New Paper: Parameter redundancy in Jolly‐Seber tag loss models

Diana along with Wei Cai, Stephanie Yurchak and Laura Cowen have published the paper:

Parameter redundancy in Jolly‐Seber tag loss models

in Ecology and Evolution


1. Capture–recapture experiments are conducted to estimate population parameters such as population size, survival rates, and capture rates. Typically, individuals are captured and given unique tags, then recaptured over several time periods with the assumption that these tags are not lost. However, for some populations, tag loss cannot be assumed negligible. The Jolly‐Seber tag loss model is used when the no‐tag‐loss assumption is invalid. Further, the model has been extended to incorporate group heterogeneity, which allows parameters to vary by group membership. Many mark–recapture models become overparameterized resulting in the inability to independently estimate parameters. This is known as parameter redundancy.
2. We investigate parameter redundancy using symbolic methods. Because of the complex structure of some tag loss models, the methods cannot always be applied directly. Instead, we develop a simple combination of parameters that can be used to investigate parameter redundancy in tag loss models.
3. The incorporation of tag loss and group heterogeneity into Jolly‐Seber models does not result in further parameter redundancies. Furthermore, using hybrid methods we studied the parameter redundancy caused by data through case studies and generated tag histories with different parameter values.
4. Smaller capture and survival rates are found to cause parameter redundancy in these models. These problems resolve when applied to large populations.


New Paper by Emily and Byron: Integrated modelling of insect population dynamics at two temporal scales

Emily and Byron, along with Marc Kery, Armin Coray, Michael Schaub and Bruno Baur have published the paper:

Integrated modelling of insect population dynamics at two temporal scales

in Ecological Modelling.


Population size of species with birth-pulse life-cycles varies both within and between seasons, but most population dynamics models assume that a population can be characterised adequately by a single number within a season. However, within-season dynamics can sometimes be too substantial to be ignored when modelling dynamics between seasons. Typical examples are insect populations or migratory animals. Numerous models for only between-season dynamics exist, but very few have combined dynamics at both temporal scales.

In a new approach, we extend appreciably the models of Dennis et al. (2016b): we show how to adapt them for a generation time  year and fit an integrated population model for multiple data types, by maximising a joint likelihood for population counts of unmarked individuals and capture–recapture data from a study with marked individuals. We illustrate the approach using annual monitoring data for the endangered flightless beetle Iberodorcadion fuliginator from 18 populations in the Upper Rhine Valley for 1998–2016, with a 2-year life cycle. Standard likelihood methods are used for model fitting and comparison, and a concentrated (profile) likelihood approach provides computational efficiency.

Additional information from the capture–recapture data makes the population model more robust and, importantly, enables true, rather than relative, abundance to be estimated. A dynamic stopover model provides estimates of both survival and phenology parameters within a season, and also of productivity between seasons. For  I. fuliginator, we demonstrate a population decline since 1998 and how this links with productivity, which is affected by temperature. A delayed mean emergence date in recent years is also shown.

A main point of interest is the focus on the two temporal scales at which perhaps most animal populations vary: in the short-term, a population is seldom truly closed within a single season, and in the long-term (between seasons) it never is. Hence our models may serve as a template for a general description of population dynamics in many species. This includes rare species with limited data sets, for which there is a general lack of population dynamic models, yet conservation actions may greatly benefit from this kind of models.


PhD Project: Modelling Citizen-science Data for Butterflies and Moths: Where and When are they Flying?

Eleni Matechou, Emily Dennis, Richard Fox, Byron Morgan and Diana Cole are offering the statistics PhD project: Modelling Citizen-science Data for Butterflies and Moths: Where and When are they Flying?
At a time of biodiversity loss, including widely reported insect declines, citizen science data play a vital role in measuring changes in species’ populations and distributions and in seeking to understand the pressures influencing such changes. Butterflies and moths respond quickly to habitat and climatic change, and hence are valuable biodiversity indicators. In the UK, millions of species occurrence records for Lepidoptera have been gathered by two large citizen science recording schemes, of which the full potential has not been fully realized.

Analysing recording data of this nature presents unique challenges relating to their vast quantity but also associated sampling biases. Using cutting edge modelling, this project will maximise these valuable datasets to enhance our understanding of species’ phenology (flight periods), distribution and range dynamics to help inform future conservation delivery and policy for UK butterflies and moths.

The candidate will undertake new statistical model developments applied to citizen science data. The research will involve:

  • Critically assessing sampling design to determine how much data are needed to reliably estimate species’ occurrence trends – can occupancy models be used for rare species with small ranges?
  • Modelling species’ phenology from citizen science data to provide new insights on variation over space and time.
  • Applying state-of-the-art variable selection techniques to better describe drivers of species’ range and distribution change through suitable spatial and environmental covariates.

