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Survival Analysis of Prostate Cancer Patients Using Cox Regression
Model and Log-Logistic Model
Samuel Olayemi Olanrewaju
1
,Ezekiel Kehinde Adeniran
2
Department of Statistics, University of Abuja, Abuja, Nigeria
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.150100051
Received: 16 January 2026; Accepted: 23 January 2026; Published: 06 February 2026
ABSTRACT
Survival time analysis focuses on the time until an event occurs and is used to identify risks in survival data.
This study employs Non-Parametric (Kaplan-Meier) methods to assess median survival time, Log-rank tests to
compare hazard and survivor functions, Semi-Parametric (Cox Proportional Hazards), and Parametric
approaches to determine the best-fitting distribution. Prostate Cancer (PC) is the second most common
malignancy in men worldwide, with 1,276,106 new cases and 358,989 deaths in 2018 (Rawla, 2019). The
incidence and mortality of prostate cancer increase with age, with the average diagnosis age being 66 years.
African-American men have higher incidence rates (158.3 new cases per 100,000 men) and nearly double the
mortality rate compared to White men (Capece et al., 2020). This study found that both the age of patients and
the year of admission were consistently significant factors. The Log-logistic model was identified as the
bestfitting model with an AIC value of 302.7047, compared to the Cox Regression model's AIC value of
434.0985.
INTRODUCTION
Prostate cancer is the second most frequent malignancy in men worldwide, with 1,276,106 new cases and
358,989 deaths in 2018 (Rawla, 2019). The incidence and mortality rates of prostate cancer correlate with
increasing age, with the average diagnosis age being 66 years. African-American men have higher incidence
rates compared to White men, with 158.3 new cases diagnosed per 100,000 men and nearly double the
mortality rate (Capece et al., 2020). This disparity may be due to social, environmental, and genetic factors. By
2040, 2,293,818 new cases are estimated, with a small variation in mortality (Tosoian et al., 2015).
Prostate cancer often presents with no symptoms and progresses slowly, with common complaints including
nocturia and difficulty urinating. Male-specific antigens (PSA > 4 ng/mL) are used for identifying prostate
malignancies, though a biopsy is required for confirmation. Dietary and exercise factors significantly influence
the onset and spread of prostate cancer (Capurso & Vendemiale, 2017). The disease continues to pose a
significant public health challenge in Nigeria, where its burden appears increasingly substantial due to both
rising incidence and late clinical presentation. National evidence suggests that prostate cancer incidence and
prevalence have been increasing over time, with a wide range of hospital-based prevalence estimates and high
mortality observed within three years of diagnosis, reflecting advanced disease at presentation in many
Nigerian settings (Epidemiology of prostate cancer in Nigeria, 2024).
In community and clinical settings, studies consistently reveal low levels of knowledge about prostate cancer
and very limited screening uptake. For example, in Sokoto State only 5% of surveyed men were aware of
prostate cancer and just 1.3% knew about available screening tests, with no participant reporting prior
screening, largely due to lack of awareness (Awosan et al., 2018). Similarly, in Ido-Ekiti only 18.2% of men
aged 40 and above reported ever having a prostate cancer screening test, despite moderate awareness,
underscoring systemic gaps in screening access (Adewoye et al., 2023). Further research in Bayelsa
communities reported low knowledge about prostate cancer symptoms, prevention, and management,
highlighting the broad need for education and early detection programs (Awareness of Prostate Cancer…,
2025).
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Collectively, this body of Nigerian evidence highlights critical gaps in awareness, screening, and early
detection, and supports the need for rigorous survival analyses that reflect the real-world experiences of
Nigerian prostate cancer patients. These findings provide essential context for survival studies, as late
presentation, often due to low screening uptake and poor knowledge, is intrinsically linked with poorer
survival outcomes in low-resource settings.
Statement of Problem
Prostate cancer (PC) is the fourth major cancer globally, accounting for 1.3 million cases (7.1% of the overall
cancer incidence) (Shafique et al., 2012). It is the second most frequently diagnosed cancer disease in men,
with varying incidence rates globally (Sebastiao & Peter, 2018). In southwestern Nigeria, PC is prevalent
among men aged 46-99 years, with a peak incidence in those ≥ 70 years old (Van Wijk & Simonsson, 2022).
Risk factors for PC include both non-modifiable factors (age, race, family history) and modifiable factors
(alcohol consumption, obesity, smoking, sedentary lifestyle, prostatitis history, high cholesterol) (Balogun et
al., 2020). This study examines the risk factors associated with prostate cancer in Nigeria and models the
survival pattern of prostate cancer patients.
Aim and Objectives
This study aims to compare Cox regression and parametric models for the survival of prostate cancer patients.
The specific research objectives are to:
i. Estimate the survival time with respect to variables of interest.
ii. Compare hazard and survivor functions of different variables of interest.
iii. Fit the Cox proportional hazard model along with different parametric models to the prostate cancer
data.
iv. Choose the best-fitted model for the prostate cancer data.
RESEARCH METHODOLOGY
Source of Data
The data for this study are secondary data obtained from the records of six hundred and sixty-one (661)
registered prostate cancer patients at the University of Ilorin Teaching Hospital. The data collected is based on
the length of stay, age, gender, and outcomes over a twelve-year period (2011-2022). To facilitate computation
and make the best use of the statistical tools applied in this research, the covariates were categorized as follows:
Table 3.0: Categories of Covariates
Covariates
Categories
Gender
Male and Female
Age (Years)
5-19, 20-39, 40-59, 60-79 and 80-99
Year of Admission
2011-2022
Outcome
Dead or Alive
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RESEARCH METHODOLOGY
Kaplan Meier Estimator
The technique used to estimate the survivor function when censoring is present is called the Kaplan-Meier or
the product-limit estimator of the survivor function (or survival probability). Equation 1 below displays the
Kaplan-Meier estimator of survival at time t. In this case, dj is the number of failures at time tj, rj is the
number of people at risk at time tj, and tj, j = 1, 2..., n is the whole set of failure times recorded (with t+ the
greatest failure time).
󰇛󰇜 =


