Analysis of tips received in restaurants using a GAMLSS approach


  • Elias Sabe Universidade Federal de Lavras
  • Viviane Silva Universidade Federal de Lavras
  • Luiz Nakamura Escola Superior de Agricultura "Luiz de Queiroz" - Universidade de São Paulo
  • Andréa Konrath Universidade Federal de Santa Catarina
  • Thiago Ramires Universidade Tecnológica Federal do Paraná


Distributional regression, Variability, Waiter


The main aim of this paper is to analyse the amount of tips obtained by a waiter in a restaurant, in dollars, while taking into account other collected variables within the establishment. In total, 244 observations were analysed, and six extra covariates were obtained in addition to the received tips (target variable), one numerical covariate (total bill in dollars) and five factors: payer gender (two levels: male; female), smoking status (two levels: yes; no), day of the week (four levels: Thursday; Friday; Saturday; Sunday), meal type (two levels: lunch; dinner), and number of people at the table (three levels: 1 or 2; 3; 4 or more). The generalised additive models for location, scale, and shape (GAMLSS) were considered due to its high flexibility. Given the asymmetric nature of the tip variable, three distributions were considered to explain the response: gamma, inverse Gaussian, and Box-Cox Cole and Green (BCCG). A stepwise-based procedure was utilised in the covariate selection process for each of the distribution parameters, and the best models for each distribution were compared using the Akaike information criterion (AIC). The model based on the BCCG distribution returned the lowest AIC and based on a residual analysis, it was found to be suitable for explaining the dataset under study.


AKAIKE, H. A new look at the statistical model identification. IEEE Transactions on Automatic Control, v. 19, n. 6, p. 716--723, 1974.

BANN, D.; WRIGHT, L.; COLE, T. J. Risk factors relate to the variability of health outcomes as well as the mean: a GAMLSS tutorial. eLife, v. 11, p. e72357, 2022.

BRYANT, P. G.; SMITH, M. A. Practical Data Analysis. Chicago: Irwin. 1995.

COLE, T. J.; GREEN, P. J. Smoothing reference centile curves: the LMS method and penalized likelihood. Statistics in Medicine, v. 11, n. 10, p. 1305--1319, 1992.

DE BASTIANI, F.; RIGBY, T. A.; STASINOPOULOS, D. M.; CYSNEIROS, A. H. M. A.; URIBE-OPAZO, M. A. Gaussian Markov random field spatial models in GAMLSS. Journal of Applied Statistics, v. 45, n. 1, p. 168--186, 2018.

DUNN, P. K.; SMYTH, G. K. Randomized quantile residuals. Journal of Computational and Graphical Statistics, v. 5, n. 3, p. 236--244, 1996.

EILERS, P. H. C.; MARX, B. D. Flexible smoothing with $B$-splines and penalties. Statistical Science, v. 11, n. 2, p. 89--121, 1996.

HASTIE, T. J.; TIBSHIRANI, R. J. Generalized Additive Models. Boca Raton: CRC Press. 1990.

HELLER, G. Z.; ROBLEDO, K. P.; MARSCHNER, I. C. Distributional regression in clinical trials: treatment effects on parameters other than the mean. BMC Medical Research Methodology, v. 22, p. 56, 2022.

MIRUGWE, A. Restaurant Tipping Linear Regression Model. Disponível em: url{ 2020. Acesso em: 30/09/2023.

NAKAMURA, L. R.; RAMIRES, T. G.; RIGHETTO, A. J.; PESCIM, R. R.; ROQUIM, F. V.; SAVIAN, T. V.; STASINOPOULOS, D. M. Cattle reference growth curves based on centile estimation: a GAMLSS approach. Computers and Electronics in Agriculture, v. 192, p. 106572, 2022a.

NAKAMURA, L. R.; RAMIRES, T. G.; RIGHETTO, A. J.; SILVA, V. C.; KONRATH, A. C. Using the Box-Cox family of distributions to model censored data:a distributional regression approach. Brazilian Journal of Biometrics, v. 40, n. 4, p. 407--414, 2022b.

