Ørnulf Borgan

• Kvamme, Håvard & Borgan, Ørnulf (2023). The Brier Score under Administrative Censoring: Problems and a Solution. Journal of machine learning research. ISSN 1532-4435. 24(2), p. 1–26. Full text in Research Archive Show summary

  The Brier score is commonly used for evaluating probability predictions. In survival analysis, with right-censored observations of the event times, this score can be weighted by the inverse probability of censoring (IPCW) to retain its original interpretation. It is common practice to estimate the censoring distribution with the Kaplan-Meier estimator, even though it assumes that the censoring distribution is independent of the covariates. This paper investigates problems that may arise for the IPCW weighting scheme when the covariates used in the prediction model contain information about the censoring times. In particular, this may occur for administratively censored data if the distribution of the covariates varies with calendar time. For administratively censored data, we propose an alternative version of the Brier score. This administrative Brier score does not require estimation of the censoring distribution and is valid also when the censoring times can be predicted from the covariates.

• Aalen, Odd O.; Andersen, Per K.; Borgan, Ørnulf; Gill, Richard D. & Keiding, Niels (2022). Martingales in Survival Analysis. In Mazliak, Laurent & Shafer, Glenn (Ed.), The Splendors and Miseries of Martingales. Their History from the Casino to Mathematics. Birkhäuser Verlag. ISSN 9783031059872. p. 295–320. doi: 10.1007/978-3-031-05988-9. Full text in Research Archive Show summary

  The paper traces the development of the use of martingale methods in survival analysis from the mid 1970s to the early 1990s. This development was initiated by Aalen’s Berkeley Ph.D.-thesis in 1975, progressed in the late 1970s and early 1980s through work on the estimation of Markov transition probabilities, non-parametric tests and Cox’s regression model, and was consolidated in the early 1990s with the publication of the monographs by Fleming and Harrington and by Andersen, Borgan, Gill and Keiding. The development was made possible by an unusually fast technology transfer of pure mathematical concepts, primarily from French probability, into practical biostatistical methodology, and we attempt to outline some of the personal relationships that helped this happen. We also point out that survival analysis was ready for this development since the martingale ideas inherent in the deep understanding of temporal development so intrinsic to the French theory of processes were already quite close to the surface in survival analysis.

• Borgan, Ørnulf; Keogh, Ruth H. & Njøs, Aleksander (2022). Use of multiple imputation in supersampled nested case-control and case-cohort studies. Scandinavian Journal of Statistics. ISSN 0303-6898. doi: 10.1111/sjos.12624. Full text in Research Archive Show summary

  Nested case-control and case-cohort studies are useful for studying associations between covariates and time-to-event when some covariates are expensive to measure. Full covariate information is collected in the nested case-control or case-cohort sample only, while cheaply measured covariates are often observed for the full cohort. Standard analysis of such case-control samples ignores any full cohort data. Previous work has shown how data for the full cohort can be used efficiently by multiple imputation of the expensive covariate(s), followed by a full-cohort analysis. For large cohorts this is computationally expensive or even infeasible. An alternative is to supplement the case-control samples with additional controls on which cheaply measured covariates are observed. We show how multiple imputation can be used for analysis of such supersampled data. Simulations show that this brings efficiency gains relative to a traditional analysis and that the efficiency loss relative to using the full cohort data is not substantial.

• Kvamme, Håvard & Borgan, Ørnulf (2021). Continuous and discrete-time survival prediction with neural networks. Lifetime Data Analysis. ISSN 1380-7870. 27(4), p. 710–736. doi: 10.1007/s10985-021-09532-6. Full text in Research Archive Show summary

  Abstract Due to rapid developments in machine learning, and in particular neural networks, a number of new methods for time-to-event predictions have been developed in the last few years. As neural networks are parametric models, it is more straightforward to integrate parametric survival models in the neural network framework than the popular semi-parametric Cox model. In particular, discrete-time survival models, which are fully parametric, are interesting candidates to extend with neural networks. The likelihood for discrete-time survival data may be parameterized by the probability mass function (PMF) or by the discrete hazard rate, and both of these formulations have been used to develop neural network-based methods for time-to-event predictions. In this paper, we review and compare these approaches. More importantly, we show how the discrete-time methods may be adopted as approximations for continuous-time data. To this end, we introduce two discretization schemes, corresponding to equidistant times or equidistant marginal survival probabilities, and two ways of interpolating the discrete-time predictions, corresponding to piecewise constant density functions or piecewise constant hazard rates. Through simulations and study of real-world data, the methods based on the hazard rate parametrization are found to perform slightly better than the methods that use the PMF parametrization. Inspired by these investigations, we also propose a continuous-time method by assuming that the continuous-time hazard rate is piecewise constant. The method, named PC-Hazard, is found to be highly competitive with the aforementioned methods in addition to other methods for survival prediction found in the literature.

• Lingjærde, Camilla; Lien, Tonje Gulbrandsen; Borgan, Ørnulf; Bergholtz, Helga & Glad, Ingrid K. (2021). Tailored graphical lasso for data integration in gene network reconstruction. BMC Bioinformatics. 22(1). doi: 10.1186/s12859-021-04413-z. Full text in Research Archive Show summary

  Background Identifying gene interactions is a topic of great importance in genomics, and approaches based on network models provide a powerful tool for studying these. Assuming a Gaussian graphical model, a gene association network may be estimated from multiomic data based on the non-zero entries of the inverse covariance matrix. Inferring such biological networks is challenging because of the high dimensionality of the problem, making traditional estimators unsuitable. The graphical lasso is constructed for the estimation of sparse inverse covariance matrices in such situations, using L1-penalization on the matrix entries. The weighted graphical lasso is an extension in which prior biological information from other sources is integrated into the model. There are however issues with this approach, as it naïvely forces the prior information into the network estimation, even if it is misleading or does not agree with the data at hand. Further, if an associated network based on other data is used as the prior, the method often fails to utilize the information effectively. Results We propose a novel graphical lasso approach, the tailored graphical lasso, that aims to handle prior information of unknown accuracy more effectively. We provide an R package implementing the method, tailoredGlasso. Applying the method to both simulated and real multiomic data sets, we find that it outperforms the unweighted and weighted graphical lasso in terms of all performance measures we consider. In fact, the graphical lasso and weighted graphical lasso can be considered special cases of the tailored graphical lasso, and a parameter determined by the data measures the usefulness of the prior information. We also find that among a larger set of methods, the tailored graphical is the most suitable for network inference from high-dimensional data with prior information of unknown accuracy. With our method, mRNA data are demonstrated to provide highly useful prior information for protein–protein interaction networks. Conclusions The method we introduce utilizes useful prior information more effectively without involving any risk of loss of accuracy should the prior information be misleading.

