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  • 1.
    Björkwall, Susanna
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Esbjörn, Ohlsson
    Stockholm University, Faculty of Science, Department of Mathematics.
    Bootstrapping the separation method in claims reserving.2010In: Astin Bulletin: Actuarial Studies in Non-Life Insurance, ISSN 0515-0361, E-ISSN 1783-1350, Vol. 40, no 2, p. 845-869Article in journal (Refereed)
    Abstract [en]

    The separation method was introduced by Verbeek (1972) in order to forecast numbers of excess claims and it was developed further by Taylor (1977) to be applicable to the average claim cost.The separation method differs from the chain-ladder in that when the chain-ladder only assumes claim proportionality between the development years, the separation method also separates the claim delay distribution from influences affecting the calendar years, e.g. inflation. Since the inflation contributes to the uncertainty in the estimate of the claims reserve it is important to consider its impact in the context of risk management, too.

    In this paper we present a method for assessing the prediction error distribution of the separation method. To this end we introduce a parametric framework within the separation model which enables joint resampling of claim counts and claim amounts. As a result, the variability of Taylor's predicted reserves can be assessed by extending the parametric bootstrap techniques of Björkwall et al. (2009). The performance of the bootstrapped separation method and chain-ladder is compared for a real data set.

  • 2.
    Björkwall, Susanna
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ohlsson, Esbjörn
    Stockholm University, Faculty of Science, Department of Mathematics.
    Non-parametric and parametric bootstrap techniques for age-to-age development factor methods in stochastic claims reserving.2009In: Scandinavian Actuarial Journal, ISSN 0346-1238, E-ISSN 1651-2030, no 4, p. 306-331Article in journal (Refereed)
    Abstract [en]

    In the literature, one of the main objects of stochastic claims reserving is to find models underlying the chain-ladder method in order to analyze the variability of the outstanding claims, either analytically or by bootstrapping. In bootstrapping these models are used to find a full predictive distribution of the claims reserve, even though there is a long tradition of actuaries calculating the reserve estimate according to more complex algorithms than the chain-ladder, without explicit reference to an underlying model. In this paper we investigate existing bootstrap techniques and suggest two alternative bootstrap procedures, one non-parametric and one parametric, by which the predictive distribution of the claims reserve can be found for other age-to-age development factor methods than the chain-ladder, using some rather mild model assumptions. For illustration, the procedures are applied to three different development triangles.

  • 3.
    Björkwall, Susanna
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ohlsson, Esbjörn
    Verrall, Richard
    A generalized linear model with smoothing effects for claims reserving2011In: Insurance, Mathematics & Economics, ISSN 0167-6687, E-ISSN 1873-5959, Vol. 49, no 1, p. 27-37Article in journal (Refereed)
    Abstract [en]

    In this paper, we continue the development of the ideas introduced in England and Verrall (2001) by suggesting the use of a reparameterized version of the generalized linear model (GLM) which is frequently used in stochastic claims reserving. This model enables us to smooth the origin, development and calendar year parameters in a similar way as is often done in practice, but still keep the GLM structure. Specifically, we use this model structure in order to obtain reserve estimates and to systemize the model selection procedure that arises in the smoothing process. Moreover, we provide a bootstrap procedure to achieve a full predictive distribution.

  • 4. Diaz-Pachón, Daniel Andrés
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Assessing, Testing and Estimating the Amount of Fine-Tuning by Means of Active Information2022In: Entropy, E-ISSN 1099-4300, Vol. 24, no 10, article id 1323Article in journal (Refereed)
    Abstract [en]

    A general framework is introduced to estimate how much external information has been infused into a search algorithm, the so-called active information. This is rephrased as a test of fine-tuning, where tuning corresponds to the amount of pre-specified knowledge that the algorithm makes use of in order to reach a certain target. A function f quantifies specificity for each possible outcome x of a search, so that the target of the algorithm is a set of highly specified states, whereas fine-tuning occurs if it is much more likely for the algorithm to reach the target as intended than by chance. The distribution of a random outcome X of the algorithm involves a parameter θ that quantifies how much background information has been infused. A simple choice of this parameter is to use θf in order to exponentially tilt the distribution of the outcome of the search algorithm under the null distribution of no tuning, so that an exponential family of distributions is obtained. Such algorithms are obtained by iterating a Metropolis–Hastings type of Markov chain, which makes it possible to compute their active information under the equilibrium and non-equilibrium of the Markov chain, with or without stopping when the targeted set of fine-tuned states has been reached. Other choices of tuning parameters θ are discussed as well. Nonparametric and parametric estimators of active information and tests of fine-tuning are developed when repeated and independent outcomes of the algorithm are available. The theory is illustrated with examples from cosmology, student learning, reinforcement learning, a Moran type model of population genetics, and evolutionary programming.

  • 5. Díaz-Pachón, Daniel Andrés
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Marks II, Robert J.
    Is cosmological tuning fine or coarse?2021In: Journal of Cosmology and Astroparticle Physics, E-ISSN 1475-7516, no 7, article id 020Article in journal (Refereed)
    Abstract [en]

    The fine-tuning of the universe for life, the idea that the constants of nature (or ratios between them) must belong to very small intervals in order for life to exist, has been debated by scientists for several decades. Several criticisms have emerged concerning probabilistic measurement of life-permitting intervals. Herein, a Bayesian statistical approach is used to assign an upper bound for the probability of tuning, which is invariant with respect to change of physical units, and under certain assumptions it is small whenever the life-permitting interval is small on a relative scale. The computation of the upper bound of the tuning probability is achieved by first assuming that the prior is chosen by the principle of maximum entropy (MaxEnt). The unknown parameters of this MaxEnt distribution are then handled in such a way that the weak anthropic principle is not violated. The MaxEnt assumption is maximally noncommittal with regard to missing information. This approach is sufficiently general to be applied to constants of current cosmological models, or to other constants possibly under different models. Application of the MaxEnt model reveals, for example, that the ratio of the universal gravitational constant to the square of the Hubble constant is finely tuned in some cases, whereas the amplitude of primordial fluctuations is not.

  • 6. Díaz-Pachón, Daniel Andrés
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Marks II, Robert J.
    Sometimes Size Does Not Matter2023In: Foundations of physics, ISSN 0015-9018, E-ISSN 1572-9516, Vol. 53, no 1, article id 1Article in journal (Refereed)
    Abstract [en]

    Recently Díaz, Hössjer and Marks (DHM) presented a Bayesian framework to measure cosmological tuning (either fine or coarse) that uses maximum entropy (maxent) distributions on unbounded sample spaces as priors for the parameters of the physical models (https://doi.org/10.1088/1475-7516/2021/07/020). The DHM framework stands in contrast to previous attempts to measure tuning that rely on a uniform prior assumption. However, since the parameters of the models often take values in spaces of infinite size, the uniformity assumption is unwarranted. This is known as the normalization problem. In this paper we explain why and how the DHM framework not only evades the normalization problem but also circumvents other objections to the tuning measurement like the so called weak anthropic principle, the selection of a single maxent distribution and, importantly, the lack of invariance of maxent distributions with respect to data transformations. We also propose to treat fine-tuning as an emergence problem to avoid infinite loops in the prior distribution of hyperparameters (common to all Bayesian analysis), and explain that previous attempts to measure tuning using uniform priors are particular cases of the DHM framework. Finally, we prove a theorem, explaining when tuning is fine or coarse for different families of distributions. The theorem is summarized in a table for ease of reference, and the tuning of three physical parameters is analyzed using the conclusions of the theorem.

