Parametric Regression Modeling for White in Stage IV

The estimated parametric regression model for White in stage IV under watchful waiting with the corrected factor is given by:

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The results of residual analysis produced a residual mean, standard error, and variances are 1.5826, 2.3698, and 5.6160, respectively. Based on the regression coefficients and the corrected regression model, we could estimate the survival time by taking the

exponential of the equation (4.16) and using the dataset. Different classical distributions were fitted to the estimated survival times. The best fitted probability distribution

function that characterized the estimated survival times was the gamma distribution with approximate maximum likelihood estimates ̂ and ̂ . The

corresponding E(x), median, and 95% confident limit are 6.30, 5.05, and (0.52, 19.10) respectively. Figure 4.19 showed the estimated survival function for White under watchful waiting based on the survival model.

Figure 4.19 Estimated Survival Function for Whites under Watchful Waiting in Stage IV

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 19%. Also, the probability of a given patient that will survive the expected survival time of 6 years is approximately 42%.

20 15 10 5 0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Time in Year S u rv iv a l P ro b a b ili ty

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The estimated parametric regression model for White in stage IV receiving radiation therapy with the corrected factor is given by:

(4.17).

The results of residual analysis produced a residual mean, standard error, and variances are 1.0528, 1.8523, and 3.4310, respectively. Based on the regression coefficients and the corrected regression model, we could estimate the survival time by taking the

exponential of the equation (4.17) and using the dataset. Different classical distributions were fitted to the estimated survival times. The best fitted probability distribution

function that characterized the estimated survival times was the gamma distribution with approximate maximum likelihood estimates ̂ and ̂ . The

corresponding E(x), median, and 95% confident limit are 10.22, 8.95, and (1.80, 25.80) respectively. Figure 4.20 showed the estimated survival function for White receiving radiation therapy based on the survival model.

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Figure 4.20 Estimated Survival Function for Whites receiving Rad in Stage IV

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 43%. Also, the probability of a given patient that will survive the expected survival time of 10 years is approximately 43%.

Case II: Surgery

The corrected parametric regression model for White in stage IV undergoing surgery is given by:

(4.18). The results of residual analysis produced a residual mean, standard error, and variances are 0.0715, 1.3395, and 1.7943, respectively. Based on the regression coefficients and the corrected regression model, we could estimate the survival time by taking the

exponential of the equation (4.18) and using the dataset. Different classical distributions 20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 Time in Year S u rv iv a l P ro b a b ili ty

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were fitted to the estimated survival times. The best fitted probability distribution

function that characterized the estimated survival times was the gamma distribution with the approximate maximum likelihood estimates ̂ and ̂ . The corresponding E(x), median, and 95% confident limit are 10.64, 9.72, and (2.76, 23.74) respectively. A graphical display of the estimated survival function for White undergoing surgery based on the survival model is showed by Figure 4.21.

Figure 4.21 Estimated Survival Function for Whites undergoing Surg in Stage IV

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 48%. Also, the probability of a given patient that will survive the expected survival time of 11 years is approximately 41%.

Case III: Combination of Radiation and Surgery

The parametric regression model for White in stage IV receiving both radiation therapy and surgery with the corrected factor is given by:

20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 Time in Year S u rv iv a l P ro b a b ili ty

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(4.19).

The results of residual analysis produced a residual mean, standard error, and variances are 0.0921, 2.3918, and 5.7207, respectively. Based on the regression coefficients and the corrected regression model, we could estimate the survival time by taking the

exponential of the equation (4.19) and using the dataset. Different classical distributions were fitted to the estimated survival times. The best fitted probability distribution

function that characterized the estimated survival times was the gamma distribution. The approximate maximum likelihood estimates ̂ and ̂ . The

corresponding E(x), median, and 95% confident limit are 11.51, 10.47, and (2.86, 26.07) respectively. A graphical display of the estimated survival function for White receiving both radiation and surgery based on the survival model is showed by Figure 4.22.

Figure 4.22 Estimated Survival Function for Whites receiving both Rad & Surg in Stage IV

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; the probability of the patient will survive in 10

20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 Time in Year S u rv iv a l P ro b a b ili ty

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years is approximately 53%. Also, the probability of a given patient that will survive the expected survival time of 11 years is approximately 46%.

Combined these cases, a graphical display of the estimated survival functions for White men undergoing different treatment of prostate cancer is given by Figure 4.23.

Figure 4.23 Stage IV: Estimated Survival Function for Whites by Treatment

It appears patients undergoing surgery, radiation therapy, or both radiation and surgery have better survivorship than the patients under watchful waiting. Also, patients undergoing combination of radiation and surgery have higher survivorship compared to the patients receiving radiation therapy or undergoing surgery. However, it could not distinguish the differences in survivorship between radiation therapy and surgery. We proceed to evaluate survival probability in different treatment by discretizing the time points and the results are shown in Table 4.6.

