Improved estimation of hunting harvest using covariates at the hunting management precinct level

Linköping University | Department of Physics, Chemistry and Biology
 Type of thesis, 60 hp | Educational Program: Biology
 Autumn term 2020 and Spring term 2021 | LITH-IFM-A-EX—21/3955--SE

Improved estimation of hunting
harvest using covariates at the
hunting management precinct level

Paula Jonsson

Examinator: Karl-Olof Bergman
Supervisor: Tom Lindström

Table of content
1. Introduction ........................................................................................................................ 4
2. Material & Methods ........................................................................................................... 5
 2.1 Game species ............................................................................................................... 5
 2.2 Data collection ............................................................................................................. 8
 Reporting game management ............................................................................... 8
 Actual hunting land .............................................................................................. 9
 Climate data........................................................................................................ 10
 Wildlife accidents ............................................................................................... 10
 Geographical distribution ................................................................................... 10
 2.3 Model specification ................................................................................................... 11
 Previous model (Null model) ............................................................................. 11
 Proposed model (Covariate model) .................................................................... 12
 Prior elicitation ................................................................................................... 13
 2.4 Model selection.......................................................................................................... 13
 2.5 Prediction ................................................................................................................... 14
 2.6 Computation .............................................................................................................. 14
 Leave one out cross-validation ........................................................................... 15
3. Results .............................................................................................................................. 15
 3.1 Model selection.......................................................................................................... 15
 Evaluation of predictive performance ................................................................ 16
 Evaluation of the 95 % credible interval ............................................................ 16
 3.2 Posterior prediction sampling .................................................................................... 18
 Predicted median among HMPs ......................................................................... 20
 Predicted variation among HMPs ...................................................................... 23
 Predictive variation at high hunting levels ......................................................... 25
4. Discussion ........................................................................................................................ 27
 4.1 Conclusion ................................................................................................................. 30
5. Societal & ethical considerations ..................................................................................... 31
6. Acknowledgements .......................................................................................................... 31
7. References ........................................................................................................................ 32
8. Appendix A ...................................................................................................................... 36

Abstract
In Sweden, reporting is voluntary for most common felled game, and the number of voluntary
reports can vary between hunting teams, HMP, and counties. In 2020, an improved harvest
estimation model was developed, which reduced the sensitivity to low reporting. However,
there were still some limits to the model, where large, credible intervals were estimated.
Therefore, additional variables were considered as the model does not take into account
landcover among HMPs, [2] the impact of climate, [4] wildlife accidents, and [4] geographical
distribution, creating the covariate model. This study aimed to compare the new model with the
covariate model to see if covariates would reduce the large, credible intervals. Two hypothesis
tests were performed: evaluation of predictive performance using leave one out cross-validation
and evaluation of the 95 % credible interval. Evaluation of predictive performance was
performed by examining the difference in expected log-pointwise predictive density (ELPD)
and standard error (SE) for each species and model. The results show that the covariates model
ranked highest for all ten species, and out of the ten species, six had an (ELPD) difference of
two to four, which implies that there is support that the covariate model will be a better predictor
for other datasets than this one. At least one covariate had an apparent effect on harvest
estimates for nine out of ten species. Finally, the covariate model reduced the large
uncertainties, which was an improvement of the null model, indicating that harvest estimates
can be improved by taking covariates into account.

Keywords: Bayesian statistics, covariates, land cover, credible interval, model selection

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1. Introduction
Reliable harvest estimates and hunting regulation are cornerstones of fact-based wildlife
management (Bergqvist et al., 2016). Wildlife management can contribute to the rehabilitation
of regional biodiversity and, due to its direct (interspecific interactions) and indirect functions
(food web and community structure), increase the health of ecosystems (Tsunoda, 2020).
However, managing wildlife is a challenge worldwide since it needs to balance societal and
ecological aspects while considering species' long-term survival (Pelikka et al., 2007; Dressel
et al., 2018). Reliable harvest estimates could reduce the challenges of managing wildlife since
they can indicate changes in population size (monitoring long-term survival of wildlife species)
(Bergqvist et al., 2016). Furthermore, harvest should be continuously adapted to changes in
ecosystems and society, making harvest estimates a fundamental and vital component in most
wildlife management programs (Danell & Bergström, 2010).

Game management and harvest reporting vary among countries. For example, in Cyprus,
Greece, and Great Britain, reporting is entirely voluntary, whereas, in Norway, Iceland, and
Germany, reporting is completely mandatory (Aubry et al., 2020). In Sweden, voluntary
reporting is applied for game species with an open season. The landowner holds the hunting
rights, which means that felled game is always owned by the landowner or a hunter(s) to whom
the landowner has authorized hunting. However, reporting is mandatory for game species with
pre-determined harvest quotas such as moose (Alces alces), red deer (Cercus elaphus), wolf
(Canis lupus), bear (Ursus arctos), lynx (Lynx lynx), and wolverine (Gulo gulo). Harvest of
these species are regulated by licenses or management area plans, and the harvest must be
reported to the County Administrative Boards.