• Nedkvitne, Eirik Nøst; Borgan, Ørnulf; Eriksen, Dag Øistein & Rui, Haakon Marius Vatten (2021). Variation in chemical composition of MSWI fly ash and dry scrubber residues. Waste Management. ISSN 0956-053X. 126, p. 623–631. doi: 10.1016/j.wasman.2021.04.007. Full text in Research Archive Show summary

  Our society generates extensive amounts of municipal solid waste (MSW), which are mainly incinerated for volume reduction and energy recovery. Though, MSW incineration generates hazardous air pollution control (APC) residues that must be treated and deposited in appropriate landfills. An alternative to landfilling is material recovery, leading to regeneration of valuable products and reducing hazardous waste amounts. The chemical composition of APC residues, stemming from MSW, makes the waste attractive for metal and salt recovery, but its variation makes the development of material recovery processes challenging. This study investigates results from 895 X-ray fluorescence analyses of fly ash and dry scrubber residue samples originating from Norway and Sweden between 2006 and 2020 to explore variation in chemical composition within and between different incineration plants. The average relative standard deviation of elemental concentration in APC residue was estimated to 30% within plants. The variation in elemental concentration between grate fired incineration plants is about half of the average variation within the plants. The study also clarifies compositional differences from APC residues originating from fluidized bed incinerators and grate incinerators. Also, reported concentrations of APC residues from other countries than Sweden and Norway showed significant differences in chemical composition. The presented variations clarifies the importance of holistic approaches for waste valorization processes which can substitute stabilization processes for landfilling.

• Riiser, Even Sannes; Haverkamp, Thomas Hendricus Augustus; Varadharajan, Srinidhi; Borgan, Ørnulf; Jakobsen, Kjetill Sigurd & Jentoft, Sissel [Show all 7 contributors for this article] (2020). Metagenomic Shotgun Analyses Reveal Complex Patterns of Intra- and Interspecific Variation in the Intestinal Microbiomes of Codfishes. Applied and Environmental Microbiology. ISSN 0099-2240. 86(6), p. 1–16. doi: 10.1128/AEM.02788-19. Full text in Research Archive
• Pladsen, Arne Valebjørg; Nilsen, Gro; Rueda, Oscar M.; Aure, Miriam Ragle; Borgan, Ørnulf & Liestøl, Knut [Show all 18 contributors for this article] (2020). DNA copy number motifs are strong and independent predictors of survival in breast cancer. Communications Biology. 3. doi: 10.1038/s42003-020-0884-6. Full text in Research Archive Show summary

  Somatic copy number alterations are a frequent sign of genome instability in cancer. A precise characterization of the genome architecture would reveal underlying instability mechanisms and provide an instrument for outcome prediction and treatment guidance. Here we show that the local spatial behavior of copy number profiles conveys important information about this architecture. Six filters were defined to characterize regional traits in copy number profiles, and the resulting Copy Aberration Regional Mapping Analysis (CARMA) algorithm was applied to tumors in four breast cancer cohorts (n = 2919). The derived motifs represent a layer of information that complements established molecular classifications of breast cancer. A score reflecting presence or absence of motifs provided a highly significant independent prognostic predictor. Results were consistent between cohorts. The nonsite-specific occurrence of the detected patterns suggests that CARMA captures underlying replication and repair defects and could have a future potential in treatment stratification.

• Riiser, Even Sannes; Haverkamp, Thomas Hendricus Augustus; Varadharajan, Srinidhi; Borgan, Ørnulf; Jakobsen, Kjetill Sigurd & Jentoft, Sissel [Show all 7 contributors for this article] (2019). Switching on the light: using metagenomic shotgun sequencing to characterize the intestinal microbiome of Atlantic cod. Environmental Microbiology. ISSN 1462-2912. 21(7), p. 2576–2594. doi: 10.1111/1462-2920.14652. Full text in Research Archive Show summary

  Atlantic cod (Gadus morhua) is an ecologically important species with a wide‐spread distribution in the North Atlantic Ocean, yet little is known about the diversity of its intestinal microbiome in its natural habitat. No geographical differentiation in this microbiome was observed based on 16S rRNA amplicon analyses, yet such finding may result from an inherent lack of power of this method to resolve fine‐scaled biological complexity. Here, we use metagenomic shotgun sequencing to investigate the intestinal microbiome of 19 adult Atlantic cod individuals from two coastal populations in Norway–located 470 km apart. Resolving the species community to unprecedented resolution, we identify two abundant species, Photobacterium iliopiscarium and Photobacterium kishitanii, which comprise over 50% of the classified reads. Interestingly, the intestinal P. kishitanii strains have functionally intact lux genes, and its high abundance suggests that fish intestines form an important part of its ecological niche. These observations support a hypothesis that bioluminescence plays an ecological role in the marine food web. Despite our improved taxonomical resolution, we identify no geographical differences in bacterial community structure, indicating that the intestinal microbiome of these coastal cod is colonized by a limited number of closely related bacterial species with a broad geographical distribution.

• Kvamme, Håvard; Borgan, Ørnulf & Scheel, Ida (2019). Time-to-Event Prediction with Neural Networks and Cox Regression. Journal of machine learning research. ISSN 1532-4435. 20(129), p. 1–30. Full text in Research Archive Show summary

  New methods for time-to-event prediction are proposed by extending the Cox proportional hazards model with neural networks. Building on methodology from nested case-control studies, we propose a loss function that scales well to large data sets and enables fitting of both proportional and non-proportional extensions of the Cox model. Through simulation studies, the proposed loss function is verified to be a good approximation for the Cox partial log-likelihood. The proposed methodology is compared to existing methodologies on real-world data sets and is found to be highly competitive, typically yielding the best performance in terms of Brier score and binomial log-likelihood. A python package for the proposed methods is available at https://github.com/havakv/pycox.

• Borgan, Ørnulf & Keilman, Nico (2019). Do Japanese and Italian Women Live Longer than Women in Scandinavia? European Journal of Population. ISSN 0168-6577. 35(1), p. 87–99. doi: 10.1007/s10680-018-9468-2. Full text in Research Archive Show summary

  Life expectancies at birth are routinely computed from period life tables. When mortality is falling, such period life expectancies will typically underestimate real life expectancies, that is, life expectancies for birth cohorts. Hence, it becomes problematic to compare period life expectancies between countries when they have different historical mortality developments. For instance, life expectancies for countries in which the longevity improved early (like Norway and Sweden) are difficult to compare with those in countries where it improved later (like Italy and Japan). To get a fair comparison between the countries, one should consider cohort data. Since cohort life expectancies can only be computed for cohorts that were born more than a hundred years ago, in this paper we suggest that for younger cohorts one may consider the expected number of years lost up to a given age. Contrary to the results based on period data, our cohort results then indicate that Italian women may expect to lose more years than women in Norway and Sweden, while there are no indications that Japanese women will lose fewer years than women in Scandinavia. The large differences seen for period data may just be an artefact due to the distortion that period life tables imply in times of changing mortality.

