Survival analysis part II: multivariate data analysis – an introduction to concepts and methods
Open accessAbstract
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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The proportional hazards assumption was assessed using the log(−log(survival)) plot.
“The proportional hazards assumption was assessed using the log(−log(survival)) plot, and those that failed the assumption or that were deemed to be time dependent were entered as continuous time-dependent covariates (Bradburn et al, 2003).”
From Cohort study of adherence to adjuvant endocrine therapy, breast cancer recurrence and mortality (2013) · cites this paper as Bradburn et al, 2003
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MethodMultivariate analysis of NMR‐based metabolomic data
Debik, J., Sangermani, M., Wang, F., Madssen, T. S., & Giskeødegård, G. F. (2021) · NMR in Biomedicine · doi:10.1002/nbm.4638 · Cited by 74 · see citing works
“Survival analysis involves analyzing the effect of one or more covariates on the expected duration of time until an event such as death, disease occurrence, or recurrence.(28) The Cox regression model may be expressed mathematically for individual i i1⁄41,.”
BackgroundAubry, L. M., Cam, E., Koons, D. N., Monnat, J., & Pavard, S. (2010) · Journal of Animal Ecology · doi:10.1111/j.1365-2656.2010.01784.x · Cited by 71 · see citing works
“Each covariate within the model is assumed to act multiplicatively (i.e. proportionally) on the baseline mortality hazard at each time step (e.g. Bradburn et al. 2003), such as hðt;XiÞ ¼ h0ðtÞ expð Pp i¼1 biXiÞ where h0 refers to the baseline hazard (i.e. hazard’s value when all covariate values…”
BackgroundFoster, T., Willetts, J., Lane, M., Thomson, P., Katuva, J., & Hope, R. (2018) · The Science of The Total Environment · doi:10.1016/j.scitotenv.2017.12.302 · Cited by 69 · see citing works
“The techniques that fall within the survival analysis umbrella have been used extensively in medical literature due to their ability to accommodate right-censored data – that is, subjects that have not yet undergone the event of interest (such as disease onset, remission or death) by the end of an observation period (…”
Gray, J. E., Ahn, M., Oxnard, G. R., Shepherd, F. A., Imamura, F., Cheng, Y., Okamoto, I., Cho, B. C., Lin, M., Wu, Y.-L. L. C., et al. (2023) · Clinical Cancer Research · doi:10.1158/1078-0432.ccr-22-3146 · Cited by 71 · see citing works
Incidence of anaemia among HIV‐infected patients treated with highly active antiretroviral therapy*
Curkendall, S., Richardson, J., Emons, M., Fisher, A., & Everhard, F. (2007) · HIV Medicine · doi:10.1111/j.1468-1293.2007.00500.x · Cited by 68 · see citing works
Practical methods for incorporating summary time-to-event data into meta-analysis: updated guidance
Tierney, J. F., Burdett, S., & Fisher, D. J. (2025) · Systematic Reviews · doi:10.1186/s13643-025-02752-z · Cited by 67 · see citing works
“The coefficient for the effect of the research treatment from such a model provides an alternative “direct” estimate of the log HR [7], with an associated standard error (SE) and p value.”
Survival analysis: Part I — analysis of time-to-event
In, J., & Lee, D. K. (2018) · Korean Journal of Anesthesiology · doi:10.4097/kja.d.18.00067 · Cited by 66 · see citing works
“This method does not presuppose any specific distribution and can analyze any variables that may influence the occurrence, unlike Kaplan-Meier survival analysis, and is thus regarded as a semiparametric method [16,17].”
MethodHOX D13 expression across 79 tumor tissue types
Cantile, M., Franco, R., Tschan, A., Baumhoer, D., Zlobec, I., Schiavo, G., Forte, I. M., Bihl, M., Liguori, G., Botti, G., et al. (2009) · International Journal of Cancer · doi:10.1002/ijc.24438 · Cited by 65 · see citing works
Dhanorkar, S., Siemsen, E., & Linderman, K. (2017) · Management Science · doi:10.1287/mnsc.2017.2748 · Cited by 63 · see citing works
Biostatistics series module 9: Survival analysis
Hazra, A., & Gogtay, N. (2017) · Indian Journal of Dermatology · doi:10.4103/ijd.ijd_201_17 · Cited by 63 · see citing works
The initial development and assessment of an automatic alert warning of acute kidney injury
Thomas, M., Sitch, A. J., & Dowswell, G. (2010) · Nephrology Dialysis Transplantation · doi:10.1093/ndt/gfq710 · Cited by 62 · see citing works
“Weibull survival analysis [17,18] was carried out with Stata software (Stata Corp.”