In Sweden, all voluntary reporting is performed through a standardized form or directly in the
database “Viltdata” owned by The Swedish Association for Hunting and Wildlife Management
(SAHWM) (Viltdata, w.y). All harvest reporting and harvest estimations are carried out within
Hunting Management Precincts (HMP, Swedish: jaktvårdskrets), with 313 HMPs throughout
Sweden during the hunting year 2018/2019. Harvest estimates are based on voluntary reporting
from hunting teams within each HMP. The previous harvest estimation method, used by
SAHWM, extrapolates linearly from harvest reports, based on the assumption that harvest rate
occurs uniformly within HMPs (Lindström & Bergqvist, 2020). Furthermore, as the method
does not provide any measures of uncertainty, there was a need to develop a new method in
order to overcome these potential drawbacks.

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Since reporting is voluntary, it is not possible to verify if hunting teams report their harvest of
all species or just some species. SAHWM continuously informs and encourages hunting teams
to report their total harvest, regardless of size, species abundance, or hunters' game preferences.
According to SAHWM, it is easier to reach large hunting teams, and because of that, hunting
teams with large areas may be over-represented in the reporting compared with smaller hunting
teams (G. Bergqvist, personal communication, May 4, 2021). However, this is not entirely
certain. Since there are major uncertainties about hunting and reporting preferences, these
uncertainties should be considered in the estimation.

During 2019-2020, Lindström and Bergqvist (2020) introduced a new model for estimating
hunting harvest, using a Hierarchical Bayesian framework. Their framework suggested that
there could be random variability among hunting teams and that there could be an effect of
hunting area per hunting team on harvest rate. Further, their results showed that hunting teams
that hunt in larger areas harvest less per unit area, suggesting that the harvest rate does not scale
linearly with hunting areas (Lindström & Bergqvist, 2020). Their framework improved the
former estimation method since it presented inherent uncertainties while reducing sensitivity to
low reporting rates.

Even if Lindström and Bergqvist’s model improved the former harvest estimation method,
there were still large uncertainties (large credible interval) for some species. The finest spatial
resolution considered from harvest reports used in their model was HMPs. Therefore, I
wanted to investigate if the uncertainty could be reduced by adding covariates to the model. I
will extend Lindström and Bergqvist’s model by adding fixed effects at the HMP level, such
as [1] landcover among HMPs, [2] the impact of climate, and [3] geographical distribution. I
will, like them, use a Hierarchical Bayesian model, with the aim of developing a more robust
model for estimating hunting harvest.

2. Material & Methods
2.1 Game species
This study included ten different game species. Like Lindström and Bergqvist’s (2020) model,
red fox (Vulpes vulpes), Eurasian beaver (Castor fiber), wild boar (Sus scrofa), and European
pine marten (Martes martes) was included as model species. These four species were included
since they exemplify game harvest at high or low numbers and uniformly or variably across
Sweden. In addition, six additional species for which landcover, climate variables, and
geographical distribution were expected to have a substantial impact were included. These six

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species are mountain hare (Lepus timidus), western capercaillie (Tetrao urogallus), rock
ptarmigan (Lagopus muta), common eider (Somateria mollissima), American mink (Neovison
vison), and greylag goose (Anser anser).

The red fox is a highly adapted generalist mesopredator and inhabits most parts of Sweden.
Populations of red fox are favoured by increasing agricultural land and infrastructure since such
areas can provide good food resources such as garbage and roadkill (Elmhagen et al., 2015;
Walton & Walton, 2018). Since the early 2000s, red foxes’ populations have increased in the
Arctic tundra, favoring the increase of reindeer carcasses (Elmhagen et al., 2015; Walton &
Walton, 2018).

Populations of Eurasian beaver have been close to extinction, mainly caused by hunting, but
during the latest decades, the populations have been reintroduced in their native range in
Sweden. Beavers are herbivores and live in freshwater throughout Sweden and use mainly
deciduous trees for foraging (Hartman et al., 2008; Willby et al., 2018). They are favoured by
warmer seasons since they can fill their cache during a more extended warmer period before
the winter (Willby et al., 2018).

Wild boars are a widespread mammal that during the latest decades has increased exceedingly,
both in numbers and distribution. They are mainly located in Sweden's south and central parts
(In Swedish: Götaland and Svealand) (Bergqvist & Elmhagen, 2019). The main reason for this
high increase is their high reproductive rate, where one sow can give birth to up to 5-6 offspring
per litter (Malmsten & Dalin, 2016; Vetter et al., 2020). Furthermore, juveniles benefit from
climate change since they survive the winter season due to a higher winter temperature and
greater food availability (Vetter et al., 2020). Wild boars are one of the most common felled
game species (Bergqvist & Elmhagen, 2016).

The European pine marten is a small omnivore that occupies conifer and deciduous forests in
most parts of Sweden (Birks et al., 2005). Their food resource during the winter season consists
of mainly small mammals, especially microtine rodents. During the spring season, pine martens
mainly forage on berries and invertebrates (Helldin, 2000).

Since 1980, mountain hare has decreased in Sweden, possibly due to interactions with the
European hare (Elmhagen et al., 2015). Today, the mountain hare is classified as near threatened
by the Swedish red list (Thurfjell et al., 2020). The mountain hare inhabits a wide range of
biotas, from taiga to tundra and can survive under poor environmental conditions. However,
they are rarely found in agricultural land (Angerbjörn & Flux, 1995; Bergqvist & Elmhagen,

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2020). Mountain hares change their food preferences during each semester. During spring, they
forage on herbs and grass, and during winter, they forage on shrubs and branches from
deciduous trees (Bergqvist & Elmhagen, 2020).

