Cited by 741Statistical Methods and InferenceProbability and Statistical ResearchStatistical Distribution Estimation and Applications

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Survival Analysis Part II: Multivariate data analysis – an introduction to concepts and methods

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Bradburn, M J, Clark, T G, Love, S B, Altman, D G·2003·British Journal of Cancer

DOI 10.1038/sj.bjc.6601119PMC PMC2394368

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Abstract

Survival analysis involves the consideration of the time between a fixed starting point (e.g. diagnosis of cancer) and a terminating event (e.g. death). The key feature that distinguishes such data from other types is that the event will not necessarily have occurred in all individuals by the time the study ends, and for these patients, their full survival times are unknown. For instance, in studies that measure the length of survival after diagnosis of cancer, it is common for a proportion of individuals to remain alive and disease-free at the end of the follow-up period, and for these patients, we know only a lower limit on their actual time to event. Thus, special methods are required for these type of data. The explanation and demonstration of some of the methods proposed to analyse such data are the basis of this series.

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Figure 1 Example of (non-) proportional hazards (groups (c) and (d) only have proportional hazards) using the Weibull distribution. For the Weibull survival model, the hazard function h(t)=λs(λt)s−1 for λ, s>0: (a) increasing hazard (λ=0.5, s=1.25); (b) decreasing hazard (λ=0.25, s=0.75); (c) decreasing hazard (λ=0.5, s=0.5); (d) decreasing hazard (λ=0.25, s=0.5).CC BY · PubMed Central

Figure 2 Predicted 5-year survival of ovarian cancer patients by age and FIGO stage.CC BY · PubMed Central

Figure 3 Illustration of the AFT model: (——), S0(t) the baseline survival function; (·······), S(t1)=S0(ϕt) for ϕ<1; (- - - - - -), S(t2)=S(ϕt) for ϕ>1.CC BY · PubMed Central

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Regression Models and Life-Tables
Cox, D. R. (1972) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1972.tb00899.x · Cited by 39,793
Survival Analysis
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Modelling for Survival Data in Medical Research
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Survival Analysis Part I: Basic concepts and first analyses
Clark, T. G., Bradburn, M. J., Love, S., & Altman, D. G. (2003) · British Journal of Cancer · doi:10.1038/sj.bjc.6601118 · Cited by 1,045 · cited 11×
“Examples of survival and hazard functions derived from some of these parametric models were presented in the previous paper (Clark et al , 2003).”
InfluentialSelf-citation
Linear regression with censored data
Buckley, J. D., & JAMES, I. R. (1979) · Biometrika · doi:10.1093/biomet/66.3.429 · Cited by 1,032 · cited 1×
“Alternative methods include the method of Buckley and James (1979), which is discussed by Stare et al (2000), and semiparametric AFT models, in which the baseline survivor function is estimated nonparame-trically (see Wei, 1992, for an overview), but have not yet been widely implemented in…”
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“…include the method of Buckley and James (1979), which is discussed by Stare et al (2000), and semiparametric AFT models, in which the baseline survivor function is estimated nonparame-trically (see Wei, 1992, for an overview), but have not yet been widely implemented in statistical software.”
A linear regression model for the analysis of life times
Aalen, O. O. (1989) · Statistics in Medicine · doi:10.1002/sim.4780080803 · Cited by 725 · cited 1×
“These important issues will be addressed in the subsequent paper in this series.”
Multivariate survival analysis using Cox's regression model
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“This information could then be displayed via tabular or Christensen (1987) based on the Cox model, but can also be used when fitting fully parametric models.”