• Borgan, Ørnulf & Samuelsen, Sven Ove (2018). Cohort Sampling for Time-to-Event Data: An Overview. In Borgan, Ørnulf; Breslow, Norman E.; Chatterjee, Nilanjan; Gail, Mitchell H.; Scott, Alastair & Wild, Christopher J. (Ed.), Handbook of Statistical Methods for Case-Control Studies. CRC Press. ISSN 9781498768580. p. 285–301. Full text in Research Archive
• Ergon, Torbjørn; Borgan, Ørnulf; Nater, Chloé Rebecca & Vindenes, Yngvild (2018). The utility of mortality hazard rates in population analyses. Methods in Ecology and Evolution. 9, p. 2046–2056. doi: 10.1111/2041-210X.13059. Full text in Research Archive
• Borgan, Ørnulf (2018). Nested Case-Control Studies: A Counting Process Approach. In Borgan, Ørnulf; Breslow, Norman E.; Chatterjee, Nilanjan; Gail, Mitchell H.; Scott, Alastair & Wild, Christopher J. (Ed.), Handbook of Statistical Methods for Case-Control Studies. CRC Press. ISSN 9781498768580. p. 329–349. Full text in Research Archive
• Riiser, Even Sannes; Haverkamp, Thomas Hendricus Augustus; Borgan, Ørnulf; Jakobsen, Kjetill Sigurd; Jentoft, Sissel & Star, Bastiaan (2018). A single vibrionales 16S rRNA oligotype dominates the intestinal microbiome in two geographically separated Atlantic cod populations. Frontiers in Microbiology. 9. doi: 10.3389/fmicb.2018.01561. Full text in Research Archive Show summary

  Atlantic cod (Gadus morhua) provides an interesting species for the study of host-microbe interactions because it lacks the MHC II complex that is involved in the presentation of extracellular pathogens. Nonetheless, little is known about the diversity of its microbiome in natural populations. Here, we use high-throughput sequencing of the 16S rRNA V4 region, amplified with the primer design of the Earth Microbiome Project (EMP), to investigate the microbial composition in gut content and mucosa of 22 adult individuals from two coastal populations in Norway, located 470 km apart. We identify a core microbiome of 23 OTUs (97% sequence similarity) in all individuals that comprises 93% of the total number of reads. The most abundant orders are classified as Vibrionales, Fusobacteriales, Clostridiales, and Bacteroidales. While mucosal samples show significantly lower diversity than gut content samples, no differences in OTU community composition are observed between the two geographically separated populations. All specimens share a limited number of abundant OTUs. Moreover, the most abundant OTU consists of a single oligotype (order Vibrionales, genus Photobacterium) that represents nearly 50% of the reads in both locations. Our results suggest that these microbiomes comprise a limited number of species or that the EMP V4 primers do not yield sufficient resolution to confidently separate these communities. Our study contributes to a growing body of literature that shows limited spatial differentiation of the intestinal microbiomes in marine fish based on 16S rRNA sequencing, highlighting the need for multi-gene approaches to provide more insight into the diversity of these communities.

• Lien, Tonje Gulbrandsen; Borgan, Ørnulf; Reppe, Sjur; Gautvik, Kaare M & Glad, Ingrid Kristine (2018). Integrated analysis of DNA-methylation and gene expression using high-dimensional penalized regression: a cohort study on bone mineral density in postmenopausal women. BMC Medical Genomics. 11(24). doi: 10.1186/s12920-018-0341-2. Full text in Research Archive Show summary

  Background Using high-dimensional penalized regression we studied genome-wide DNA-methylation in bone biopsies of 80 postmenopausal women in relation to their bone mineral density (BMD). The women showed BMD varying from severely osteoporotic to normal. Global gene expression data from the same individuals was available, and since DNA-methylation often affects gene expression, the overall aim of this paper was to include both of these omics data sets into an integrated analysis. Methods The classical penalized regression uses one penalty, but we incorporated individual penalties for each of the DNA-methylation sites. These individual penalties were guided by the strength of association between DNA-methylations and gene transcript levels. DNA-methylations that were highly associated to one or more transcripts got lower penalties and were therefore favored compared to DNA-methylations showing less association to expression. Because of the complex pathways and interactions among genes, we investigated both the association between DNA-methylations and their corresponding cis gene, as well as the association between DNA-methylations and trans-located genes. Two integrating penalized methods were used: first, an adaptive group-regularized ridge regression, and secondly, variable selection was performed through a modified version of the weighted lasso. Results When information from gene expressions was integrated, predictive performance was considerably improved, in terms of predictive mean square error, compared to classical penalized regression without data integration. We found a 14.7% improvement in the ridge regression case and a 17% improvement for the lasso case. Our version of the weighted lasso with data integration found a list of 22 interesting methylation sites. Several corresponded to genes that are known to be important in bone formation. Using BMD as response and these 22 methylation sites as covariates, least square regression analyses resulted in R2=0.726, comparable to an average R2=0.438 for 10000 randomly selected groups of DNA-methylations with group size 22. Conclusions Two recent types of penalized regression methods were adapted to integrate DNA-methylation and their association to gene expression in the analysis of bone mineral density. In both cases predictions clearly benefit from including the additional information on gene expressions.

• Strohmaier, Susanne; Røysland, Kjetil; Hoff, Rune; Borgan, Ørnulf; Pedersen, Terje Rolf & Aalen, Odd O. (2015). Dynamic path analysis - a useful tool to investigate mediation processes in clinical survival trials. Statistics in Medicine. ISSN 0277-6715. 34(29), p. 3866–3887. doi: 10.1002/sim.6598. Full text in Research Archive
• Borgan, Ørnulf & Keogh, Ruth (2015). Nested case–control studies: should one break the matching? Lifetime Data Analysis. ISSN 1380-7870. 21(4), p. 517–541. doi: 10.1007/s10985-015-9319-y. Full text in Research Archive
• Borgan, Ørnulf (2015). Aalen–Johansen Estimator. In Balakrishnan, N.; Brandimarte, Paolo; Everitt, Brian S.; Molenberghs, Geert; Piegorsch, Walter W. & Ruggeri, Fabrizio (Ed.), Wiley StatsRef: Statistics Reference Online. John Wiley & Sons. ISSN 9781118445112. Full text in Research Archive
• Gran, Jon Michael; Lie, Stein Atle; Øyeflaten, Irene Larsen; Borgan, Ørnulf & Aalen, Odd O. (2015). Causal inference in multi-state models-sickness absence and work for 1145 participants after work rehabilitation. BMC Public Health. 15:1082. doi: 10.1186/s12889-015-2408-8. Full text in Research Archive Show summary

