## Tensor-variate finite mixture modeling for the analysis of university professor remuneration.(English)Zbl 1477.62376

Summary: There has been a long-standing interest in the analysis of university professor salary data. The vast majority of the publications on the topic employ linear regression models in an attempt to predict individual salaries. Indeed, the administration of any academic institution is interested in adequately compensating the faculty to attract and keep the best specialists available on the market. However, higher administration and legislators are not concerned with the matter of individual compensation and need to have a bigger picture for developing university strategies and policies. This paper is the first attempt to model university compensation data at the institutional level. The analysis of university salary patterns is a challenging problem due to the heterogeneous, skewed, multiway and temporal nature of the data. This paper aims at addressing all the above-mentioned issues by proposing a novel tensor regression mixture model and applying it to the data set obtained from the American Association of University Professors. The utility of the developed model is illustrated on addressing several important questions related to gender equity and peer institution comparison.

### MSC:

 62P20 Applications of statistics to economics 62H30 Classification and discrimination; cluster analysis (statistical aspects)

MatTransMix
Full Text:

### References:

 [1] Ashraf, J. and Shabbir, T. (2006). Are there racial differences in faculty salaries? J. Econ. Finance 30 306-316. [2] Banfield, J. D. and Raftery, A. E. (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics 49 803-821. · Zbl 0794.62034 [3] Basser, P. J. and Pajevic, S. (2003). A normal distribution for tensor-valued random variables: Applications to diffusion tensor MRI. IEEE Trans. Med. Imag. 22 785-794. [4] Basu, S., Banerjee, A. and Mooney, R. J. (2004). Active semi-supervision for pairwise constrained clustering. In Proceedings of the Fourth SIAM International Conference on Data Mining 333-344. SIAM, Philadelphia, PA. [5] Becker, G. S. (1975). Front matter, human capital: A theoretical and empirical analysis, with special reference to education. In Human Capital: A Theoretical and Empirical Analysis, with Special Reference to Education 1-22 2nd ed. NBER. [6] Becker, W. E. and Toutkoushian, R. K. (2003). Measuring gender bias in the salaries of tenured faculty members. New Directions for Institutional Research 117 5-20. [7] Biernacki, C., Celeux, G. and Govaert, G. (2003). Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models. Comput. Statist. Data Anal. 413 561-575. · Zbl 1429.62235 [8] Box, G. E. P. and Cox, D. R. (1964). An analysis of transformations. (With discussion). J. Roy. Statist. Soc. Ser. B 26 211-252. · Zbl 0156.40104 [9] Browne, R. P. and McNicholas, P. D. (2015). A mixture of generalized hyperbolic distributions. Canad. J. Statist. 43 176-198. · Zbl 1320.62144 [10] Cohn, E. (1973). Factors affecting variations in faculty salaries and compensation in institutions of higher education. The Journal of Higher Education 44 124-136. [11] DeLorme, C. D. J., Hill, R. C. and Wood, N. J. (1979). Analysis of a quantitative method of determining faculty salaries. Journal of Economic Education 11 20-25. [12] Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1-38. · Zbl 0364.62022 [13] Fairweather, J. S. (1993). Faculty reward structures: Toward institutional and professional homogenization. Res. High. Educ. 34 603-623. [14] Ferreira, A. P. (2013). Are all faculty members being compensated equally? A multi-method approach to investigating faculty salary. Open Access Dissertation. [15] Franczak, B. C., Browne, R. P. and McNicholas, P. D. (2014). Mixtures of shifted asymmetric Laplace distributions. IEEE Trans. Pattern Anal. Mach. Intell. 36 1149-1157. [16] Friedman, M. and Kuznets, S. (1945). Income from Independent Professional. National Bureau of Economic Research, New York. [17] Gallaugher, M. P. B. and McNicholas, P. D. (2017). A matrix variate skew-$$t$$ distribution. Stat 6 160-170. [18] Gallaugher, M. P. B. and McNicholas, P. D. (2018). Finite mixtures of skewed matrix variate distributions. Pattern Recognit. 80 83-93. [19] Gallaugher, M. P. B. and McNicholas, P. D. (2019). Package MatSkew: Matrix skew-$$t$$ parameter estimation. [20] Gallaugher, M. P. B. and McNicholas, P. D. (2019c). Three skewed matrix variate distributions. Statist. Probab. Lett. 145 103-109. · Zbl 1414.62173 [21] Hearn, J. C. (1999). Pay and performance in the university: An examination of faculty salaries. The Review of Higher Education 22 391-410. [22] Hexter, H. (1990). Faculty salaries in perspective. Research Briefs 1. [23] Hill, M. O. (1973). Diversity and evenness: A unifying notation and its consequences. Ecology 54 427-432. [24] Lee, S. X. and McLachlan, G. J. (2013a). Model-based clustering and classification with non-normal mixture distributions. Stat. Methods Appl. 22 427-454. · Zbl 1332.62209 [25] Lee, S. X. and McLachlan, G. J. (2013b). On mixtures of skew normal and skew $$t$$-distributions. Adv. Data Anal. Classif. 