Western capercaillie is a grouse that inhabits mature forests (Storch, 2000). Populations of
western capercaillie are highly susceptible to habitat and structure changes, and populations are
declining due to fragmented areas and loss of suitable habitats (Graf et al., 2007). During the
hunting year of 2018/2019, there was a decreasing trend of capercaillie harvest in the south of
Sweden, while harvest still is relatively stable throughout the rest of Sweden (Bergqvist et al.,
2020).

Rock ptarmigan is an herbivorous bird that is inhabiting arctic regions above the treeline. Due
to global warming, there are concerns about rock ptarmigan since there is a loss of valuable
habitats (an increase of shrubs and trees is replacing open areas) in the arctic regions (Watson
& Moss, 2008; Elmhagen et al., 2015).

The common eider is a marine-benthic diver that occupies the marine environment all year-
round throughout Sweden (Robertson, 2016; Houle et al., 2017), where they prey mainly on
molluscs (Fenstad et al., 2016). The common eider has declined dramatically in the last decades,
as has hunting of the species (Ekroos et al., 2012; Bergqvist et al., 2020). The species is reported
as endangered on the Swedish red list (SLU, 2010).

The American mink was introduced to Europe in the early 1900s, mainly for fur farming, and
is classified as an invasive species (Bonesi & Palazon, 2007). However, during the latest years,
there is a decrease in both felled game and mink populations in Sweden, which is seen as
desirable due to an invasive species' potentially ecological risks (Bergqvist et al., 2020). The
American mink is a semi-aquatic carnivore and is often found near lakes, rivers, and/or near the
coast. They eat primarily small vertebrates and crustaceans (Bonesi et al., 2004).

Greylag goose has increased tremendously in the last decades, which is a result of increased
agricultural land and stabilized harvest (Ebbinge, 1991; Kampe-Persson, 2002; Fox &
Abraham, 2017; Bergqvist et al., 2020). Greylag geese are often found in and around lakes
covered by small woods and agricultural land where they cause significant damage (Buij et al.,
2017). Hunting of greylag goose occurs throughout the country, mostly in southern Sweden
(Liljebäck et al., 2019).

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2.2 Data collection
 Reporting game management
Each hunting year (July 1 to June 30), hunting teams are free to voluntarily report felled game
through the database Viltdata, owned by SAHWM. In Sweden, there were 313 HMPs during
the hunting year 2018/2019, and in total 7,482 hunting teams reported their harvest, covering
an area of 9,561,156 hectares, corresponding to 29,2 % of the total huntable area. For wildlife
species with voluntary reporting, each report contains the hunting area, the number of felled
individuals of each species, and the HMP in which game were felled.

Each HMP were linked to different covariates, which include landcover (deciduous forest and
not temporary forest, arable land, wetland, water area, and coastline), climate variables, wildlife
accidents (only applies for wild boar), and geographical distribution (Table 1). The
incorporations were performed using R (R Core Team, 2019) with the GIStools-package
(rgeos, rgdal, raster). HMPs that are not included in SAHWM harvest estimation were
excluded from the analysis. These include urban HMPs (Stockholm and Uppsala) and HMPs
above the north's cultivation limit (Vilhelminafjällen and Tärna).

Table 1. Covariates.

Thematic class (combined)
Wetland
Farmland
Open area with vegetation
Open area without vegetation
Infrastructure
Water (actual huntable water)
Not temporary forest
Coniferous forest (will be used as” zero habitat” and not included as a
covariate)
Deciduous forest
Temperature (K)
Precipitation (kg/m2)
Humidity (%)
Coastline lake and river
Coastline sea
Wildlife accidents (only for wild boar)
West and East gradients
South and North gradients

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Actual hunting land
The Swedish landcover map (SLC) was downloaded from Naturvårdsverket (Naturvårdsverket,
2020), and it represents landcover as pixels, where each pixel (10*10-meter, 0.01 ha) includes
a thematic class (Naturvårdsverket, 2018). All 313 HMPs specified by SAHWM during the
game period 2018/2019 were merged with SLC, and landcover per HMP was extracted. In order
to extract landcover classes per HMP, all national parks, water areas not suitable for hunting,
and governmentally owned land in each HMP were removed. For all covariate data collection,
each thematic class was calculated as the proportion of total landcover (Jonsson et al., 2020).

The water area suitable for hunting was determined by adding an inner buffer area of 25 m
(representing the range of a hunting shotgun) for each sea, lake, and river per HMP. The rest of
the water areas were excluded. Furthermore, coastline for each sea, lake and river per HMP
were calculated as the circumference of each lake/river and sea.

There are 29 National parks in Sweden, and out of the 313 HMPs, 34 HMPs were located in a
Swedish National Park. A shapefile over national parks in Sweden was collected from the
Swedish data portal by Naturvårdsverket (Naturvårdsverket, 2021), and these were removed (as
no hunting is allowed there). Furthermore, 14 HMPs were located in the Swedish mountain
range and above the cultivation limit and were therefore treated differently. The HMPs that
were included in the mountain chain were, Kiruna, Jokkmokk, Gällivare, Arjeplog, Dorotea,
Tärna, Vilhelminafjällen, Sorsele, Strömsund, Hammerdal, Krokom, Västjämtland, Berg and
Härjedalen HMP.