󰇛
󰇜
, for all
(1)
It is simply the empirical probability of surviving past certain times in the sample (taking into account
censoring). When there is inappropriate censoring, the Kaplan Meier method is not appropriate. The
general formula for a KM survival probability at failure time t(j ) is shown below;
S(t(j))=󰇛󰇛 1󰇜󰇜 Pr󰇟 󰇛󰇜 󰇛󰇜󰇠 (2)
This formula gives the probability of surviving past the previous failure time t(j−1), multiplied by the
conditional probability of surviving past time t (j), given survivalto at least time t (j). The above KM formula
can also be expressed as a product limit if the survival probability Sˆ(t( j−1)) is substituted, the product of all
fractions that estimate the conditional probabilities for failure times t( j−1) and earlier. This is expressed as;
󰇛󰇛 󰇜󰇜 П 󰇟 󰇛󰇜 󰇛󰇜󰇠 (3)
󰇛󰇜 
󰇛
󰇜
 0 (4)
Equation 4 is the empirical probability of surviving past certain times in the sample (taking into account
censoring).
Log Rank Test
The Log rank test also known as Mantel- Haenszel test, is a large sample chi-square test that uses as its test
criterion a statistic that provides an overall comparism of the KM curves being compared (Sebastiao and Peter,
2018). It is applicable to data where there is progressive censoring and gives equal weight to early and late
failures. The test statistic can be expressed as;
2
󰇛

