The Dynamics of Death in Prostate Cancer

Anatomic Pathology / Hazard in Prostate Cancer

The Dynamics of Death in Prostate Cancer Robin T. Vollmer, MD, MS Key Words: Hazard function; Histology; Prostate cancer; Survival DOI: 10.1309/AJCPJK9V9LUMUETV

Abstract The hazard function provides the instantaneous probability of death (or other key end point) at various times after diagnosis. Unlike the survival curve, the hazard function illustrates graphically or through calculations when deaths are common or uncommon. In this study, hazard functions were derived for prostate cancer by using survival data on large numbers of patients with prostate cancer with data in the Surveillance, Epidemiology and End Results (SEER) database. The results demonstrate a form of prostate cancer that rapidly evolves to cause death within 5 years, and this form of tumor is only partly identified by routine prognostic variables such as serum prostatespecific antigen (PSA) level, histologic grade, and quantity of tumor. The results also validate the presence of a reservoir of nonfatal prostate cancers that have increased rapidly during the PSA era, and they demonstrate that the incidence of fatal prostate cancers has declined.

The ASCP is accredited by the Accreditation Council for Continuing Medical Education to provide continuing medical education for physicians. The ASCP designates this journal-based CME activity for a maximum of 1 AMA PRA Category 1 Credit ™ per article. Physicians should claim only the credit commensurate with the extent of their participation in the activity. This activity qualifies as an American Board of Pathology Maintenance of Certification Part II Self-Assessment Module. The authors of this article and the planning committee members and staff have no relevant financial relationships with commercial interests to disclose. Questions appear on p 1010. Exam is located at www.ascp.org/ajcpcme.

Prostate-specific antigen (PSA) was discovered in the 1970s,1 purified and named as such by 1979,2 and tested as a serum marker in the 1980s.3-6 In 1992, the US Food and Drug Administration approved PSA as a screening test for prostate cancer, and it was hailed as “the most useful tumor marker”7 and lauded for its ability to enhance the detection of early prostate cancer.8 Subsequently, the number of PSA tests soared, as did the number of needle biopsies of prostate. For example, ❚Figure 1❚ shows the percentage of men without prostate cancer who were older than 65 years and who had at least 1 PSA test during the calendar year on the abscissa (dots for white men and line for black men).9 Nevertheless, as pathologists fretted about overlooking small foci of tumor10-12 and as urologists opined that the PSA thresholds for performing biopsies should be lowered,13-16 epidemiologists and statisticians began to argue that in the PSA era, prostate cancer was being overdiagnosed, that too many men were being treated, and that PSA-based screening for prostate cancer offered an unproven benefit.9,17-20 Although their conclusions were based on mathematical models and computer simulations, 2 prospective trials (US Prostate, Lung, Colorectal, and Ovarian Cancer Screening trial and the European Randomized Study of Screening for Prostate Cancer trial) demonstrated that PSA-based screening produces overdiagnosis and overtreatment.21,22 Taken collectively, the trials showed at most a modest reduction in prostate cancer–specific mortality. Nevertheless, both trial results were considered preliminary because they were based on interim analyses that included just 634 patients observed to die of prostate cancer. Recently, Welch and Black23 emphasized that overdiagnosis in tumors is closely linked to the dynamics of tumor

© American Society for Clinical Pathology

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DOI: 10.1309/AJCPJK9V9LUMUETV

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CME/SAM

Upon completion of this activity you will be able to: • be able to define the hazard function and know how it demonstrates peaks and valleys in the timing of death. • describe how the hazard function for prostate cancer identifies an aggressive form of tumor that rapidly evolves to cause death. • recognize how screening for prostate cancer with serum prostatespecific antigen has decreased hazard for death largely by diluting fatal cases with newly diagnosed nonfatal cases.