This project has been shortlisted for Aries funding. More details can be found at:



New paper: A Generic Method for Estimating and Smoothing Multispecies Biodiversity Indicators Using Intermittent Data

The paper, A Generic Method for Estimating and Smoothing Multispecies Biodiversity Indicators Using Intermittent Data, by Stephen Freeman, Nicholas Isaac, Panagiotis Besbeas, Emily Dennis and Byron Morgan has just been published in the Journal of Agricultural, Biological and Environmental Statistics



Biodiversity indicators summarise extensive, complex ecological data sets and are important in influencing government policy. Component data consist of time-varying indices for each of a number of different species. However, current biodiversity indicators suffer from multiple statistical shortcomings. We describe a state-space formulation for new multispecies biodiversity indicators, based on rates of change in the abundance or occupancy probability of the contributing individual species. The formulation is flexible and applicable to different taxa. It possesses several advantages, including the ability to accommodate the sporadic unavailability of data, incorporate variation in the estimation precision of the individual species’ indices when appropriate, and allow the direct incorporation of smoothing over time. Furthermore, model fitting is straightforward in Bayesian and classical implementations, the latter adopting either efficient Hidden Markov modelling or the Kalman filter. Conveniently, the same algorithms can be adopted for cases based on abundance or occupancy data—only the subsequent interpretation differs. The procedure removes the need for bootstrapping which can be prohibitive. We recommend which of two alternatives to use when taxa are fully or partially sampled. The performance of the new approach is demonstrated on simulated data, and through application to three diverse national UK data sets on butterflies, bats and dragonflies. We see that uncritical incorporation of index standard errors should be avoided.



The statistical ecology group from Kent (SE@K) attended the first virtual international statistical ecology conference (vISEC) from 22nd June to 26th June.

Fabian, Katie Oscar and James gave contributed talks. Fabian talked about Objective Priors from Scoring rules for N-mixture models. Katie  gave a talk on Mark-recapture modelling to inform conservation management in Mauritius. Oscar talked about Assessing the Spatio-Temporal distribution of invasive species. James gave the talk Modelling butterfly lifespans using citizen-science count data.

Speed talks were given by Diana, Eleni and Ulrike. Diana  talked about Inference with parameter redundant models: reparameterisation, constraints, robust design and integrated models. Eleni talked about Efficient Bayesian variable selection in ecological models. Ulrike’s talk was on Adjusting for misclassified sex observation in a capture recapture study of Telfair skinks.

Rachel was co-chair of the scientific committee.


New paper: Duration of female parental care and their survival in the little auk Alle alle – are these two traits linked?

The paper: Duration of female parental care and their survival in the little auk Alle alle – are these two traits linked? by: Katarzyna Wojczulanis-Jakubas, Marina Jiménez-Muñoz, Dariusz Jakubas, Dorota Kidawa, Nina Karnovsky, Diana Cole and Eleni Matechou, has just been published in Behavioral Ecology and Sociobiology.


Desertion of offspring before its independence by one of the parents is observed in a number of avian species with bi-parental care but reasons for this strategy are not fully understood. This behaviour is particularly intriguing in species where bi-parental care is crucial to raise the brood successfully. Here, we focus on the little auk, Alle alle, a small seabird with intensive bi-parental care, where the female deserts the brood at the end of the chick rearing period. The little auk example is interesting as most hypotheses to explain desertion of the brood by females (e.g. “re-mating hypothesis”, “body condition hypothesis”) have been rejected for this species. Here, we analysed a possible relationship between the duration of female parental care over the chick and her chances to survive to the next breeding season. We performed the study in two breeding colonies on Spitsbergen with different foraging conditions – more favourable in Hornsund and less favourable in Magdalenefjorden. We predicted that in Hornsund females would stay for shorter periods of time with the brood and would have higher survival rates in comparison with birds from Magdalenefjorden. We found that indeed in less favourable conditions of Magdalenefjorden, females stay longer with the brood than in the more favourable conditions of Hornsund. Moreover, female survival was negatively affected by the length of stay in the brood. Nevertheless, duration of female parental care over the chick was not related to their parental efforts, earlier in the chick rearing period, and survival of males and females was similar. Thus, although females brood desertion and winter survival are linked, the relationship is not straightforward.



New Paper: Size‐ and stage‐dependence in cause‐specific mortality of migratory brown trout

The paper Size‐ and stage‐dependence in cause‐specific mortality of migratory brown trout by Chloé R. Nater, Yngvild Vindenes, Per Aass, Diana Cole, Øystein Langangen, S. Jannicke Moe, Atle Rustadbakken, Daniel Turek, Leif Asbjørn Vøllestad and Torbjørn Ergon was published in Journal of Animal Ecology.