󰇜
2
1 2
2
1  (5)
󰇛󰇜
The log-logistic model is discussed below because it is appropriate for cancer survival data where the risk of
death may vary over time rather than remain constant. In prostate cancer, mortality risk often increases after
diagnosis and treatment before stabilizing among longer-term survivors, a pattern the log-logistic model can
capture through its flexible hazard function. Unlike models that assume monotonic hazards, log-logistic
distribution allows for this non-monotonic behaviour. Its suitability in this study is further supported by lower
AIC and BIC values compared with the Cox proportional hazards model, indicating a better fit to the data.
Log-Logistic Model
The log-logistic (LL) distribution (branded as the Fisk distribution in economics) possesses a rather supple
functional form. The LL distribution is among the class of survival time parametric models where the hazard
rate initially increases and then decreases and at times can be hump-shaped (Adelian et al., 2015). The hazard
rate function of LL model is given below:
ℎ󰇛  󰇜

(6)
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󰇟1󰇛
󰇜 󰇠
The log-logistic (LL) model is expressed mathematically as;
11

󰇛
1
1
2
2

󰇜 (7)
Where the mean and the variance of the log-logistic model is given as:

Mean =
󰇜
 1  . (8)
󰇛
Cox-proportional Hazard Model
The Cox-proportional hazards model possess the property that different individuals have hazard functions that
are proportional i.e. [h(t|x1)/h(t|x2)], the ratio of the hazard functions for two individuals with prognostic
factors or covariates x1=(
11
21
1
󰇜′, and x2=(
12
22
2
󰇜 is a constant (does not vary with time t). This
means that the ratio of the risk of dying of two individuals is the same no matter how long they survive. The
log hazard's linear-like model must also be specified for this model. The exponential distribution can be used
to parameterize a parametric model in the following way:
log ℎ
󰇛󰇜 =
1
11
+
2
12
+ … +
1
(9)
In this case, the constant represents the log-baseline since ℎ
󰇛󰇜 = , when all the x’s are zero. The Cox
proportional hazards model is a semi-parametric model where the baseline hazard (t) is allowed to vary
with time.
RESULTS AND DISCUSSION
This chapter discusses the analysis of data collected on prostate cancer patients using three methods:
Nonparametric methods (including Kaplan-Meier and Log-Rank Statistic), Semi-parametric (Cox
Proportional Hazard), and a parametric model (Log-normal). The results are present
Kaplan-Meier Survival Curves
The figures below show the general Kaplan-Meier survival curve for all prostate cancer patients as well as for
each covariate considered in this study.
0.00 0.25 0.50 0.75 1.00
0 50 100 150
analysis time
Kaplan-Meier survival estimate
0.00 0.25 0.50 0.75 1.00
0 50 100 150
analysis time
Male
Female
Kaplan-Meier survival estimates
Figure 4.1: General survival curve for prostate cancer patients. Figure 4.2: Survival curve on sex of
prostate cancer patients with log-rank p-value = 0.6446.
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0.00 0.25 0.50 0.75 1.00
0 50 100 150
analysis time
1-10 years 11-20 years
21-30 years 31-40 years
41-50 years 51-60 years
61-70 years 71-80 years
81-90 years
Kaplan-Meier survival estimates
0.00 0.25 0.50 0.75 1.00
0 50 100 150
analysis time
2011 2012
2013 2014
2015 2016
2017 2018
2019 2020
2021 2022
Kaplan-Meier survival estimates
Figure 4.3: Survival curve on age prostate cancer patients Figure 4.4: Survival curve on year
distribution of prostate with log-rank p-value = 0.0179. cancer patients with log-
rank p-value = 0.0372.
Figure 4.1 shows the survivorship function based on the total number of patients’ data collected with the
number at risk at different time points. Figure 4.2 reveals the survivorship function based on the sex of prostate
cancer patients. While Figure 4.3 indicates the survivorship function based on age categories of prostate cancer
patients and Figure 4.4 shows the survivorship function based on the year of admission of prostate cancer
patients with the number at risk at different time points.
Log-Rank Test for Equality of Survivor Functions
4.2.1 Log-rank Statistic
󰇛