Vollmer / Hazard in Prostate Cancer

In addition, I used survival data from a previously published study of 663 American Veterans Affairs patients diagnosed with prostate cancer and who were followed up to death or for a median of 5 years.26 I obtained the numbers of practicing urologists in the United States from http://www.census. gov/prod/3/98pubs/98statab/ (accessed August 4, 2011) and the prevalence of testing for serum PSA from Etzioni et al.9

% of Men Tested for PSA

40

30

20

10

0 1980

1990

1985

1995

Year

❚Figure 1❚ The percentage of men without prostate cancer who are older than 65 years and who had at least 1 serum prostate-specific antigen (PSA) test in the year noted on the x-axis. The dots represent white men, and the line, black men. The data are from Etzioni et al.9

growth and progression. Rapidly growing tumors cause clinical symptoms, can be diagnosed without screening, and often progress to cause death. By contrast, slowly growing tumors may not cause symptoms and progress so slowly that many patients die of other causes. The slowly progressive tumors comprise a reservoir of asymptomatic cases detectable by screening tests such as PSA. In this manner, screening and overdiagnosis of any tumor are directly linked to the dynamics of the tumor’s progression. A probabilistic function that in turn is closely linked to the dynamics of a tumor’s progression is the hazard function,24,25 which provides the instantaneous probability of death at any time after diagnosis. In what follows, I derive hazard functions for death in prostate cancer using large numbers of patients with known outcomes in the Surveillance, Epidemiology and End Results (SEER) database. The hazard-derived results help identify an aggressive form of prostate cancer that existed before and after the PSA era, and, with these mature historical data, they validate the phenomenon of overdiagnosis.

Materials and Methods Study Data The primary data for this study are values for incidence and survival, and these were obtained from the SEER database (http://www.seer.cancer.gov/csr/1975_2007; accessed August 1, 2010). Specifically, 11 survival curves were obtained for patients diagnosed during the following 11 periods: 1975-1979 (median at 1977), 1980-1984 (median at 1982), 1985-1989 (median at 1987), 1990, 1991, 1992, 1993, 1994, 1995, 1996, and 1997. The follow-up for the patients represented in these data was such that there were at least 10 years of usable survival results for each survival curve. 958 958

Am J Clin Pathol 2012;137:957-962 DOI: 10.1309/AJCPJK9V9LUMUETV

The Hazard Function Kaplan-Meier plots are plots of the survival probability, S(t), at various times, t, after diagnosis, and when there are sufficient numbers of patients, S(t) can be considered a continuous function of t. Thus, in what follows, we will take advantage of this feature of S(t). The hazard function, h(t), is often called the force of mortality because higher levels of h(t) coincide with times of frequent deaths, and h(t) is related to S(t) as follows: ❚Equation 1❚

h(t) = – ∂ log (S(t))/∂ t Here, log stands for the natural logarithm, and ∂ symbolizes the partial differential with respect to time.27 Through integration, the relationship between S(t) and h(t) can also be written as: ❚Equation 2❚

S(t) = exp (– ∫ h(x) dx) Here, exp stands for exponentiation, and the integration limits for x are from 0 to t. The integral ∫ h(x) dx is often called the total hazard. Equation 2 implies that total fatality from 0 to any time, t, can be calculated as 1 – exp ( – ∫ h(x) dx) with the integration limits being from 0 to t. Finally, in this study I used the sum of 2 γ functions for the mathematical form of h(t), because this form allowed for a near perfect fit of the survival data. (See ❚Appendix 1❚ and the “Results” section for details.) There are at least 2 advantages for examining h(t). First, graphs of h(t) show times of peak hazard and times of low hazard. Second, knowing the mathematical form for the hazard allows accurate estimation of the cumulative probability of death for any period during follow-up. Thus, in what follows, the derived mathematical hazard function will allow calculation of observed probabilities of death by 5 years after diagnosis and for all times of follow-up.

Results The Fit of the Derived Hazard Function to SEER Data ❚Figure 2❚ shows a plot of observed values of S(t) (on the x-axis) vs that predicted by the derived hazard function h(t) and Equation 2 (on the y-axis) for all 11 SEER survival curves. The line on the plot shows where perfect agreement would occur, and the fact that the points lie close to the line indicates that the γ function model is an accurate model of the hazard © American Society for Clinical Pathology

Anatomic Pathology / Original Article

0.08

0.9 Hazard Function

Predicted Survival Probability

1.0

0.8 0.7 0.6

1982

0.06

0.04

0.02 1993

0.5 0.00 0.4

0.5

0.6

0.7

0.8

0.9

0

1.0

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Observed Survival Probability

❚Figure 2❚ Observed values of survival probability, S(t), on the x-axis vs that predicted by the hazard model and Equation 2 on the y-axis. The data combine the results for all Surveillance, Epidemiology and End Results database survival curves studied, and the line on the plot shows where perfect agreement would occur.

