third of bi- and multi-racial students, based upon self-reported racial ethnic status on the admission application form.1 In contrast to studies that examine samples from multiple institutions, this study is designed to capture the rich details of students’ experiences at a single institution with multiple data points and merges of various types of institutional data, usually unavailable in other studies. The final sample for both cohorts included 1536 students (602 whites, 290 Asians, 340 blacks, 237 Latinos, and 67 bi- or multi-racial). Respondents were surveyed in the summer preceding college enrollment. About 77 percent of sample members (n = 1181) completed the mail questionnaire and about 91 percent of these respondents provided signed release to their institutional records as well. Refusals were low at 1.8 percent of sample members. Response rates to in-college waves, administered by mail and web (senior year only) were 71 percent for the first year, 65 percent for the second year, and 59 percent for the senior year. Our analysis sub-sample for this paper includes all students who participated in at least one wave and provided signed release to their institutional records (n=1255). Elsewhere we have provided detailed comparisons of non-response bias, possible drop-out bias, and patterns of missing data, and generally conclude that the effects are quite small (XXXXX 2005).2 The Appendix provides further information. Measures
Dependent Measures Our dependent measure is semester GPA as taken from Registrar’s records for the end of the spring semester of the first, second and fourth college years. The in-college surveys were administered at the start of the spring semester. A majority of those who responded had done so by Spring Break; over 95 percent of those who responded had done so before the end of the semester and the reporting of final grades. This should minimize causal order problems with independent measures taken during early in the semester relative to grades assigned at the end of the semester. In using grade performance as a dependent variable we do not assume a perfect measure of intellectual and cognitive development and learning. Grades are an imperfect measure in other respects as well. Grade standards and use might vary from academic department to department. Grades reflect a range of student inputs from prior academic achievement to personal traits, and even situational sources of variation. We use grades because they are sociologically important. As Pascarella and Terenzini (2005: 396) note, grades are the lingua franca of the undergraduate academic world. First, they are instrumental in determining student academic standing, honors, admissions to some programs, and degree completion. Second, grades represent important signals for access to post-graduate education programs and professional schools, and for job interviews and labor market placement. They are socially recognized as indicators of potential in future role positions. Third, grades potentially feed-back on a range of social psychological outcomes such as self-confidence in achievement settings, and expectations and aspirations for future educational, occupational and income achievements. Figure 1 reports the semester GPAs (on a four-point scale) by racial ethnic group for CLL respondents. Several patterns are apparent. First, CLL respondents replicate well-know national differences in college grades (Bowen and Bok 1999; Massey et al. 2003). In the first college year, Asian and white students score from one-tenth to two-tenths of a letter grade higher than bi-multi-racial and Latino students, and four-to-five tenths of a letter grade higher compared with black students. Second, racial ethnic differences in grades are at their maximum in the first college year, and the gaps narrow progressively over the remainder of the college career. The decline in the gap is quite dramatic. For example, by the second semester of the senior year, the achievement gap between black and white students has narrowed from .45 of a letter grade to .18 of a letter grade, a reduction of 60 percent. Gaps between other groups have narrowed by the senior year. Third, most of the narrowing of the gap occurs in the second and third college years, a time when students have settled into college majors, minors and certificate programs, and students’ experiences with large classes have declined or largely ended at Duke. These differences form the basis for the comparisons that follow. Time-Invariant Measures Table 1 reports descriptive statistics for the independent measures, both fixed and changing (i.e., measured at multiple time points in college), by racial ethnic group. Given the breakdown by racial ethnic category, the data in Table 1 are unweighted; model estimates that follow use weighted data. We include a standard set of fixed controls, including race, sex, mother’s and father’s education, US citizenship, and SAT mathematics score. SAT verbal scores are highly correlated with SAT mathematics scores so we use the mathematics score alone. Other time-invariant measures include four social psychological indicators of pre-college academic capital: good student identity, a two-item scale for self-perceived ability (in last challenging high school mathematics and English courses), academic self-confidence mathematics and academic self-confidence English. Also, we include a measure of time management taken in the first college