A Smooth Test of Goodness-of-Fit for the Baseline Hazard Function for Time-to-First Occurrence in Recurrent Events: An Application to HIV Retention Data

Collins Odhiambo; John Odhiambo; Bernard Omolo

doi:10.6000/1929-6029.2017.06.03.2

Authors

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 and Computer Science, University of South Carolina-Upstate, 800 University Way, Spartanburg, South Carolina, USA

DOI:

https://doi.org/10.6000/1929-6029.2017.06.03.2

Keywords:

BBS model, Hazard function, Loss to follow-up, Neymanâ€™s smooth test, Recurrent events, Retention in HIV care

Abstract

Motivated by HIV retention, we present an application of the smooth test of goodness-of-fit under right-censoring to time to first occurrence of a recurrent event. The smooth test applied here is an extension of Neyman’s smooth test to a class of hazard functions for the initial distribution of a recurrent failure-time event. We estimate the baseline hazard function of time-to-first loss to follow-up, using a Block, Borges and Savits (BBS) minimal repair model of the data (n = 2,987,72% censored). Simulations were conducted at various percentages of censoring to assess the performance of the smooth test. Results show that the smooth test performed well under right-censoring.

References

Aalen OO, Husebye E. Statistical Analysis of Repeated Events Forming Renewal Processes. Statistics in Medicine. 1991; 10(8): 1227-1240. https://doi.org/10.1002/sim.4780100806 DOI: https://doi.org/10.1002/sim.4780100806

Andersen PK, Borgan O, Gill RD, Keiding N. Statistical models based on counting processes. Springer Science & Business Media 2012.

Cook RJ, Lawless JF. Analysis of repeated events. Statistical Methods in Medical Research 2002; 11(2): 141-166. https://doi.org/10.1191/0962280202sm278ra DOI: https://doi.org/10.1191/0962280202sm278ra

Nelson WB. Recurrent events data analysis for product repairs, disease recurrences, and other applications. SIAM 2003. https://doi.org/10.1137/1.9780898718454 DOI: https://doi.org/10.1137/1.9780898718454

Cook RJ, Lawless J. The statistical analysis of recurrent events. Springer Science & Business Media 2007.

Prentice RL, Williams BJ, Peterson AV. On the regression analysis of multivariate failure time data. Biometrika 1981; pp. 373-379. https://doi.org/10.1093/biomet/68.2.373 DOI: https://doi.org/10.1093/biomet/68.2.373

Cox DR, Oakes D. Analysis of survival data. vol. 21. CRC Press 1984.

Huang CY, Luo X, Follmann DA. A model checking method for the proportional hazards model with recurrent gap time data. Biostatistics 2011; 12(3): 535-547. https://doi.org/10.1093/biostatistics/kxq071 DOI: https://doi.org/10.1093/biostatistics/kxq071

Jahn-Eimermacher A, Ingel K, Ozga AK, Preussler S, Binder H. Simulating recurrent event data with hazard functions defined on a total time scale. BMC Medical Research Methodology 2015; 15(1): 16. https://doi.org/10.1186/s12874-015-0005-2 DOI: https://doi.org/10.1186/s12874-015-0005-2

Baxter LA, Kijima M, Tortorella M. A Point Process Model for the Reliability of a Maintained System Subject to General Repair. Stochastic Models 1996; 12(1): 12-11. https://doi.org/10.1080/15326349608807372 DOI: https://doi.org/10.1080/15326349608807372

Agustin MZN, Peña EA. A basis Approach to Goodness-of-fit Testing in Recurrent Event Models. Journal of Statistical Planning and Inference 2005; 133(2): 285-303. https://doi.org/10.1016/j.jspi.2004.03.022

Agustin MZN, Pena EA. Order Statistic Properties, Random Generation, and Goodness-of-fit Testing for a Minimal Repair Model. Journal of the American Statistical Association 1999; 94(445): 266-272. https://doi.org/10.1080/01621459.1999.10473842 DOI: https://doi.org/10.1080/01621459.1999.10473842

Block HW, Borges WS, Savits TH. Age-dependent Minimal Repair. Journal of Applied Probability 1985; pp. 370-385. https://doi.org/10.1017/S0021900200037839 DOI: https://doi.org/10.2307/3213780

Brown M, Proschan F. Imperfect Repair. Journal of Applied Probability 1983; pp. 851-859. https://doi.org/10.1017/S0021900200024153 DOI: https://doi.org/10.2307/3213596