The cultivation limit in the mountain chain divide Kiruna, Gällivare, Jokkmokk, Arjeplog,
Dorotea, Tärna, Vilhelminafjällen, and Sorsele HMP into an upper and lower limit. Subparts of
HMPs in the lower limit were accepted as huntable land, and parts of HMPs in the upper limit
as not huntable land. The whole area of Tärna and Vilhelminfjällen HMP is located in the upper
part of the cultivation limit and, therefore, not handled as huntable land. However, for Kiruna,
Gällivare, Jokkmokk, Arjeplog, Dorotea, and Sorsele HMP, the upper limit was further divided
into privately and governmentally owned parts, where the privately owned land was accepted
as huntable land.

The first step for Kiruna, Gällivare, Jokkmokk, Arjeplog, Dorotea, and Sorsele HMP was to
subtract national parks from both the upper limits (privately owned land) and lower limit.
Second, excess water was removed, and coastal lines and huntable water areas were calculated
for each sea, lake, or river. These conditions also apply to Strömsund, Hammerdal, Krokom,

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Västjämtland, Berg, but instead of the cultivation limit, all HMPs were divided by the reindeer
grazing limit. All landcover above reindeer grazing limit is governmentally owned and was
therefore not accepted as huntable land.

 Climate data
Climate data were collected from the Swedish Meteorological and Hydrological Institute
Application Programming Interface (SMHI API) database. Data contains entries for various
weather parameters, including temperature, relative humidity, and precipitation, recorded every
hour and is presented in a grib square of 2.5*2.5 km2. Each grib square consists of a climate
measuring station with associated longitude and latitude coordinates. Data were read in Python
3.6.3, using the package eccodes. Parameters were extracted from each grib-file with
associated longitude and latitude coordinates and saved as the sum of precipitation (kg/m2),
mean of relative humidity (%), and mean of temperature (K) per year.

Out of the 313 HMPs, seven HMPs (Bokenäset, Lidingö, Danmark-Funbo-Vaksala, Solna,
Sundbyberg, Ljungskile and Södra Skärgården Göteborg) were handled differently, as they did
not overlap between HMP and any climate measuring station, which means that climatic
measurement could not be extracted from these HMPs. As these HMPs did not overlap with
any grib squares, parameter values from adjacent HMP were used to estimate a relative
measurement for each special-case HMP. A convex hull was created to receive parameter
values around all the adjacent HMPs, using R-package sf. Parameter values from every grib
square in adjacent HMPs were extracted, and extracted parameter values from adjacent HMP
were saved as a relative measurement.

 Wildlife accidents

All data used for analysing wildlife accidents were collected from the National wildlife accident
council (viltolycka.se). Each wildlife accident with corresponding coordinates, documented as
SWEREF99, was matched with the associated HMP, receiving an actual number of wildlife
accidents per species, year, county, and HMP. The received data also included species, year,
county, and HMPs when there were zero accidents. For this study, only wildlife accidents for
wild boar were included.

 Geographical distribution

To better estimate where the model species are located in the country and if this could affect
the posterior distribution, a north/south/west/east- gradient was created. The gradient, with

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north (N) and east (E) coordinates (SWEREF99), was based at the central point in Sweden. The
central point was denoted to function as a zero-point and was performed using the function
gUnaryUnion from package rgeos. A centroid was then placed in each HMP with the function
gCentroid (package rgeos) and was saved with associated coordinates (N and E coordinates).
Coordinates for each HMP-centroid were then saved in a data frame by subtracting the zero-
point coordinates with the centroid coordinates of each HMP. The last step was to standardize
each coordinate for the respective HMP. Standardizing coordinates was performed by
subtracting the easternmost point from the westernmost point and then divide the E-coordinates
for each HMP with the subtracted value. The same was performed for the north coordinates by
subtracting the northernmost point with the southernmost point and divide the N-coordinates
for each HMP with the subtracted value.

2.3 Model specification
A Hierarchical Bayesian framework estimates the posterior distribution parameters while
allowing parameters to vary probabilistically, and it has become an important tool for many
ecologists (Hooten & Hobbs, 2015). The modeling structure exploits different information
levels to draw conclusions about complex relationships (Clark, 2005). For this study, the
hierarchical model consists of multiple levels, where the available information (hunting teams
that have reported felled game and area) is used for unobserved levels (hunting teams that have
not reported felled game). Therefore, it is an appropriate tool to estimate harvest for unreported
hunting areas.

 Previous model (Null model)

Lindström and Bergqvist improved the formerly used estimation method by taking into account
possible inherent uncertainties. The model consisted of three steps: area analysis, analysing
reported harvest, and posterior prediction.

The first step was to model a set of reported hunting areas (i.e., area of reported hunting teams),
which estimates the distribution of the hunting area for different HMPs and counties. This was
performed by using the mean and standard deviation of reported hunting areas in a Log-Normal
distribution. Exploratory of data, Lindström and Bergqvist revealed that there could be variation
within HMPs in hunting area per hunting team, and within-HMP variation could vary between
counties. Area analysis was performed on a log scale, where the standard deviation is a scale-
free measurement of the variability in hunting teams within each HMP. The results of the area
analysis gave a compilation of the estimated average area per hunting team in each HMP.