  Background Multi-state models, as an extension of traditional models in survival analysis, have proved to be a flexible framework for analysing the transitions between various states of sickness absence and work over time. In this paper we study a cohort of work rehabilitation participants and analyse their subsequent sickness absence using Norwegian registry data on sickness benefits. Our aim is to study how detailed individual covariate information from questionnaires explain differences in sickness absence and work, and to use methods from causal inference to assess the effect of interventions to reduce sickness absence. Examples of the latter are to evaluate the use of partial versus full time sick leave and to estimate the effect of a cooperation agreement on a more inclusive working life. Methods Covariate adjusted transition intensities are estimated using Cox proportional hazards and Aalen additive hazards models, while the effect of interventions are assessed using methods of inverse probability weighting and G-computation. Results Results from covariate adjusted analyses show great differences in sickness absence and work for patients with assumed high risk and low risk covariate characteristics, for example based on age, type of work, income, health score and type of diagnosis. Causal analyses show small effects of partial versus full time sick leave and a positive effect of having a cooperation agreement, with about 5 percent points higher probability of returning to work. Conclusions Detailed covariate information is important for explaining transitions between different states of sickness absence and work, also for patient specific cohorts. Methods for causal inference can provide the needed tools for going from covariate specific estimates to population average effects in multi-state models, and identify causal parameters with a straightforward interpretation based on interventions.

• Borgan, Ørnulf & Zhang, Ying (2015). Using Cumulative Sums of Martingale Residuals for Model Checking in Nested Case-Control Studies. Biometrics. ISSN 0006-341X. 71(3), p. 696–703. doi: 10.1111/biom.12308. Full text in Research Archive
• Günther, Clara-Cecilie; Tvete, Ingunn Fride; Aas, Kjersti; Hagen, Jørgen Andreas; Kvifte, Lars & Borgan, Ørnulf (2014). Predicting Future Claims Among High Risk Policyholders Using Random Effects. In Silvestrov, Dmitrii & Martin-Löf, Anders (Ed.), Modern Problems in Insurance Mathematics. Springer. ISSN 9783319066523. p. 171–186. doi: 10.1007/978-3-319-06653-0_11. Full text in Research Archive Show summary

  Insurance claims are often modelled by a standard Poisson model with fixed effects. With such a model, no individual adjustments are made to account for unobserved heterogeneity between policyholders. A Poisson model with random effects makes it possible to detect policyholders with a high or low individual risk. The premium can then be adjusted accordingly. Others have applied such models without much focus on the model’s prediction performance. As the usefulness of an insurance claims model typically is measured by its ability to predict future claims, we have chosen to focus on this aspect of the model. We model insurance claims with a Poisson random effects model and compare its performance with the standard Poisson fixed effects model. We show that the random effects model both fits the data better and gives better predictions for future claims for high risk policy holders than the standard model.

• Günther, Clara-Cecilie; Tvete, Ingunn Fride; Aas, Kjersti; Sandnes, Geir Inge & Borgan, Ørnulf (2014). Modelling and predicting customer churn from an insurance company. Scandinavian Actuarial Journal. ISSN 0346-1238. p. 58–71. doi: 10.1080/03461238.2011.636502. Full text in Research Archive Show summary

  Within a company's customer relationship management strategy, finding the customers most likely to leave is a central aspect. We present a dynamic modelling approach for predicting individual customers’ risk of leaving an insurance company. A logistic longitudinal regression model that incorporates time-dynamic explanatory variables and interactions is fitted to the data. As an intermediate step in the modelling procedure, we apply generalised additive models to identify non-linear relationships between the logit and the explanatory variables. Both out-of-sample and out-of-time prediction indicate that the model performs well in terms of identifying customers likely to leave the company each month. Our approach is general and may be applied to other industries as well.

• Nilsen, Gro; Borgan, Ørnulf; Liestøl, Knut & Lingjærde, Ole Christian (2013). Identifying clusters in genomics data by recursive partitioning. Statistical Applications in Genetics and Molecular Biology. 12(5), p. 637–652. doi: 10.1515/sagmb-2013-0016. Full text in Research Archive
• Borgan, Ørnulf & Samuelsen, Sven Ove (2013). Nested Case-Control and Case-Cohort Studies. In Klein, John P; Houwelingen, Hans C van; Ibrahim, Joseph G & Scheike, Thomas H (Ed.), Handbook of Survival Analysis. CRC Press. ISSN 9781466555662. p. 343–367. Full text in Research Archive
• Paulsen, Jonas; Lien, Tonje Gulbrandsen; Sandve, Geir Kjetil; Holden, Lars; Borgan, Ørnulf & Glad, Ingrid Kristine [Show all 7 contributors for this article] (2013). Handling realistic assumptions in hypothesis testing of 3D co-localization of genomic elements. Nucleic Acids Research (NAR). ISSN 0305-1048. 41(10), p. 5164–5174. doi: 10.1093/nar/gkt227. Full text in Research Archive Show summary

  The study of chromatin 3D structure has recently gained much focus owing to novel techniques for detecting genome-wide chromatin contacts using next-generation sequencing. A deeper understanding of the architecture of the DNA inside the nucleus is crucial for gaining insight into fundamental processes such as transcriptional regulation, genome dynamics and genome stability. Chromatin conformation capture-based methods, such as Hi-C and ChIA-PET, are now paving the way for routine genome-wide studies of chromatin 3D structure in a range of organisms and tissues. However, appropriate methods for analyzing such data are lacking. Here, we propose a hypothesis test and an enrichment score of 3D co-localization of genomic elements that handles intra- or interchromosomal interactions, both separately and jointly, and that adjusts for biases caused by structural dependencies in the 3D data. We show that maintaining structural properties during resampling is essential to obtain valid estimation of P-values. We apply the method on chromatin states and a set of mutated regions in leukemia cells, and find significant co-localization of these elements, with varying enrichment scores, supporting the role of chromatin 3D structure in shaping the landscape of somatic mutations in cancer.