MethodTang, T., Hwang, A., Liu, L.-K., Lee, W., Chen, L.-Y., Wu, Y., Huang, C., Hung, C.-H., Wang, C., Lin, M.-H., et al. (2017) · The Journals of Gerontology Series A · doi:10.1093/gerona/glx148 · Cited by 60 · see citing works
“Survival analysis by the Cox proportional hazard model was used to estimate the association of FNIH-defined sarcopenia with composite adverse outcomes and all-cause mortality, and to obtain the hazard ratios and their 95% confidence intervals (38,39).”
MethodDwivedi, A. K., & Shukla, R. (2019) · Cancer Reports · doi:10.1002/cnr2.1211 · Cited by 58 · see citing works
Brunet‐Ratnasingham, E., Anand, S. P., Gantner, P., Dyachenko, A., Moquin‐Beaudry, G., Brassard, N., Beaudoin-Bussières, G., Pagliuzza, A., Gasser, R., Benlarbi, M., et al. (2021) · Science Advances · doi:10.1126/sciadv.abj5629 · Cited by 53 · see citing works
Youth sport dropout according to the Process-Person-Context-Time model: a systematic review
Moulds, K., Galloway, S., Abbott, S., & Cobley, S. (2022) · International Review of Sport and Exercise Psychology · doi:10.1080/1750984x.2021.2012817 · Cited by 38 · see citing works
“…ts of latter measures, like hazard rates, is their capability to determine dropout occurrence during a standardised exposure time-period (Blagoev et al., 2012), while accounting for sample sizes in respective groups (e.g. Hazard Ratios - Bradburn et al., 2003); so, should be encouragingly adopted.”
BackgroundAuthier, M., Bentaleb, I., Ponchon, A., Martin, C., & Guinet, C. (2012) · PLoS ONE · doi:10.1371/journal.pone.0032026 · Cited by 48 · see citing works
“We estimated 4 individual parameters describing a broken-stick model and subsequently used them as predictors in an Accelerated Failure Time (AFT) model [50,51].”
MethodMoazemi, S., Erle, A., Lütje, S., Gaertner, F. C., Essler, M., & Bundschuh, R. A. (2021) · Diagnostics · doi:10.3390/diagnostics11020186 · Cited by 57 · see citing works
“the Kaplan–Meier estimator [20] is used to analyze and visualize the survival prediction results.”
MethodRaquin, V., Merkling, S. H., Gausson, V., Moltini-Conclois, I., Frangeul, L., Varet, H., Dillies, M., Saleh, M., & Lambrechts, L. (2017) · PLoS neglected tropical diseases · doi:10.1371/journal.pntd.0006152 · Cited by 56 · see citing works
“A Cox regression model including dsRNA injection and DENV exposure as covariates was used to compare mosquito survival across treatments [67].”
Freeman, D. E. (2018) · Equine Veterinary Journal · doi:10.1111/evj.12817 · Cited by 56 · see citing works
Santos, J. A., Leite, A., Soares, P., Duarte, R., & Nunes, C. (2021) · BMC Public Health · doi:10.1186/s12889-021-12245-y · Cited by 55 · see citing works
“Cox proportional hazards model [29] was used to estimate the effect of the clinical and sociodemographic variables on the time until the event of interest.”
MethodCoronavirus disease 2019 (COVID-19): survival analysis using deep learning and Cox regression model
Atlam, M., Torkey, H., El-Fishawy, N. A., & Salem, H. (2021) · Pattern Analysis and Applications · doi:10.1007/s10044-021-00958-0 · Cited by 54 · see citing works
Wong, S. H., Petrie, A., & Plant, G. T. (2016) · Journal of Neuro-Ophthalmology · doi:10.1097/wno.0000000000000350 · Cited by 54 · see citing works
“The guideline for number of predictors to include in a model is 10% of the outcome of interest in each of the response categories, that is, in this case, 10% of 19 patients who developed GMG within 2 years (for the logistic model) or 30 patients for the survival analyses (16,32).”