❚Figure 3❚ The hazard function for Surveillance, Epidemiology and End Results data on cases diagnosed in 1982 (before the serum prostate-specific antigen [PSA] level was used) vs cases diagnosed in 1993 (after the serum PSA level was commonly used).

function for the observed SEER data. The median difference between observed S(t) and that predicted from the derived hazards was 0 (range, –0.015 to 0.008). Values of α averaged 0.0758 (range 0.02 to 0.235). Values of β averaged 1.31 (range, 0.627 to 4.93). Values of λ averaged 0.00228 (range, 0.0002170.00732), and values of δ averaged 0.318 (range, 0.126-0.991).

score, tumor length, fraction of positive cores, and percentage of tumor were all significantly higher in patients observed to die within 5 years (P ~0; Wilcoxon tests). Nevertheless, the overlapping ranges for these variables suggest that none of them reliably discriminates aggressive tumors from the less aggressive tumors. Logistic regression analysis of these data showed that 2 variables, log(PSA) and primary Gleason grade, provided additive information about the probability of

Typical Hazard Functions ❚Figure 3❚ shows plots of 2 hazard functions: the first derived from SEER cases diagnosed in 1982 (upper curve) and the second derived from SEER cases diagnosed in 1993 (lower curve). The plots show that both hazard functions peak in the first 5 years after diagnosis and then gradually fall to near zero. All of the hazard functions in the SEER data manifest these early peaks, with an average peak at 1.2 years after diagnosis (range, 0.2-1.8 years). Furthermore, for all diagnostic years except 1992, the early peak was also the time for the highest hazard. The early peak in hazard implied that many deaths in prostate cancer occur during the first 5 years after diagnosis. For example, the percentage of total deaths in the SEER data occurring in the first 5 years averaged 44% (range, 20%-90%). Thus, the early peaks in hazard function seem due to an aggressive form of prostate cancer that rapidly evolves to cause death. After 5 years, the fall in hazard function implies a lower death rate at scattered subsequent times. Variables Related to Aggressive Prostate Cancer The foregoing results suggest that death within 5 years of diagnosis is an end point that is closely related to an aggressive form of prostate cancer that rapidly evolves to cause death. ❚Table 1❚ shows how several prognostic variables relate to this end point. For example, median values of PSA, Gleason

❚Table 1❚ Relationship Between Variables at the Time of Diagnosis and Death at Five Years* Variable PSA level, ng/mL (μg/L) Median Range Gleason score Median Range Tumor length (mm) Median Range Fraction of positive cores Median Range Percentage of tumor Median Range Predictive probability Median Range

Fatal

Nonfatal

P

13.6 0.2-2,691

7.2 0.4-201

~0

7 5-10

6 4-10

~0

10.6 0.02-167

5.0 0.02-136

~0

0.50 0.1-1

0.33 0.06-1

~0

15 0.4-90

6 0.1-95

~0

0.17 0.02-0.93

0.09 0.01-0.68

See text

PSA, prostate-specific antigen. * The data represent 663 American Veterans. All variables were observed at the time of diagnosis, and the histologic variables were obtained from the diagnostic biopsy specimens. The fraction of positive cores stands for the ratio of cores with tumor to the total number of cores. P values were calculated by using the nonparametric Wilcoxon test. The predictive probability comes from a logistic regression analysis using a 2-variable model with log(PSA) and the primary Gleason grade (see text).


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early death, with P values, respectively, of approximately 0 for log(PSA) and .001 for the primary Gleason grade. After these 2, none of the remaining variables in Table 1, including full Gleason score, provided additional prognostic information (P > .2). Nevertheless, the predictive probability of early death formulated from the 2-variable logistic model once again demonstrated so much overlap of values between the rapidly evolving vs the more slowly evolving tumors that it also cannot be considered a reliable discriminating model.

implied a higher total death rate, which for cases diagnosed in 1982 was 57% compared with 14% for cases diagnosed in 1993. ❚Figure 4❚ shows that there was a steady decline in the hazard peak during the years before and after PSA testing was used (significant, P ~ 0; linear regression). Total mortality also dropped from 61% in 1977 to 2% in 1997 (P ~ 0; linear regression). Nevertheless, the times of peak hazard did not change (P > .1; linear regression), implying that a component of rapidly progressing tumor remained present throughout the period.

