# A test for proportional hazards assumption within the class of exponential conditional mean models

- 106 Downloads
- 2 Citations

## Abstract

Two classes of econometric estimators are popular for modeling outcomes with idiosyncratic characteristics such as those present in medical costs data: (1) estimators based on the exponential conditional mean models where the mean function of the outcome is equal to exponential of the linear predictor and (2) estimators based on the proportional hazard assumption where hazard function of the outcome is equal to exponential of the linear predictor. Recent work has provided guidance both on choosing between the two classes of estimators and also on choosing among alternative estimators within the exponential conditional mean framework. The present work extends this literature by proposing a test for identifying the proportional hazards assumption within the class of exponential conditional mean models, thereby eliminating the need to run both classes of models in order to make informative choices. We implement this test using the generalized gamma regression model, thereby allowing the analyst to select between both parametric alternatives and also the semi-parametric Cox model from one cohesive framework. Our simulation results indicate that the proposed test perform as well as the traditional test of proportional hazards assumption following a Cox regression based on power and Type I error under a variety of data generating mechanisms. We illustrate its use in an analysis of physician visits.

## Keywords

Health econometrics Skewed outcomes Generalized gamma regression Proportional hazards assumption Cox regression## JEL Classification

C1 Econometric and Statistical Methods: General C5 Econometric Modeling## Notes

### Acknowledgments

This research was supported in part by the National Institute on Alcohol Abuse and Alcoholism (NIAAA) grant 1 RO1 AA12664-01 A2. We would like to thank John Mullahy and two anonymous reviewers for very helpful suggestions. All remaining errors are ours.

