A Smooth Test of Goodness-of-Fit for the Weibull Distribution: An Application to an HIV Retention Data


  • Collins Odhiambo Strathmore Institute of Mathematical Sciences, Strathmore University, Ole Sangale Road, Nairobi, Kenya
  • John Odhiambo Strathmore Institute of Mathematical Sciences, Strathmore University, Ole Sangale Road, Nairobi, Kenya
  • Bernard Omolo Division of Mathematics & Computer Science, University of South Carolina-Upstate, 800 University Way, Spartanburg, South Carolina, USA




Goodness-of-fit, Loss to follow-up, Neyman's smooth test, Retention in HIV care, Weibull distribution.


In this study, we fit the two-parameter Weibull distribution to an HIV retention data and assess the fit using a smooth test of goodness-of-fit. The smooth test described here is a score test and is derived as an extension of the Neyman’s smooth test. Simulations are conducted to compare the power of the smooth test with the power of each of three empirical goodness-of-fit tests for the Weibull distribution. Results show that the smooth tests of order three and four are more powerful than the three empirical goodness-of-fit tests. For validation, we used retention data from an HIV care setting in Kenya.


Kenya AIDS Response Progress Report 2014: Progress towards Zero; 2014.

Rachlis B, Cole DC, van Lettow M, Escobar M, Muula AS, Ahmad F, et al. Follow-Up Visit Patterns in an Antiretroviral Therapy (ART) Programme in Zomba, Malawi. PloS one 2014; 9(7): e101875. DOI: https://doi.org/10.1371/journal.pone.0101875

Rasschaert F, Koole O, Zachariah R, Lynen L, Manzi M, Van Damme W. Short and long term retention in antiretroviral care in health facilities in rural Malawi and Zimbabwe. BMC Health Services Research 2012; 12(1): 444. https://doi.org/10.1186/1472-6963-12-444 DOI: https://doi.org/10.1186/1472-6963-12-444

Asiimwe SB, Kanyesigye M, Bwana B, Okello S, Muyindike W. Predictors of dropout from care among HIV-infected patients initiating antiretroviral therapy at a public sector HIV treatment clinic in sub-Saharan Africa. BMC Infectious Diseases 2016; 16(1): 43. https://doi.org/10.1186/s12879-016-1392-7 DOI: https://doi.org/10.1186/s12879-016-1392-7

Haddow L, Edwards S, Sinka K, Mercey D. Patients lost to follow up: experience of an HIV clinic. Sexually Transmitted Infections 2003; 79(4): 349-350. https://doi.org/10.1136/sti.79.4.349-b DOI: https://doi.org/10.1136/sti.79.4.349-b

Hahn GJ, Shapiro SS. Statistical models in engineering. In: Statistical models in engineering. John Wiley & Sons 1968.

Cousineau D. Fitting the three-parameter Weibull distribution: Review and evaluation of existing and new methods. IEEE Transactions on Dielectrics and Electrical Insulation 2009; 16(1): 281-288. https://doi.org/10.1109/TDEI.2009.4784578 DOI: https://doi.org/10.1109/TDEI.2009.4784578

Neyman J. Smooth test for goodness of fit. Scandinavian Actuarial Journal 1937; 1937(3-4): 149-199. https://doi.org/10.1080/03461238.1937.10404821 DOI: https://doi.org/10.1080/03461238.1937.10404821

Rayner JC, Thas O, Best DJ. Smooth tests of goodness of fit: using R. John Wiley & Sons 2009. DOI: https://doi.org/10.1002/9780470824443

Ledwina T. Data-driven version of Neyman's smooth test of fit. Journal of the American Statistical Association 1994; 89(427): 1000-1005. https://doi.org/10.1080/01621459.1994.10476834 DOI: https://doi.org/10.1080/01621459.1994.10476834

Kopecky KJ, Pierce DA. Efficiency of smooth goodness-of-fit tests. Journal of the American Statistical Association 1979; 74(366a): 393-397. DOI: https://doi.org/10.1080/01621459.1979.10482525

Bargal AI, Thomas DR. Smooth goodness of fit tests for the Weibull distribution with singly censored data. Communications in Statistics-Theory and Methods 1983; 12(12): 1431-1447. https://doi.org/10.1080/03610928308828542 DOI: https://doi.org/10.1080/03610928308828542

Janic-Wróblewska A. Data-driven smooth tests for the extreme value distribution. Statistics 2004; 38(5): 413-426. https://doi.org/10.1080/02331880410001692967 DOI: https://doi.org/10.1080/02331880410001692967

Rayner J, Best D. Smooth tests of goodness of fit: an overview. International Statistical Review/Revue Internationale de Statistique 1990; pp. 9-17. DOI: https://doi.org/10.2307/1403470

Rayner J, Best D. Neyman-type smooth tests for location-scale families. Biometrika 1986; pp. 437-446. https://doi.org/10.1093/biomet/73.2.437 DOI: https://doi.org/10.1093/biomet/73.2.437

Rayner J, Thas O, De Boeck B. A generalized Emerson recurrence relation. Australian & New Zealand Journal of Statistics 2008; 50(3): 235-240. https://doi.org/10.1111/j.1467-842X.2008.00514.x DOI: https://doi.org/10.1111/j.1467-842X.2008.00514.x

Krit M. Goodness-of-fit tests in reliability: Weibull distribution and imperfect maintenance models. Université de Grenoble 2014.

