Advertisement

Statistical Methods & Applications

, Volume 28, Issue 4, pp 749–767 | Cite as

Zero-one augmented beta and zero-inflated discrete models with heterogeneous dispersion for the analysis of student academic performance

  • Hildete P. PinheiroEmail author
  • Rafael P. Maia
  • Eufrásio A. Lima Neto
  • Mariana Rodrigues-Motta
Original Paper
  • 141 Downloads

Abstract

The purpose of this work is to present suitable statistical methods to study the performance of undergraduate students based on the incidence/proportion of failed courses/subjects. Three approaches are considered: first, the proportion of failed subjects is modeled considering a zero-one augmented beta distribution; second, discrete models are used to model the probability of failing subjects with logit link; third, incidence is modeled using regression for count data with log link and the logarithm of the total number of subjects as an offset. Zero-inflated versions are used to account for the excess of zeros in the data when appropriate and we also considered the heterogeneous dispersion parameter, when applicable. Overall, the zero-inflated negative binomial and zero inflated beta-binomial models, with regression on the mean and the dispersion parameters, present good measures of goodness of fit to the data. The database consists of records of Engineering major students who entered the State University of Campinas, Brazil, from 2000 to 2005. Entrance exam scores and demographic variables as well as socio-economic status are considered as covariates in the models.

Keywords

Academic performance Heteroscedasticity Quantile residuals Residual analysis Overdispersion Zero-inflated discrete models Zero-one augmented beta models 

Mathematics Subject Classification

62-07 62J12 

Notes

Acknowledgements

The authors are grateful to the two anonymous referees and the Associate Editor for their careful revision, valuable suggestions, and comments which improved this paper. We also would like to thank the Espaço da Escrita - Pró-Reitoria de Pesquisa/UNICAMP - by the language services provided, Conselho Nacional de Desenvolvimento Cientíífico e Tecnológico (CNPq) [grant 308583/2015-9 to H.P.P.] and Fundação de Amparo à Pesquisa do Estado de São Paulo (Fapesp) [grants 11/15047-7 and 14/03043-5 to H.P.P.].

References

  1. Adelfio G, Boscaino G (2016) Degree course change and student performance: a mixed-effect model approach. J Appl Stat 43(1):3–15MathSciNetCrossRefGoogle Scholar
  2. Adelfio G, Boscaino G, Capursi V (2014) A new indicator for higher education student performance. High Educ 68:653–668CrossRefGoogle Scholar
  3. Attanasio M, Boscaino G, Capursi V, Plaia A (2013) Can students’ career be helpful in predicting an increase in universities income? In: Giudici P, Ingrassia A, Vichi M (eds) Statistical models for data analysis. Studies in classification, data analysis and knowledge organization. Springer, Switzerland, pp 9–16Google Scholar
  4. Bayer F, Cribari-Neto F (2017) Model selection criteria in beta regression with varying dispersion. Commun Stat Simul Comput 46(1):729–746MathSciNetCrossRefGoogle Scholar
  5. Bianconcini S, Cagnone S (2012) Multivariate latent growth models for mixed data with covariate effects. Commun Stat Theory Methods 41:3079–3093MathSciNetCrossRefGoogle Scholar
  6. Birch ER, Miller P (2006) Student outcome at university in australia: a quantile regression approach. Aust Econ Pap 45:1–17CrossRefGoogle Scholar
  7. Chambers JM, Cleveland WS, Kleiner B, Tukey PA (1983) Tutorial on methods for interval-censored data and their implementation in R. In: Graphical methods for data Analysis. Wadsworth, Belmont, CAGoogle Scholar
  8. Dunn PK, Smyth GK (1996) Randomized quantile residuals. J Comput Graph Stat 5:236–244Google Scholar
  9. Ferrari SLP, Cribari-Neto F (2004) Beta regression for modeling rates and proportions. J Appl Stat 7:799–815CrossRefGoogle Scholar
  10. Grilli L, Rampichini C, Varriale A (2015) Binomial mixture modelling of university credits. Commun Stat Theory Methods 44:4866–4879CrossRefGoogle Scholar
  11. Grilli L, Rampichini C, Varriale A (2016) Statistical modelling of gained university credits to evaluate the role of pre-enrollment assessment tests: an approach based on quantile regression for counts. Stat Model 16:47–66CrossRefGoogle Scholar
  12. Maia RP, Pinheiro HP, Pinheiro A (2016) Academic performance of students from entrance to graduation via quasi u-statistics: a study at a brazilian research university. J Appl Stat 43(1):72–86MathSciNetCrossRefGoogle Scholar
  13. Masserini L, Bini M, Pratesi M (2017) Effectiveness of non-selective evaluation test scores for predicting first-year performance in university career: a zero-inflated beta regression approach. Qual Quant Int J Methodol 51(2):693–708CrossRefGoogle Scholar
  14. McCullagh P, Nelder JA (1992) Generalized linear models. Chapman & Hall, LondonzbMATHGoogle Scholar
  15. Murray-Harvey R (1993) Identifying characteristics of successful tertiary students using path analysis. Aust Educ Res 20:63–81CrossRefGoogle Scholar
  16. Murtagh F, Legendre P (2014) Ward’s hierarchical agglomerative clustering method: which algorithms implement ward’s criterion? J Classif 31:274–295MathSciNetCrossRefGoogle Scholar
  17. Ospina R, Ferrari SLP (2010) Inflated beta distributions. Stat Pap 51:111–126MathSciNetCrossRefGoogle Scholar
  18. Ospina R, Ferrari SLP (2012) A general class of zero-or-one inflated beta regression models. Comput Stat Data Anal 56:1609–1623MathSciNetCrossRefGoogle Scholar
  19. Pedrosa RHL, Dachs JNW, Maia RP, Andrade CY, Carvalho BS (2007) Academic performance, student’s background and affirmative action at a Brazilian research university. High Educ Manag Policy 19(3):1–20CrossRefGoogle Scholar
  20. Pinheiro A, Sen PK, Pinheiro HP (2009) Decomposability of high-dimensional diversity measures: quasi u-statistics, martingales and nonstandard asymptotics. J Multivar Anal 100(8):1645–1656MathSciNetCrossRefGoogle Scholar
  21. Pinheiro A, Sen PK, Pinheiro HP (2011) A class of asymptotically normal degenerate quasi u-statistics. Ann Inst Math Stat 63:1165–1182MathSciNetCrossRefGoogle Scholar
  22. Pinheiro HP, Sen PK, Pinheiro A, Kiihl S (2018) A nonparametric approach to asses undergraduate performance. arXiv:181000678 [statME] (submitted)
  23. R Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/
  24. Rigby RA, Stasinopoulos DM (2005) Generalized additive models for location, scale and shape. J R Stat Soc Ser C (Appl Stat) 54(3):507–554MathSciNetCrossRefGoogle Scholar
  25. Rigby RA, Stasinopoulos DM, Heller G, Voudouris V (2014) The distribution toolbox of GAMLSS. http://www.gamlss.org/
  26. Salehi M, Roudbari M (2015) Zero inflated poisson and negative binomial regression models: application in education. Med J Islam Repub Iran 29:297–297Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of CampinasCampinasBrazil
  2. 2.Department of StatisticsFederal University of ParaíbaJoão PessoaBrazil

Personalised recommendations