year. Finally, we control for college major (natural science/engineering, social science, humanities, and undeclared; the latter category refers to a handful of students who left college prior to declaring an academic major). If a student had multiple majors that include a natural science or engineering discipline, we coded major as natural science/engineering; in other cases we coded the first major listed. Time-Varying Measures Table 1 also includes the focal nine indicators of within-college capital as measured in each survey year. Self-assessed academic skills are measured with an 8-item scale (Cronbach’s alpha: .79, .78 and .79 for the first, second and senior year scales). The items include remembering factual knowledge; understanding fundamental concepts or theories; applying knowledge, concepts or theories to a specific situation; analyzing ideas, arguments; synthesizing and integrating information; conducting research in a specific field; oral expression; and writing skills. The response scale was 5 points, ranging from very low to very high. White and bi-multi-racial students rate themselves the highest at each survey year. Interestingly, Asian student rate themselves the lowest even though they have the highest SAT scores by a clear margin, and attain higher first semester GPA than other racial ethnic groups. Self-assessed academic skills increase by two-thirds to three-quarters of a standard deviation over the college career for all racial ethnic groups, increasing the most for black students (by .85 of the senior year standard deviation). Academic self-confidence is measured by a single question in the most challenging class set of questions in each survey year (5-point scale from not all confident to extremely confident). Bi-multi-racial students report the highest levels in each survey year, while blacks and Asians in the senior year report the lowest. All groups gain in confidence over college, although the gain for Asian students is minimal. Work hard attributions also come from a single question in the most challenging class loop in each survey year. The distributions show only minor variations over time and by racial ethnic group. Active learning behaviors were measured by six binary items in the most challenging class loop (first and second year surveys only). These scales included items like studying with students from the class, receiving informal tutoring, use of review sessions, and receiving tutoring from the university’s academic skills center. Black students were somewhat more likely to engage in these behaviors; Asian students were the least likely. Good student identity was measured on a five point scale (ranging from not at all important to extremely important). Black and bi-multi-racial students consider this identity to be the most important while white students report the lowest levels by a clear margin. The importance of the good student identity declines progressively over the college career for all racial ethnic groups. Global self-esteem was measured with three items from the Rosenberg self-esteem scale (Rosenberg 1989) (Cronbach’s alpha: .66, .70 and .71 for the first, second and senior year scales). As found in other studies, black students tend to report somewhat higher levels of self-esteem compared with other racial ethnic groups. Academic (specific) self-esteem was measured with two questions asked in each wave. Self-assessed ability refers to students’ judgments about their ability compared with others in most challenging class taken in the previous semester. Self-assessed smartness refers to students judgments in each wave about how smart they are compared to the average Duke student. These two items are generally similar to academic self-concept measures used by Rosenberg and colleagues (1995) and Morgan and Mehta (2004). Finally, hours spent studying was measured with a single item (referent: a typical non-exam week) in each survey year, and was part of a larger set of time-use items. Asian and bi-multiracial students spent more time studying outside of class, particularly in the first college year; Latino students generally report the lowest levels in each survey year. Methods
For the longitudinal analyses, we estimated both latent growth curve models and fixed effects models. The latent growth curve models include a latent intercept and latent slope, which allow individuals to vary on their initial GPA and their rate of change over time. In a latent growth curve model (also known as a random coefficients model), time-specific individual-level measures are assumed to contain input from two sources: the latent process under consideration and random error. If we assume that the process of interest follows a linear pattern over time, the individual measures can be modeled with an individual-specific intercept and slope across time plus error. With time-varying covariates, the Level 1 equation is: 
where  is the response variable for individual i at time t;  i is a subject-specific intercept term;  is the subject-specific slope multiplied by time;  is the time-specific influence of covariate w for individual i at time t; and  is the disturbance for individual i at time t. This portion of the model captures within individual change over time and is equivalent to the Level 1 sub-model in the HLM framework. In a structural equation modeling framework, the variance of the errors can be fixed or forced to be equal across time. We allow them to vary.