7 241-266. · Zbl 1273.62115 [26] Lin, T.-I., Ho, H. J. and Lee, C.-R. (2014). Flexible mixture modelling using the multivariate skew-$$t$$-normal distribution. Stat. Comput. 24 531-546. · Zbl 1325.62113 [27] Lo, K. and Gottardo, R. (2012). Flexible mixture modeling via the multivariate $$t$$ distribution with the Box-Cox transformation: An alternative to the skew-$$t$$ distribution. Stat. Comput. 22 33-52. · Zbl 1322.62173 [28] Manceur, A. M. and Dutilleul, P. (2013). Maximum likelihood estimation for the tensor normal distribution: Algorithm, minimum sample size, and empirical bias and dispersion. J. Comput. Appl. Math. 239 37-49. · Zbl 1255.65029 [29] Manly, B. F. J. (1976). Exponential data transformations. Biometrics Unit 25 37-42. [30] Melguizo, T. and Strober, M. H. (2007). Faculty salaries and the maximization of prestige. Res. High. Educ. 48 633-668. [31] Melnykov, V., Melnykov, I. and Michael, S. (2016). Semi-supervised model-based clustering with positive and negative constraints. Adv. Data Anal. Classif. 10 327-349. · Zbl 1414.62255 [32] Melnykov, V. and Zhu, X. (2018). On model-based clustering of skewed matrix data. J. Multivariate Anal. 167 181-194. · Zbl 1395.62165 [33] Melnykov, V. and Zhu, X. (2019). Studying crime trends in the USA over the years 2000-2012. Adv. Data Anal. Classif. 13 325-341. · Zbl 1459.62220 [34] Mincer, J. (1958). Investment in human capital and personal income distribution. J. Polit. Econ. 66 281-302. [35] Mohanty, D., Dodder, R. and Karman, T. (1986). Faculty salary analysis by region, rank, and discipline from 1977-1978 to 1983-1984. Res. High. Educ. 24 304-317. [36] O’Hagan, A., Murphy, T. B., Gormley, I. C., McNicholas, P. D. and Karlis, D. (2016). Clustering with the multivariate normal inverse Gaussian distribution. Comput. Statist. Data Anal. 93 18-30. · Zbl 1468.62151 [37] Perna, L. W. (2001). Sex differences in faculty salaries: A cohort analysis. The Review of Higher Education 24 283-307. [38] Rippner, J. A. and Toutkoushian, R. K. (2015). The ‘Big Bang’ in public and private faculty salaries. Journal of Education Finance 41 103-123. [39] Sarkar, S., Melnykov, V. and Zhu, X. (2021). Supplement to “Tensor-variate finite mixture modeling for the analysis of university professor remuneration.” https://doi.org/10.1214/20-AOAS1420SUPP [40] Sarkar, S., Zhu, X., Melnykov, V. and Ingrassia, S. (2020). On parsimonious models for modeling matrix data. Comput. Statist. Data Anal. 142 106822, 26. · Zbl 07135537 [41] Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461-464. · Zbl 0379.62005 [42] Shental, N., Bar-Hillel, A., Hertz, T. and Weinshall, D. (2003). Computing Gaussian mixture models with EM using equivalence constraints. In Advances in NIPS 15. · Zbl 1161.68775 [43] Simpson, W. B. (1981). Faculty salary structure for a college or university. The Journal of Higher Education 52 219-236. [44] Snyder, J. K., Hyer, P. B. and McLaughlin, G. W. (1994). Faculty salary equity: Issues and options. Res. High. Educ. 35 1-19. [45] Toutkoushian, R. K., Bellas, M. L. and Moore, J. V. (2007). The interaction effects of gender, race, and marital status on faculty salaries. The Journal of Higher Education 78 572-601. [46] Umbach, P. D. (2007). Gender equity in the academic labor market: An analysis of academic disciplines. Res. High. Educ. 48 169-192. [47] Viroli, C. (2011a). Finite mixtures of matrix normal distributions for classifying three-way data. Stat. Comput. 21 511-522. · Zbl 1221.62083 [48] Viroli, C. (2011b). Model based clustering for three-way data structures. Bayesian Anal. 6 573-602. · Zbl 1330.62262 [49] Viroli, C. (2012). On matrix-variate regression analysis. J. Multivariate Anal. 111 296-309. · Zbl 1259.62060 [50] White, A. and Murphy, T. B. (2016). Exponential family mixed membership models for soft clustering of multivariate data. Adv. Data Anal. Classif. 10 521-540. · Zbl 1414.62284 [51] Yeo, I.-K. and Johnson, R. A. (2000). A new family of power transformations to improve normality or symmetry. Biometrika 87 954-959. · Zbl 1028.62010 [52] Zhu, X. and Melnykov, V. (2018). Manly transformation in finite mixture modeling. Comput. Statist. Data Anal. 121 190-208. · Zbl 1469.62184 [53] Zhu, X. and Melnykov, V. (2020). MatTransMix: An R package for clustering matrices. R package version 0.1.9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.