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Coelho‐Silva, J. L., Silveira, D. R., Pereira‐Martins, D. A., Rojas, C. A. O., Lucena-Araujo, A. R., Rego, E. M., Machado‐Neto, J. A., Bendit, I., Rocha, V., & Traina, F. (2021) · Scientific Reports · doi:10.1038/s41598-020-80918-6 · Cited by 3
Measuring Patient Survival on Hemodialysis
Brunelli, S. M. (2009) · Journal of the American Society of Nephrology · doi:10.1681/asn.2009070689 · Cited by 3
Survival analysis approaches for prostate cancer.
Alhasawi, E. (2015) · Lu Zone Ul (Laurentian University) · Cited by 3
Human papillomavirus and World Health Organization type III nasopharyngeal carcinoma: Multicenter study from an endemic area in Southern China
Tham, T. (2018) · Cancer · doi:10.1002/cncr.31819 · Cited by 3
miRDriver: A Tool to Infer Copy Number Derived miRNA-Gene Networks in CancerPreprint
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“We computed the hazard ratio (HR) of higher degree selected genes (genes targeted by four or more miRNAs) and lower degree selected genes (genes targeted by three or fewer miRNAs) using multivariate Cox regression analysis [46].”
Method
Market ResiliencePreprint
Danı́elsson, J., Panayi, E., Peters, G. W., & Zigrand, J. (2018) · SSRN Electronic Journal · doi:10.2139/ssrn.3169755 · Cited by 3
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Keno, H. A., Lou, Z., Kong, N., Landry, S. J., & Callahan, C. M. (2018) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2018.2876673 · Cited by 2
“, yields more precise estimate through smaller standard errors [36].”
Summary Measures and Measures of Effect: Summarizing and Comparing Outcomes in Rehabilitation Research. Part 3: Time‐to‐Event Outcomes, Unequal Follow‐up Times, and Survival Analysis
Speckman, R. A. (2020) · PM&R · doi:10.1002/pmrj.12535 · Cited by 2
“Are there other variables that could cause confounding bias? Are there other variables of interest that might be associated with the outcome? Can any of these other variables change values over time? Multiple regression is a large class of statistical techniques that allow researchers to take additional variables into…”
Pattern electroretinogram, blue-yellow visual evoked potentials and the risk of developing visual field defects in glaucoma suspects: a longitudinal “survival” analysis with a very long follow-up
Huchzermeyer, C. R., Lämmer, R., Mardin, C. Y., Kruse, F. E., Kremers, J., & Horn, F. K. (2024) · Graefe s Archive for Clinical and Experimental Ophthalmology · doi:10.1007/s00417-023-06364-y · Cited by 2
“Furthermore, survival analysis is superior for longitudinal analyses when patients are lost to follow-up [37, 38].”
Background
Towards the recalibration of US dollar’s international dominance
Iliyasu, J., Mamman, S. O., & Abubakar, A. B. (2024) · Journal of Financial Economic Policy · doi:10.1108/jfep-08-2023-0206 · Cited by 2
Uncovering gene expression signatures and diagnostic – Biomarkers in hepatocellular carcinoma through multinomial logistic regression analysis
Park, I., Lee, H. B., Kim, N., Lee, S., Park, K., Son, M., Cho, H., & Kim, D. (2024) · Journal of Biotechnology · doi:10.1016/j.jbiotec.2024.09.003 · Cited by 2
Impact of COVID-19 Pandemic on Airbnb Listings in New York City: Challenges and Opportunities for Urban Housing Sustainability
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Reproductive Performance of Black and White Rhinoceroses in Relation to Rainfall, Greenness and Density
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Ghisays, V., Khajouei, E., Piras, I. S., Malek‐Ahmadi, M. H., Protas, H. D., Goradia, D. D., Chen, Y., Naymik, M., Saner, D., Veatch, O. J., et al. (2025) · Alzheimer s & Dementia · doi:10.1002/alz.70245 · Cited by 2
“We used Cox proportional hazards ratio (HR) and 95% confidence intervals(CIs) 28,29 withdatafromparticipantswitheachofthesegenotypes and chi-squared to compare APOE4 allelic frequency in the ethnora-cialandsexgroups.”
Sex as a major determinant of pro-longevity drug efficacy: a review of two decades of the NIA Interventions Testing Program
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Development and validation of a nomogram for predicting suicide risk factors in thyroid cancer patients following diagnosis: a population-based retrospective study
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“…hazards model is suitable for analyzing survival data over time (26), and it can be used to adjust for baseline differences between groups and provide hazard ratios (calculation of con ﬁ dence intervals by 1000 iterations) to quantify the effect of any single factor on the survival curve (27, 28).”
Epidemiology and Prognostic Role of HPV Infection in Head and Neck Cancer: A Population‐Based Study of the SEER Database
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“For variables violating this assumption, stratified Cox models were applied to allow for non-proportional hazards across strata [23, 27].”
Methodological quality and reporting using time-to-event data in sports science and medicine: a scoping review
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Kim, H., Pan, L., Talati, A., Gameroff, M. J., Weissman, M. M., & Simpson, H. B. (2026) · JCPP Advances · doi:10.1002/jcv2.70117 · Cited by 2
“We utilized the semi - parametric Cox regression model, which makes fewer assumptions about survival function than other parametric models (e.g., Weibull, exponential) (Bradburn et al., 2003; Collett, 2023), making it a practical and appropriate choice.”
Equity or Equality? Which Approach Brings More Satisfaction in a Kidney-Exchange Chain?
