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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.
    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.

  • 5.
    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/).

  • 6.
    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.

  • 7.
    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.

  • 8.
    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.

  • 9.
    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. 

  • 10.
    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.

     

  • 11.
    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. 

  • 12. 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.))
  • 13. 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.

  • 14. 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)
  • 15. 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.

  • 16. 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.

  • 17.
    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.

  • 18.
    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.

  • 19.
    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).

  • 20.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Spatial autocorrelations for subdivided populations with invariant migration schemes2012Report (Other academic)
  • 21.
    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.

  • 22.
    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.

  • 23.
    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)
  • 24.
    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.

  • 25.
    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.

  • 26.
    Hössjer, Ola
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nyberg, Tommy
    Petrovic, Stefan
    Sjöberg, Frank
    Öberg, Tommy
    Online distributed detection by local sensors2012Report (Other academic)
  • 27.
    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.

  • 28.
    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.

  • 29.
    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)
  • 30.
    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.

  • 31.
    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.

  • 32.
    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.

  • 33. Kurbasic, Azra
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    A general method of linkage disequilibrium correction for multipoint linkage and association2008In: Genetic Epidemiology, Vol. 32, p. 647-657Article in journal (Refereed)
  • 34.
    Laikre, Linda
    et al.
    Stockholm University, Faculty of Science, Department of Zoology.
    Olsson, Fredrik
    Stockholm University, Faculty of Science, Department of Mathematics.
    Jansson, Eeva
    Stockholm University, Faculty of Science, Department of Zoology.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ryman, Nils
    Stockholm University, Faculty of Science, Department of Zoology.
    Metapopulation effective size and conservation genetic goals for the Fennoscandian wolf (Canis lupus) population2016In: Heredity, ISSN 0018-067X, E-ISSN 1365-2540, Vol. 117, no 4, p. 279-289Article in journal (Refereed)
    Abstract [en]

    The Scandinavian wolf population descends from only five individuals, is isolated, highly inbred and exhibits inbreeding depression. To meet international conservation goals, suggestions include managing subdivided wolf populations over Fennoscandia as a metapopulation; a genetically effective population size of N-e >= 500, in line with the widely accepted long-term genetic viability target, might be attainable with gene flow among subpopulations of Scandinavia, Finland and Russian parts of Fennoscandia. Analytical means for modeling N-e of subdivided populations under such non-idealized situations have been missing, but we recently developed new mathematical methods for exploring inbreeding dynamics and effective population size of complex metapopulations. We apply this theory to the Fennoscandian wolves using empirical estimates of demographic parameters. We suggest that the long-term conservation genetic target for metapopulations should imply that inbreeding rates in the total system and in the separate subpopulations should not exceed Delta f = 0.001. This implies a meta-Ne of N-eMeta >= 500 and a realized effective size of each subpopulation of N-eRx >= 500. With current local effective population sizes and one migrant per generation, as recommended by management guidelines, the meta-Ne that can be reached is similar to 250. Unidirectional gene flow from Finland to Scandinavia reduces meta-N-e to similar to 130. Our results indicate that both local subpopulation effective sizes and migration among subpopulations must increase substantially from current levels to meet the conservation target. Alternatively, immigration from a large (N-e >= 500) population in northwestern Russia could support the Fennoscandian metapopulation, but immigration must be substantial (5-10 effective immigrants per generation) and migration among Fennoscandian subpopulations must nevertheless increase.

  • 35. Lekman, Magnus
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Andrews, Peter
    Källberg, Henrik
    Uvehag, Daniel
    Charney, Dennis
    Manji, Husseini
    Rush, John A.
    McMahon, Francis J.
    Moore, Jason H.
    Kockum, Ingrid
    The genetic interacting landscape of 63 candidate genes in Major Depressive Disorder: an explorative study2014In: BioData Mining, ISSN 1756-0381, E-ISSN 1756-0381, Vol. 7, p. 19-Article in journal (Refereed)
    Abstract [en]