NAKAMURA, L. R.; RIGBY, R. A.; STASINOPOULOS, D. M.; LEANDRO, R. A.; VILLEGAS, C.; PESCIM, R. R. Modelling location, scale and shape parameters of the Birnbaum-Saunders generalized $t$ distribution. Journal of Data Science, v. 15, n. 2, p. 221--237, 2017.

NELDER, J. A.; WEDDERBURN, R. W. M. Generalized linear models. Journal of the Royal Statistical Society: Series A, v. 135, n. 3, p. 370--384, 1972.

R CORE TEAM. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. 2023. ISBN 3-900051-07-0, URL url{

RAMIRES, T. G.; NAKAMURA, L. R.; RIGHETTO, A. J.; KONRATH, A. C.; PEREIRA, C. A. B. Incorporating clustering techniques into GAMLSS. Stats, v. 2021, n. 4, p. 916--930, 2021.

RAMIRES, T. G.; NAKAMURA, L. R.; RIGHETTO, A. J.; PESCIM, R. R.; MAZUCHELI, J.; CORDEIRO, G. M. A new semiparametric Weibull cure rate model: fitting different behaviors within GAMLSS. Journal of Applied Statistics, v. 46, n. 15, p. 2744--2760, 2019.

RAMIRES, T. G.; NAKAMURA, L. R.; RIGHETTO, A. J.; PESCIM, R. R.; MAZUCHELI, J.; RIGBY, R. A.; STASINOPOULOS, D. M. Validation of stepwise-based procedure in GAMLSS. Journal of Data Science, v. 19, n. 1, p. 96--110, 2021.

RIBEIRO, L. F. N. Gorjetas: uma Breve Análise Jurídica sob o Prisma do Direito Consumerista e Trabalhista. Trabalho de Conclusão de Curso (Graduação em Direito) -- Faculdade de Direito do Recife, Universidade Federal de Pernambuco, p. 60. 2017.

RIBEIRO, T. F.; SEIDEL, E. J.; GUERRA, R. R.; PEÑA-RAMÍREZ, F. A.; SILVA, A. M. Soybean production value in the Rio Grande do Sul under the GAMLSS framework. Communications in Statistics: Case Studies, Data Analysis and Applications, v. 7, n. 2, p. 146--165, 2021.

RIGBY, R. A.; STASINOPOULOS, D. M. Generalized additive models for location, scane and shape. Journal of the Royal Statistical Society -- Series C: Applied Statistics, v. 54, n. 3, p. 507--554, 2005.

RIGBY, R. A.; STASINOPOULOS, M. D.; HELLER, G. Z.; DE BASTIANI, F. Distributions for Modeling Location, Scale, and Shape: Using GAMLSS in R. Boca Raton: CRC Press. 2019.

SANVOZO, C. C. Discurso polêmico em torno da lei da gorjeta. Discursividades, v. 12, n. 1, p. e-121230, 2023.

STASINOPOULOS, D. M.; RIGBY, R. A.; DE BASTIANI, F. Principal component regression in GAMLSS applied to Greek–German government bond yield spreads. Statistical Modelling, v. 22, n. 1--2, p. 127--145, 2022.

STASINOPOULOS, M. D.; RIGBY, R. A.; HELLER, G. Z.; VOUDOURIS, V.; DE BASTIANI, F. Flexible Regression and Smoothing: Using GAMLSS in R. Boca Raton: CRC Press. 2017

VAN BUUREN, S.; FREDRIKS, M. Worm plot: a simple diagnostic device for modelling growth reference curves. Statistics in Medicine, v. 20, n. 8, p. 1259--1277, 2001.

WADA, Y.; GOTO, N.; KITAGUCHI, Y.; YASUO, M.; HANAOKA, M. Referential equations for pulmonary diffusing capacity using GAMLSS models derived from Japanese individuals with near-normal lung function. PLOS ONE, v. 17, n. 7, p. e0271129, 2022.



How to Cite

Sabe, E., Silva, V., Nakamura, L., Konrath, A., & Ramires, T. (2023). Analysis of tips received in restaurants using a GAMLSS approach. Sigmae, 12(3), 82–92. Retrieved from