The Hazard Functions Before and After Serum PSA The dramatic differences in curve height and enclosed area for the 2 hazard functions illustrated in Figure 3 are a small sample of a downward trend in hazard during the PSA era. In Figure 3, the hazard function for cases diagnosed in 1982 (before PSA testing was used) is clearly higher than for cases diagnosed in 1993 (when PSA testing was commonly used). The areas under these 2 curves correspond to the integral in Equation 2, so that the larger area for the 1982 hazard

Incidence of Nonfatal and Fatal Forms of Prostate Cancer The incidence rates for nonfatal and fatal cases of prostate cancer were obtained as products of the incidence of prostate cancer for each diagnostic year and the probabilities of nonfatality and fatality, respectively, for those years as follows:

Maximum Hazard

0.10 0.08 0.06 0.04 0.02

1980

1985

1990

1995

Year

❚Figure 4❚ Peak hazard for Surveillance, Epidemiology and End Results data from the years of diagnosis 1975 to 1997. 250

Incidence

200 Nonfatal

150 100

❚Equation 3❚

Nonfatal Cases = Incidence * [exp (– ∫ h(x) dx)] ❚Equation 4❚

Fatal Cases = Incidence * [1 – exp (– ∫ h(x) dx)] Here, the hazard function used was specific to the year of diagnosis, and the integration limits were from 0 to ∞. ❚Figure 5❚ shows plots of the incidence of nonfatal and fatal prostate cancers vs the year of diagnosis. Whereas the incidence of nonfatal cases rose during the PSA era (P = .001; linear regression analysis of logarithm of incidence vs time), the incidence of fatal cases fell (P < .006 for linear regression analysis of incidence vs time and time2). ❚Figure 6❚ shows that during this time, the number of practicing urologists in the United States also rose steadily in nearly a linear manner (P ~ 0; linear regression analysis). Thus, the incidence of nonfatal cases was positively associated with the number of practicing urologists (P = .0002 for linear regression analysis of logarithm of incidence vs number of urologists), and the incidence of fatal cases dropped as the number of practicing urologists rose (P < .002 by linear regression for a parabolic relationship between the incidence of fatal cases and the number of urologists). Although these results do not prove a causative relationship between the number of urologists and the incidence of nonfatal cases of prostate cancer, the results support the concept of Welch and Black23 of a reservoir of subclinical prostate cancers.

50 Fatal

Discussion

0 1980

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Year

❚Figure 5❚ The expected incidence of nonfatal cases of prostate cancer (upper curve) and the expected incidence of fatal cases (lower curve) for the years of diagnosis given on the x-axis.

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This study demonstrates that the hazard function provides useful information about prostate cancer, and it demonstrates that the effects of PSA screening on the types of tumors detected could have been predicted from a careful analysis of mature SEER survival data, that is, without using the results of prospective randomized trials. Deriving © American Society for Clinical Pathology

the hazard function requires neither complex modeling nor assumptions about stage progressions, and it requires no computer simulations. Its derivation comes straightforwardly from the calculus of survival, and its analysis requires nothing other than mature survival data. The results also documented a form of rapidly evolving fatal prostate cancer that persists into the PSA era, even though it has been diluted by many cases of nonfatal prostate cancer. This form of tumor relates to the observation that an average of 44% of deaths occur during the first 5 years after diagnosis. Because the hazard function peaks at less than 2 years after diagnosis, this form of fatal tumor may evolve so quickly that it may not be affected by screening or localized treatment. And it is not accurately recognized by routine prognostic factors. The results support conclusions about the rising incidence of a slowly evolving and nonfatal prostate cancer during the PSA era and document that the incidence of this reservoir of nonfatal tumors is directly related to the number of practicing urologists. Finally, the results demonstrated a decline in fatal forms of prostate cancer and showed that this decline is associated with the use of serum PSA testing and the number of practicing urologists. The reasons for the decline in fatal forms of prostate cancer are probably multiple. The decline may be due in part to earlier diagnosis and effective treatment of prostate cancers that evolve at a sufficiently slow rate that they can be detected by PSA screening and treated before they spread. The decline in fatality may also be due in part to how prostatectomy removes hyperplastic tissue, thereby preventing obstructive uropathy and renal failure. And the decline may relate to how serum PSA testing allows more accurate attribution of death as being due to prostate cancer. Before serum PSA testing, some men’s death may have been attributed to prostate cancer, when, in fact, their level of PSA, had it been measured, would have been found to be too low. Now with serum PSA testing, such deaths are attributed to other causes. From the Department of Laboratory Medicine, VA Medical Center, and Department of Pathology, Duke University Medical Center, Durham, NC. Address reprint requests to Dr Vollmer: Laboratory Medicine 113, VA Medical Center, 508 Fulton St, Durham, NC 27705.

