Log Normal Regression and its Application

Ni Luh Sri Diantini; Calvin Riswandi

doi:10.20956/j.v21i3.42579

Authors

Ni Luh Sri Diantini Matana University
Calvin Riswandi Matana University

DOI:

https://doi.org/10.20956/j.v21i3.42579

Keywords:

Log normal regression, maximum likelihood estimation, maximum likelihood ratio test, number of poor people

Abstract

Log Normal Regression and its Application

Ni Luh Sri Diantini¹, Calvin Riswandi¹

¹Statistics Department Matana University

Email: ¹niluh.sridiantini@matanauniversity.ac.id,

¹calvin.riswandi@student.matanauniversity.ac.id

Received: 19 December 2024, revised: 17 March 2025, accepted: 24 March 2025

Abstract

The log-normal distribution represents a type of continuous probability distribution that is characterized by a positive skew, which signifies a long tail on the right. A log-normal distribution describes a statistical distribution of values that have been logarithmically transformed from a related normal distribution. In situations where predictor variables affect positive outcomes, log-normal regression becomes significant. This research will construct a regression model that utilizes a continuous response variable following a log-normal distribution, known as log-normal regression (LNR). The purpose of developing the LNR model is to overcome the assumptions in classical regression that are often unmet, such as normality, since not all data can full fill these assumptions. Therefore, need an alternative method that does not require the normality assumption. One method that can be employed is a regression with a specific distribution approach, such as the log-normal distribution, known as log-normal regression. The LNR model will be developed through the maximum likelihood estimation (MLE) approach to achieve parameter estimation using a numerical method based on the Newton-Raphson iteration. Following this, hypothesis testing will be performed using the maximum likelihood ratio test (MLRT) and a partial test that employs the Wald test. The ultimate objective of this research is to illustrate how to apply the proposed LNR model to real data. LNR model will be applied to analysis the number of poor people in Indonesia, examining the factors that contribute to this issue. The results obtained in this study that variables human development index, unemployment rate, percentage of gross regional domestic product, and minimum wage in provinces influencing significance the number of poor people in Indonesia.

References

[1] Adji, A., Hidayat, T., Tuhiman, H., Kurniawati, S., dan Maulana, A., 2020. Pengukuran Garis Kemiskinan di Indonesia: Tinjauan Teoretis dan Usulan Perbaikan. Jakarta: Tim Nasional Percepatan Penanggulangan Kemiskinan.

[2] BPS Indonesia, S. I., 2023. Catalog : 1101001. Statistik Indonesia 2023, 1101001, 790. https://www.bps.go.id/publication/2020/04/29/e9011b3155d45d70823c141f/statistik-indonesia-2020.html

[3] Budinirmala, K., Purhadi, & Choiruddin, A., 2024. Parameter Estimation and Hypothesis Testing on Bivariate Log-Normal Regression Models. KnE Life Sciences, 2024, 177–185. https://doi.org/10.18502/kls.v8i1.15546

[4] Diantini, N. L. S., Purhadi, & Choiruddin, A., 2023. Parameter estimation and hypothesis testing on three parameters log normal regression. The 8Th International Conference and Workshop on Basic and Applied Science (Icowobas) 2021, 2554, 030024. https://doi.org/10.1063/5.0104443

[5] Diwakar, R., 2017. An evaluation of normal versus lognormal distribution in data description and empirical analysis. Practical Assessment, Research and Evaluation, 22(13), 1–15.

[6] Gustavsson, S., Fagerberg, B., Sallsten, G., & Andersson, E. M. (2014). Regression models for log-normal data: Comparing different methods for quantifying the association between abdominal adiposity and biomarkers of inflammation and Insulin Resistance. International Journal of Environmental Research and Public Health, 11(4), 3521–3539. https://doi.org/10.3390/ijerph110403521

[7] Gustavsson, S. M., Johannesson, S., Sallsten, G., & Andersson, E. M., 2012. Linear Maximum Likelihood Regression Analysis for Untransformed Log-Normally Distributed Data. Open Journal of Statistics, 02(04), 389–400. https://doi.org/10.4236/ojs.2012.24047

[8] Ibrahim, S., Al-Kassab, M. M., & Al-Awjar, M. Q., 2023. Investigating Multicollinearity in Factors Affecting Number of Born Children in Iraq. Springer Proceedings in Mathematics and Statistics, 418(2), 81–91. https://doi.org/10.1007/978-981-99-0447-1_7

[9] Krishnamoorthy, K., 2006. Handbook of statistical distributions with applications. Chapman and Hall/CRC.

[10] Moral, J., & Rubia, D., 2024. Lognormal distribution for social researchers : A probability classic. 16(June), 10–25. https://doi.org/10.5897/IJPC2024.0702

[11] Nadhim, A. I., & Al-Jilawi, A. S., 2022. The bridge between Newton’s method and Newton-Raphson’s method in numerical optimization. International Journal of Health Sciences, 6(April), 5249–5267. https://doi.org/10.53730/ijhs.v6ns1.6042

[12] Neamvonk, J., & Phuenaree, B., 2022. Assessment of Anderson-Darling and their Modified Tests for right skewed distribution. International Journal of Mathematics and Computer Science, 17(3), 1327–1339.

[13] Praja, R. B., Muchtar, M., & Sihombing, P. R., 2023. Analisis Pengaruh Indeks Pembangunan Manusia, Laju Pertumbuhan Penduduk, dan Tingkat Pengangguran Terbuka terhadap Kemiskinan di DKI Jakarta. Ecoplan, 6(1), 78–86. https://doi.org/10.20527/ecoplan.v6i2.656

[14] Roosyidah, N. A. N., Choiruddin, A., & Zain, I., 2024. Predicting household per capita expenditure by using log-normal regression in Poso Regency Central Sulawesi. In Procedia Computer Science (Vol. 234, pp. 512–519). https://doi.org/10.1016/j.procs.2024.03.034

[15] Sambuaga, O. R., Kalangi, J. B., & Siwu, H. F. D., 2024. Analisis Faktor-Faktor Yang Mempengaruhi Jumlah Penduduk Miskin Provinsi Sulawesi Utara. Jurnal Berkala Ilmiah Efisiensi, 24(7), 1-14.

[16] Shrestha, N., 2020. Detecting Multicollinearity in Regression Analysis. American Journal of Applied Mathematics and Statistics, 8(2), 39–42. https://doi.org/10.12691/ajams-8-2-1

[17] Siegrist, K., 2017. Probability, mathematical statistics, stochastic processes.

[18] Wang, C., & Zhu, H., 2024. Tests of fit for the power function lognormal distribution. PLoS ONE, 19(2 February), 1–18. https://doi.org/10.1371/journal.pone.0298309

[19] Zuanetti, D. A., Diniz, C., & Leite, J. G., 2006. a Lognormal Model for Insurance Claims Data. REVSTAT-Statistical Journal, 4(2), 131–142.