󰇛
󰇜
2
󰇜, i = 1, 2, 3… ~
2
󰇛󰇜
i = 1, 2, 3…
: Survival curves for covariates are the same
Vs
1
: Survival curves for covariates are not the same
Decision rule: Reject
if p-value < -level (0.05). Otherwise, do not reject
.
Table 4.1: Log-rank Test for the Sex of the Prostate Cancer (PC) Patients
SEX
EVENT OBSERVED
Male
43
Female
3
Total
46
2
󰇛
1
󰇜
= 0.21 Pr>chi2 = 0.0410
Decision: Since the p-value= 0.0410< α=0.05. The null hypothesis (
) is therefore rejected.
CONCLUSION
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Since the null hypothesis has been rejected. It can therefore be concluded that patients’ sex has significant
different K-M survival curves. This means that sex of prostate cancer patients does affect their survival.
Table 4.2: Log-rank Test for the Different Age Categories of Prostate Cancer Patients
AGE
EVENT OBSERVED
1-10 years
2
11-20 years
0
21-30 years
0
31-40 years
3
41-50 years
12
51-60 years
11
61-70 years
14
71-80 years
4
81-90 years
0
Total
46
2
󰇛
8
󰇜
= 5.35 Pr>chi2 = 0.0179
Decision: Since the p-value= 0.0179< α=0.05. The null hypothesis (
) is therefore rejected.
CONCLUSION
Since the null hypothesis has been rejected. It can therefore be concluded that different categories of the
patients’ age have significant different K-M survival curves. This means that patients’ age of prostate cancer
patients does affect their survival.
Table 4.3: Log-rank Test for the Years of Admission of Prostate Cancer Patients
YEARS OF ADMISSION
EVENT OBSERVED
EVENT EXPECTED
2011
5
4.27
2012
4
3.02
2013
6
7.19
2014
0
2.20
2015
4
3.47
2016
6
5.62
2017
2
1.49
2018
0
1.81
2019
2
3.36
2020
5
1.49
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2021
6
7.16
2022
6
4.91
Total
46
46.00
2
󰇛
11
󰇜
= 15.06 Pr>chi2 = 0.0372
Decision: Since the p-value= 0.0372< α=0.05. The null hypothesis (
) is therefore rejected.
CONCLUSION
Since the null hypothesis has been rejected. It can therefore be concluded that different years of admission
have significant different K-M survival curves. This means that patients’ year of admission do affect patients’
survival rate.
Test Of Proportional Hazards Assumption
: The Proportional Hazard assumption is not violated
Vs
1
: The Proportional Hazard assumption is violated
Time: log(t)
Table 4.4: Proportional Hazard Assumption
Covariates
Rho
chi2
df
prob>chi2
SEX
-
-
-
-
Female
0.05524
0.15
1
0.0248
AGE Categories
-
-
1
-
11-20 years
-
-
1
-
21-30 years
-
-
1
-
31-40 years
-
-
1
-
41-50 years
-0.17575
1.23
1
0.2671
51-60 years
-0.32579
3.52
1
0.0607
61-70 years
-0.24246
1.8
1
0.1800
71-80 years
-0.27858
2.61
1
0.1059
81-90 years
-0.30277
3.3
1
0.0692
Years of Admission
-
-
-
-
2012
0.11455
0.61
1
0.4366
2013
-0.07376
0.28
1
0.5965
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2014
-
-
1
-
2015
0.18845
1.57
1
0.2108
2016
-0.02635
0.03
1
0.8570
2017
0.07262
0.26
1
0.6085
2018
-
-
1
-
2019
0.09379
0.37
1
0.5413
2020
-0.03495
0.06
1
0.8046
2021
0.02286
0.02
1
0.8757
2022
0.05076
0.13
1
0.7220
Global Test
-
9.95
15
0.8227
The test Table 4.4 do not suggest violation of the PH assumption for any of the covariates with their p-values
greater than 0.05. thus, the cox model will be fitted to the prostate cancer data.
COX PROPORTIONAL HAZARD MODEL
No. of subjects = 309 Number of observations = 309
No. of failures = 46 LR
2
(15) = 20.17
Time at risk = 3902 Prob >
2
= 0.1656
Log likelihood = -202.04923
Table 4.5 Cox Regression Model
Covariates
Coefficient
()
Hazard
Ratio
Std.
Err.
z
p>