References 1. Ablin RJ, Soanes WA, Bronson P, et al. Precipitating antigens of the normal human prostate. J Reprod Fertil. 1970;22:573-574. 2. Wang MC, Valenzuela LA, Murphy GP, et al. Purification of a human prostate specific antigen. Invest Urol. 1979;17: 159-163. 3. Papsidero LD, Wang MC, Valenzuela LA, et al. A prostate antigen in the sera of prostatic cancer patients. Cancer Res. 1980;40:2428-2432.

No. of Urologists (in thousands)

Anatomic Pathology / Original Article

8.5 8.0 7.5 7.0 6.5 6.0 5.5 1980

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❚Figure 6❚ The number of practicing urologists in the United States vs calendar year. The data come from the US Census statistical abstracts (see the “Materials and Methods” section).

4. Killian CS, Yang N, Emrich LJ, et al. Prognostic importance of prostate-specific antigen for monitoring patients with stages B2 to D1 prostate cancer. Cancer Res. 1985;45:886-891. 5. Stamey TA, Yang N, Hay AR, et al. Prostate-specific antigen as a serum marker for adenocarcinoma of the prostate. N Engl J Med. 1987;317:909-916. 6. Hudson MA, Bahnson RR, Catalona WJ. Clinical use of prostate specific antigen in patients with prostate cancer. J Urol. 1989;142:1011-1017. 7. Andriole GL. Serum prostate specific antigen: the most useful tumor marker. J Clin Oncol. 1992;10:1205-1207. 8. Catalona WJ, Richie JP, Ahmann FR, et al. Comparison of digital rectal examination and serum prostate specific antigen in the early detection of prostate cancer: results of a multicenter clinical trial of 6,630 men. J Urol. 1994;151:1283-1290. 9. Etzioni R, Penson DF, Legler JM, et al. Overdiagnosis due to prostate-specific antigen screening: lessons from US prostate cancer incidence trends. J Natl Cancer Inst. 2002;94:981-990. 10. Epstein JI. Diagnostic criteria of limited adenocarcinoma of the prostate on needle biopsy. Hum Pathol. 1995;26:233-239. 11. Kronz JD, Milord R, Wilentz R, et al. Lesions missed on prostate biopsies by pathologists [abstract]. Mod Pathol. 2001;14:114A. 12. Griffiths RC, Epstein JI. Review of prostate needle biopsies by an expert [abstract]. Mod Pathol. 2001;14:110A. 13. Smith DS, Carvalhal GF, Mager DE, et al. Use of lower prostate specific antigen cutoffs for prostate cancer screening in black and white men. J Urol. 1998;160:1734-1738. 14. Lodding P, Aus G, Bergdahl S, et al. Characteristics of screening detected prostate cancer in men 50 to 66 years old with 3 to 4 ng/mL prostate specific antigen. J Urol. 1998;159:899-903. 15. Catalona WJ, Ramos CG, Carvalhal GF, et al. Lowering the PSA cutoffs to enhance detection of curable prostate cancer. Urology. 2000;55:791-795. 16. Punglia RS, D’Amico AV, Catalona WJ, et al. Effect of verification bias on screening for prostate cancer by measurement of prostate-specific antigen. N Engl J Med. 2003;349:335-342. 17. Cowen ME, Chartrand M, Weitzel WF. A Markov model of the natural history of prostate cancer. J Clin Epidemiol. 1994;47:3-21.


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❚Appendix 1❚ Estimating the Hazard Function To estimate the hazard function, h(t), I first obtained raw values for h(t) as follows. If the survival probabilities in a survival curve at times t1 and t2 are S(t1) and S(t2), respectively, then Equation 1 implies that h(t) can be approximated for a median time, tm, between t1 and t2 as:

diagnosed between 1980 and 1984, and the smooth curve comes from the fitted h(t) as used in Equation 2. See the “Results” section for how well the γ model for h(t) worked for all SEER data. 0.10

❚Equation A1❚ h(tm) = – log [S(t2)/S(t1)]/(t2 – t1)