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The analysis of reported harvest uses the data collected from the area analysis to obtain the
posterior distribution (harvest rate for each hunting team and HMP). Hunting follows a Poisson
process with a specific hunting team rate, where the specific team rate is assumed to be gamma
distributed, which results in a Gamma-Poisson mixture distribution.

The model, which will henceforth be denoted as the null model ( 0 ) accounts for (1) random
variability among hunting teams, (2) rate-variability association, and (3) the effect of hunting
area on harvest rate (Figure 1). However, Lindström and Bergqvist’s results showed that there
were computational problems with the parameter (2), rate-variability association, and that the
importance of the parameter was dubious. Posterior estimates showed that when harvest rate
was low, there was a greater variability between hunting teams in an HMP. Furthermore, it is
likely that game is not present in all areas for species at low densities. Therefore, a reduced null
model, 0 , was included in this study, where (2) rate-variability association was excluded.
Details are found in Lindström and Bergqvist (2020).

 0 0

 (2) Rate- (3) Effect of (3) Effect of
 (1) Variation hunting area (1) Variation hunting area
 variability
 within HMPs on harvest within HMPs on harvest
 association
 rate rate

Figure 1: Parameters included in the null model ( 0 ) and the reduced null model ( 0 ).

 Proposed model (Covariate model)

Like Lindström and Bergqvist, this study also consisted of the following three steps: area
analysis, analysis of reported harvest, and posterior prediction for unreported hunting teams.
Lindström and Bergqvist's model considers random effects of nationwide, county, and HMP
effect, calculating harvest per HMP k and county l.

 µ , = exp⁡( + + , ⁡) (1)

To be able to add the effect of covariates to the null model ( 0 ), an additional parameter was
introduced, , , for HMP k and county l.

 µ , = exp⁡( + + , + , ) (2)

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 , are coefficients per covariate (bn ) multiplicated with the covariate value of each covariate
in HMP k and county l (X n,k,l ).

 , = ⁡ b1⁡ ∗ ⁡ X1, , + ⁡ b2⁡ ∗ ⁡ X2, , + ⁡ …⁡b ⁡ ∗ ⁡ , , ⁡⁡ (3)

Creating the covariate model, which adds the effect of covariates to the null model (equation
3). Since Lindström and Bergqvist's results showed that the inclusion of the rate-variability
association parameter was dubious, this parameter was included in this study to see if there
would be an effect when including covariates at the precinct level. This study, therefore,
included one full and one reduced covariate model ( , ⁡ ), which either includes all
three parameters (Figure 1) or excludes (2), the rate-variability association. Then, the full null
model was compared with the full covariate model, and the reduced null model was compared
with the reduced covariates model. This was done to see which of the models ranked highest in
terms of predictive density.

 Prior elicitation

All priors used in this study were equivalent to the ones in Lindström and Bergqvist’s model.
However, an additional prior was added (β), which was determined to be proportional to 1.

2.4 Model selection
This study performed two hypothesis tests: first, a covariate model was selected for each species
based on the 95 % credible interval evaluation. Second, leave one out cross-validation was
performed to evaluate predictive ability between models (covariate model compared to the null
model). Two covariate models were tested, one full covariate model (including rate-variability
association), M and the reduced covariate model (excluding rate-variability association),
M . Each model was always compared to the null model, that is M was compared with
M0 and M ⁡was compared with M 0 .

Model selection was performed for each species. However, for wild boar, two different models
were performed: wild boar including wildlife accidents, where one extra covariate was added,
that is, wildlife accidents for wild boar, and wild boar excluding wildlife accidents. This was
done to see if wildlife accidents would give a better or worse posterior density than when
excluding wildlife accidents. Moreover, all species started with 15 (16 for wild boar including
wildlife accidents) covariates for each model, aiming that in the end, only some covariates
would remain, that is, covariates that actually have an effect on hunting estimate.

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Removal of covariates started after the first MCMC simulation. The first step was to determine
which of the 95 % credible interval for each covariate overlapped with zero (i.e., no strong
effect of that covariate on hunting estimate). The second step was to investigate if any non-
influential covariates correlated with another (i.e., determine if there was any covariance
between covariates). Removal of correlated covariates was performed since there may be
problems with identifiability when several covariates show covariation and that it may be
unclear which of the parameters has an effect. If so, removing of one correlated covariate out
of two was determined by removing the covariate with its standard deviation (s.d) closest to
zero from its mean. To summarise, if the 95 % credible interval overlapped with zero, and if
there was a correlation between covariates, only one covariate was removed. This process was
repeated, for each species, until the remaining covariates did not overlap with zero, and the
remaining covariates for each species only positively or negatively affected the amount of
hunting harvest per HMP. The third step was to determine (for each species and model M ,
M , M0 , M 0 ) which of the four models ranked highest in the form of predictive ability.
This was performed by examining the difference in expected log-pointwise predictive density
(ELPD) and standard error (SE) for each species and model, see (exact LOO-CV) - see chapter
Leave one out cross-validation.

2.5 Prediction

Posterior predictive sampling for unreported areas was performed by sampling hunting
team areas with harvest rates for associated hunting teams until all hunting teams
covered all areas in each HMP, and unreported and reported areas were merged.