• Aldrin, Magne; Lyngstad, Trude Marie; Kristoffersen, Anja Bråthen; Storvik, Bård; Borgan, Ørnulf & Jansen, Peder Andreas (2011). Modelling the spread of infectious salmon anaemia among salmon farms based on seaway distances between farms and genetic relationships between infectious salmon anaemia virus isolates. Journal of the Royal Society Interface. ISSN 1742-5689. 8(62), p. 1346–1356. doi: 10.1098/rsif.2010.0737. Full text in Research Archive
• Bøvelstad, Hege & Borgan, Ørnulf (2011). Assessment of evaluation criteria for survival prediction from genomic data. Biometrical Journal. ISSN 0323-3847. 53(2), p. 202–216. doi: 10.1002/bimj.201000048. Full text in Research Archive
• Moger, Tron Anders; Haugen, Marion; Yip, Benjamin H. K.; Gjessing, Håkon & Borgan, Ørnulf (2011). A hierarchical frailty model applied to two-generation melanoma data. Lifetime Data Analysis. ISSN 1380-7870. 17(3), p. 445–460. doi: 10.1007/s10985-010-9188-3. Full text in Research Archive Show summary

  We present a hierarchical frailty model based on distributions derived from non-negative Lévy processes. The model may be applied to data with several levels of dependence, such as family data or other general clusters, and is an alternative to additive frailty models. We present several parametric examples of the model, and properties such as expected values, variance and covariance. The model is applied to a case-cohort sample of age at onset for melanoma from the Swedish Multi-Generation Register,organized in nuclear families of parents and one or two children. We compare the genetic component of the total frailty variance to the common environmental term, and estimate the effect of birth cohort and gender.

• Aalen, Odd O.; Andersen, Per Kragh; Borgan, Ørnulf; Gill, Richard D. & Keiding, Niels (2009). History of applications of martingales in survival analysis. Journal Électronique d'Histoire des Probabililtés et de la Statistique. 5(1), p. 1–28. Full text in Research Archive
• Bøvelstad, Hege; Nygård, Ståle & Borgan, Ørnulf (2009). Survival prediction from clinico-genomic models - a comparative study. BMC Bioinformatics. 10. doi: 10.1186/1471-2105-10-413. Full text in Research Archive Show summary

  Background Survival prediction from high-dimensional genomic data is an active field in today's medical research. Most of the proposed prediction methods make use of genomic data alone without considering established clinical covariates that often are available and known to have predictive value. Recent studies suggest that combining clinical and genomic information may improve predictions, but there is a lack of systematic studies on the topic. Also, for the widely used Cox regression model, it is not obvious how to handle such combined models. Results We propose a way to combine classical clinical covariates with genomic data in a clinico-genomic prediction model based on the Cox regression model. The prediction model is obtained by a simultaneous use of both types of covariates, but applying dimension reduction only to the high-dimensional genomic variables. We describe how this can be done for seven well-known prediction methods: variable selection, unsupervised and supervised principal components regression and partial least squares regression, ridge regression, and the lasso. We further perform a systematic comparison of the performance of prediction models using clinical covariates only, genomic data only, or a combination of the two. The comparison is done using three survival data sets containing both clinical information and microarray gene expression data. Matlab code for the clinico-genomic prediction methods is available at http://www.med.uio.no/imb/stat/bmms/software/clinico-genomic/ webcite. Conclusions Based on our three data sets, the comparison shows that established clinical covariates will often lead to better predictions than what can be obtained from genomic data alone. In the cases where the genomic models are better than the clinical, ridge regression is used for dimension reduction. We also find that the clinico-genomic models tend to outperform the models based on only genomic data. Further, clinico-genomic models and the use of ridge regression gives for all three data sets better predictions than models based on the clinical covariates alone.

• Nygård, Ståle; Borgan, Ørnulf; Lingjærde, Ole Christian & Størvold, Hege Leite (2008). Partial least squares Cox regression for genome-wide data. Lifetime Data Analysis. ISSN 1380-7870. 14, p. 179–195. doi: 10.1007/s10985-007-9076-7. Full text in Research Archive
• Moger, Tron Anders; Pawitan, Yudi & Borgan, Ørnulf (2008). Case-cohort methods for survival data on families from routine registers. Statistics in Medicine. ISSN 0277-6715. 27, p. 1062–1074. doi: 10.1002/sim.3004. Full text in Research Archive
• Borgan, Ørnulf & Langholz, Bryan (2007). Using martingale residuals to assess goodness-of-fit for sampled risk set data. In Nair, Vijay (Eds.), Advances in statistical modeling and inference. Essays in honor of Kjell A. Doksum. World Scientific. ISSN 9789812703699. p. 65–90. Full text in Research Archive
• Borgan, Ørnulf (2007). Event history analysis: An overview and some areas of current research. In Østreng, Willy (Eds.), Consilience. Centre for Advanced Study at the Norwegian Academy of Science and Letters. ISSN 9788299636742. p. 60–64. Full text in Research Archive
• Borgan, Ørnulf; Fiaccone, RL; Henderson, R & Barreto, ML (2007). Dynamic analysis of recurrent event data with missing observations, with application to infant diarrhoea in Brazil. Scandinavian Journal of Statistics. ISSN 0303-6898. 34, p. 53–69. doi: 10.1111/j.1467-9469.2006.00525.x. Full text in Research Archive
• Bøvelstad, Hege; Nygård, Ståle; Størvold, Hege Leite; Aldrin, Magne; Borgan, Ørnulf & Frigessi, Arnoldo [Show all 7 contributors for this article] (2007). Predicting survival from microarray data - a comparative study. Bioinformatics. ISSN 1367-4803. 23, p. 2080–2087. doi: 10.1093/bioinformatics/btm305. Full text in Research Archive
• Gunnes, Nina; Borgan, Ørnulf & Aalen, Odd O. (2007). Estimating stage occupation probabilities in non-Markov models. Lifetime Data Analysis. ISSN 1380-7870. 13, p. 211–240. doi: 10.1007/s10985-007-9034-4. Full text in Research Archive
• Langerod, A; Zhao, H; Borgan, Ørnulf; Nesland, Jahn M; Bukholm, IR & Ikdahl, T [Show all 9 contributors for this article] (2007). TP53 mutation status and gene expression profiles are powerful prognostic markers of breast cancer. Breast Cancer Research. ISSN 1465-5411. 9, p. R30–R30. doi: 10.1186/bcr1675. Full text in Research Archive Show summary

  Introduction Gene expression profiling of breast carcinomas has increased our understanding of the heterogeneous biology of this disease and promises to impact clinical care. The aim of this study was to evaluate the prognostic value of gene expression-based classification along with established prognostic markers and mutation status of the TP53 gene (tumour protein p53) in a group of breast cancer patients with long-term (12 to 16 years) follow-up. Methods The clinical and histopathological parameters of 200 breast cancer patients were studied for their effects on clinical outcome using univariate/multivariate Cox regression. The prognostic impact of mutations in the TP53 gene, identified using temporal temperature gradient gel electrophoresis and sequencing, was also evaluated. Eighty of the samples were analyzed for gene expression using 42 K cDNA microarrays and the patients were assigned to five previously defined molecular expression groups. The strength of the gene expression based classification versus standard markers was evaluated by adding this variable to the Cox regression model used to analyze all samples. Results Both univariate and multivariate analysis showed that TP53 mutation status, tumor size and lymph node status were the strongest predictors of breast cancer survival for the whole group of patients. Analyses of the patients with gene expression data showed that TP53 mutation status, gene expression based classification, tumor size and lymph node status were significant predictors of survival. Breast cancer cases in the 'basal-like' and 'ERBB2+' gene expression subgroups had a very high mortality the first two years, while the 'highly proliferating luminal' cases developed the disease more slowly, showing highest mortality after 5 to 8 years. The TP53 mutation status showed strong association with the 'basal-like' and 'ERBB2+' subgroups, and tumors with mutation had a characteristic gene expression pattern. Conclusion TP53 mutation status and gene-expression based groups are important survival markers of breast cancer, and these molecular markers may provide prognostic information that complements clinical variables. The study adds experience and knowledge to an ongoing characterization and classification of the disease.