  • 7.
    Ekheden, Erland
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Analysis of the Stochasticity of Mortality Using Variance Decomposition2014In: Modern Problems in Insurance Mathematics / [ed] Dmitri Silvestrov and Anders Martin-Löf, Springer Publishing Company, 2014, p. 199-222Chapter in book (Refereed)
    Abstract [en]

    We analyse the stochasticity in mortality data from the USA, the UK and Sweden, and in particular to which extent mortality rates are explained by systematic variation, due to various risk factors, and random noise. We formalise this in terms of a mixed regression model with a logistic link function, and decomposethe variance of the observations into three parts: binomial risk, the variance due to random mortality variation in a finite population, systematic risk explained by the covariates and unexplained systematic risk, variance that comes from real changes in mortality rates, not captured by the covariates. The fraction of unexplained variance caused by binomial risk provides a limit in terms of the resolution that can be achieved by a model. This can be used as a model selection tool for selecting the number of covariates and regression parameters of the deterministic part of the regression function, and for testing whether unexplained systematic variation should be explicitly modelled or not. We use a two-factor model with ageand calendar year as covariates, and perform the variance decomposition for a simple model with a linear time trend on the logit scale. The population size turns out to be crucial, and for Swedish data, the simple model works surprisingly well, leaving only a small fraction of unexplained systematic risk, whereas for the UK and the USA, the amount of unexplained systematic risk is larger, so that more elaborate models might work better.

  • 8.
    Ekheden, Erland
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Multivariate Time Series Modeling, Estimation and Prediction of MortalitiesArticle in journal (Refereed)
    Abstract [en]

    We introduce a mixed regression model for morality data whichcan be decomposed into a deterministic trend component explainedby the covariates age and calendar year, a multivariate Gaussian timeseries part not explained by the covariates, and binomial risk. Datacan be analyzed by means of a simple logistic regression model whenthe multivariate Gaussian time series component is absent and there isno overdispersion, as in Ekheden and Hössjer (2014). In this paper werather allow for overdispersion and the mixed regression model is ttedto mortality data from the United States and Sweden, with the aim toprovide prediction and condence intervals for future mortality, as wellas smoothing historical data, using the best linear unbiased predictor.We nd that the form of the Gaussian time series has a large impact onthe width of the prediction intervals, and it poses some new questionson proper model selection.

  • 9.
    Ekheden, Erland
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Multivariate time series modeling, estimation and prediction of mortalities2015In: Insurance, Mathematics & Economics, ISSN 0167-6687, E-ISSN 1873-5959, Vol. 65, p. 156-171Article in journal (Refereed)
    Abstract [en]

    We introduce a mixed regression model for mortality data which can be decomposed into a deterministic trend component explained by the covariates age and calendar year, a multivariate Gaussian time series part not explained by the covariates, and binomial risk. Data can be analyzed by means of a simple logistic regression model when the multivariate Gaussian time series component is absent and there is no overdispersion. In this paper we rather allow for overdispersion and the mixed regression model is fitted to mortality data from the United States and Sweden, with the aim to provide prediction and intervals for future mortality and annuity premium, as well as smoothing historical data, using the best linear unbiased predictor. We find that the form of the Gaussian time series has a large impact on the width of the prediction intervals, and it poses some new questions on proper model selection.

    This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

  • 10.
    Ekheden, Erland
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Pricing catastrophe risk in life (re)insurance2014In: Scandinavian Actuarial Journal, ISSN 0346-1238, E-ISSN 1651-2030, Vol. 2014, no 4, p. 352-367Article in journal (Refereed)
    Abstract [en]

    What is the catastrophe risk a life insurance company faces? What is the correct price of a catastrophe cover? During a review of the current standard model, due to Strickler, we found that this model has some serious shortcomings. We therefore present a new model for the pricing of catastrophe excess of loss cover (Cat XL). The new model for annual claim cost C is based on a compound Poisson processof catastrophe costs. To evaluate the distribution of the cost of each catastrophe, we use the Peaks Over Threshold model for the total number of lost lives in each catastrophe and the beta binomial model for the proportion of these corresponding to customers of the insurance company. To be able to estimate the parameters of the model, international and Swedish data were collected and compiled,listing accidents claiming at least twenty and four lives, respectively. Fitting the new model to data, we find the fit to be good. Finally we give the price of a Cat XL contract and perform a sensitivity analysis of how some of the parameters affect the expected value and standard deviation of the cost and thus the price.

  • 11.
    Grunewald, Maria
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    A General Statistical Framework for Multistage Designs2012In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 39, no 1, p. 131-152Article in journal (Refereed)
    Abstract [en]

    The efficiency of observational studies may be increased by applying multistage sampling designs. It is, however, not always transparent how to construct such a design to obtain increased efficiency. We here present a general statistical framework for describing and constructing multistage designs. We also provide tools for efficiency and cost-efficiency comparisons, to facilitate the choice of sampling scheme. The comparisons are based on Fisher information matrices and the results are presented in graphs, where either efficiency or cost-adjusted efficiency is plotted against a normalized measure of cost. The former curve resides in the unit square and is analogous to the receiver operating characteristic curve used for testing.

  • 12.
    Grünewald, Maria
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Humphreys, Keith
    Karolinska institutet.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    A Stochastic EM Type Algorithm for Parameter Estimation in Models with Continuous Outcomes, under Complex Ascertainment2010In: The International Journal of Biostatistics, ISSN 1557-4679, E-ISSN 1557-4679, Vol. 6, no 1, p. Article 23-Article in journal (Refereed)
    Abstract [en]

    Outcome-dependent sampling probabilities can be used to increase efficiency in observational studies. For continuous outcomes, appropriate consideration of sampling design in estimating parameters of interest is often computationally cumbersome. In this article, we suggest a Stochastic EM type algorithm for estimation when ascertainment probabilities are known or estimable. The computational complexity of the likelihood is avoided by filling in missing data so that an approximation of the full data likelihood can be used. The method is not restricted to any specific distribution of the data and can be used for a broad range of statistical models. 

  • 13.
    Grünewald, Maria
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    A General Statistical Framework for Multistage Designs2010Manuscript (preprint) (Other academic)
    Abstract [en]

    The efficiency of observational studies may be increased by applying multistage sampling designs. It is however not always transparent how to construct such a design in order to obtain increased efficiency. We here present a general statistical framework for describing and con- structing multistage designs. We also provide tools for efficiency and cost-efficiency comparisons, to facilitate the choice of sampling scheme. The comparisons are based on Fisher information matrices and the results are suggested being presented in graphs, where either efficiency or cost adjusted efficiency is plotted against a normalized measure of cost. The former curve resides in the unit square and is analogous to the receiver operating characteristic curve used for testing.