BackgroundPérez-Navarro, L. M., Restrepo, B. I., Fuentes-Domínguez, F. J., Duggirala, R., Morales‐Romero, J., López-Alvarenga, J. C., Comas, I., & Zenteno‐Cuevas, R. (2017) · Tuberculosis · doi:10.1016/j.tube.2017.01.006 · Cited by 53 · see citing works
Xu, W. V., Puligandla, M., Halbert, B. T., Haas, N. B., Flaherty, K. T., Uzzo, R. G., Dutcher, J. P., DiPaola, R. S., Sabbisetti, V. S., & Bhatt, R. S. (2021) · Clinical Cancer Research · doi:10.1158/1078-0432.ccr-21-0025 · Cited by 52 · see citing works
“Similar associations between KIM-1 and DFS were seen when using the imputed low KIM-1 values (Supplementary Table S1).”
MethodInfluentialNater, A., Martin, A. R., Sahgal, A., Choi, D., & Fehlings, M. G. (2017) · PLoS ONE · doi:10.1371/journal.pone.0171507 · Cited by 52 · see citing works
Big Data Toolsets to Pharmacometrics: Application of Machine Learning for Time‐to‐Event Analysis
Gong, X., Hu, M., & Zhao, L. Y. (2018) · Clinical and Translational Science · doi:10.1111/cts.12541 · Cited by 53 · see citing works
“With approach i, six groups of survival data sets were generated with different relationships between the predictor variables in the hazard function: (1) linear, (2) nonlinear, (3) interaction, (4) nonlinear + interaction, (5) nonlinear + interaction + correlation, and (6) high-dimension.”
MethodAbraham, J. A., Weaver, M. J., Hornick, J. L., Zurakowski, D., & Ready, J. E. (2012) · Journal of Bone and Joint Surgery · doi:10.2106/jbjs.k.00460 · Cited by 48 · see citing works
“AJCC stage* Ia 2 (2) 0 (0) Ib 1 (1) 0 (0) IIa 6 (5) 1 (17) IIb 24 (21) 8 (33) IIc 4 (3) 1 (25) III 60 (52) 36 (60) IV 18 (16) 14 (78)”
Recurrent low-grade gliomas: the role of fractionated stereotactic re-irradiation
Combs, S. E., Ahmadi, R., Schulz–Ertner, D., Thilmann, C., & Debus, J. (2005) · Journal of Neuro-Oncology · doi:10.1007/s11060-004-2029-z · Cited by 47 · see citing works
Jaskot, A., Silveyra, A. C., Plantinga, A., Flury, S., Hayes, M. J., Chisholm, J., Heckman, T., Pentericci, L., Schaerer, D., Trebitsch, M., et al. (2024) · Astrophysical Journal · doi:10.3847/1538-4357/ad58b9 · Cited by 33 · see citing works
Wolfe, M. L., Koons, D. N., Stoner, D. C., Terletzky, P. A., Gese, E. M., Choate, D. M., & Aubry, L. M. (2014) · Biological Conservation · doi:10.1016/j.biocon.2014.12.008 · Cited by 46 · see citing works
“Rather, each covariate within the model is assumed to act multiplicatively (proportionally) on the time-specific baseline mortality hazard and across covariate levels (Bradburn et al., 2003), such as hi(t) = h0(t) exp (biXi) where h0 refers to the baseline hazard (i.”
BackgroundProportional hazards models of infrastructure system recovery
Barker, K., & Baroud, H. (2014) · Reliability Engineering & System Safety · doi:10.1016/j.ress.2013.12.004 · Cited by 45 · see citing works
“The accelerated failure time model is used especially when an underlying hazard function is known, or when the baseline hazard function would vary from observation to observation [33].”