  1. Evidence‐based management of natural populations under strong human influence frequently requires not only estimates of survival but also knowledge about how much mortality is due to anthropogenic vs. natural causes. This is the case particularly when individuals vary in their vulnerability to different causes of mortality due to traits, life history stages, or locations.
  2. Here, we estimated harvest and background (other cause) mortality of landlocked migratory salmonids over half a century. In doing so, we quantified among‐individual variation in vulnerability to cause‐specific mortality resulting from differences in body size and spawning location relative to a hydropower dam.
  3. We constructed a multistate mark–recapture model to estimate harvest and background mortality hazard rates as functions of a discrete state (spawning location) and an individual time‐varying covariate (body size). We further accounted for among‐year variation in mortality and migratory behaviour and fit the model to a unique 50‐year time series of mark–recapture–recovery data on brown trout (Salmo trutta) in Norway.
  4. Harvest mortality was highest for intermediate‐sized trout, and outweighed background mortality for most of the observed size range. Background mortality decreased with body size for trout spawning above the dam and increased for those spawning below. All vital rates varied substantially over time, but a trend was evident only in estimates of fishers’ reporting rate, which decreased from over 50% to less than 10% throughout the study period.
  5. We highlight the importance of body size for cause‐specific mortality and demonstrate how this can be estimated using a novel hazard rate parameterization for mark–recapture models. Our approach allows estimating effects of individual traits and environment on cause‐specific mortality without confounding, and provides an intuitive way to estimate temporal patterns within and correlation among different mortality sources.

Parameter Redundancy and Identifiability Book Published

Diana Cole’s book Parameter Redundancy and Identifiability has been publish by Chapman and Hall/CRC

Book Synopsis

Statistical and mathematical models are defined by parameters that describe different characteristics of those models. Ideally it would be possible to find parameter estimates for every parameter in that model, but, in some cases, this is not possible. For example, two parameters that only ever appear in the model as a product could not be estimated individually; only the product can be estimated. Such a model is said to be parameter redundant, or the parameters are described as non-identifiable. This book explains why parameter redundancy and non-identifiability is a problem and the different methods that can be used for detection, including in a Bayesian context. Key features of this book:

  • Detailed discussion of the problems caused by parameter redundancy and non-identifiability
  • Explanation of the different general methods for detecting parameter redundancy and non-identifiability, including symbolic algebra and numerical methods
  • Chapter on Bayesian identifiability
  • Throughout illustrative examples are used to clearly demonstrate each problem and method. Maple and R code are available for these examples
  • More in-depth focus on the areas of discrete and continuous state-space models and ecological statistics, including methods that have been specifically developed for each of these areas

This book is designed to make parameter redundancy and non-identifiability accessible and understandable to a wide audience from masters and PhD students to researchers, from mathematicians and statisticians to practitioners using mathematical or statistical models.

Book website: https://www.routledge.com/Parameter-Redundancy-and-Identifiability/Cole/p/book/9781498720878

Code for book available at: https://www.kent.ac.uk/smsas/personal/djc24/parameterredundancy.html


New publication: Statistical Development of Animal Density Estimation Using Random Encounter Modelling

New publication in JABES: Statistical Development of Animal Density Estimation Using Random Encounter Modelling by Natoya Jourdain, Diana Cole, Martin Ridout and Marcus Rowcliffe.


Camera trapping is widely used in ecological studies to estimate animal density, although these studies are largely restricted to animals that can be identified to the individual level. The random encounter model, developed by Rowcliffe et al. (J Anal Ecol 45(4):1228–1236, 2008), estimates animal density from camera-trap data without the need to identify animals. Although the REM can provide reliable density estimates, it lacks the potential to account for the multiple sources of variance in the modelling process. The density estimator in REM is a ratio, and since the variance of a ratio estimator is intractable, we examine and compare the finite sample performance of many approaches for obtaining confidence intervals via simulation studies. We also propose an integrated random encounter model as a parametric alternative, which is flexible and can incorporate covariates and random effects. A data example from Whipsnade Wild Animal Park, Bedfordshire, south England, is used to demonstrate the application of these methods.




Congratulation to Marina whose poster won silver at STEM for Britain.

On 13th March Marina went to Parliament to present her poster on How do bird populations vary across Britain? Spatially-explicit integrated population models, as part of STEM for Britain competition. As stated on the STEM for Britain website, “STEM for BRITAIN Awards are made on the basis of the very best research work and results by an early-stage or early-career researcher together with their ability to communicate their work to a lay audience.” Marina’s poster won silver in the Mathematics category. Well done Marina on this amazing achievement.