     

  • 14.
    Grünewald, Maria
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Efficient ascertainment schemes for maximum likelihood estimation2010In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 140, no 7, p. 2078-2088Article in journal (Refereed)
    Abstract [en]

    A well chosen sampling scheme can substantially increase the efficiency of a study. However, it is not always obvious how to sample well. Neyman (1938) presents the possibility of two-stage sampling to increase efficiency in field sampling, and concludes that two-stage sampling sometimes, but not always, reduces the variance of estimates of means. Since then various authors have investigated the effects of two-stage and multistage sampling in different settings, most of which focus on binary outcome variables. In some special cases, such as case-control studies, there are rules of thumb to follow with regards to efficiency, see for example Maydrech and Kupper (1978), but in most other settings more elaborate calculations are necessary to discriminate between different options. Multistage sampling is described in the context of genetic epidemiology by, among others, Whittemore and Halpern (1997): Case-control status of prostate cancer is first ascertained and then more expensive measures such as family history of disease and DNA samples are collected. Asymptotic variances of Horvitz–Thompson estimates are derived. Reilly (1996) investigates optimal allocation of available resources for two-stage data with binary outcomes. Complete information is there available from variables sampled in Stage 1, while Stage 2 variables are sampled more sparsely with probabilities determined by Stage 1 data. Cost is allowed to differ between sampling in Stage 1 and sampling in Stage 2. The author emphasizes the usefulness of pilot studies to obtain information needed to find the optimal allocation. Zhou et al. (2007) investigate outcome dependent sampling where the outcome variable is continuous. Power of tests based on a semi-parametric estimator are compared with the power of an inverse probability weighted estimator and the power of a maximum likelihood estimator based on a simple random sample. 

  • 15. Hartman, Linda
    et al.
    Humphreys, Keith
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Utilizing identity-by-descent probabilities for genetic fine-mapping in population based samples, via spatial smoothing of haplotype effects.2009In: Computational Statistics & Data Analysis, ISSN 0167-9473, E-ISSN 1872-7352, Vol. 53, no 5, p. 1802-1817Article in journal (Refereed)
    Abstract [en]

    Genetic fine mapping can be performed by exploiting the notion that haplotypes that are structurally similar in the neighbourhood of a disease predisposing locus are more likely to harbour the same susceptibility allele. Within the framework of Generalized Linear Mixed Models this can be formalized using spatial smoothing models, i.e. inducing a covariance structure for the haplotype risk parameters, such that risks associated with structurally similar haplotypes are dependent. In a Bayesian procedure a local similarity measure is calculated for each update of the presumed disease locus. Thus, the disease locus is searched as the place where the similarity structure produces risk parameters that can best discriminate between cases and controls. From a population genetic perspective the use of an identity-by-descent based similarity metric is theoretically motivated. This approach is then compared to other more intuitively motivated models and other similarity measures based on identity-by-state, suggested in the literature.

  • 16. Hartman, Linda
    et al.
    Humphreys, Keith
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Utilizing identity-by-descent probabilities for genetic fine-mapping in population based samples, via spatial smoothing of haplotype effects2007Report (Other (popular science, discussion, etc.))
  • 17. Hartman, Linda
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Fast kriging of large data sets with Gaussian Markov random fields models2008In: Computational Statistics and Data Analysis, Vol. 52, no 5, p. 2331-2349Article in journal (Refereed)
  • 18. Hedström, Anna Karin
    et al.
    Bellocco, Rino
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ye, Weimin
    Trolle Lagerros, Ylva
    Åkerstedt, Torbjörn
    Stockholm University, Faculty of Social Sciences, Department of Psychology, Stress Research Institute. Karolinska Institutet, Sweden.
    The relationship between nightmares, depression and suicide2021In: Sleep Medicine, ISSN 1389-9457, E-ISSN 1878-5506, Vol. 77, p. 1-6Article in journal (Refereed)
    Abstract [en]

    Objective: Previous studies investigating the association between nightmares and suicide have yielded different results. We aimed to investigate whether nightmares, directly or indirectly, influence the incidence of suicide.

    Methods: We used a prospective cohort study, based on 40,902 participants with a mean follow-up duration of 19.0 years. Cox proportional hazards models with attained age as time-scale were fitted to estimate hazard ratios (HR) of suicide with 95% confidence intervals (CI) as a function of the presence or absence of depression and nightmares. Mediation analysis was used to asses to what extent the relationship between nightmares and the incidence rate of suicide could be mediated by depression.

    Results: No association was observed between nightmares and the incidence of suicide among participants without depression. Compared with non-depressed participants without nightmares, the incidence of suicide among participants with a diagnosis of depression was similar among those with and without nightmares (HR 12.3, 95% CI 5.55-27.2 versus HR 13.2, 95% CI 7.25-24.1). The mediation analysis revealed no significant effects of nightmares on suicide incidence. However, the incidence of depression during follow-up was higher among those who suffered from nightmares than among those who did not (p < 0.001).

    Conclusions: Our findings indicate that nightmares have no influence on the incidence rate of suicide, but may reflect pre-existing depression. This is supported by a recent discovery of a strong genetic correlation of nightmares with depressive disorders, with no evidence that nightmares would predispose to psychiatric illness or psychological problems. Interventions targeting both depression and nightmares, when these conditions co-occur, may provide additional therapeutic benefit.

  • 19. Hedström, Anna Karin
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Bellocco, Rino
    Ye, Weimin
    Trolle Lagerros, Ylva
    Åkerstedt, Torbjörn
    Stockholm University, Faculty of Social Sciences, Department of Psychology, Stress Research Institute. Karolinska Institutet, Sweden.
    Insomnia in the context of short sleep increases suicide risk2021In: Sleep, ISSN 0161-8105, E-ISSN 1550-9109, Vol. 44, no 4, article id zsaa245Article in journal (Refereed)
    Abstract [en]

    Study objectives: The relationship between insomnia and suicide risk is not completely understood. We aimed to investigate the influence of insomnia on suicide risk, taking both sleep duration and depression into consideration.

    Methods: The present study is based on a Swedish prospective cohort study of 38,786 participants with a mean follow-up time of 19.2 years. Cox proportional hazards models with attained age as time-scale were used to estimate hazard ratios (HRs) of death by suicide with 95% confidence intervals (CI) for participants categorized by frequency of insomnia symptoms. Causal mediation analysis was performed to assess to what extent the relationship between insomnia and suicide risk is mediated by depression.

    Results: Insomnia was only associated with suicide risk among short sleepers, whereas no significant association was observed among those who slept 7 h/night or more. The total effect of insomnia in the context of short sleep on suicide risk, expressed on the HR scale, was 2.85 (95% CI 1.42-5.74). The direct effect was 2.25 (95% CI 1.12-4.54) and the indirect effect, mediated by depression, was 1.27 (95% CI 1.05-1.53). Of the total effect, 32% was mediated by depression. The association between insomnia and suicide risk became more pronounced with decreasing depressive symptoms (p value for trend <0.05).

    Conclusions: Insomnia in the context of short sleep increases suicide risk, both directly and indirectly by affecting the risk of depression. Abnormalities of sleep duration and insomnia symptoms should be evaluated when assessing suicide risk.

  • 20. Hedström, Anna Karin
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hillert, Jan
    Stridh, Pernilla
    Kockum, Ingrid
    Olsson, Tomas
    Alfredsson, Lars
    The influence of human leukocyte antigen-DRB1*15:01 and its interaction with smoking in MS development is dependent on DQA1*01:01 status2020In: Multiple Sclerosis Journal, ISSN 1352-4585, E-ISSN 1477-0970, Vol. 26, no 13, p. 1638-1646Article in journal (Refereed)
    Abstract [en]

    Background: HLA-DRB1*15:01, absence of HLA-A*02:01, and smoking interact to increase multiple sclerosis (MS) risk.