Berheto TM, Haile DB, Mohammed S, et al. Predictors of loss to follow-up in patients living with HIV/AIDS after initiation of antiretroviral therapy. North American Journal of Medical Sciences 2014; 6(9): 453. https://doi.org/10.4103/1947-2714.141636 DOI: https://doi.org/10.4103/1947-2714.141636

Sengayi M, Dwane N, Marinda E, Sipambo N, Fairlie L, Moultrie H. Predictors of loss to follow-up among children in the first and second years of antiretroviral treatment in Johannesburg, South Africa. Glob Health Action 2013; 6: 19248. https://doi.org/10.3402/gha.v6i0.19248 DOI: https://doi.org/10.3402/gha.v6i0.19248

Kimber A. Tests for the exponential, Weibull and Gumbel distributions based on the stabilized probability plot. Biometrika 1985; pp. 661-663. https://doi.org/10.1093/biomet/72.3.661 DOI: https://doi.org/10.1093/biomet/72.3.661

Team RC. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria 2013; 2014.

Kallenberg WC, Ledwina T. Consistency and Monte Carlo simulation of a data driven version of smooth goodness-of-fit tests. The Annals of Statistics 1995; pp. 1594-1608. https://doi.org/10.1214/aos/1176324315 DOI: https://doi.org/10.1214/aos/1176324315

Lemeshko BY, Lemeshko S, Postovalov S. Comparative analysis of the power of goodness-of-fit tests for near competing hypotheses. I. The verification of simple hypotheses. Journal of Applied and Industrial Mathematics 2009; 3(4): 462. https://doi.org/10.1134/S199047890904005X DOI: https://doi.org/10.1134/S199047890904005X

Sürücü B. A power comparison and simulation study of goodness-of-fit tests. Computers & Mathematics with Applications 2008; 56(6): 1617-1625. https://doi.org/10.1016/j.camwa.2008.03.010 DOI: https://doi.org/10.1016/j.camwa.2008.03.010

Kang SI. Performance of generalized Neyman smooth goodness of fit tests 1978.

Megerso A, Garoma S, Tolosa Eticha TW, Daba S, Tarekegn M, Habtamu Z. Predictors of loss to follow-up in antiretroviral treatment for adult patients in the Oromia region, ethiopia. HIV/AIDS (Auckland, NZ) 2016; 8: 83. https://doi.org/10.2147/hiv.s98137 DOI: https://doi.org/10.2147/HIV.S98137

Wang B, Losina E, Stark R, Munro A, Walensky RP, Wilke M, et al. Loss to follow-up in a community clinic in South Africa: roles of gender, pregnancy and CD4 count. SAMJ: South African Medical Journal 2011; 101(4): 253-257. https://doi.org/10.7196/SAMJ.4078 DOI: https://doi.org/10.7196/SAMJ.4078

Ramadhani HO, Thielman NM, Landman KZ, Ndosi EM, Gao F, Kirchherr JL, et al. Predictors of incomplete adherence, virologic failure, and antiviral drug resistance among HIV-infected adults receiving antiretroviral therapy in Tanzania. Clinical Infectious Diseases 2007; 45(11): 1492-1498. https://doi.org/10.1086/522991 DOI: https://doi.org/10.1086/522991

Brinkhof M, Spycher B, Yiannoutsos C, Weigel ea R. Adjusting Mortality for Loss to Follow-up: Analysis of Five ART Programmes in Sub-Saharan Africa. PLoS ONE 2010; 5(11): e14149. DOI: https://doi.org/10.1371/journal.pone.0014149

Wu J. Power and Sample Size for Randomized Phase III Survival Trials under the Weibull Model. J Biopharm Stat 2015; 25(1): 16-28. https://doi.org/10.1080/10543406.2014.919940 DOI: https://doi.org/10.1080/10543406.2014.919940




How to Cite

Odhiambo, C., Odhiambo, J., & Omolo, B. (2017). A Smooth Test of Goodness-of-Fit for the Weibull Distribution: An Application to an HIV Retention Data. International Journal of Statistics in Medical Research, 6(2), 68–78. https://doi.org/10.6000/1929-6029.2017.06.02.2



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