2.6 Computation
All analyses were performed with the programming language R (R Core Team, 2019), where
all Bayesian modeling were performed with Stan, using the R-package Rstan (Stan Develop
Team, 2020)

The posterior distributions of the considered models do not have a standard form, and, therefore,
numerical methods were required. One of the most frequently used numerical methods for
Hierarchical Bayesian interference is Markov chain Monte Carlo (MCMC) (Monnahan et al.,
2017). An MCMC simulation makes use of random sampling from a target distribution (in this,
case the posterior) (Kurt, 2019). Within the MCMC family, Hamiltonian Monte Carlo (HMC)
is an algorithm that facilitates efficient sampling and avoids inefficient random walk sampling
(Monnahan et al., 2017). However, the HMC algorithm requires expert tuning, and Stan’s

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Hamiltonian Monte Carlo algorithm is a flexible modeling software that often outperforms
other samplers (Monnahan et al., 2017). Therefore, for HMC algorithms, I used Stan for faster
computation.

 Leave one out cross-validation

Leave one out cross-validation (LOO-CV) is a statistical method for evaluating predictive
ability, where one observation at a time (in this study, harvest per report) is excluded to later
predict the remaining data of a model (Bürkner et al., 2020). Predictive performance was
compared between models (in this study, the null model was compared to the covariate model).
The comparison was performed through expected log-pointwise density (ELPD), which shows
the relative predictive ability.

Since LOO-CV can be computationally expensive, Pareto Smoothed Importance Sampling
Leave One Out Cross-validation (PSIS-LOO-CV) was used to approximate out of sample
predictive density. This method makes it possible to use LOO-CV, where all the observations,
in this case, reports, were included. Furthermore, PSIS-LOO-CV also flags observations where
the approximation of predictive density is deemed unreliable (i.e., flagged observations- in this
study, hunting reports). Based on Vehtari et al. (2017) recommendations, exact LOO-CV was
performed on observations with a Pareto-k value higher than 0.7. Furthermore, to reduce high
computational demands, exact LOO-CV was only performed if the number of flagged
observations did not exceed ten observations per model.

Evaluation of predictive performance was performed by examining the difference in expected
log-pointwise predictive density (ELPD) and standard error (SE) for each species and model.
A higher ELPD difference is a safer range to ensure that ELPD differences are valid with other
datasets than this one. Two SE differences were considered weak support, and a difference of
four or more SE was considered strong support.

 3. Results
3.1 Model selection
During model selection, Pareto Smoothed Importance Sampling Leave One Out Cross-
Validation (PSIS-LOO-CV), approximated out of sample predictive density where LOO-CV
were not reliable (flagged observations, in this case, reports). Out of ten species (eleven species
including models for wild boar including wildlife accidents), seven species had less than ten
flagged observations with a Pareto k-value higher than 0.7. Rock ptarmigan, American mink,

 15

and greylag goose had more than ten flagged observations with a Pareto k-value higher than 0.7
and were further excluded from exact leave one out cross-validation.

 Evaluation of predictive performance

Evaluation of predictive performance was performed by examining the difference in expected
log-pointwise predictive density (ELPD) and standard error (SE) (ELPD/SE). The higher
difference between ELPD and SE, the more valid the model is for other datasets. For nine out
of ten species, the full covariate model, M , ranked highest of the four models (Table 2). The
reduced covariate model (M ) ranked highest for greylag goose.

The full covariate model only had a pronounced effect for beaver and wild boar (excluding
wildlife accidents) with a SE four times smaller than ELPD (Table 2). For wild boar (including
wildlife accidents), red fox, mountain hare, and common eider, with a SE two to three times
smaller than ELPD, there was weak support that the covariate model would perform better with
other datasets. Finally, for nine out of ten species, at least one covariate had an effect on harvest
estimates. For rock ptarmigan, covariates did not have an apparent effect on harvest estimates
(Table 2).

Table 2. Table shows covariate model predictive performance (column two) and the number of
covariates with an effect on hunting rate (column three).
 Species ELPD (SE) Nr of covariates
 Wild boar -72.5 (29) 10
 Wild boar (incl. wildlife accidents) -99.5 (30.2) 12
 Red fox -41.1 (17) 5
 Greylag goose -5.3 (4.6) 4
 Western capercaillie -16.6 (10.4) 7
 Mountain hare -22.7 (11.4) 5
 European pine marten -2.9 (2.7) 1
 American mink -11.5 (14.2) 3
 Eurasian beaver -40.9 (9) 2
 Common eider -15.2 (5.6) 1
 Rock ptarmigan - 0

 Evaluation of the 95 % credible interval
All game species were initially tested with all 15 covariates (16 for wild boar, including wildlife
accidents). However, if any credible interval overlapped with zero or there was a correlation
between covariates, these covariates were removed.

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For nine out of ten species, at least one covariate either had a positive or negative effect on
harvest estimate (Figure 2, Table 2). For example, a positive effect of temperature on harvest
estimate indicates that the higher temperature within an HMP, the more harvest there is for
respective species. A positive effect of West/East (gradients) indicates that the further east in
Sweden, the more harvest there is for separate species. If the credible interval is close to zero,
there is an effect of that covariate on harvest estimate, however vague. If the credible interval
is further away from zero, there is a strong effect on the covariate on harvest estimate.

