Maximum approximate Bernstein likelihood estimation in proportional hazard model for interval‐censored data

Abstract

Maximum approximate Bernstein likelihood estimates of the baseline density function and the regression coefficients in the proportional hazard regression models based on interval‐censored event time data result in smooth estimates of the survival functions which enjoys an almost n1/2‐rate of convergence faster than the n1/3‐rate for the existing estimates. The proposed method was shown by a simulation to have better finite sample performance than its main competitors. Some examples including real data are used to illustrate the usage of the proposed method.

Maximum approximate likelihood estimation in accelerated failure time model for interval‐censored data

“…36 The estimated survival curves are shown in Figure 1 where SE and SB represent, respectively, the NPMLE and the MABLE of S based on each of the two samples, and ŜB (⋅ | x) represents the proposed estimate based on the combined sample using AFT model or the estimate using PH model. 28 From this figure it can be seen that, although the two models give similar estimates for radiation and chemotherapy and for radiation only up to about 45 months, the AFT model fit the data better for months 45 through 60. It is reasonable to believe that the survival probability at month 60 is significantly larger than 0 due to nature and the high percentage of right-censored observations.…”

“…The estimated coefficient are (0.542, 0.015, 0.004) with standard errors (0.193, 0.006, 0.003) and model degree $$$ m=12 $$$ chosen by the change‐point method. Using the method of Guan 28 for PH model we obtained coefficient estimates (0.486, $$$ -0.024 $$$, 0.0116) with standard errors (0.260, 0.004, 0.004) and model degree $$$ m=22 $$$ chosen by the change‐point method. This is very close to the semiparametric estimates for PH and PO models.…”

“…The estimated survival curves are shown in Figure 1 where $$$ {\tilde{S}}_{\mathrm{E}} $$$ and $$$ {\tilde{S}}_{\mathrm{B}} $$$ represent, respectively, the NPMLE and the MABLE of $$$ S $$$ based on each of the two samples, and $$$ {\hat{S}}_{\mathrm{B}}\left(\cdotp \kern0.3em |\kern0.3em \boldsymbol{x}\right) $$$ represents the proposed estimate based on the combined sample using AFT model or the estimate using PH model 28 . From this figure it can be seen that, although the two models give similar estimates for radiation and chemotherapy and for radiation only up to about 45 months, the AFT model fit the data better for months 45 through 60.…”

“…This model has been successfully applied to grouped, contaminated, multivariate, and interval censored data. [25][26][27] Particularly, Guan 28 applied this model to obtain maximum likelihood estimation in Cox's proportional hazards regression model. Due to a better approximation of the unknown underlying baseline density function, not only a smooth estimate of the survival function but improved estimates of regression coefficients can be resulted.…”