    Objective: To analyze whether MS-associated human leukocyte antigen (HLA) alleles, apart from DRB1*15:01 and absence of A*02:01, interact with smoking in MS development, and to explore whether the established HLA-smoking interaction is affected by the DQA1*01:01 allele, which confers a protective effect only in the presence of DRB1*15:01.

    Methods: In two Swedish population-based case-control studies (5838 cases, 5412 controls), subjects with different genotypes and smoking habits were compared regarding MS risk, by calculating odds ratios with 95% confidence intervals employing logistic regression. Interaction on the additive scale between different genotypes and smoking was evaluated.

    Results: The DRB1*08:01 allele interacted with smoking to increase MS risk. The interaction between DRB1*15:01 and both the absence of A*02:01 and smoking was confined to DQA1*01:01 negative subjects, whereas no interactions occurred among DQA1*01:01 positive subjects.

    Conclusion: Multifaceted interactions take place between different class II alleles and smoking in MS development. The influence of DRB1*15:01 and its interaction with the absence of A*02:01 and smoking is dependent on DQA1*01:01 status which may be due to differences in the responding T-cell repertoires.

  • 21. Hedström, Anna Karin
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Katsoulis, Michail
    Kockum, Ingrid
    Olsson, Tomas
    Alfredsson, Lars
    Organic solvents and MS susceptibility Interaction with MS risk HLA genes2018In: Neurology, ISSN 0028-3878, E-ISSN 1526-632X, Vol. 91, no 5, p. E455-E462Article in journal (Refereed)
    Abstract [en]

    Objective We hypothesize that different sources of lung irritation may contribute to elicit an immune reaction in the lungs and subsequently lead to multiple sclerosis (MS) in people with a genetic susceptibility to the disease. We aimed to investigate the influence of exposure to organic solvents on MS risk, and a potential interaction between organic solvents and MS risk human leukocyte antigen (HLA) genes. Methods Using a Swedish population-based case-control study (2,042 incident cases of MS and 2,947 controls), participants with different genotypes, smoking habits, and exposures to organic solvents were compared regarding occurrence of MS, by calculating odds ratios with 95% confidence intervals using logistic regression. A potential interaction between exposure to organic solvents and MS risk HLA genes was evaluated by calculating the attributable proportion due to interaction. Results Overall, exposure to organic solvents increased the risk of MS (odds ratio 1.5, 95% confidence interval 1.2-1.8, p = 0.0004). Among both ever and never smokers, an interaction between organic solvents, carriage of HLA-DRB1*15, and absence of HLA-A*02 was observed with regard to MS risk, similar to the previously reported gene-environment interaction involving the same MS risk HLA genes and smoke exposure. Conclusion The mechanism linking both smoking and exposure to organic solvents to MS risk may involve lung inflammation with a proinflammatory profile. Their interaction with MS risk HLA genes argues for an action of these environmental factors on adaptive immunity, perhaps through activation of autoaggressive cells resident in the lungs subsequently attacking the CNS.

  • 22. Hedström, Anna Karin
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Klareskog, Lars
    Alfredsson, Lars
    Interplay between alcohol, smoking and HLA genes in RA aetiology2019In: RMD Open, E-ISSN 2056-5933, Vol. 5, no 1, article id e000893Article in journal (Refereed)
    Abstract [en]

    Objectives The relationship between alcohol consumption and risk for rheumatoid arthritis (RA) is incompletely understood. We aimed to determine the influence of alcohol on anticitrullinated protein antibody (ACPA) positive and ACPA-negative RA and investigate potential interactions between alcohol consumption, smoking and the presence of human leucocyte antigen (HLA)-DRB1-shared epitope (SE).

    Methods A Swedish population-based case-control study with incident cases of RA was used (3353 cases, 2836 matched controls). Subjects with different HLA-DRB1-SE status, smoking and alcohol consumption were compared regarding risk of ACPA-positive and ACPA-negative RA, by calculating OR with 95% CI employing logistic regression. Interaction on the additive scale between alcohol, HLA-DRB1-SE and smoking was estimated by calculating the attributable proportion (AP) due to interaction.

    Results Compared with non-drinking, low and moderate alcohol consumption was dose dependently associated with a reduced risk of ACPA-positive and ACPA-negative RA. Independent of smoking habits, non-drinking and the presence of HLA-DRB1-SE interacted to increase the risk of ACPA-positive RA. Among HLA-DRB1-SE positive subjects, there was also a significant interaction between non-drinking and smoking with regard to risk for ACPA-positive RA. A three-way interaction was observed between alcohol, smoking and HLA-DRB1-SE with regard to risk for ACPA-positive RA (AP 0.7, 95% CI 0.6 to 0.8) that remained significant when the influence from the two-way interactions was removed (AP 0.4, 95% CI 0.2 to 0.6).

    Conclusions Our findings emphasize the need to investigate complex interactions between several environmental and genetic factors in order to better understand the etiology of RA. Whereas of great interest in an aetiological perspective, the finding of a protective role of alcohol on risk for RA must, however, be interpreted with caution in a clinical and public health perspective.

  • 23. Hedström, Anna Karin
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trolle Lagerros, Ylva
    Åkerstedt, Torbjorn
    Stockholm University, Faculty of Social Sciences, Stress Research Institute. Karolinska Institutet, Sweden.
    Short- and long-term mortality following hypnotic use2020In: Journal of Sleep Research, ISSN 0962-1105, E-ISSN 1365-2869, Vol. 29, no 4, article id e13061Article in journal (Refereed)
    Abstract [en]

    Potential long-term consequences of hypnotics remain controversial. We used the prospective Swedish National March Cohort, a study based on 41,695 participants with a mean follow-up duration of 18.9 years. Logistic regression models and Cox proportional hazards models with attained age as timescale were used to assess associations of hypnotic use with short- and long-term mortality. The proportion of subjects who initiated or discontinued hypnotic use during follow-up was substantial. All groups of hypnotics were associated with increased mortality within 2 years after a first prescription, with an overall OR of 2.38 (95% CI, 2.13-2.66). The association was more pronounced among subjects younger than 60 years (OR, 6.16; 95% CI, 3.98-9.52). There was no association between hypnotic use and long-term mortality. The association between hypnotic use and increased mortality was thus restricted to a relatively short period after treatment initiation, and may be explained in terms of confounding by indication.

  • 24. Hedström, Anna Karin
    et al.
    Katsoulis, Michail
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Bomfim, Izaura L.
    Oturai, Annette
    Bach Sondergaard, Helle
    Sellebjerg, Finn
    Ullum, Henrik
    Wegner Thørner, Lise
    Wendel Gustavsen, Marte
    Harbo, Hanne F.
    Obradovic, Dragana
    Gianfrancesco, Milena A.
    Barcellos, Lisa F.
    Schaefer, Catherine A.
    Hillert, Jan
    Kockum, Ingrid
    Olsson, Tomas
    Alfredsson, Lars
    The interaction between smoking and HLA genes in multiple sclerosis: replication and refinement2017In: European Journal of Epidemiology, ISSN 0393-2990, E-ISSN 1573-7284, Vol. 32, no 10, p. 909-919Article in journal (Refereed)
    Abstract [en]

    Interactions between environment and genetics may contribute to multiple sclerosis (MS) development. We investigated whether the previously observed interaction between smoking and HLA genotype in the Swedish population could be replicated, refined and extended to include other populations. We used six independent case-control studies from five different countries (Sweden, Denmark, Norway, Serbia, United States). A pooled analysis was performed for replication of previous observations (7190 cases, 8876 controls). Refined detailed analyses were carried out by combining the genetically similar populations from the Nordic studies (6265 cases, 8401 controls). In both the pooled analyses and in the combined Nordic material, interactions were observed between HLA-DRB*15 and absence of HLA-A*02 and between smoking and each of the genetic risk factors. Two way interactions were observed between each combination of the three variables, invariant over categories of the third. Further, there was also a three way interaction between the risk factors. The difference in MS risk between the extremes was considerable; smokers carrying HLA-DRB1*15 and lacking HLA-A*02 had a 13-fold increased risk compared with never smokers without these genetic risk factors (OR 12.7, 95% CI 10.8-14.9). The risk of MS associated with HLA genotypes is strongly influenced by smoking status and vice versa. Since the function of HLA molecules is to present peptide antigens to T cells, the demonstrated interactions strongly suggest that smoking alters MS risk through actions on adaptive immunity.