The second level of the model allows the random intercepts and slopes to be a function of covariates. In this model, the random intercept and slope are allowed to correlate. The Level 2 equations are: 
In these equations, αi and βi are the intercept and slope for individual i and Although the distribution of the GPA variable is skewed, violating the assumption of multivariate normal distribution, we use a linear model. We tried several alternative analytic strategies, including different modeling techniques and different transformations of GPA, but the findings were similar across models. Figure 2 provides a graphic depiction of the model we estimate. The models were run with weights and robust standard errors using M Plus 5.0 (Muthén and Muthén 2007). Factor loadings on the intercept were fixed to 1.0, while the factor loadings on the slope were fixed at 0, 1, 2 and 3 for the four years of college. Within-college human capital variables were included as time-varying covariates (w), while demographics, final major, and pre-college human capital variables were included as time-invariant covariates (X). The time-invariant covariates affected both the latent intercept and the latent slope of GPA, while the time-varying covariates examined the effect of within-college capital variables on GPA, above and beyond the random growth process of GPA (Bollen and Curran 2006). We first estimated an unconditional linear growth model to examine whether the characteristics of individual trajectories of GPA varied across subjects and then we estimated the model with only racial ethnic group indicators (results not shown) to assess racial ethnic differences in GPA without any explanatory variables and whether there was significant residual variance (e.g., was there a growth process to explain?). Finally, we tested the model with the human capital variables as time-varying covariates to examine their association with deviation from each subject’s predicted GPA trajectory (Bryk and Raudenbush 1992; Curran and Hussong 2002; Curran, Muthén, and Harford 1998; Bollen and Curran 2006). We used full-information maximum likelihood estimation for our model (FIML) because not all of the subjects responded at every wave. FIML requires the assumption that data are missing at random (MAR)3. Using all available data, FIML computes a case-wise likelihood function using only those variables that are observed for individual i. Additionally, data from partially complete cases contribute to the estimation of parameters that involve missing data. Simulations have shown that under the MAR assumption, FIML performs better than list-wise deletion and multiple imputation in terms of both bias and efficiency (Enders and Bandalos 2001). We include several measures of fit (RMSEA and standardized root mean square error) and both indicate good model fit. Because random effects models assume that the disturbances from the Level 2 equations are uncorrelated with the predictors, we also present fixed effects models of GPA on the time-varying covariates. Fixed effects models do not require this assumption and are preferred in cases where correlation exists (Allison 2005; Halaby 2004).4 Our model is based on the equation Yit = αi + β1X1it + β2X2it + …+ βkXkit + eit where αi is a unique constant for each individual i that controls for time-invariant characteristics. An individual’s GPA at each time point is treated as a deviation from that individual’s mean GPA across all years of college: 
Because this treatment examines difference, we cannot include time-invariant covariates, as the term will reduce to 0; however the fixed effects account for all time-invariant characteristics of individuals. The coefficients from this model describe the within-subject effects of the time-varying covariates. Unlike the growth curve model, the fixed effect model employs list-wise deletion on a person-year basis. Therefore, we used a multiple imputation procedure to impute the missing data before we ran the fixed effect model.5 FE models were run in Stata 10 (StataCorp 2007) and include probability weights and robust standard errors. FINDINGS Growth Curve Model Table 2 presents the parameters for the random effects growth curve model for GPA. The table includes three types of parameter estimates for the effects of independent variables, both fixed and changing: estimates for effects on intercept (a starting value for GPA), estimates for overall effects on slope of GPA (how a variable affects changes in GPA over college), and in the time varying equations, effects of time varying independent variables for within-college capital on GPA in the first, second year and fourth college year, independent of expected GPA trajectory. That is, the effect of time-varying covariates can be interpreted as disturbances that shift trajectories up or down. The intercept estimates are analogous to the coefficients one sees in models of race differences in academic achievement in a cross-sectional design (or panel design that does not explicitly model change). Net of other variables in the equation, blacks start college with a .28 letter grade deficit in GPA compared with whites, and Latino and bi-multi-racial students have a one-eighth of letter grade deficit. Most of the other variables, including pre-college capital variables increase GPA at the start of the college career, including the effects of being female, higher SAT (math) scores, having high self-assessed academic abilities, and stronger time management skills. Also, eventual social science majors begin college with higher GPAs compared with eventual natural science majors, by about one-tenth of a letter grade, net of other variables in the equation. Perhaps the only surprise is the effect of pre-college self-confidence in challenging mathematics courses, which