• Stavem, K; Bjortuft, O; Borgan, Ørnulf; Geiran, O & Boe, J (2006). Lung transplantation in patients with chronic obstructive pulmonary disease in a national cohort is without obvious survival benefit. The Journal of Heart and Lung Transplantation. ISSN 1053-2498. 25. Full text in Research Archive
• Fosen, Johan; Ferkingstad, Egil; Borgan, Ørnulf & Aalen, Odd O. (2006). Dynamic path analysis - a new approach to analyzing time-dependent covariates. Lifetime Data Analysis. ISSN 1380-7870. 12, p. 143–167. Full text in Research Archive
• Fosen, Johan; Borgan, Ørnulf; Weedon-Fekjaer, H & Aalen, Odd Olai (2006). Dynamic analysis of recurrent event data using the additive hazard model. Biometrical Journal. ISSN 0323-3847. 48, p. 381–398. Full text in Research Archive
• Aalen, Odd Olai; Fosen, Johan; Weedon-Fekjær, Harald; Borgan, Ørnulf & Husebye, Einar (2004). Dynamic analysis of multivariate failure time data. Biometrics. ISSN 0006-341X. 60, p. 764–773. Full text in Research Archive
• Borgan, Ørnulf & Samuelsen, Sven Ove (2003). A review of cohort sampling designs for Cox's regression model: Potentials in epidemiology. Norsk Epidemiologi. ISSN 0803-2491. 13(2), p. 239–248. Full text in Research Archive
• Borgan, Ørnulf (2002). Estimation of covariate-dependent Markov transition probabilites from nested case-control data. Statistical Methods in Medical Research. ISSN 0962-2802. 11, p. 183–202. Full text in Research Archive
• Aalen, Odd Olai; Borgan, Ørnulf & Weedon-Fekjær, Harald (2001). Covariate adjustment of event histories estimated from Markov chains: The additive approach. Biometrics. ISSN 0006-341X. 57, p. 993–1001. Full text in Research Archive
• Borgan, Ørnulf; Langholz, Bryan; Samuelsen, Sven Ove; Goldstein, Larry & Pogoda, Janice (2000). Exposure stratified case-cohort designs. Lifetime Data Analysis. ISSN 1380-7870. 6, p. 39–58. Full text in Research Archive
• Borgan, Ørnulf & Olsen, Espen F. (1999). The efficiency of simple and counter-matched nested case-control sampling. Scandinavian Journal of Statistics. ISSN 0303-6898. 26, p. 493–509. Full text in Research Archive
• Zhang, Jian & Borgan, Ørnulf (1999). Aalen's linear model for sampled risk set data: a large sample study. Lifetime Data Analysis. ISSN 1380-7870. 5, p. 345–363. Full text in Research Archive
• Borgan, Ørnulf & Langholz, Bryan (1998). Risk set sampling for proportional hazards models. In Everitt, Brian S. & Dunn, Graham (Ed.), Statistical Analysis of Medical Data: New Developments. Arnold, London. ISSN 9780340677759. p. 75–100. Full text in Research Archive
• Langholz, Bryan & Borgan, Ørnulf (1997). Estimation of absolute risk from nested case-control data. Biometrics. ISSN 0006-341X. 53, p. 767–774. Full text in Research Archive
• Borgan, Ørnulf & Langholz, Bryan (1997). Estimation of excess risk from case-control data using Aalen's linear regression model. Biometrics. ISSN 0006-341X. 53, p. 690–697. Full text in Research Archive
• Grønnesby, Jon Ketil & Borgan, Ørnulf (1996). A method for checking regression models in survival analysis based on the risk score. Lifetime Data Analysis. ISSN 1380-7870. 2, p. 315–328. Full text in Research Archive
• Borgan, Ørnulf; Goldstein, Larry & Langholz, Bryan (1995). Methods for the analysis of sampled cohort data in the Cox proportional hazards model. Annals of Statistics. ISSN 0090-5364. 23, p. 1749–1778. Full text in Research Archive
• Langholz, Bryan & Borgan, Ørnulf (1995). Counter-matching: A stratified nested case-control sampling method. Biometrika. ISSN 0006-3444. 82, p. 69–79. Full text in Research Archive
• Borgan, Ørnulf & Langholz, Bryan (1993). Nonparametric estimation of relative mortality from nested case-control studies. Biometrics. ISSN 0006-341X. 49, p. 593–602. Full text in Research Archive
• Borgan, Ørnulf & Liestøl, Knut (1990). A note on confidence intervals and bands for the survival function based on transformations. Scandinavian Journal of Statistics. ISSN 0303-6898. 17, p. 35–41. Full text in Research Archive

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• Borgan, Ørnulf; Borge, Inger Christin; Engeseth, John; Heir, Odd; Moe, Håvard & Norderhaug, Tea Toft [Show all 7 contributors for this article] (2022). Matematikk S2. Aschehoug & Co. ISBN 9788203408991. 334 p. Full text in Research Archive
• Borgan, Ørnulf; Borge, Inger Christin; Engeseth, John; Heir, Odd; Moe, Håvard & Norderhaug, Tea Toft [Show all 7 contributors for this article] (2021). Matematikk S1. Aschehoug & Co. ISBN 9788203408946. 375 p. Full text in Research Archive
• Borgan, Ørnulf; Borge, Inger Christin; Engeseth, John; Heir, Odd; Moe, Håvard & Norderhaug, Tea Toft [Show all 7 contributors for this article] (2021). Matematikk R1. Aschehoug & Co. ISBN 9788203408847. 414 p. Full text in Research Archive
• Borgan, Ørnulf; Breslow, Norman E.; Chatterjee, Nilanjan; Gail, Mitchell H.; Scott, Alastair & Wild, Christopher J. (2018). Handbook of Statistical Methods for Case-Control Studies. CRC Press. ISBN 9781498768580. 536 p. Full text in Research Archive
• Aalen, Odd O.; Borgan, Ørnulf & Gjessing, Håkon K. (2008). Survival and Event History Analysis: A Process Point of View. Springer. ISBN 9780387202877. 539 p. Full text in Research Archive
• Andersen, Per; Borgan, Ørnulf; Gill, Richard D. & Keiding, Niels (1993). Statistical models based on counting processes. Springer-Verlag, New York. 767 p. Full text in Research Archive

