In order to perform the parametric survival analysis, we need to identify appropriate probability distribution function that characterizes the behavior of the survival times. Different classical distributions were fitted to the observed data. The observed survival times follow a two parameter gamma distribution.
The two parameter gamma probability distribution function with shape parameter and location parameter is given by:
∫
The corresponding survival function is given by:
∫ ( )
where ∫ is the lower incomplete gamma function and the hazard function given by:
is the cumulative density function of . The survival function has no closed-form expression. However, there exist algorithms for its computation.
The approximate maximum likelihood estimates (MLE) for White men under watchful waiting are ̂ and ̂ [20]. From the identified probability distribution function and their estimates, the E(x), median, and 95% confidence limits of the survival time of a given prostate cancer patient from such a population are 7.99, 7.17,
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and (1.76, 18.84), respectively. The analytical form of the survival function with the MLE is given by:
∫ ( )
With the maximum likelihood estimates, the survival function can be reduced to
( )
The graphical form is given by Figure 3.6.
Figure 3.6 Stage I: Survival Function for Whites under watchful waiting
Notice that the time t=0 year: patients were diagnosed with prostate cancer and under watchful waiting. Thus, a physician could be able to answer questions from a patient under watchful waiting; the probability of the patient will survive in 10 years is
approximately 30%. Also, the probability of a given patient that will survive the expected survival time of 8 years is approximately 37%.
20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 Time in Year Su rv iv al P ro ba bi lit y
35 Case I: Radiation Therapy
The best fitted probability distribution function that characterizes the behavior of the survival times in Stage I for White men receiving radiation therapy was the gamma distribution. The approximate maximum likelihood estimates (MLE) for White men receiving radiation therapy are ̂ and ̂ [20]. From the identified probability distribution function and their estimates, the E(x), median, and 95% confidence limits of the survival time of a given prostate cancer patient from such as population are 9.27, 8.65 and (2.99, 19.04), respectively. The analytical form of the survival function with the MLE is given by:
∫ ( )
With the maximum likelihood estimates, the survival function can be reduced to
( )
The graphical form is given by Figure 3.7.
Figure 3.7 Stage I: Survival Function for Whites receiving Radiation Therapy
20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 Time in Year Su rv iv al P ro ba bi lit y
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Notice that the time t=0 year: patients were diagnosed with prostate cancer and proceed with radiation therapy. Thus, a physician could be able to answer questions from a patient after radiation therapy; the probability of the patient will survive in 10 years is
approximately 38%. Also, the probability of a given patient that will survive the expected survival time of 9.3 years is approximately 40%.
Case II: Surgery
The best fitted probability distribution function that characterizes the behavior of the survival times in Stage I for White men undergoing surgery was the gamma distribution with the approximate maximum likelihood estimates ̂ and ̂ . The corresponding E(x), median, and 95% confident limit are 9.57, 8.66, and (2.25, 22.10), respectively. The analytical form of the survival function with the MLE is given by:
∫ ( )
With the maximum likelihood estimates, the survival function can be reduced to
( )
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Figure 3.8 Stage I: Survival Function for Whites undergoing Surgery
Notice that the time t=0 year: patients were diagnosed with prostate cancer and proceed with surgery. Thus, a physician could be able to answer questions from a patient after surgery; the probability of the patient will survive in 10 years is approximately 41%. Also, the probability of a given patient that will survive the expected survival time of 9.6 years is approximately 42%.
Case III: Combination of Radiation and Surgery
The best fitted probability distribution function that characterizes the behavior of the survival times in Stage I for White men undergoing surgery and radiation therapy was the gamma distribution with the approximate maximum likelihood estimates ̂ and ̂ . The corresponding E(x), median, and 95% confident limit are 9.92, 9.12, and (2.75, 21.61), respectively. The analytical form of the survival function with the MLE is given by:
∫ ( )
With the maximum likelihood estimates, the survival function can be reduced to
20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 Time in Year Su rv iv al P ro ba bi lit y
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( )
The graphical form is given by Figure 3.9.
Figure 3.9 Stage I: Survival Function for Whites receiving both Surgery and Radiation Therapy
Notice that the time t=0 year: patients were diagnosed with prostate cancer and proceed with both surgery and radiation. Thus, a physician could be able to answer questions from a patient after both surgery and radiation therapy; the probability of the patient will survive in 10 years is approximately 45%. Also, the probability of a given patient that will survive the expected survival time of 9.9 years is approximately 45%.
Combined the results, a graphical display of the survival functions for white men undergoing different treatment of prostate cancer is given by Figure 3.10.