is negative. This might reflect that those with the highest mathematics skills (and confidence) often take advanced, demanding mathematics courses in the first and second college semesters, which might produce somewhat lower grades. Only one time-invariant independent variable exerts a significant effect on GPA slope (or the growth in GPA) over the college years. As we saw in Figure 1, blacks increased GPA at a faster rate compared with whites, while Asians increased at a somewhat slower rate. Black students increase GPA by .04 of a letter grade per year compared with whites. While seemingly small, the effect indexes the effects of being black on changes in GPA per year, about .16 of a letter grade when taken over the college career. This effect is comparable to the corresponding estimate taken from a fixed effects model (see below). Further, this effect is net of time-invariant, within-college capital variables as they might operate for black students. Most important to our theorizing are the time-varying equations that provide estimates of each changing within-college capital variable by year (first, second and fourth). Six of nine within-college capital variables have significant effects on GPA for at least one time point. The effects for four variables are consistent with theorizing in the literature. Importantly, most of the effects are small-to-modest in size, so within college capital does not provide an overwhelming explanation for the changes in GPA. Changes in self-reported academic skills affect GPA in the second and fourth college years but the effect is negative, not in the expected direction. Recall, most of the change in GPA is in earlier college years. Also, active learning behaviors have a significant negative effect on GPA only in the first year, contrary to our theorizing. The active learning behaviors measure was taken from a set of questions on the most challenging class, as identifying by students, in the previous semester (first and second year surveys only). It could be that the measure is detecting these behaviors for students who are struggling in the most challenging class, and hence our measure is not a particularly effective one of broader active learning behaviors. Attributions that students worked hard in challenging class situations have an expected positive effect on GPA in the first college year. Reported hard work increases changes in GPA by one-tenth of a letter grade compared with students who did not report the hard work attribution in challenging class situations, controlling for other variables. This effect is not statistically significant at conventional levels at later points in the college career. More strongly ascribing to the good student identity increases GPA. A one unit increase in the importance of being a good student is associated with .04 of a letter-grade increase in the first college year, a .03 increase in the second college year, and a positive but not significant increase in the senior year. This is consistent with our theorizing. The size of the coefficients may not seem large, but these effects are above and beyond the regular growth process in GPA, while controlling for other variables.
Changes in time spent studying outside of class do not yield significant GPA gains in any college year. This could be because our measure of time use is not specific enough. It could reflect the fact that time spent studying is fairly stable over the college years for most students (as distinct from between-student differences in study time). Recall, time management efficiency had a fairly strong positive effect on the intercept for GPA, but small negative effects on change in GPA over college. Finally, we investigated whether changes in within-college capital variables might operate differently for black students compared with other students. This seemed plausible given the rather dramatic increase in black student GPA over the college career. None of the coefficients for interaction terms for black by each within-college capital variable were significant at conventional levels (results not shown). Changes in within-college capital appear to operate similarly for black students and those from other racial ethnic groups. Table 4 summarizes the black-white achievement gap by college year, displaying the unadjusted grade gap, the adjusted gap as predicted with the growth curve model, the difference between unadjusted and adjusted grade gaps, and difference as a percentage of the unadjusted grade gap. The percentage offers one indication of the extent to which the model (and fixed and changing capital variables) accounts for the gap. We focus on the black-white comparison as this one has been most prominent in the literature and in policy debate. The unadjusted gap declines steadily through the junior year of college, and then dramatically shrinks from the end of the junior to the end of the senior year of college. Under the growth curve model, one-third to 45 percent of the gap is accounted for by the model by the junior year of college. The model is fairly efficacious in helping us understand the gap and its narrowing. By the end of the senior year, the model is much less efficacious, the gross gap has closed considerably, and the model accounts for less than 10 percent of the gap. In one sense, the dramatically smaller gap provides much less for the model to explain. Yüklə 449,5 Kb. Dostları ilə paylaş:
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and
are the means of the intercept and slope when the x variables equal zero. The remaining part of each equation sums for K time-invariant variables, the effect of each predictor on the random intercept and slope and includes a disturbance term representing deviation from the mean intercept and slope for individual i, respectively.