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Figure 2: Boxplot over covariate effect on hunting of A. wild boar, B. wild boar including
wildlife accidents, C. red fox, D. greylag goose, E. western capercaillie, F. mountain hare, G.
European pine marten, H. American mink, I. Eurasian beaver, and J. common eider. The plot
shows the 95 % credible intervals, where the box represents 50 % of all values and the
distance between the percentiles. The black line inside the box indicates the median, and the
dashed line outside the box represents outliers. A negative/positive effect indicates less/more
hunting for the respective covariate. Abbreviations: veg = vegetation, temp. = temporary, CL
= Coastline.

3.2 Posterior prediction sampling
During posterior prediction sampling on unreported areas, all parameters from earlier steps
were used to predict how much has been hunted in the unreported areas. Results are shown with
a 95 % credible interval.

The posterior prediction sampling was performed for the covariate and null model for nine out
of ten game species (excluding rock ptarmigan were no covariates had a clear effect). The 95
% credible interval for the null model and covariate model is shown in figure 3, where a smaller
bar indicates smaller credible interval. The red bar indicates the covariate model, and the grey
bar indicates the null model. By looking at the figures in Figure 3, the credible interval for the
two models differed. In general, the credible interval between the covariate model was smaller

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than the null model, and the covariate model estimated a lower median than the null model.
However, this does not apply for all species. For example, European pine marten, where there
was no pronounced difference in the credible interval between models. For American mink, the
credible interval seems to differ for almost every county. By looking at the figures for red fox
and greylag geese, the credibility intervals differ more in counties in northern Sweden than in
counties in southern Sweden. As for common eider, which was only found in three counties,
the credible interval for the covariate model was larger than the credible interval for the null
model (county 14, Figure 3J). However, reports for common eider were only found in three
counties. A list of counties is found in Appendix A.

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Figure 3: Variation between models shown with the 95 % credible interval of A. wild boar, B.
wild boar including wildlife accidents, C. red fox, D. greylag goose, E. western capercaillie, F.
mountain hare, G. European pine marten, H. American mink, I. Eurasian beaver, and J.
common eider. Numbers on the x-axis represent the respective county. The upper and lower
limit indicates the 2.5 % and 97.5 % percentiles, and the middle line indicates the median of a
95% credible interval.

 Predicted median among HMPs

During posterior predictive sampling, each model and species received an estimated median for
each HMP. Then, a comparison between models was performed by dividing the estimated
median for the null and covariate model (Median of null model/Median of covariate model)
(Figure 4). A positive value (blue colour) implies that the estimated median for the null model
was higher than the covariate model, and a negative value (red colour) implies that the covariate
model estimated a higher median than the null model. The white colour implies that the ratio is
one.

By looking at the figures, the estimated median for the null model was higher than the covariate
model in northern parts of Sweden for greylag goose and Eurasian beaver. As for common
eider, which is only felled in three counties, one cannot determine which model estimated a
higher median by looking at the figure. There was no apparent difference in the estimated
median throughout Sweden for the rest of the game species, and no clear spatial patterns were
discovered.

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21

Figure 4. Map over Swedish HMP estimated median of A. wild boar, B. wild boar including
wildlife accidents, C. red fox, D. greylag goose, E. western capercaillie, F. mountain hare, G.
European pine marten, H. American mink, I. Eurasian beaver, and J. common eider. Blue areas
represent, where the estimated median was higher for the null model, and orange represents
where the estimated median was higher for the covariate model. Values are shown in a ratio
on log scale, and the white area represents where the ratio is 1, indicating no difference in the
estimated median between the two models.

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Predicted variation among HMPs

The coefficient of variation among HMPs was used to visualize the difference in variation in a
scale-free measurement. The coefficient of variation (CV), calculated from each HMPs
predictive posterior mean and standard deviation for both the covariate and null model, was
compared for each HMP and county (Figure 5). The comparison was made by evaluating the
ratio between CV between the two models (CV of null model/ CV of covariate model), where
a positive value implies that the CV was higher for the null model (red colour), and a negative
value implies that the CV was higher for the covariate model (green colour).

The total number of HMPs with a smaller ratio of the CV of the covariate model was for wild
boar (169/243), wild boar (including wildlife accidents) (186/243), red fox (265/306), greylag
goose (278/298), Western capercaillie (202/271), American mink (298/306), Eurasian beaver
(151/249), common eider (55/76), mountain hare (154/306), and European pine marten
(179/301).

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Figure 5. Map over Swedish HMP showing coefficient of variation of A. wild boar, B. wild boar
including wildlife accidents, C. red fox, D. greylag goose, E. western capercaillie, F. mountain
hare, G. European pine marten, H. American mink, I. Eurasian beaver, and J. common eider.
Red areas represent where the ratio of CV was smaller for the covariate model green represents
where the ratio of CV was smaller for the null model. Values are on semi-log scale, and the
white area indicates that the ratio is 1, indicating no difference in CV between the two models.

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Predictive variation at high hunting levels

Prediction variation at high hunting levels was performed by calculating the coefficient of
variation (CV) difference between the covariate and null models ( - 0 ) and plotting
these together with each county's predictive median. A positive slope indicates an increased
harvest (higher median) coupled to smaller variation (i.e., smaller predictive variation in CV)
for the covariate model. For western capercaillie, mountain hare, European pine marten,
Eurasian beaver, and common eider, there is a positive slope, which implies that the CV for the
covariate model decreases when harvest per county (positive slope) increases (Figure 5).
However, harvest of common eider is only reported for three counties, and therefore it could be
challenging to say with certainty that the covariate model's predictive variation in harvest at
high levels is smaller or greater than the null model. For wild boar (both models), red fox,
greylag goose, and American mink the CV for the covariate model increases when harvest per
county (negative slope) increased (Figure 5).

