Different causes of closure of Small Business Enterprises: alternative models for competing risks survival analysis


We examine the time until closure of Small Business Enterprises in Umbria, Italy between 2008 and 2013, and the factors that influence it. Earlier analysis, using Cox regression, considered failure (closure) from any cause. However, there are different reasons for inactivity: voluntary winding-up (1808 of
15184 firms in our data, 59.3% of the 3049 failures); bankruptcy (236, 7.7%); and closure without action by creditors or courts (1005, 33.0%). While the earlier analysis provides a valuable overall picture, it is also interesting to ex-
amine the separate causes, their rates of occurrence and which factors influence them separately. We do this using competing risks analyses, employing both of the regression methods that are prominent in the literature, based
on cause-specific and sub-distribution hazard functions (Fine-Gray model). Furthermore, a proportional odds model was used to estimate cumulative incidences of failure by cause. Data included the firm's year of foundation, location, legal form and sector of activity. Financial indexes were constructed
from annual balance sheets. The date and reason for closure were recorded if the firrm ceased activity. Findings included major differences between types of firm; for example, cooperatives had greatly increased hazards for winding-up
(HR of 2.44 and 2.61 in the two approaches) but greatly reduced hazards for closure (0.48 and 0.45) compared to publicly traded companies. All-causes analysis averaged these strong effects into an insignicant one (1.05). Coefficients from the proportional odds model were similar to those from the
Fine-Gray model, but have the advantage of interpretability.

Keywords: bankruptcy; survival analysis; competing risks; cause-specific hazards; sub-distribution hazards; proportional odds


Amendola, A., Restaino, M., and Sensini, L. (2015). An analysis of the determinants of financial distress in Italy: A competing risks approach. International Review of Economics & Finance, 37:33-41.

Andersen, P. K. and Keiding, N. (2012). Interpretability and importance of functionals in competing risks and multistate models. Statistics in Medicine, 31(11-12):1074-1088.

Austin, P. C. and Fine, J. P. (2017). Practical recommendations for reporting Fine-Gray model analyses for competing risk data. Statistics in Medicine, 36(27):4391-4400.

Bakoyannis, G. and Touloumi, G. (2012). Practical methods for competing risks data: a review. Statistical Methods in Medical Research, 21(3):257-272.

Balcaen, S., Manigart, S., Buyze, J., and Ooghe, H. (2012). Firm exit after distress: differentiating between bankruptcy, voluntary liquidation and M & A. Small Business Economics, 39(4):949-975.

Balcaen, S. and Ooghe, H. (2006). 35 years of studies on business failure: an overview of the classic statistical methodologies and their related problems. The British Account-

ing Review, 38(1):63-93.

Banasik, J., Crook, J. N., and Thomas, L. C. (1999). Not if but when will borrowers default. Journal of the Operational Research Society, 50(12):1185-1190.

Bhattacharjee, A., Higson, C., Holly, S., and Kattuman, P. (2009). Macroeconomic instability and business exit: Determinants of failures and acquisitions of UK firms. Economica, 6(301):108-131.

Chancharat, N., Tian, G., Davy, P., McCrae, M., and Lodh, S. (2010). Multiple states of financially distressed companies: Tests using a competing-risks model. Australasian Accounting, Business and Finance Journal, 4(4):27-44.

Collett, D. (2015). Modelling survival data in medical research. Chapman and Hall/CRC.

Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society: Series B (Methodological), 34(2):187-202.

Crowder, M. J. (2001). Classical competing risks. Chapman and Hall/CRC.

David, H. A. and Moeschberger, M. L. (1978). The theory of competing risks. C. Griffin.

Dignam, J. J., Zhang, Q., and Kocherginsky, M. (2012). The use and interpretation of competing risks regression models. Clinical Cancer Research, 18(8):2301-2308.

Elsayed, E. and Chan, C. (1990). Estimation of thin-oxide reliability using proportional hazards models. IEEE Transactions on Reliability, 39(3):329-335.

Eriksson, F., Li, J., Scheike, T., and Zhang, M.-J. (2015). The proportional odds cumulative incidence model for competing risks. Biometrics, 71(3):687-695.