MethodRisk of tuberculin conversion among healthcare workers and the adoption of preventive measures
Baussano, I., Bugiani, M., Carosso, A., Df, M., Barocelli, A. P., Tagna, M., Cascio, V., Piccioni, P., & Arossa, W. (2006) · Occupational and Environmental Medicine · doi:10.1136/oem.2006.028068 · Cited by 45 · see citing works
“[23,24] The following covariates were separately investigated with respect to their influence on the HR for being infected with Mycobacterium tuberculosis: age, gender, duration of employment as HCW (categorised as <1 and >1 years), working in a hospital setting rather than a community setting, occupation, workplace a…”
MethodCommitted anchor investment and IPO survival – The roles of cornerstone and strategic investors
Espenlaub, S. K., Khurshed, A., Mohamed, A., & Saadouni, B. (2016) · Journal of Corporate Finance · doi:10.1016/j.jcorpfin.2016.08.016 · Cited by 45 · see citing works
“The time ratio is calculated as the exponential of the AFT coefficient (Bradburn et al. 2003).”
Ebert, M. A., Gulliford, S., Acosta, O., de Crevoisier, R., McNutt, T., Heemsbergen, W. D., Witte, M. G., Palma, G., Rancati, T., & Fiorino, C. (2021) · Physics in Medicine and Biology · doi:10.1088/1361-6560/ac0681 · Cited by 44 · see citing works
Self-image and suicide in a Swedish national eating disorders clinical register
Runfola, C. D., Thornton, L. M., Pisetsky, E. M., Bulik, C. M., & Birgegård, A. (2013) · Comprehensive Psychiatry · doi:10.1016/j.comppsych.2013.11.007 · Cited by 44 · see citing works
“The Cox regression model allows for unequal follow-up time of participants and includes information from individuals whose suicide attempt status is unknown [64,65].”
Biron, V. L., Lentsch, E. J., Gerry, D. R., & Bewley, A. F. (2014) · Head & Neck · doi:10.1002/hed.23686 · Cited by 43 · see citing works
Batson, S., Greenall, G., & Hudson, P. (2016) · PLoS ONE · doi:10.1371/journal.pone.0154870 · Cited by 43 · see citing works
“The results of the current review suggest that it is likely that Cox regression models are routinely chosen by trialists due to its widespread application, and to aid comparability with results of other trials [37].”
ResultRoessiger, J., Griess, V. C., Härtl, F., Clasen, C., & Knoke, T. (2013) · Ecological Modelling · doi:10.1016/j.ecolmodel.2013.01.019 · Cited by 42 · see citing works
Radiochemotherapy in Patients With Primary Glioblastoma Comparing Two Temozolomide Dose Regimens
Combs, S. E., Wagner, J., Bischof, M., Welzel, T., Edler, L., Rausch, R., Wagner, F., Bois, A. Z.-D., Debus, J., & Schulz-Ertner, D. (2008) · International Journal of Radiation Oncology*Biology*Physics · doi:10.1016/j.ijrobp.2007.11.064 · Cited by 42 · see citing works
“using the univariate and multivariate Cox proportional regression model (12, 13).”
MethodKim, I.-H., Castillo, J. C., Aryan, A., Martín-Martín, I., Nouzová, M., Noriega, F. G., Barletta, A. B. F., Calvo, E., Adelman, Z. N., Ribeiro, J. M. C., et al. (2020) · PLoS Pathogens · doi:10.1371/journal.ppat.1008288 · Cited by 40 · see citing works
“The data were analyzed using the log-rank test as well as a fit to an accelerated time failure model with a Weibull distribution [15].”
MethodLalos, A., Tülek, A., Tosti, N., Mechera, R., Wilhelm, A., Soysal, S. D., Däster, S., Kancherla, V., Weixler, B., Spagnoli, G. C., et al. (2021) · Scientific Reports · doi:10.1038/s41598-020-80382-2 · Cited by 40 · see citing works
Osborne, J. L., Hindt, L. A., Lutz, N. M., Hodgkinson, N., & Leon, S. C. (2021) · Children and Youth Services Review · doi:10.1016/j.childyouth.2021.106000 · Cited by 39 · see citing works
- Authors
- Bradburn, M. J., Clark, T. G., Love, S. B., & Altman, D. G.
- Affiliations
- Cancer Research UK (GB)
- Corresponding author
- Bradburn, M J
- Venue
- British Journal of Cancer
- Published
- 2003-07-29
- Volume
- 89
- Issue
- 3
- Pages
- 431-436
- ISSN
- 0007-0920 (ISSN-L)
- 1532-1827
- 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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