0.08 Hazard

An example of a plot of such raw values for the hazard function of 1 Surveillance, Epidemiology and End Results (SEER) survival curve is shown as the points in ❚Figure 7❚. Note that the raw estimates of hazard rise to a peak before 5 years and then gradually fall. This early peak in raw hazards occurred at an average of 1.8 years after diagnosis across the 11 SEER survival curves (range, 0.5-3.5 years), and the pattern of an early rise in hazard followed by a fall suggested that the sum of 2 γ functions would be a suitable mathematical form for h(t). For example, the smooth curve in the plot comes from the sum of 2 such γ functions, h1(t) and h2(t), given as follows:

0.06 0.04 0.02 0.00 0

❚Equation A2❚ h1(t) = 0.115 * t * exp( – 0.646 * t)

❚Equation A3❚

h2(t) = 0.0027 * t2 * exp( – 0.212 * t)

Consequently, the following general form for h(t) was adopted:

5

10

15

20

Years

❚Figure 7❚ Raw hazard estimates (points) for 1 Surveillance, Epidemiology and End Results survival curve. The smooth line comes from a hazard function comprising the sum of 2 γ functions.

❚Equation A4❚

1.0

h1(t) = α * t * exp( – β * t)

❚Equation A5❚

h2(t) = λ * t2 * exp( – δ * t)

18. Draisma G, Boer R, Otto SJ, et al. Lead times and overdetection due to prostate-specific antigen screening: estimates from the European Randomized Study of Screening for Prostate Cancer. J Natl Cancer Inst. 2003;95:868-878. 19. Davidov O, Zelen M. Overdiagnosis in early detection programs. Biostatistics. 2004;5:603-613. 20. Welch HG, Albertsen PC. Prostate cancer diagnosis and treatment after introduction of prostate-specific antigen screening: 1986-2005. J Natl Cancer Inst. 2009;101:1325-1329. 21. Andriole GL, Crawford ED, Grubb RL III, et al. Mortality results from a randomized prostate-cancer screening trial. N Engl J Med. 2009;360:1310-1319. 22. Schröder FH, Hugosson J, Roobol MJ, et al. Screening and prostate-cancer mortality in a randomized European study. N Engl J Med. 2009;360:1320-1328.

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0.8 0.6 S(t)

with α, β, λ, and δ representing coefficients to be adjusted so that h(t) as used in Equation 2 would fit the observed values of S(t). The form for h1 allows for an early peak in hazard at a time equaling 1/β, while the form for h2 allows for a later peak at a time equaling 2/δ. In fact, initial values for β and δ were obtained for each survival curve using the locations of the first and secondary peaks in the raw hazards, and values of α and λ were obtained from the heights of the raw hazard peaks. Final values for α, β, λ, and δ were obtained by the iterative nonlinear least squares fitting routine nls in SPLUS (MathSoft, Seattle, WA). Thus, for each of the SEER survival curves, h(t) was derived in a stepwise manner: first plotting raw estimates of h(t), then obtaining values of α, β, λ, and δ to roughly match h(t), and then finally using these values and nonlinear leastsquares fitting to obtain final values of α, β, λ, and δ. The goal in this process was to derive a mathematical form for h(t) so that the right side of Equation 2 would provide a close fit to observed values of S(t). ❚Figure 8❚ shows an example of how well this process worked for one of the survival curves. In Figure 8, the points are the observed survival values from the SEER survival curve for cases

0.4 0.2 0.0 0

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❚Figure 8❚ Plot of observed values of the survival probability, S(t) (points), for 1 Surveillance, Epidemiology and End Results survival curve. The smooth line comes from the γ model of the hazard function, h(t), used in Equation 2 after the values of α, β, λ, and δ were derived from a least-squares fit of the survival points.

23. Welch HG, Black WC. Overdiagnosis in cancer. J Natl Cancer Inst. 2010;102:605-613. 24. Simes RJ, Zelen M. Exploratory data analysis and the use of the hazard function for interpreting survival data: an investigator’s primer. J Clin Oncol. 1985;3:1418-1431. 25. Yakovlev AY, Tsodikov AD, Boucher K, et al. The shape of the hazard function in breast cancer: curability of disease revisited. Cancer. 1999;85:1789-1798. 26. Vollmer RT. Tumor length in prostate cancer. Am J Clin Pathol. 2009;131:86-91. 27. Harrell FE Jr. Regression Modeling Strategies With Applications to Linear Models, Logistic Regression, and Survival Analysis. New York, NY: Springer-Verlag; 2001:392-398.