20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 Time in Year Su rv iv al P ro ba bi lit y
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Figure 3.10 Stage I: Survival Function for Whites by Treatment
Table 3.4 Survival Probability for Whites in Stage I by Treatment
Survival Probability
Time in Year NT Rad Surg Comb
1 0.9948 0.9997 0.9978 0.9993 2 0.9649 0.9948 0.9822 0.9913 3 0.9041 0.9746 0.9455 0.9669 4 0.8190 0.9308 0.8879 0.9219 5 0.7198 0.8618 0.8140 0.8572 6 0.6164 0.7725 0.7300 0.7778 7 0.5164 0.6712 0.6419 0.6897 8 0.4245 0.5665 0.5548 0.5989 9 0.3433 0.4659 0.4723 0.5104 10 0.2739 0.3742 0.3967 0.4277 11 0.2158 0.2944 0.3293 0.3531 12 0.1683 0.2273 0.2705 0.2876 13 0.1300 0.1726 0.2201 0.2315 14 0.0996 0.1292 0.1776 0.1843 15 0.0757 0.0954 0.1423 0.1453 16 0.0572 0.0695 0.1132 0.1136 17 0.0429 0.0502 0.0895 0.0881 18 0.0321 0.0358 0.0703 0.0678 19 0.0238 0.0254 0.0550 0.0518 20 0.0176 0.0178 0.0428 0.0394 25 20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 Time in Year Su rv iv al P ro ba bi lit y NT Rad Surg Comb Variable
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Figure 3.10 shows that patients receiving radiation therapy, surgery, or combination of radiation and surgery have better survivorship than the patients under watchful waiting. However, it could not distinguish the survivorship among the three types of treatments. We proceed to evaluate survival probability in different treatment by discretizing the time points and the results are shown in Table 3.4.
In determining the treatment response for each year, we performed a pairwise comparison between two treatments at each time point is defined as the survival probability residuals. Let ̂ represent the survival probability residual between no treatment and radiation therapy:
̂ ̂ ̂
̂ represents the survival probability residual between no treatment and surgery. That is:
̂ ̂ ̂
̂ represents the survival probability residual between no treatment and combination of radiation and surgery. That is:
̂ ̂ ̂
̂ represents the survival probability residual between radiation therapy and surgery. That is:
̂ ̂ ̂
̂ represents the survival probability residual between radiation therapy and combination of radiation and surgery. That is:
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̂ ̂ ̂
̂ represents the survival probability residual between surgery and combination of radiation and surgery. That is:
̂ ̂ ̂
The estimated mean probability residual between watchful waiting and radiation therapy is approximately -0.065, between watchful waiting and surgery is approximately -0.075, and between watchful waiting and combination of radiation and surgery is approximately -0.093, between radiation therapy and surgery is approximately -0.010, between radiation therapy and combination of radiation and surgery is approximately -0.029, and between surgery and combination of radiation and surgery is approximately -0.019. A series of hypothesis tests was performed to compare the significant differences between two treatments. The results were found significant between watchful waiting and radiation therapy (p < 0.0001), between watchful waiting and surgery (p < 0.0001), between watchful waiting and combination of radiation and surgery (p < 0.0001), between radiation therapy and combination of radiation and surgery (p < 0.0001), and between surgery and combination of radiation and surgery (p < 0.0001). However, it is not significant between radiation therapy and surgery (p = 0.2). The nonparametric test also verified our decision and the results are consistent.
3.5.2 Stage II: Evaluation of Survivorship for Whites by Treatment
Case 0: No Treatment (Watchful Waiting)
The best fitted probability distribution function that characterizes the behavior of the survival times in Stage II for White men under watchful waiting was the gamma distribution. The approximate maximum likelihood estimates (MLE) for White men
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under watchful waiting are ̂ and ̂ . From the identified probability distribution function and their estimates, the E(x), median, and 95% confidence limits of the survival time of a given prostate cancer patient from such as population are 7.31, 6.54, and (1.56, 17.43) respectively. The analytical form of the survival function with the MLE is given by:
∫ ( )
With the maximum likelihood estimates, the survival function can be reduced to
( )
Its graphical form is given by Figure 3.11.
Figure 3.11 Stage II: Survival Function for Whites under watchful waiting
Notice that the time t=0 year: patients were diagnosed with prostate cancer under watchful waiting. Thus, a physician could be able to answer questions from a patient under watchful waiting; the probability of the patient will survive in 10 years is
20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 Time in Year Su rv iv al P ro ba bi lit y
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approximately 22%. Also, the probability of a given a patient that will survive the expected survival time of 7.3 years is approximately 35%.