  • 25.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    A SPATIO-TEMPORAL POINT PROCESS MODEL FOR PARTICLE GROWTH2019In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 56, no 1, p. 23-38Article in journal (Refereed)
    Abstract [en]

    A spatio-temporal model of particle or star growth is defined, whereby new unit masses arrive sequentially in discrete time. These unit masses are referred to as candidate stars, which tend to arrive in mass-dense regions and then either form a new star or are absorbed by some neighbouring star of high mass. We analyse the system as time increases, and derive the asymptotic growth rate of the number of stars as well as the size of a randomly chosen star. We also prove that the size-biased mass distribution converges to a Poisson-Dirichlet distribution. This is achieved by embedding our model into a continuous-time Markov process, so that new stars arrive according to a marked Poisson process, with locations as marks, whereas existing stars grow as independent Yule processes. Our approach can be interpreted as a Hoppe-type urn scheme with a spatial structure. We discuss its relevance for and connection to models of population genetics, particle aggregation, image segmentation, epidemic spread, and random graphs with preferential attachment.

  • 26.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Coalescence theory for a general class of structured populations with fast migration2011In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 43, no 4, p. 1027-1047Article in journal (Refereed)
    Abstract [en]

    In this paper we study a general class of population genetic models where the total population is divided into a number of subpopulations or types. Migration between subpopulations is fast. Extending the results of Nordborg and Krone (2002) and Sagitov and Jagers (2005), we prove, as the total population size N tends to infinity, weak convergence of the joint ancestry of a given sample of haploid individuals in the Skorokhod topology towards Kingman's coalescent with a constant change of time scale c. Our framework includes age-structured models, geographically structured models, and combinations thereof. We also allow each individual to have offspring in several subpopulations, with general dependency structures between the number of offspring of various types. As a byproduct, explicit expressions for the coalescent effective population size N/c are obtained.

  • 27.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the eigenvalue effective size of structured populations2015In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 71, no 3, p. 595-646Article in journal (Refereed)
    Abstract [en]

    A general theory is developed for the eigenvalue effective size () of structured populations in which a gene with two alleles segregates in discrete time. Generalizing results of Ewens (Theor Popul Biol 21:373-378, 1982), we characterize in terms of the largest non-unit eigenvalue of the transition matrix of a Markov chain of allele frequencies. We use Perron-Frobenius Theorem to prove that the same eigenvalue appears in a linear recursion of predicted gene diversities between all pairs of subpopulations. Coalescence theory is employed in order to characterize this recursion, so that explicit novel expressions for can be derived. We then study asymptotically, when either the inverse size and/or the overall migration rate between subpopulations tend to zero. It is demonstrated that several previously known results can be deduced as special cases. In particular when the coalescence effective size exists, it is an asymptotic version of in the limit of large populations.

  • 28.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Spatial Autocorrelation for Subdivided Populations with Invariant Migration Schemes2014In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 16, no 4, p. 777-810Article in journal (Refereed)
    Abstract [en]

    For populations with geographic substructure and selectively neutral genetic data, the short term dynamics is a balance between migration and genetic drift. Before fixation of any allele, the system enters into a quasi equilibrium (QE) state. Hossjer and Ryman (2012) developed a general QE methodology for computing approximations of spatial autocorrelations of allele frequencies between subpopulations, subpopulation differentiation (fixation indexes) and variance effective population sizes. In this paper we treat a class of models with translationally invariant migration and use Fourier transforms for computing these quantities. We show how the QE approach is related to other methods based on conditional kinship coefficients between subpopulations under mutation-migration-drift equilibrium. We also verify that QE autocorrelations of allele frequencies are closely related to the expected value of Moran's autocorrelation function and treat limits of continuous spatial location (isolation by distance) and an infinite lattice of subpopulations. The theory is illustrated with several examples including island models, circular and torus stepping stone models, von Mises models, hierarchical island models and Gaussian models. It is well known that the fixation index contains information about the effective number of migrants. The spatial autocorrelations are complementary and typically reveal the type of migration (local or global).

  • 29.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Spatial autocorrelations for subdivided populations with invariant migration schemes2012Report (Other academic)
  • 30.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Alfredsson, Lars
    Hedström, Anna Karin
    Lekman, Magnus
    Kockum, Ingrid
    Olsson, Tomas
    Quantifying and estimating additive measures of interaction from case-control data2017In: Modern stochastics: theory and applications, ISSN 2351-6054, Vol. 4, no 2, p. 109-125Article in journal (Refereed)
    Abstract [en]

    In this paper we develop a general framework for quantifying how binary risk factors jointly influence a binary outcome. Our key result is an additive expansion of odds ratios as a sum of marginal effects and interaction terms of varying order. These odds ratio expansions are used for estimating the excess odds ratio, attributable proportion and synergy index for a case-control dataset by means of maximum likelihood from a logistic regression model. The confidence intervals associated with these estimates of joint effects and interaction of risk factors rely on the delta method. Our methodology is illustrated with a large Nordic meta dataset for multiple sclerosis. It combines four studies, with a total of 6265 cases and 8401 controls. It has three risk factors (smoking and two genetic factors) and a number of other confounding variables.

  • 31.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Bechly, Günter
    Gauger, Ann
    On the waiting time until coordinated mutations get fixed in regulatory sequences2021In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 524, article id 110657Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the time evolution of a population of size N with overlapping generations, in the vicinity of m genes. We assume that this population is subject to point mutations, genetic drift, and selection. More specifically, we analyze the statistical distribution of the waiting time T-m until the expression of these genes have changed for all individuals, when transcription factors recognize and attach to short DNA-sequences (binding sites) within regulatory sequences in the neighborhoods of the m genes. The evolutionary dynamics is described by a multitype Moran process, where each individual is assigned a m x L regulatory array that consists of regulatory sequences with L nucleotides for all m genes. We study how the waiting time distribution depends on the number of genes, the mutation rate, the length of the binding sites, the length of the regulatory sequences, and the way in which the targeted binding sites are coordinated for different genes in terms of selection coefficients. These selection coefficients depend on how many binding sites have appeared so far, and possibly on their order of appearance. We also allow for back mutations, whereby some acquired binding sites may be lost over time. It is further assumed that the mutation rate is small enough to warrant a fixed state population, so that all individuals have the same regulatory array, at any given time point, until the next successful mutation arrives in some individual and spreads to the rest of the population. We further incorporate stochastic tunneling, whereby successful mutations get mutated before their fixation. A crucial part of our approach is to divide the huge state space of regulatory arrays into a small number of components, assuming that the array component varies as a Markov process over time. This implies that Tm is the time until this Markov process hits an absorbing state, with a phase-type distribution. A number of interesting results can be derived from our general setup, for instance that the expected waiting time increases exponentially with m, for a selectively neutral model, when back-mutations are possible. (C) 2021 The Author(s). Published by Elsevier Ltd.