## References

- Basu, A., Rathouz, P.J.: Estimating marginal and incremental effects on health outcomes using flexible link and variance function models. Biostatistics
**6**(1), 93–109 (2005)PubMedCrossRefGoogle Scholar - Basu, A., Manning, W.G., Mullahy, J.: Comparing alternative models: log vs. Cox proportional hazards? Health Econ.
**13**(8), 749–765 (2004)PubMedCrossRefGoogle Scholar - Blough, D.K., Madden, C.W., Hornbrook, M.C.: Modeling risk using generalized linear models. J. Health Econ.
**18**, 153–171 (1999)PubMedCrossRefGoogle Scholar - Buntin, M.B., Zaslavsky, A.M.: Too much ado about two-part models and transformation? Comparing methods of modeling Medicare expenditures. J. Health Econ.
**23**, 525–542 (2004)PubMedCrossRefGoogle Scholar - Chang, J.K., Calligaro, K.D., Lombardi, J.P., et al.: Factors that predict prolonged length of stay after aortic surgery. J. Vasc. Surg.
**38**(2), 335–339 (2003)PubMedCrossRefGoogle Scholar - Cox, D.R.: Regression models and life-tables. J. Roy. Stat. Soc. B
**34**, 187–200 (1972)Google Scholar - Cox, D.R.: Partial Likelihood. Biometrika
**62**(2), 269–276 (1975)CrossRefGoogle Scholar - Dudley, R.A., Harrell, F.E. Jr., Smith, L.R., et al.: Comparison of analytic models for estimating the effect of clinical factors on the cost of coronary artery bypass graft surgery. J. Clin. Epidemiol.
**46**(3), 261–271 (1993)PubMedCrossRefGoogle Scholar - Efron, B.: The efficiency of Cox’s likelihood function for censored data. J. Am. Stat. Assoc.
**76**, 312–319 (1977)CrossRefGoogle Scholar - Etzioni, R.D., Feuer, E.J., Sullivan, S.D., et al.: On the use of survival analysis techniques to estimate medical care costs. J. Health Econ.
**18**, 365–380 (1999)PubMedCrossRefGoogle Scholar - Fenn, P., McGuire, A., Phillips, V., et al.: The analysis of censored treatment cost data in economic evaluation. Med. Care
**33**(8), 851–863 (1995)PubMedCrossRefGoogle Scholar - Grambsch, P.M., Therneau, T.M.: Proportional hazards tests and diagnostics based on weighted residuals. Biometrika
**81**, 515–526 (1994)CrossRefGoogle Scholar - Grambsch, P.M., Therneau, T.M.: Modeling Survival Data: Extending the Cox Model. Statistics for Biology & Health Series, Springer-Verlag (2000)Google Scholar
- Greene, W.: Econometric Analysis, 4th edn. Prentice Hall (2000)Google Scholar
- Hallstrom, A., Sullivan, S.D.: On estimating costs for economic evaluation in failure time studies. Med. Care
**36**(3), 433–436 (1998)PubMedCrossRefGoogle Scholar - Hosmer, D.W., Lemeshow, S.: Applied Logistic Regression, 2nd edn. John Wiley & Sons, New York (1995)Google Scholar
- Lin, D.Y., Etzioni, R., Feuer, E.J., Wax, Y.: Estimating medical costs from incomplete follow-up data. Biometrics
**53**, 419–434 (1997)PubMedCrossRefGoogle Scholar - Lin, D.Y.: Linear regression analysis of censored medical costs. Biostatistics
**1**(1), 35–47 (2000a)CrossRefGoogle Scholar - Lin, D.Y.: Proportional means regression for censored medical costs. Biometrics
**56**(3), 775–778 (2000b)CrossRefGoogle Scholar - Lin, D.Y.: Regression analysis of incomplete medical cost data. Stat. Med.
**22**(7), 1181–1200 (2003)PubMedCrossRefGoogle Scholar - Lipscomb, J., Ancukiewicz, M., Parmigiani, G., et al.: Predicting the cost of illness: a comparison of alternative models applied to stroke. Med. Decision Making
**18**(2), S39–S56 (1998)Google Scholar - Manning, W.G.: The logged dependent variable, heteroscedasticity, and the retransformation problem. J. Health Econ.
**17**, 283–295 (1998)PubMedCrossRefGoogle Scholar - Manning, W.G., Basu, A., Mullahy, J.: Generalized approaches to risk adjustment on skewed outcomes data. NBER Technical Paper Series t0293 (2003)Google Scholar
- Manning, W.G., Basu, A., Mullahy, J.: Generalized approaches to risk adjustment on skewed outcomes data. J. Health Econ.
**24**(3), 465–488 (2005)PubMedCrossRefGoogle Scholar - Manning, W.G., Mullahy, J.: Estimating log models: to transform or not to transform? J. Health Econ.
**20**(4), 461–494 (2001)PubMedCrossRefGoogle Scholar - Mullahy, J.: Much ado about two: reconsidering retransformation and the two-part model in health econometrics. J. Health Econ.
**17**, 247–281 (1998)PubMedCrossRefGoogle Scholar - Murkofsky, R.L., Phillips, R.S., McCarthy, E.P., et al.: Length of stay in home care before and after the 1997 Balanced Budget Act. J. Am. Med. Assoc.
**289**, 2841–2848 (2003)CrossRefGoogle Scholar - Pettitt, A. N., Bin Daud, I.: Investigating time dependence in Cox’s proportional hazards model. J. Roy. Stat. Soc. B (Appl. Stat.)
**39**, 313–329 (1990)Google Scholar - Pregibon, D.: Goodness of link tests for generalized linear models. Appl. Stat.
**29**, 15–24 (1980)CrossRefGoogle Scholar - Schoenfeld, D.: Partial residuals for the proportional hazards model. Biometrika
**69**, 239–241 (1982)CrossRefGoogle Scholar - Stevens, A., Hammer, K., Buchkremer, G.: A statistical model for length of psychiatric in-patient treatment and an analysis of contributing factors. Acta Psychiatr. Scand.
**103**(3), 203–211 (2001)PubMedCrossRefGoogle Scholar