    Background: Genetic contributions to major depressive disorder (MDD) are thought to result from multiple genes interacting with each other. Different procedures have been proposed to detect such interactions. Which approach is best for explaining the risk of developing disease is unclear. This study sought to elucidate the genetic interaction landscape in candidate genes for MDD by conducting a SNP-SNP interaction analysis using an exhaustive search through 3,704 SNP-markers in 1,732 cases and 1,783 controls provided from the GAIN MDD study. We used three different methods to detect interactions, two logistic regressions models (multiplicative and additive) and one data mining and machine learning (MDR) approach. Results: Although none of the interaction survived correction for multiple comparisons, the results provide important information for future genetic interaction studies in complex disorders. Among the 0.5% most significant observations, none had been reported previously for risk to MDD. Within this group of interactions, less than 0.03% would have been detectable based on main effect approach or an a priori algorithm. We evaluated correlations among the three different models and conclude that all three algorithms detected the same interactions to a low degree. Although the top interactions had a surprisingly large effect size for MDD (e. g. additive dominant model P-uncorrected = 9.10E-9 with attributable proportion (AP) value = 0.58 and multiplicative recessive model with P-uncorrected = 6.95E-5 with odds ratio (OR estimated from beta 3) value = 4.99) the area under the curve (AUC) estimates were low (< 0.54). Moreover, the population attributable fraction (PAF) estimates were also low (< 0.15). Conclusions: We conclude that the top interactions on their own did not explain much of the genetic variance of MDD. The different statistical interaction methods we used in the present study did not identify the same pairs of interacting markers. Genetic interaction studies may uncover previously unsuspected effects that could provide novel insights into MDD risk, but much larger sample sizes are needed before this strategy can be powerfully applied.

  • 36. Lekman, Magnus
    et al.
    Karlsson, Robert
    Graae, Lisette
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kockum, Ingrid
    A significant risk locus on 19q13 for bipolar disorder identified using a combined genome-wide linkage and copy number variation analysis2015In: BioData Mining, ISSN 1756-0381, E-ISSN 1756-0381, Vol. 8, article id 42Article in journal (Refereed)
    Abstract [en]

    Background: The genetic background to bipolar disorder (BPD) has been attributed to different genetic and genomic risk factors. In the present study we hypothesized that inherited copy number variations (CNVs) contribute to susceptibility of BPD. We screened 637 BP-pedigrees from the NIMH Genetic Initiative and gave priority to 46 pedigrees. In this subsample we performed parametric and non-parametric genome-wide linkage analyses using similar to 21,000 SNP-markers. We developed an algorithm to test for linkage restricted to regions with CNVs that are shared within and across families. Results: For the combined CNV and linkage analysis, one region on 19q13 survived correction for multiple comparisons and replicates a previous BPD risk locus. The shared CNV map to the pregnancy-specific glycoprotein (PSG) gene, a gene-family not previously implicated in BPD etiology. Two SNPs in the shared CNV are likely transcription factor binding sites and are linked to expression of an F-box binding gene, a key regulator of neuronal pathways suggested to be involved in BPD etiology. Conclusions: Our CNV-weighted linkage approach identifies a risk locus for BPD on 19q13 and forms a useful tool to future studies to unravel part of the genetic vulnerability to BPD.

  • 37.
    Malmberg, Hannes
    et al.
    Stockholm University, Faculty of Social Sciences, Institute for International Economic Studies.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Argmax over continuous indeces of random variables - an approach using random fields2012Report (Other academic)
  • 38.
    Malmberg, Hannes
    et al.
    Stockholm University, Faculty of Social Sciences, Institute for International Economic Studies.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Probabilistic choice with an infinite set of options: An Approach Based on Random Sup Measures2014In: Modern Problems in Insurance Mathematics / [ed] Dmitrii Silvestrov, Anders Martin-Löf, London: Springer, 2014, p. 291-312Chapter in book (Refereed)
    Abstract [en]

    This chapter deals with probabilistic choice when the number of options is infinite. The choice space is a compact set S⊆R k   and we model choice over S  as a limit of choices over triangular sequences {x n1 ,…,x nn }⊆S  as n→∞  . We employ the theory of random sup measures and show that in the limit when n→∞  , people behave as though they are maximising over a random sup measure. Thus, our results complement Resnick and Roy’s [18] theory of probabilistic choice over infinite sets. They define choice as a maximisation over a stochastic process on S  with upper semi-continuous (usc) paths. This connects to our model as their random usc function can be defined as a sup-derivative of a random sup measure, and their maximisation problem can be transformed into a maximisation problem over this random sup measure. One difference remains though: with our model the limiting random sup measures are independently scattered, without usc paths. A benefit of our model is that we provide a way of connecting the stochastic process in their model with finite case distributional assumptions, which are easier to interpret. In particular, when choices are valued additively with one deterministic and one random part, we explore the importance of the tail behaviour of the random part, and show that the exponential distribution is an important boundary case between heavy-tailed and light-tailed distributions.