Fine, J. P. and Gray, R. J. (1999). A proportional hazards model for the subdistribution of a competing risk. Journal of the American Statistical Association, 94(446):496-509.

Grambauer, N., Schumacher, M., and Beyersmann, J. (2010). Proportional subdistribution hazards modeling offers a summary analysis, even if misspecified. Statistics in Medicine, 29(7-8):875-884.

Gray, B. (2019). cmprsk": Subdistribution analysis of competing risks. R package version 2.2-9. Available online at: https://cran.r-project.org/web/packages/cmprsk/


Haller, B., Schmidt, G., and Ulm, K. (2013). Applying competing risks regression models: an overview. Lifetime Data Analysis, 19(1):33-58.

Hosmer, D. W., Lemeshow, S., and May, S. (2008). Applied survival analysis: regression modeling of time-to-event data. Wiley-Interscience.

Hougaard, P. (1999). Fundamentals of survival data. Biometrics, 55(1):13-22.

Hutton, J. and Monaghan, P. (2002). Choice of parametric accelerated life and proportional hazards models for survival data: asymptotic results. Lifetime Data Analysis, 8(4):375-393.

Kohl, M., Plischke, M., Leffondre, K., and Heinze, G. (2015). Pshreg: a SAS macro for proportional and nonproportional subdistribution hazards regression. Computer Methods and Programs in Biomedicine, 118(2):218-233.

Kwon, S. and Hahn, S. B. (2010). Duration analysis of corporate bankruptcy in the presence of competing risks. Applied Economics Letters, 17(15):1513-1516.

Latouche, A., Allignol, A., Beyersmann, J., Labopin, M., and Fine, J. P. (2013). A competing risks analysis should report results on all cause-specific hazards and cumulative incidence functions. Journal of Clinical Epidemiology, 66(6):648-653.

Lau, B., Cole, S. R., and Gange, S. J. (2009). Competing risk regression models for epidemiologic data. American Journal of Epidemiology, 170(2):244-256.

Ma, Z. and Krings, A. W. (2008). Competing risks analysis of reliability, survivability, and prognostics and health management (PHM). In 2008 IEEE Aerospace Conference, pages 1-21. IEEE.

Mao, L. and Lin, D. (2017). Efficient estimation of semiparametric transformation models for the cumulative incidence of competing risks. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(2):573-587.

Narain, B. (1992). Survival analysis and the credit granting decision. In: Thomas, L.C., Crook, J.N., Edelman, D.B. (eds.), Credit Scoring and Credit Control, Oxford University Press, 109-122.

Oakes, D. (2013). An introduction to survival models: in honor of Ross Prentice. Lifetime Data Analysis, 19(4):442-462.

Ohlson, J. A. (1980). Financial ratios and the probabilistic prediction of bankruptcy. Journal of Accounting Research, 18(1):109-131.

Pierri, F. and Caroni, C. (2017). Bankruptcy prediction by survival models based on current and lagged values of time-varying financial data. Communications in Statistics: Case Studies, Data Analysis and Applications, 3(3-4):62-70.

Putter, H., Fiocco, M., and Geskus, R. B. (2007). Tutorial in biostatistics: competing risks and multi-state models. Statistics in Medicine, 26(11):2389-2430.

Scheike, M. T. (2019). timereg": Flexible regression models for survival data. R package version 1.9.4. Available online at: https://cran.r-project.org/web/packages/timereg/timereg.pdf.

Scheike, T. H. and Zhang, M.-J. (2011). Analyzing competing risk data using the R timereg package. Journal of Statistical Software, 38(2):1-16.

Scrucca, L., Santucci, A., and Aversa, F. (2007). Competing risk analysis using R: an easy guide for clinicians. Bone Marrow Transplantation, 40(4):381.

Shumway, T. (2001). Forecasting bankruptcy more accurately: A simple hazard model. The Journal of Business, 74(1):101-124.

Sohn, S. Y. and Jeon, H. (2010). Competing risk model for technology credit fund for small and medium-sized enterprises. Journal of Small Business Management, 48(3):378-394.

Therneau, T. M. and Grambsch, P. M. (2013). Modeling survival data: extending the Cox model. Springer Science & Business Media.

Full Text: pdf

Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.