Sydney Rosen CJG Matthew P Fox. Patient Retention in Antiretroviral Therapy Programs in Sub-Saharan Africa: A Systematic Review. PLoS Medicine 2007; 4(10): e298. https://doi.org/10.1371/journal.pmed.0040298 DOI: https://doi.org/10.1371/journal.pmed.0040298

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

Pena EA. Smooth goodness-of-fit tests for the baseline hazard in Cox's proportional hazards model. Journal of the American Statistical Association 1998; 93(442): 673-692. https://doi.org/10.2307/2670118 DOI: https://doi.org/10.1080/01621459.1998.10473720

Pena EA. Smooth Goodness-Of-Fit Tests for Composite Hypothesis in Hazard Based Models. The Annals of Statistics 1998; 26(5): 1935-1971. https://doi.org/10.1214/aos/1024691364 DOI: https://doi.org/10.1214/aos/1024691364

Agustin MZN, Peña EA. Goodness-of-fit of the Distribution of Time-to-first occurrence in Recurrent Event Models. Lifetime Data Analysis 2001; 7(3): 289-306. https://doi.org/10.1023/A:1011692525705 DOI: https://doi.org/10.1023/A:1011692525705

Agustin MZN, Peña EA. A basis approach to goodness-of-fit testing in recurrent event models. Journal of Statistical Planning and Inference 2005; 133(2): 285-303. https://doi.org/10.1016/j.jspi.2004.03.022 DOI: https://doi.org/10.1016/j.jspi.2004.03.022

Dorado CA. On a General Repair Model for Repairable Systems 1995.

Dorado C, Hollander M, Sethuraman J. Nonparametric Estimation for a General Repair Model. The Annals of Statistics 1997; pp. 1140-1160. DOI: https://doi.org/10.1214/aos/1069362741

Hollander M, Presnell B, Sethuraman J. Nonparametric Methods for Imperfect Repair Models. The Annals of Statistics 1992; pp. 879-896. https://doi.org/10.1214/aos/1176348661 DOI: https://doi.org/10.1214/aos/1176348661

Kijima M. Some Results for Repairable Systems with General Repair. Journal of Applied Probability 1989; pp. 89-102. https://doi.org/10.1017/S0021900200041826 DOI: https://doi.org/10.2307/3214319

Odhiambo, Collins J, Odhiambo, Omolo B. Validation of the Smooth Test of Goodness-of-Fit for Proportional Hazards in Cancer Survival Studies. International Journal of Statistics in Medical Research 2017; 6(2): 49-67. DOI: https://doi.org/10.6000/1929-6029.2017.06.02.1

Kenya A. Response progress report 2014: Progress towards zero. Nairobi, Kenya: Government Printer 2014.

UNAIDS. 90-90-90: An ambitious treatment target to help end the AIDS epidemic. UNAIDS Geneva 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. https://doi.org/10.1017/S0021900200041826 DOI: https://doi.org/10.1371/journal.pone.0101875

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

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

Gwynn RC, Fawzy A, Viho I, Wu Y, Abrams EJ, Nash D. Risk factors for loss to follow-up prior to ART initiation among patients enrolling in HIV care with CD4+ cell count 200 cells/µL in the multi-country MTCT-Plus Initiative. BMC health Services Research 2015; 15(1): 1. https://doi.org/10.1186/s12913-015-0898-9 DOI: https://doi.org/10.1186/s12913-015-0898-9

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

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

Odhiambo, Collins J, Odhiambo, Omolo B. 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 2017; 6(1): 68-78. DOI: https://doi.org/10.6000/1929-6029.2017.06.02.2

Assembly UG. Political Declaration on HIV and AIDS: Intensifying our Efforts to Eliminate HIV and AIDS. 65th Session New York: UN General Assembly (UNGASS) 2011.

Programme JUN, HIV/AIDS o. Getting to Zero: 2011-2015 strategy. UNAIDS, Geneva 2010.

Stricker SM, Fox KA, Baggaley R, Negussie E, de Pee S, Grede N, et al. Retention in care and adherence to ART are critical elements of HIV care interventions. AIDS and Behavior 2014; 18(5): 465-475. https://doi.org/10.1007/s10461-013-0598-6 DOI: https://doi.org/10.1007/s10461-013-0598-6