  • 32.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Díaz-Pachón, Daniel Andrés
    Rao, J. Sunil
    A Formal Framework for Knowledge Acquisition: Going beyond Machine Learning2022In: Entropy, E-ISSN 1099-4300, Vol. 24, no 10, article id 1469Article in journal (Refereed)
    Abstract [en]

    Philosophers frequently define knowledge as justified, true belief. We built a mathematical framework that makes it possible to define learning (increasing number of true beliefs) and knowledge of an agent in precise ways, by phrasing belief in terms of epistemic probabilities, defined from Bayes’ rule. The degree of true belief is quantified by means of active information I+: a comparison between the degree of belief of the agent and a completely ignorant person. Learning has occurred when either the agent’s strength of belief in a true proposition has increased in comparison with the ignorant person (I+>0), or the strength of belief in a false proposition has decreased (I+<0). Knowledge additionally requires that learning occurs for the right reason, and in this context we introduce a framework of parallel worlds that correspond to parameters of a statistical model. This makes it possible to interpret learning as a hypothesis test for such a model, whereas knowledge acquisition additionally requires estimation of a true world parameter. Our framework of learning and knowledge acquisition is a hybrid between frequentism and Bayesianism. It can be generalized to a sequential setting, where information and data are updated over time. The theory is illustrated using examples of coin tossing, historical and future events, replication of studies, and causal inference. It can also be used to pinpoint shortcomings of machine learning, where typically learning rather than knowledge acquisition is in focus.

  • 33.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Eriksson, Bengt
    Järnmalm, Kajsa
    Ohlsson, Esbjörn
    Stockholm University, Faculty of Science, Department of Mathematics.
    Assessing individual unexplained variation in non-life insurance.2009In: Astin Bulletin: Actuarial Studies in Non-Life Insurance, ISSN 0515-0361, E-ISSN 1783-1350, Vol. 39, no 1, p. 249-273Article in journal (Refereed)
    Abstract [en]

    We consider variation of observed claim frequencies in non-life insurance, modeled by Poisson regression with overdispersion. In order to quantify how much variation between insurance policies that is captured by the rating factors, one may use the coefficient of determination, R2, the estimated proportion of total variation explained by the model. We introduce a novel coefficient of individual determination (CID), which excludes noise variance and is defined as the estimated fraction of total individual variation explained by the model. We argue that CID is a more relevant measure of explained variation than R2 for data with Poisson variation. We also generalize previously used estimates and tests of overdispersion and introduce new coefficients of individual explained and unexplained variance.Application to a Swedish three year motor TPL data set reveals that only 0.5% of the total variation and 11% of the total individual variation is explained by a model with seven rating factors, including interaction between sex and age. Even though the amount of overdispersion is small (4.4% of the noise variance) it is still highly significant. The coefficient of variation of explained and unexplained individual variation is 29% and 81% respectively.

  • 34.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hartman, Linda
    Humphreys, Keith
    Ancestral recombination graphs under nonrandom ascertainment, with applications to gene mapping.2009In: Statistical Applications in Genetics and Molecular Biology, ISSN 1544-6115, E-ISSN 1544-6115, Vol. 8, no 1Article in journal (Refereed)
  • 35.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Jorde, Per Erik
    Ryman, Nils
    Stockholm University, Faculty of Science, Department of Zoology.
    Quasi equilibrium approximations of the fixation index under neutrality: The finite and infinite island models2013In: Theoretical Population Biology, ISSN 0040-5809, E-ISSN 1096-0325, Vol. 84, p. 9-24Article in journal (Refereed)
    Abstract [en]

    The fixation index FST and the coefficient of gene differentiation GST are analyzed for the finite island model under short time spans, ignoring mutations. Dividing the reproduction cycle into the three steps–gamete formation, fertilization, and migration–we develop a new approach for computing quasi equilibrium formulas for FST (and GST). Our formulas generalize earlier ones and reveal that the equilibrium value of FST is influenced not only by the migration rate and local effective population size, Ne, but also by the local census size N, particularly so when the migration rate is high. The order of migration and fertilization is found to have a smaller effect on FST. A major advantage compared to previous approaches is that stochastic allele frequency of migrants is easily accommodated, thereby avoiding underestimation of FST for large migration rates.

  • 36.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Karlsson, Måns
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the use of L-functionals in regression models2023In: Open Mathematics, ISSN 2391-5455, Vol. 21, no 1, article id 20220597Article, review/survey (Refereed)
    Abstract [en]

    In this article, we survey and unify a large class or L -functionals of the conditional distribution of the response variable in regression models. This includes robust measures of location, scale, skewness, and heavytailedness of the response, conditionally on covariates. We generalize the concepts of L -moments (G. Sillito, Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample, Biometrika 56 (1969), no. 3, 641–650.), L -skewness, and L -kurtosis (J. R. M. Hosking, L-moments: analysis and estimation of distributions using linear combinations or order statistics, J. R. Stat. Soc. Ser. B Stat. Methodol. 52 (1990), no. 1, 105–124.) and introduce order numbers for a large class of L -functionals through orthogonal series expansions of quantile functions. In particular, we motivate why location, scale, skewness, and heavytailedness have order numbers 1, 2, (3,2), and (4,2), respectively, and describe how a family of L -functionals, with different order numbers, is constructed from Legendre, Hermite, Laguerre, or other types of polynomials. Our framework is applied to models where the relationship between quantiles of the response and the covariates follows a transformed linear model, with a link function that determines the appropriate class of L -functionals. In this setting, the distribution of the response is treated parametrically or nonparametrically, and the response variable is either censored/truncated or not. We also provide a framework for asymptotic theory of estimates of L -functionals and illustrate our approach by analyzing the arrival time distribution of migrating birds. In this context, a novel version of the coefficient of determination is introduced, which makes use of the abovementioned orthogonal series expansion.

  • 37.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Laikre, Linda
    Stockholm University, Faculty of Science, Department of Zoology, Population Genetics.
    Ryman, Nils
    Stockholm University, Faculty of Science, Department of Zoology, Population Genetics.
    Assessment of the Global Variance Effective Size of Subdivided Populations, and Its Relation to Other Effective Sizes2023In: Acta Biotheoretica, ISSN 0001-5342, E-ISSN 1572-8358, Vol. 71, no 3, article id 19Article in journal (Refereed)
    Abstract [en]

    The variance effective population size (N-eV) is frequently used to quantify the expected rate at which a population's allele frequencies change over time. The purpose of this paper is to find expressions for the global N-eV of a spatially structured population that are of interest for conservation of species. Since N-eV depends on allele frequency change, we start by dividing the cause of allele frequency change into genetic drift within subpopulations (I) and a second component mainly due to migration between subpopulations (II). We investigate in detail how these two components depend on the way in which subpopulations are weighted as well as their dependence on parameters of the model such a migration rates, and local effective and census sizes. It is shown that under certain conditions the impact of II is eliminated, and N-eV of the metapopulation is maximized, when subpopulations are weighted proportionally to their long term reproductive contributions. This maximal N-eV is the sought for global effective size, since it approximates the gene diversity effective size N-eGD, a quantifier of the rate of loss of genetic diversity that is relevant for conservation of species and populations. We also propose two novel versions of N-eV, one of which (the backward version of N-eV) is most stable, exists for most populations, and is closer to N-eGD than the classical notion of N-eV. Expressions for the optimal length of the time interval for measuring genetic change are developed, that make it possible to estimate any version of N-eV with maximal accuracy.