[95% Conf.
Interval]
SEX
-
-
-
-
-
-
Female
0.4033
1.4967
0.6542
0.62
0.038
-0.8789
1.6855
AGE
Categories
-
-
-
-
-
11-20 years
-43.2297
1.68e-19
-
-
-
-
21-30 years
-42.4671
3.60e-19
-
-
-
-
31-40 years
-0.1435
0.8663
1.0135
-
0.14
0.887
-2.1299
1.8429
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41-50 years
-0.4262
0.6529
0.8296
-
0.51
0.607
-2.0521
1.1997
51-60 years
-0.9063
0.4040
0.8600
-
1.05
0.292
-2.5919
0.7793
61-70 years
-0.5354
0.5854
0.8677
-
0.62
0.537
-2.2361
1.1652
71-80 years
0.0447
1.0458
0.9892
0.05
0.964
-1.8941
1.9835
81-90 years
-0.6815
0.5058
-
-
-
-
Years of
Admission
-
-
-
-
-
-
2012
0.1771
1.1937
0.6981
0.25
0.800
-1.1912
1.5453
2013
-0.4286
0.6514
0.6790
-
0.63
0.528
-1.7595
0.9022
2014
-45.2978
2.13e-20
-
-
-
-
2015
0.0996
1.1048
0.6852
0.15
0.884
-1.2435
1.4427
2016
0.0043
1.0043
0.6332
0.01
0.995
-1.2367
1.2453
2017
0.2853
1.3301
0.8848
0.32
0.747
-1.4489
2.0195
2018
-45.3221
2.07e-20
-
-
-
-
2019
-0.3839
0.6812
0.8616
-
0.45
0.656
-2.0726
1.3046
2020
1.2129
3.3634
0.6729
1.80
0.071
-0.1059
2.5318
2021
-0.3268
0.7212
0.6423
-
0.51
0.611
-1.5857
0 .9321
2022
0.0628
1.0648
0.6413
0.10
0.922
-1.1941
1.3196
The Estimated model: ĥ(t, X )=ℎ
(t)
1
1
2
2
3
3
. The Cox’s Proportional Hazard Model was employed to
determine the hazard ratio of the groups of covariates.
Key Observations:
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The hazard ratio for male relative to female is 1.4967, indicating that male patients have a longer survival time
than female patients, with female patients having a 1.4967 times higher risk of dying from prostate cancer.
The proportional hazards assumption was formally assessed using Schoenfeld residuals. The test results
indicated that none of the covariates violated the proportional hazards assumption, and the global test was also
not statistically significant, suggesting that the hazard ratios remained constant over time. This confirms that
the Cox proportional hazards model was appropriate for analyzing the survival data in this study and that the
estimated effects of the covariates can be reliably interpreted. The absence of significant violations further
strengthens the validity of the Cox model results presented.
Table 4.6:
Akaike information criterion and Bayesian information criterion
Model
Observation
11 (null)
11 (model)
df
AIC
BIC
309
-212.133
-202.0492
15
434.0985
490.0986
Table 4.6 shows the Akaike information criterion and Bayesian information criterion generated for the cox
regression model which will later be used to compare the other model fitted to the prostate cancer data. Thus,
the best model would be chosen among the fitted models.
Parametric Method (Log-logistic Model)
No. of subjects = 309 Number of observations = 309
No. of failures = 46 LR
2
(15) = 0.19
Time at risk = 3902 Prob >
2
= 0.9796
Log likelihood = -146.35234
Table 4.7: Log-logistics Model
Covariates
Coefficient
Std. Err.
z
p> 
[95% Conf. Interval]
SEX
-0.1518
0.4908
-0.31
0.001
-1.1137 0.8101
AGE
-0.0154
0.0876
-0.18
0.050
-0.1871 0.1563
YEARS OF ADMISSION
-0.0079
0.0344
-0.23
0.017
-0.0754 0.0595
_Cons
4.2609
0.8413
5.06
0.001
2.6119 5.9099
/ln_gam
-0.2845
0.1136
2.50
0.012
-0.5073 -0.0618
/gamma
0.7524
0.0855
0.6021 0.9400
The Log-logistic model fitted to the data is as follows;
1
1
 󰇛