20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 Time in Year S u rv iv a l P ro b a b ili ty NT Rad Surg Comb Variable

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Table 4.6 Survival Probability for Whites in Stage IV by Treatment

Survival Probability

Time in Year NT Rad Surg Comb

1 0.9338 0.9938 0.9991 0.9991 2 0.8265 0.9681 0.9909 0.9916 3 0.7122 0.9229 0.9680 0.9713 4 0.6034 0.8626 0.9272 0.9359 5 0.5051 0.7925 0.8697 0.8861 6 0.4189 0.7174 0.7992 0.8246 7 0.3450 0.6412 0.7205 0.7551 8 0.2825 0.5669 0.6384 0.6812 9 0.2302 0.4964 0.5568 0.6064 10 0.1868 0.4310 0.4788 0.5332 11 0.1512 0.3716 0.4065 0.4637 12 0.1219 0.3183 0.3413 0.3994 13 0.0981 0.2710 0.2836 0.3408 14 0.0788 0.2296 0.2335 0.2885 15 0.0631 0.1936 0.1907 0.2425 16 0.0505 0.1626 0.1545 0.2024 17 0.0403 0.1360 0.1244 0.1679 18 0.0321 0.1134 0.0994 0.1385 19 0.0256 0.0942 0.0791 0.1137 20 0.0203 0.0780 0.0625 0.0929

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. The estimated mean probability residual between watchful waiting and radiation therapy is approximately -0.1817, between watchful waiting and surgery is approximately - 0.2099, and between watchful waiting and combination of radiation and surgery is approximately -0.2454, between radiation therapy and surgery is approximately -0.0281, between radiation therapy and combination of radiation and surgery is approximately - 0.0637, and between surgery and combination of radiation and surgery is approximately - 0.0355. A series of hypothesis tests was performed to compare the significant differences between two treatments. The results were found significant between watchful waiting and

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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 surgery (p = 0.002), between radiation therapy and combination of radiation and surgery (p < 0.0001), and between surgery and combination of radiation and surgery (p < 0.0001). The nonparametric test also verified our decision and the results are consistent.

4.6 Conclusion

In this study, parametric survival modeling was performed to evaluate the most effective treatment for Whites undergoing different treatment at a particular stage of prostate cancer considering the risk factors: age at diagnosis, size of tumor, and the interaction between age at diagnosis and size of tumor. Actual data was revealed that the analytical behavior of survival time as a function of age for Whites is linear in every 5 years. Similarly, the analytical behavior of the survival time as a function of tumor size for Whites is linear in every 15mm. Thus, we performed parametric survival modeling, with collapsing age at diagnosis into ten groups and tumor size as three groups, to evaluate the survivorship for Whites under different treatment at each stage of prostate cancer.

In stage I, White men have a better survival receiving radiation therapy, surgery, and combination of radiation and surgery than under watchful waiting. The corresponding estimated mean probability residual are 17.7%, 21.8%, and 20.7%, respectively. Moreover, the survivorship of White men undergoing surgery and combination of radiation and surgery are better than receiving radiation therapy. The corresponding estimated mean probabilities residual are 4.1% and 3.0%. However, the

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survivorship of surgery and combination of radiation and surgery are approximately the same under 5% level of significance.

In stage II, White men have a better survival receiving radiation therapy, surgery and combination of radiation surgery than under watchful waiting. The corresponding estimated mean probability residual are 15.7%, 21.2%, and 17.4%, respectively. Moreover, the survivorship of White men undergoing surgery and combination of radiation and surgery are better than receiving radiation therapy. The corresponding estimated mean probabilities residual are 5.5%, 1.8%, respectively. Also, the survivorship of surgery is better than combination of radiation and surgery with the estimated mean probability residual of 3.7%.

In stage III, White men have a better survival receiving radiation therapy, surgery and combination of radiation surgery than under watchful waiting. The corresponding estimated mean probability residual are 15.9%, 22.2%, and 28.5%, respectively. Moreover, the survivorship of White men undergoing combination of radiation and surgery are better than receiving radiation therapy and undergoing surgery. The corresponding estimated mean probabilities residual are 12.5 %, 6.3%, respectively. Also, the survivorship of surgery is better than radiation therapy with the estimated mean probability residual of 6.2%.

In stage IV, White men have a better survival receiving radiation therapy, surgery and combination of radiation surgery than under watchful waiting. The corresponding estimated mean probability residual are 18.2%, 21.0%, and 24.5%, respectively. Moreover, the survivorship of White men undergoing combination of radiation and surgery are better than receiving radiation therapy and undergoing surgery. The

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corresponding estimated mean probabilities residual are 6.4%, 3.6%, respectively. Also, the survivorship of surgery is better than radiation therapy with the estimated mean probability residual of 2.8%.

These findings will help physicians give an estimate of the survivorship for White men under different treatment at the four different stages of prostate cancer, and provide the most effective treatment for patients depending on their age, tumor size, and the stage of prostate cancer.

4.7 Contributions

In the present chapter are answered some important questions concerning prostate cancer.

 Age at diagnosis, tumor size, and the interaction between age at diagnosis and tumor size are shown to be significantly contributed to the survival time for White men undergoing different treatment at some particular stages of prostate cancer.  We estimated the survival time using the parametric survival modeling and the

best fitted probability distribution function that characterizes the behavior of the estimated survival times for White men is gamma distribution.

 With the estimated survival time and their distribution, we have evaluated the estimated mean probability residual of Whites undergoing different treatments at a particular stage of prostate cancer. These findings will help physicians provide the most effective treatment and cost-effective strategies for white patients by knowing their age, tumor size, and the stage of prostate cancer.

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CHAPTER 5 COMPARISON BETWEEN PARAMETRIC SURVIVAL ANALYSIS