  • 38.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Laikre, Linda
    Stockholm University, Faculty of Science, Department of Zoology.
    Ryman, Nils
    Stockholm University, Faculty of Science, Department of Zoology.
    Effective sizes and time to migration-drift equilibrium in geographically subdivided populations2016In: Theoretical Population Biology, ISSN 0040-5809, E-ISSN 1096-0325, Vol. 112, p. 139-156Article in journal (Refereed)
    Abstract [en]

    Many versions of the effective population size (N-e) exist, and they are important in population genetics in order to quantify rates of change of various characteristics, such as inbreeding, heterozygosity, or allele frequencies. Traditionally, N-e was defined for single, isolated populations, but we have recently presented a mathematical framework for subdivided populations. In this paper we focus on diploid populations with geographic subdivision, and present new theoretical results. We compare the haploid and diploid versions of the inbreeding effective size (N-ei) with novel expression for the variance effective size (N-ev), and conclude that for local populations N-ev is often much smaller than both versions of Nei, whenever they exist. Global N(ev)of the metapopulation, on the other hand, is close to the haploid Net and much larger than the diploid Nei. We introduce a new effective size, the additive genetic variance effective size Neill', which is of particular interest for long term protection of species. It quantifies the rate at which additive genetic variance is lost and we show that this effective size is closely related to the haploid version of Nei. Finally, we introduce a new measure of a population's deviation from migration-drift equilibrium, and apply it to quantify the time it takes to reach this equilibrium. Our findings are of importance for understanding the concept of effective population size in substructured populations and many of the results have applications in conservation biology.

  • 39.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nyberg, Tommy
    Petrovic, Stefan
    Sjöberg, Frank
    Öberg, Tommy
    Bayesian Distributed Detection by Sensor Networks with Missing Data2012Report (Other academic)
    Abstract [en]

    We consider detection and identication of a moving target, using a network of sensors. The target emits a signal; a stationary stochastic process corrupted by additive noise, independently across sensors. Before intersensor communication, all sensors reduce external data as energy over disjoint frequency bands and time blocks. One sensor, the internal fusion center (IFC), gathers feature vectors from the other sensors, possibly after message passing. Using Bayesian decision theory, it decides for presence or absence of the target and computes a maximum posterior estimate of target (trajectory and spectral) parameters. The main novelties of the paper are: 1) To apply statistical theory of missing data to an inter-sensor communication protocol which censors weak signals before transmission and an imperfect channel in which some transmitted signals are lost. A Naive Bayes approximate detector is dened, which requires recursive computation of reception probabilities. 2) To derive asymptotic approximations of the distribution of the spectral feature vectors and a Laplace type approximation of the detector. The performance of the proposed Bayesian detector is shown in a simulation study to be only slightly inferior to that of an ideal Bayesian detector (with no missing data), as well as superior to a naive detector.

  • 40.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Olsson, Fredrik
    Stockholm University, Faculty of Science, Department of Mathematics.
    Laikre, Linda
    Stockholm University, Faculty of Science, Department of Zoology.
    Ryman, Nils
    Stockholm University, Faculty of Science, Department of Zoology.
    A new general analytical approach for modeling patterns of genetic differentiation and effective size of subdivided populations over time2014In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 258, p. 113-133Article in journal (Refereed)
    Abstract [en]

    The main purpose of this paper is to develop a theoretical framework for assessing effective population size and genetic divergence in situations with structured populations that consist of various numbers of more or less interconnected subpopulations. We introduce a general infinite allele model for a diploid, monoecious and subdivided population, with subpopulation sizes varying overtime, including local subpopulation extinction and recolonization, bottlenecks, cyclic census size changes or exponential growth. Exact matrix analytic formulas are derived for recursions of predicted (expected) gene identities and gene diversities, identity by descent and coalescence probabilities, and standardized variances of allele frequency change. This enables us to compute and put into a general framework a number of different types of genetically effective population sizes (N-e) including variance, inbreeding, nucleotide diversity, and eigenvalue effective size. General expressions for predictions (g(ST)) of the coefficient of gene differentiation G(ST) are also derived. We suggest that in order to adequately describe important properties of a subdivided population with respect to allele frequency change and maintenance of genetic variation over time, single values of g(ST) and N-e are not enough. Rather, the temporal dynamic patterns of these properties are important to consider. We introduce several schemes for weighting subpopulations that enable effective size and expected genetic divergence to be calculated and described as functions of time, globally for the whole population and locally for any group of subpopulations. The traditional concept of effective size is generalized to situations where genetic drift is confounded by external sources, such as immigration and mutation. Finally, we introduce a general methodology for state space reduction, which greatly decreases the computational complexity of the matrix analytic formulas.

  • 41.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Olsson, Fredrik
    Stockholm University, Faculty of Science, Department of Mathematics.
    Laikre, Linda
    Stockholm University, Faculty of Science, Department of Zoology.
    Ryman, Nils
    Stockholm University, Faculty of Science, Department of Zoology.
    Metapopulation inbreeding dynamics, effective size and subpopulation differentiation-A general analytical approach for diploid organisms2015In: Theoretical Population Biology, ISSN 0040-5809, E-ISSN 1096-0325, Vol. 102, p. 40-59Article in journal (Refereed)
    Abstract [en]

    Motivated by problems in conservation biology we study genetic dynamics in structured populations of diploid organisms (monoecious or dioecious). Our analysis provides an analytical framework that unifies substantial parts of previous work in terms of exact identity by descent (IBD) and identity by state (IBS) recursions. We provide exact conditions under which two structured haploid and diploid populations are equivalent, and some sufficient conditions under which a dioecious diploid population can be treated as a monoecious diploid one. The IBD recursions are used for computing local and metapopulation inbreeding and coancestry effective population sizes and for predictions of several types of fixation indices over different time horizons.

  • 42.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ryman, Nils
    Stockholm University, Faculty of Science, Department of Genetics, Microbiology and Toxicology.
    Quasi equilibrium, variance effective population size and fixation index for models with spatial structure2012Report (Other academic)
  • 43.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ryman, Nils
    Stockholm University, Faculty of Science, Department of Zoology.
    Quasi equilibrium, variance effective size and fixation index for populations with substructure2014In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 69, no 5, p. 1057-1128Article in journal (Refereed)
    Abstract [en]

    In this paper, we develop a method for computing the variance effective size , the fixation index and the coefficient of gene differentiation of a structured population under equilibrium conditions. The subpopulation sizes are constant in time, with migration and reproduction schemes that can be chosen with great flexibility. Our quasi equilibrium approach is conditional on non-fixation of alleles. This is of relevance when migration rates are of a larger order of magnitude than the mutation rates, so that new mutations can be ignored before equilibrium balance between genetic drift and migration is obtained. The vector valued time series of subpopulation allele frequencies is divided into two parts; one corresponding to genetic drift of the whole population and one corresponding to differences in allele frequencies among subpopulations. We give conditions under which the first two moments of the latter, after a simple standardization, are well approximated by quantities that can be explicitly calculated. This enables us to compute approximations of the quasi equilibrium values of , and . Our findings are illustrated for several reproduction and migration scenarios, including the island model, stepping stone models and a model where one subpopulation acts as a demographic reservoir. We also make detailed comparisons with a backward approach based on coalescence probabilities.