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• Borgan, Ørnulf & Glad, Ingrid Kristine (2024). A conversation with Nils Lid Hjort. Scandinavian Journal of Statistics. ISSN 0303-6898. doi: 10.1111/sjos.12732. Full text in Research Archive Show summary

  Professor (now emeritus) Nils Lid Hjort has through more than four decades been one of the most original and productive statisticians in Norway, contributing to a wide range of topics such as survival analysis, Bayesian nonparametrics, empirical likelihood, density estimation, focused inference, model selection, and confidence distributions. This conversation, which took place at the University of Oslo in December 2023, sheds light on how Nils Hjort's curious and open mind, coupled with a deep understanding, has enabled him to seamlessly navigate between different fields of statistics and its applications. Our aim is to encourage the statistics community to always be on the lookout for unexpected connections in statistical science and to embrace unexpected encounters with fellow statisticians from around the world.

• Borgan, Ørnulf & Glad, Ingrid Kristine (2024). A special section honoring Nils Lid Hjort. Scandinavian Journal of Statistics. ISSN 0303-6898. doi: 10.1111/sjos.12745. Full text in Research Archive
• Borgan, Ørnulf & Glad, Ingrid Kristine (2023). Nils Lid Hjort 70: A short walk through a long and splendid career. Full text in Research Archive
• Borgan, Ørnulf & Gjessing, Håkon K. (2019). Special issue dedicated to Odd O. Aalen. Lifetime Data Analysis. ISSN 1380-7870. 25(4), p. 587–592. doi: 10.1007/s10985-019-09483-z. Full text in Research Archive
• Borgan, Ørnulf (2009). Inference and prediction in multi-state models: the additive approach. 57th Session of the International Statistical Institute. Full text in Research Archive
• Borgan, Ørnulf & H., Lindqvist Bo (2007). Editorial. Scandinavian Journal of Statistics. ISSN 0303-6898. 34(1), p. 1–2. Full text in Research Archive
• Borgan, Ørnulf (2007). Discussion contribution to D. Zeng and D.Y. Lin's paper "Maximum likelihood estimation in semiparametric regression models with censored data". Journal of the Royal Statistical Society, Series B (Statistical Methodology). ISSN 1369-7412. 69(4), p. 545–545. Full text in Research Archive
• Senn, S; Keiding, N; Andrews, N; Musonda, P; Hocine, M & Moreau, T [Show all 16 contributors for this article] (2006). Semiparametric analysis of case series data - Discussion. Journal of the Royal Statistical Society, Series C (Applied Statistics). ISSN 0035-9254. 55, p. 581–594. Full text in Research Archive
• Borgan, Ørnulf; Aalen, Odd O. & Røsjø, Bjarne (2005). Bedre statistikk kan redde flere liv. [Journal]. Informasjonsblad nr 2/2005. Senter for grunnforskning. Full text in Research Archive
• Borgan, Ørnulf (2004). Sverdrup, Erling (1917-1994). In Teugels, Jozef & Sund, Ståle (Ed.), Encyclopedia of Actuarial Science. John Wiley & Sons. ISSN 9780470846766. p. 1665–1666. Full text in Research Archive
• Borgan, Ørnulf (2003). Review of "The Statistical Analysis of Failure Time Data" by J.D. Kalbfleisch and R.L. Prentice. Journal of the American Statistical Association. ISSN 0162-1459. 98(464), p. 1084–1085. Full text in Research Archive
• Borgan, Ørnulf (2001). Review of "Modeling Survival Data: Extending the Cox Model" by T. M. Therneau and P. M. Grambsch. Statistics in Medicine. ISSN 0277-6715. 20, p. 2053–2054. Full text in Research Archive
• Borgan, Ørnulf (2001). Cohort sampling in epidemiological studies. 53rd Session of The International Statistical Institute. Full text in Research Archive
• Borgan, Ørnulf (1999). Countermatched sampling. In Kotz, Samuel; Read, CB & Banks, DL (Ed.), Encyclopedia of Statistical Sciences. Wiley, New York. p. 129–131. Full text in Research Archive
• Borgan, Ørnulf (1999). Nested case-control sampling. In Kotz, Samuel; Read, CB & Banks, DL (Ed.), Encyclopedia of Statistical Sciences. Wiley, New York. p. 523–528. Full text in Research Archive
• Borgan, Ørnulf (1999). Review of "Applied survival analysis" by C.T. Lee. Statistics in Medicine. ISSN 0277-6715. 18, p. 492–494. Full text in Research Archive
• Borgan, Ørnulf (1998). Nelson-Aalen estimator. In Armitage, Peter & Colton, Theodore (Ed.), Encyclopedia of Biostatistics. Wiley, New York. ISSN 9780471975762. p. 2967–2972. Full text in Research Archive
• Borgan, Ørnulf (1998). Kaplan-Meier estimator. In Armitage, Peter & Colton, Theodore (Ed.), Encyclopedia of Biostatistics. Wiley, New York. ISSN 9780471975762. p. 2154–2160. Full text in Research Archive
• Borgan, Ørnulf (1998). Aalen-Johansen estimator. In Armitage, Peter & Colton, Theodore (Ed.), Encyclopedia of Biostatistics. Wiley, New York. ISSN 9780471975762. p. 5–10. Full text in Research Archive
• Günther, Clara-Cecilie; Tvete, Ingunn Fride; Aas, Kjersti & Borgan, Ørnulf (2010). Modellering og prediksjon av kundeavgang. Norsk Regnesentral. Full text in Research Archive
• Nygård, Ståle; Borgan, Ørnulf; Lingjærde, Ole Christian & Størvold, Hege Leite (2006). Partial least squares Cox regression on genomic data handling additional covariates. Universitetet i Oslo. Full text in Research Archive Show summary

  There exist many methods for survival prediction from high-dimensional genomic data. Most of them combine the Cox proportional hazards model with some dimension reduction estimation technique, like partial least squares (PLS). For PLS it is not obvious how it should be applied to the Cox model, and different approaches have been suggested. The perhaps most reasonable one, Park et al. (2002), uses a reformulation of the Cox likelihood to a Poisson type likelihood, thereby enabling estimation by iteratively reweighted partial least squares for generalized linear models. We present a modified version of the method of Park et al. (2002), which estimates the baseline hazard and the gene effects in separate steps. Our approach has the advantages of leading to PLS directions that have more reasonable biological interpretations, providing estimates of survival probabilities for new patients and enabling a faster and less memory-demanding estimation procedure. In addition our method allows for incorporation of lower-dimensional non-genomic variables like disease grade and tumor thickness. Applying our method to two different microarray gene expression breast cancer data sets, one with additional non-genomic covariates, shows that our method gives at least as good predictions as the method of Park et al. (2002), and that there is a lot to be gained by including gene expressions together with the clinical variables.