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“In stratified Cox regression models, the baseline hazard is stratum-specific and a common hazard ratio for each covariate is aggregated over strata (Bradburn et al. , 2003, Cochran, 1968).”
Background
Separate-and-conquer survival action rule learning
Henzel, J., Hermansa, M., Kozielski, M., Sikora, M., & Wróbel, Ł. (2023) · Knowledge-Based Systems · doi:10.1016/j.knosys.2023.110981 · Cited by 2
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A Novel Mathematical Approach for Analysis of Integrated Cell–Patient Data Uncovers a 6-Gene Signature Linked to Endocrine Therapy Resistance
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“In addition, the METABRIC cohort was used to produce univariate and multivariate Cox proportional-hazard regression models 41 that further helped to examine the predictive capabilities of the signature and allowed for comparison with other previously identi ﬁ ed gene signatures for resistance to…”
Method
2-Methoxybenzaldehyde effectively repels ants
Kay, T., Siegenthaler, G., Kench, T., & Keller, L. (2023) · Journal of Economic Entomology · doi:10.1093/jee/toad225 · Cited by 2
De-stabilizing innate immunity in COVID-19: effects of its own positive feedback and erratic viraemia on the alternative pathway of complement
REEVE, J. (2024) · Royal Society Open Science · doi:10.1098/rsos.221597 · Cited by 2
“The time to capsize was made the outcome variable in Cox Proportional Hazard models [39] that included determinants of potential interest such as pre-infection complement protein levels and the presence or absence of pulmonary clotting.”
Method
Transcriptome Sequencing Unveils a Molecular-Stratification-Predicting Prognosis of Sarcoma Associated with Lipid Metabolism
Hong, Y., Zhang, L., Lin, W., Yang, Y., Cao, Z., Feng, X., Yu, Z., & Gao, Y. (2024) · International Journal of Molecular Sciences · doi:10.3390/ijms25031643 · Cited by 2
“A novel prognostic model was constructed using multivariate Cox proportional hazards regression analysis [45].”
Method
Social and spatial disparities in individuals’ mobility response time to COVID-19: A big data analysis incorporating changepoint detection and accelerated failure time models
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Survival Probability of Tourist Accommodation Establishments in Romania in Relation to the Action of Some Relevant Factors
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An Evaluation of an Algorithm for the Selection of Flexible Survival Models for Cancer Immunotherapies: Pass or Fail?
Latimer, N. R., Taylor, K., Hatswell, A. J., Ho, S., Okorogheye, G., Chen, C. C., Kim, I., Borrill, J., & Bertwistle, D. (2024) · PharmacoEconomics · doi:10.1007/s40273-024-01429-0 · Cited by 2
“To test this assumption quantile-quantile plots are the appropriate choice [7, 55].”
Assessment of NSCLC disease burden: A survival model-based meta-analysis study
Kudryashova, N., Shulgin, B., Katuninks, N., Kulesh, V., Helmlinger, G., Zhudenkov, K., & Peskov, K. V. (2024) · Computational and Structural Biotechnology Journal · doi:10.1016/j.csbj.2024.09.012 · Cited by 2
Relationship Between Coefficients in Parametric Survival Models for Exponentially Distributed Survival Time—Registered Unemployment in Poland
Bieszk‐Stolorz, B. (2025) · Econometrics · doi:10.3390/econometrics13010001 · Cited by 2
“Research indicates that the parametric model is more informative than the Cox model (Bradburn et al., 2003).”
Data-driven prior elicitation for Bayes factors in Cox regression for nine subfields in biomedicine
Linde, M., Jochim, L., Tendeiro, J. N., & van Ravenzwaaij, D. (2025) · PLoS ONE · doi:10.1371/journal.pone.0322144 · Cited by 2
“Among the many existing forms of survival analysis - for example, Kaplan-Meier product-limit estimator [39], parametric survival analyses (e.g., Exponential, Gompertz, and Weibull), and Cox regression [38] - the latter is used most frequently within biomedicine (e.g., [10]).”
Legal traditions and the ratification of the Paris agreement
Fredriksson, P. G., Sharma, S., & Wollscheid, J. R. (2025) · Energy Economics · doi:10.1016/j.eneco.2025.108719 · Cited by 2
A Practical Guide for Statistics in Wildlife Studies
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Authors: Bradburn, M. J., Clark, T. G., Love, S. B., & Altman, D. G.
Affiliations: Cancer Research UK (GB)
Corresponding author: Bradburn, M J
ORCID: Bradburn, M J
Clark, T G
Love, S B
Altman, D G
Venue: British Journal of Cancer
Publisher: Springer Science and Business Media LLC
Published: 2003-07-29
Volume: 89
Issue: 3
Pages: 431-436
ISSN: 0007-0920 (ISSN-L)
1532-1827
DOI: 10.1038/sj.bjc.6601119
PMC: PMC2394368
OpenAlex: W2107482271
Cited by: 741
References: 16
Citations by year: 2026: 31 · 2025: 55 · 2024: 86 · 2023: 67 · 2022: 54 · 2021: 91 · 2020: 55 · 2019: 41 · 2018: 43 · 2017: 35 · 2016: 26 · 2015: 21 · 2014: 27 · 2013: 15 · 2012: 24
FWCI: 7.17
Type: journal article
Language: en
Open access: hybrid
doi.org
Licence: cc-by-nc-sa
www.springer.com/tdm (unspecified)
Locations: British Journal of Cancer (publishedVersion, cc-by-nc-sa)
PubMed Central (publishedVersion, cc-by)
PubMed (publishedVersion)
White Rose Research Online (University of Leeds, The University of Sheffield, University of York) (acceptedVersion)
LSHTM Research Online (London School of Hygiene and Tropical Medicine) (acceptedVersion)
Funders: Cancer Research UK
Topics: Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
Keywords: survival analysis, multivariate survival data, censored data, time to event, cancer survival, censoring
MeSH terms: Clinical Trials as Topic, Humans, Neoplasms, Models, Statistical, Multivariate Analysis, Survival Analysis

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