  • 44.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Sjölander, Arvid
    Sharp lower and upper bounds for the covariance of bounded random variables2022In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 182, article id 109323Article in journal (Refereed)
    Abstract [en]

    In this paper we derive sharp lower and upper bounds for the covariance of two bounded random variables which are applicable when knowledge about their expected values, variances or both is available. When only the expected values are known, our result can be viewed as an extension of the Bhatia–Davis inequality for variances. We also provide a number of different ways to standardize covariance. For a binary pair of random variables, one of these standardized measures of covariation agrees with a frequently used measure of dependence between genetic variants.

  • 45.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Tyvand, Peder A.
    A monoecious and diploid Moran model of random mating2016In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 394, p. 182-196Article in journal (Refereed)
    Abstract [en]

    An exact Markov chain is developed for a Moran model of random mating for monoecious diploid individuals with a given probability of self-fertilization. The model captures the dynamics of genetic variation at a biallelic locus. We compare the model with the corresponding diploid Wright-Fisher (WF) model. We also develop a novel diffusion approximation of both models, where the genotype frequency distribution dynamics is described by two partial differential equations, on different time scales. The first equation captures the more slowly varying allele frequencies, and it is the same for the Moran and WF models. The other equation captures departures of the fraction of heterozygous genotypes from a large population equilibrium curve that equals Hardy-Weinberg proportions in the absence of selfing. It is the distribution of a continuous time Ornstein-Uhlenbeck process for the Moran model and a discrete time autoregressive process for the WF model. One application of our results is to capture dynamics of the degree of non-random mating of both models, in terms of the fixation index f(IS). Although f(IS) has a stable fixed point that only depends on the degree of selfing, the normally distributed oscillations around this fixed point are stochastically larger for the Moran than for the WF model.

  • 46.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Tyvand, Peder A.
    Local fluctuations of genetic processes defined on two time scales, with applications to effective size estimation2020In: Theoretical Population Biology, ISSN 0040-5809, E-ISSN 1096-0325, Vol. 131, p. 79-99Article in journal (Refereed)
    Abstract [en]

    In this paper we develop a general framework for how the genetic composition of a structured population with strong migration between its subunits, evolves over time. The dynamics is described in terms of a vector-valued Markov process of allele, genotype or haplotype frequencies that varies on two time scales. The more rapid changes are random fluctuations in terms of a multivariate autoregressive process, around a quasi equilibrium fix point, whereas the fix point itself varies more slowly according to a diffusion process, along a lower-dimensional subspace. Under mild regularity conditions, the fluctuations have a magnitude inversely proportional to the square root of the population size N, and hence can be used to estimate a broad class of genetically effective population sizes N-e, with genetic data from one time point only. In this way we are able to unify a number of existing notions of effective size, as well as proposing new ones, for instance one that quantifies the extent to which genotype frequencies fluctuate around Hardy-Weinberg equilibrium. (C) 2019 Elsevier Inc. All rights reserved.

  • 47.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Tyvand, Peder A.
    Miloh, Touvia
    Exact Markov chain and approximate diffusion solution for haploid genetic drift with one-way mutation2016In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 272, p. 100-112Article in journal (Refereed)
    Abstract [en]

    The classical Kimura solution of the diffusion equation is investigated for a haploid random mating (Wright-Fisher) model, with one-way mutations and initial-value specified by the founder population. The validity of the transient diffusion solution is checked by exact Markov chain computations, using a. Jordan decomposition of the transition matrix. The conclusion is that the one-way diffusion model mostly works well, although the rate of convergence depends on the initial allele frequency and the mutation rate. The diffusion approximation is poor for mutation rates so low that the non-fixation boundary is regular. When this happens we perturb the diffusion solution around the non-fixation boundary and obtain a more accurate approximation that takes quasi-fixation of the mutant allele into account. The main application is to quantify how fast a specific genetic variant of the infinite alleles model is lost. We also discuss extensions of the quasi-fixation approach to other models with small mutation rates.

  • 48.
    Karlsson, Måns
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    A comparison between quantile regression and linear regression on empirical quantiles for phenological analysis in migratory response to climate changeManuscript (preprint) (Other (popular science, discussion, etc.))
    Abstract [en]

    It is well established that migratory birds in general have advanced their arrival times in spring, and in this paper we investigate potential ways of enhancing the level of detail in future phenological analyses. We perform single as well as multiple species analyses, using linear models on empirical quantiles, non-parametric quantile regression and likelihood-based parametric quantile regression with asymmetric Laplace distributed error terms. We conclude that non-parametric quantile regression appears most suited for single as well as multiple species analyses.

  • 49.
    Karlsson, Måns
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Classification under partial reject optionsManuscript (preprint) (Other academic)
    Abstract [en]

    We study set-valued classification for a Bayesian model where data originates from one of a finite number N of possible hypotheses. Thus we consider the scenario where the size of the classified set of categories ranges from 0 to N. Empty sets corresponds to an outlier, size 1 represents a firm decision that singles out one hypotheses, size N corresponds to a rejection to classify, whereas sizes 2…,N−1 represent a partial rejection, where some hypotheses are excluded from further analysis. We introduce a general framework of reward functions with a set-valued argument and derive the corresponding optimal Bayes classifiers, for a homogeneous block of hypotheses and for when hypotheses are partitioned into blocks, where ambiguity within and between blocks are of different severity. We illustrate classification using an ornithological dataset, with taxa partitioned into blocks and parameters estimated using MCMC. The associated reward function's tuning parameters are chosen through cross-validation.

  • 50.
    Karlsson, Måns
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Classification Under Partial Reject Options2023In: Journal of Classification, ISSN 0176-4268, E-ISSN 1432-1343, article id s00357-023-09455-xArticle in journal (Refereed)
    Abstract [en]

    In many applications there is ambiguity about which (if any) of a finite number N of hypotheses that best fits an observation. It is of interest then to possibly output awhole set of categories, that is, a scenario where the size of the classified set of categories ranges from 0 to N. Empty sets correspond to an outlier, sets of size 1 represent a firm decision that singles out one hypothesis, sets of size N correspond to a rejection to classify, whereas sets of sizes 2,..., N - 1 represent a partial rejection to classify, where some hypotheses are excluded from further analysis. In this paper, we review and unify several proposed methods of Bayesian set-valued classification, where the objective is to find the optimal Bayesian classifier that maximizes the expected reward. We study a large class of reward functions with rewards for sets that include the true category, whereas additive or multiplicative penalties are incurred for sets depending on their size. For models with one homogeneous block of hypotheses, we provide general expressions for the accompanying Bayesian classifier, several of which extend previous results in the literature. Then, we derive novel results for the more general setting when hypotheses are partitioned into blocks, where ambiguity within and between blocks are of different severity. We also discuss how well-known methods of classification, such as conformal prediction, indifference zones, and hierarchical classification, fit into our framework. Finally, set-valued classification is illustrated using an ornithological data set, with taxa partitioned into blocks and parameters estimated using MCMC. The associated reward function's tuning parameters are chosen through cross-validation.

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