1
1

2
2

󰇜 
Therefore, the fitted model is given below;
11
4260901518

00154

00079

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Also, the model fitted for the significant covariates are as follows;
11
4260901518

0015400079

Table 4.8:
Akaike information criterion and Bayesian information criterion
Model
Observation
11 (null)
11 (model)
df
AIC
BIC
309
-
146.4462
-146.3523
5
302.7047
321.3714
Table 4.8 reveals the Akaike information criterion and Bayesian information criterion generated for the
Loglogistic model which will later be used to compare Cox model fitted to the prostate cancer data. Therefore,
the model with the least AIC would be selected as the best model fitted to the prostate cancer data.
Table 4.9:
MODEL SELECTION
Model
Number of Parameters (p)
Log Likelihood
AIC
BIC
Cox Regression
3
-202.0492
434.0985
490.0986
Loglogistic
4
-146.3523
302.7047
321.3714
For the best model to be chosen, the AIC values for the fitted models are compared. Thus, the best fitted model
selected for the prostate cancer data is Loglogistic model having the least AIC and BIC value respectively.
SUMMARY OF FINDINGS
This study aimed to identify the factors that affect the survival level of prostate cancer (PC) patients. Key
findings include:
•
Kaplan-Meier Survival Estimates: Sex, age, and year of admission significantly affected the survival
rate of PC patients.
•
Log-Rank Test: The survival experience of patients differed significantly based on sex, age, and year
of admission at a 0.05 significance level.
•
Cox Regression and Loglogistic Models: Both models identified sex, age, and year of admission as
significant covariates. These factors consistently contributed to the models.
•
Best Model: The Loglogistic model was the best fit for the prostate cancer data, with the least AIC
value of 302.7047.
CONCLUSION
Based on the analysis, the following conclusions were made:
• Sex, age, and year of admission significantly affect the survival rate of PC patients at a 0.05 level of
significance.
• Both Cox and Loglogistic models identified sex, age, and year of admission as significant covariates.
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• The covariates consistently significant in both models were sex, age, and year of admission.
RECOMMENDATIONS
Based on this research, the following recommendations are made:
•
The year of admission of prostate cancer (PC) patients is crucial. People with PC should seek hospital
treatment as early as possible for better outcomes.
•
Parametric models are recommended for analyzing PC data. The Loglogistic model is the best fit for
PC data irrespective of sample sizes.
Limitation of the Study
Although this study examined key demographic and temporal covariates such as age, sex, and year of
admission, other clinically relevant factors that may influence prostate cancer survival were not included in the
analysis. These include treatment modality, disease stage at diagnosis, comorbid conditions, and socio-
economic status, which have been shown in previous studies to affect cancer outcomes. The absence of these
variables in the hospital records limited the ability to fully adjust for potential confounding effects, and their
omission should be considered when interpreting the findings.
In the Nigerian healthcare context, variations in access to diagnostic facilities, treatment availability, and
followup care may further contribute to differences in survival outcomes. Patients are often present at
advanced stages due to low screening uptake and delayed healthcare-seeking behaviour, factors that are closely
linked to both disease progression and treatment response. Future studies incorporating clinical staging
information, treatment data, and indicators of socio-economic status would provide a more comprehensive
understanding of prostate cancer survival and support more targeted interventions in Nigeria.
Suggestions for Further Study
This study focused on the survival rate of prostate cancer using data from the University of Ilorin Teaching
Hospital (UITH). Suggestions for future research include:
•
Employ different parametric models other than the Loglogistic model to compare the survival levels of
prostate cancer.
•
Compare other semi-parametric models aside from the Cox Regression Model for analyzing the survival
level of prostate cancer patients.
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