• Borgan, Ørnulf & Langholz, Bryan (2005). Using martingale residuals to assess goodness-of-fit for sampled risk set data. Universitetet i Oslo. Full text in Research Archive Show summary

  Standard use of Cox's regression model and other relative risk regression models for censored survival data requires collection of covariate information on all individuals under study even when only a small fraction of them die or get diseased. For such situations risk set sampling designs offer useful alternatives. For cohort data, methods based on martingale residuals are useful for assessing the fit of a model. Here we introduce grouped martingale residual processes for sampled risk set data, and show that plots of these processes provide a useful tool for checking model-fit. Further we study the large sample properties of the grouped martingale residual processes, and use these to derive a formal goodness-of-fit test to go along with the plots. The methods are illustrated using data on lung cancer deaths in a cohort of uranium miners.

• Borgan, Ørnulf; Fiaccone, Rosemeire L.; Henderson, Robin & Barreto, Mauricio L. (2005). Dynamic analysis of recurrent event data with missing observations, with application to infant diarrhoea in Brazil. Universitetet i Oslo. Full text in Research Archive Show summary

  This paper examines and applies methods for modelling of longitudinal binary data subject to both intermittent missingness and dropout. The paper is based around the analysis of data from a study into the health impact of a sanitation programme carried out in Salvador, Brazil. Our objective is to investigate risk factors associated with incidence and prevalence of diarrhoea in children aged up to 3 years old. In total 926 children were followed up at home twice a week from October 2000 to January 2002, from which daily occurrence of diarrhoea was recorded for each child being followed up. A challenging factor in analysing these data is the presence of between subject heterogeneity not explained by known risk factors, combined with significant loss of observed data through either intermittent missingness (average of 78 days per child) or dropout (21% of children). We discuss modelling strategies and show the advantages of taking an event history approach with an additive discrete time regression model.

• Fosen, Johan; Ferkingstad, Egil; Borgan, Ørnulf & Aalen, Odd O. (2005). Dynamic path analysis - a new approach to analyzing time-dependent covariates. Universitetet i Oslo. Full text in Research Archive Show summary

  In this article we introduce a general approach to dynamic path analysis. This is an extension of classical path analysis to the situation where variables may be time-dependent and where the outcome of main interest is a stochastic process. In particular we will focus on the survival and event history analysis setting where the main outcome is a counting process. Our approach will be especially fruitful for analyzing event history data with internal time-dependent covariates, where an ordinary regression analysis may fail. Our approach enables us to describe how the effect of a fixed covariate partly is working directly and partly indirectly through internal time-dependent covariates. For the sequence of times of event, we define a sequence of path analysis models. At each time of an event, ordinary linear regression is used to estimate the relation between the covariates, while the additive hazard model is used for the regression of the counting process on the covariates. The methodology is illustrated using data from a randomized trial on survival for patients with liver cirrhosis

• Aalen, Odd O.; Fosen, Johan; Weedon-Fekjær, Harald; Borgan, Ørnulf & Husebye, Einar (2004). Dynamic analysis of multivariate failure time data. Universitetet i Oslo. Full text in Research Archive Show summary

  We present an approach for analysing internal dependencies in counting processes. This covers the case with repeated events on each of a number of individuals, and, more generally, the situation where several processes are observed for each individual. We define dynamic covariates, i.e. covariates depending on the past of the processes. The statistical analysis is performed mainly by the nonparametric additive approach. This yields a method for analysing multivariate survival data, which is an alternative to the frailty approach. We present cumulative regression plots, statistical tests, residual plots and a hat matrix plot for studying outliers. A program in R and S-PLUS for analyzing survival data with the additive regression model is available on the web site http://www.med.uio.no/imb/english/research/groups/causal-inference-methods/software/ The program has been developed to fit the counting process framework.

• Borgan, Ørnulf (2002). Estimation of covariate-dependent Markov transition probabilities from nested case-control data. Universitetet i Oslo. Full text in Research Archive Show summary

  Multi-state models are used to describe situations where individuals may move among a finite number of states defined by specific conditions of health, including death. The transition intensities of the models are described by proportional hazards models, and it is reviewed how estimation of the regression parameters and the baseline transition intensities may be performed when only nested case-control data are available for all or some of the transitions. The regression parameter estimates and the estimates of baseline transition intensities are combined to give estimates of the integrated transition intensities for specified covariate histories, and from these estimates covariate-dependent Markov transition probabilities are derived.

• Aalen, Odd O.; Borgan, Ørnulf & Weedon-Fekjær, Harald (2001). Covariate adjustment of event histories estimated from Markov chains: the additive approach. Universitetet i Oslo. Full text in Research Archive
• Borgan, Ørnulf (2001). Cohort sampling in epidemiological studies. Universitetet i Oslo. Full text in Research Archive
• Borgan, Ørnulf; Goldstein, Larry; Langholz, Bryan; Pogoda, Janice & Samuelsen, Sven Ove (1998). Exposure stratified case-cohort designs. Universitetet i Oslo. Full text in Research Archive
• Borgan, Ørnulf & Langholz, Bryan (1997). Risk set sampling designs for proportional hazard models. Universitetet i Oslo. Full text in Research Archive
• Borgan, Ørnulf (1997). Two contributions to the update of the Encyclopedia of Statistical Science: Nested case-control sampling and Counter-matched sampling. Universitetet i Oslo. Full text in Research Archive
• Borgan, Ørnulf (1997). Three contributions to the Encyclopedia of Biostatistics: The Nelson-Aalen, Kaplan-Meier, and Aalen-Johansen. Universitetet i Oslo. Full text in Research Archive
• Langholz, Bryan & Borgan, Ørnulf (1996). Estimation of absolute risk from nested case-control data. Universitetet i Oslo. Full text in Research Archive
• Grønnesby, Jon Ketil & Borgan, Ørnulf (1996). A method for checking regression models in survival analysis based on risk score. Universitetet i Oslo. Full text in Research Archive
• Borgan, Ørnulf & Langholz, Bryan (1995). Estimation of excess risk from case-control data using Aalen's linear regression model. Universitetet i Oslo. Full text in Research Archive
• Langholz, Bryan & Borgan, Ørnulf (1993). Counter-matching: A stratified nested case-control sampling method. Universitetet i Oslo. Full text in Research Archive
• Borgan, Ørnulf (1990). Lecture notes in Life history analysis. Universitetet i Oslo. Full text in Research Archive

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