  • 39.
    Olsson, Fredrik
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Equilibrium distributions and simulation methods for age structured populations2015In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 268, p. 45-51Article in journal (Refereed)
    Abstract [en]

    A simulation method is presented for the demographic and genetic variation of age structured haploid populations. First, we use matrix analytic methods to derive an equilibrium distribution for the age class sizes conditioned on the total population size. Knowledge of this distribution eliminates the need of a burn-in time in simulations. Next, we derive the distribution of the alleles at a polymorphic locus in various age classes given the allele frequencies in the total population and the age size composition. For the time dynamics, we start by simulating the dynamics for the total population. In order to generate the inheritance of the alleles, we derive their distribution conditionally on the simulated population sizes. This method enables a fast simulation procedure of multiple loci in linkage equilibrium.

  • 40.
    Olsson, Fredrik
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Estimation of the variance effective population size in age structured populations2015In: Theoretical Population Biology, ISSN 0040-5809, E-ISSN 1096-0325, Vol. 101, p. 9-23Article in journal (Refereed)
    Abstract [en]

    The variance effective population size for age structured populations is generally hard to estimate and the temporal method often gives biased estimates. Here, we give an explicit expression for a correction factor which, combined with estimates from the temporal method, yield approximately unbiased estimates. The calculation of the correction factor requires knowledge of the age specific offspring distribution and survival probabilities as well as possible correlation between survival and reproductive success. In order to relax these requirements, we show that only first order moments of these distributions need to be known if the time between samples is large, or individuals from all age classes which reproduce are sampled. A very explicit approximate expression for the asymptotic coefficient of standard deviation of the estimator is derived, and it can be used to construct confidence intervals and optimal ways of weighting information from different markers. The asymptotic coefficient of standard deviation can also be used to design studies and we show that in order to maximize the precision for a given sample size, individuals from older age classes should be sampled since their expected variance of allele frequency change is higher and easier to estimate. However, for populations with fluctuating age class sizes, the accuracy of the method is reduced when samples are taken from older age classes with high demographic variation. We also present a method for simultaneous estimation of the variance effective and census population size.

  • 41.
    Olsson, Fredrik
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Simulation methods for age structured populationsIn: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134Article in journal (Refereed)
  • 42.
    Olsson, Fredrik
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    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.
    Characteristics of the variance effective population size over time using an age structured model with variable size2013In: Theoretical Population Biology, ISSN 0040-5809, E-ISSN 1096-0325, Vol. 90, p. 91-103Article in journal (Refereed)
    Abstract [en]

    The variance effective population size (N-ev) is a key concept in population biology, because it quantifies the microevolutionary process of random genetic drift, and understanding the characteristics of N-ev is thus of central importance. Current formulas for Nev for populations with overlapping generations weight age classes according to their reproductive values (i.e. reflecting the contribution of genes from separate age classes to the population growth) to obtain a correct measure of genetic drift when computing the variance of the allele frequency change over time. In this paper, we examine the effect of applying different weights to the age classes using a novel analytical approach for exploring N-ev. We consider a haploid organism with overlapping generations and populations of increasing, declining, or constant expected size and stochastic variation with respect to the number of individuals in the separate age classes. We define Nov, as a function of how the age classes are weighted, and of the span between the two points in time, when measuring allele frequency change. With this model, time profiles for N-ev can be calculated for populations with various life histories and with fluctuations in life history composition, using different weighting schemes. We examine analytically and by simulations when Nei, using a weighting scheme with respect to reproductive contribution of separate age classes, accurately reflect the variance of the allele frequency change due to genetic drift over time. We show that the discrepancy of N-ev, calculated with reproductive values as weights, compared to when individuals are weighted equally, tends to a constant when the time span between the two measurements increases. This constant is zero only for a population with a constant expected population size. Our results confirm that the effect of ignoring overlapping generations, when empirically assessing Nell from allele frequency shifts, gets smaller as the time interval between samples increases. Our model has empirical applications including assessment of (i) time intervals necessary to permit ignoring the effect of overlapping generations for N-ev estimation by means of the temporal method, and (ii) effects of life table manipulation on N-ev over varying time periods.

  • 43.
    Olsson, Fredrik
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Laikre, Linda
    Stockholm University, Faculty of Science, Department of Zoology.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ryman, Nils
    Stockholm University, Faculty of Science, Department of Zoology.
    GESP: A computer program for modelling genetic effective population size, inbreeding and divergence in substructured populations2017In: Molecular Ecology Resources, ISSN 1755-098X, E-ISSN 1755-0998, Vol. 17, no 6, p. 1378-1384Article in journal (Refereed)
    Abstract [en]

    The genetically effective population size (N-e) is of key importance for quantifying rates of inbreeding and genetic drift and is often used in conservation management to set targets for genetic viability. The concept was developed for single, isolated populations and the mathematical means for analysing the expected N-e in complex, subdivided populations have previously not been available. We recently developed such analytical theory and central parts of that work have now been incorporated into a freely available software tool presented here. gesp (Genetic Effective population size, inbreeding and divergence in Substructured Populations) is R-based and designed to model short- and long-term patterns of genetic differentiation and effective population size of subdivided populations. The algorithms performed by gesp allow exact computation of global and local inbreeding and eigenvalue effective population size, predictions of genetic divergence among populations (G(ST)) as well as departures from random mating (F-IS, F-IT) while varying (i) subpopulation census and effective size, separately or including trend of the global population size, (ii) rate and direction of migration between all pairs of subpopulations, (iii) degree of relatedness and divergence among subpopulations, (iv) ploidy (haploid or diploid) and (v) degree of selfing. Here, we describe gesp and exemplify its use in conservation genetics modelling.

  • 44.
    Ryman, Nils
    et al.
    Stockholm University, Faculty of Science, Department of Zoology.
    Allendorf, Fred W.
    Jorde, Per Erik
    Laikre, Linda
    Stockholm University, Faculty of Science, Department of Zoology.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Samples from subdivided populations yield biased estimates of effective size that overestimate the rate of loss of genetic variation2014In: Molecular Ecology Resources, ISSN 1755-098X, E-ISSN 1755-0998, Vol. 14, no 1, p. 87-99Article in journal (Refereed)
    Abstract [en]

    Many empirical studies estimating effective population size apply the temporal method that provides an estimate of the variance effective size through the amount of temporal allele frequency change under the assumption that the study population is completely isolated. This assumption is frequently violated, and the magnitude of the resulting bias is generally unknown. We studied how gene flow affects estimates of effective size obtained by the temporal method when sampling from a population system and provide analytical expressions for the expected estimate under an island model of migration. We show that the temporal method tends to systematically underestimate both local and global effective size when populations are connected by gene flow, and the bias is sometimes dramatic. The problem is particularly likely to occur when sampling from a subdivided population where high levels of gene flow obscure identification of subpopulation boundaries. In such situations, sampling in a manner that prevents biased estimates can be difficult. This phenomenon might partially explain the frequently reported unexpectedly low effective population sizes of marine populations that have raised concern regarding the genetic vulnerability of even exceptionally large populations.

  • 45.
    Silvestrov, Dmitrii
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Petersson, Mikael
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nonlinearly perturbed birth-death-type models2016Report (Other academic)
    Abstract [en]

    Asymptotic expansions for stationary and conditional quasi-stationary distributions of nonlinearly perturbed birth-death-type semi-Markov models are presented. Applications to models of population growth, epidemic spread and population genetics are discussed.

  • 46. Sjölander, Arvid
    et al.
    Hartman-Werner, Linda
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Humphreys, Keith
    Fine mapping of disease genes using tagging SNPs2007In: Annals of Human Genetics, Vol. 71, no 6, p. 815-827Article in journal (Other (popular science, discussion, etc.))
  • 47. Verrall, Richard
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Björkwall, Susanna
    Stockholm University, Faculty of Science, Department of Mathematics.
    Modelling claims run off with reversible jump markov chain Monte Carlo methods2012In: Astin Bulletin: Actuarial Studies in Non-Life Insurance, ISSN 0515-0361, E-ISSN 1783-1350, Vol. 42, no 1, p. 35-58Article in journal (Refereed)
    Abstract [en]

    In this paper we describe a new approach to modelling the development of claims run-off triangles. This method replaces the usual ad hoc practical process of extrapolating a development pattern to obtain tail factors with an objective procedure. An example is given, illustrating the results in a practical context, and the WinBUGS code is supplied.

  • 48.
    Verrall, Richard
    et al.
    City University, Cass Business School.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Björkwall, Susanna
    Stockholm University, Faculty of Science, Department of Mathematics.
    Modelling claims run-off with reversible jump Markov chain Monte Carlo methodsManuscript (preprint) (Other academic)
  • 49. Ängquist, Lars
    et al.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Groop, Leif
    Strategies for conditional two-locus nonparametric linkage analysis2008In: Human Heredity, Vol. 66, p. 138-156Article in journal (Refereed)
1 - 49 of 49
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