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Figure 5: Result of correlation between models in coefficient of variation (CV) and the amount
of hunting (median) of A. wild boar, B. wild boar including wildlife accidents, E. red fox, D.
greylag goose, E. western capercaillie, F. mountain hare, G. European pine marten, H.
American mink, I. Eurasian beaver, and J. common eider.

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4. Discussion
Harvest estimates are among the most important activities since it can function as an indicator
of population size changes (Skalsi et al., 2006: Bergqvist et al., 2016), and it should be
continuously adapted to species distribution, land use, and biotic/abiotic factors. However,
harvest reporting systems vary among countries, depending on local conditions and preferences
(Åhl et al., w.y). In Sweden hunting reporting is voluntary for most common felled game, which
results in a variation in the number of reports between hunting teams, HMP, and counties.
Therefore, there is a demand for new statistical methods for quantifying harvest in Sweden. To
our knowledge, the implemented method of using Bayesian modelling and covariates at the
precinct level, is a novel model for felled game.

The results from this study show that harvest estimates can be improved using covariates at the
precinct level. The covariates model ranked highest for all ten species, and out of the ten species,
six had an ELPD difference of two to four, which implies that, depending on game species,
there is both strong and week support that the covariate model will be a better predictor for
other datasets than this one. At least one covariate had an apparent effect on harvest estimates
for nine out of ten species. In addition, the covariate model reduced the large uncertainties,
which is an improvement of the null model.

For all nine species, excluding rock ptarmigan, the covariate model estimated a smaller CV
(coefficient of variation, a scale-free measurement of the variation) and a 95 % credible interval
than the null model, indicating that the large uncertainties were removed by including covariates
in the model, and hunting estimates became even more precise.

During the evaluation of the 95 % credible interval, this study found both expected and
unexpected effects. Some of the expected effects were that infrastructure and farmland had a
positive effect on harvest estimations for red fox, which implies that the more areas of
infrastructure and farmland there are in HMPs, the more red fox is felled. The effects were
expected since a landscape with increased levels of farmland and infrastructure provides good
food resources for the red fox. Second, open areas without vegetation and farmland had a
negative effect on western capercaillie harvest estimations, i.e., less western capercaillie is
felled in HMPs where there are areas of farmland and open areas without vegetation. The
western capercaillie is mainly located in northern parts of Sweden, where agricultural land is
scarce. In addition, western capercaillie is disadvantaged by exploited areas and open areas.

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Lastly, coastline for the sea had a strong positive effect on the American mink, which is
probable as the coastline is a part of the American mink’s natural habitat.

The unexpected effects from covariates include, for example, that agricultural land had a
positive effect on harvest for beaver, i.e., the more farmland land, the more beavers are felled
in HMPs. One explanation could be that there is a higher abundance of beaver populations in
southern Sweden, where there are larger agricultural land areas than in the northern parts of
Sweden. For mountain hare, there were quite contradictory results. There were two distinct
north and south gradients, precipitation and humidity. These two imply that the more
precipitation and humid it is, the more mountain hare is felled. Precipitation was an expected
effect since precipitation is higher around the mountain range and abundance for mountain hare
is higher in the northern parts of Sweden. However, it is most humid in the southern parts of
Sweden (SMHI, w.y). In addition, the results also show that more mountain hare is felled in
southern Sweden, which can be misleading as most mountain hare hunting is located in the
northern part of Sweden.

Two models were tested for wild boar, with wildlife accidents, and without wildlife accidents.
The number of covariates varied from 12 (including wildlife accidents) to 10 (excluding
wildlife accidents). One hypothesis was that including wildlife accidents in the model would
substantially affect other covariates since it is a strong predictor. If there is a high abundance
of wild boar in an HMP, it is more likely that there is an increase of wildlife accidents. However,
the wildlife accidents covariate results showed a negative effect on the harvest estimates, which
might seem like a contradiction as a higher abundance of wild boar should increase the risk for
accidents. Another contradictory result was that water had a positive effect, while coastline for
sea/lake and river had a negative effect. One possible explanation for these results (both for
mountain hare and wild boar) is that there could be a correlation between covariates, that they
are very alike, contributing to a complicated variation composition of several factors.

The null model by Lindström and Bergqvist found that the inclusion of rate variability
association was dubious. However, when covariates were added to the model, it turned out that
the inclusion of rate-variability association was valuable information to include in the model.
The full covariate model received a higher difference in ELPD and SE for nine out of ten game
species during the evaluation of predictive performance. For greylag goose, however, the
reduced covariate model ranked highest. For greylag goose, when including rate-variability
association to the model, not all MCMC chains converged. However, when excluding rate-
variability association from the model, all chains converged. Greylag goose is felled in 20
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(reported) counties and in 298 (reported) HMPs, with, during the hunting year 2018/2019, over
7000 reports for greylag goose. However, even if greylag goose is felled in large parts of the
country, a possible explanation for why parameter (2) did not converge may be due to greylag
goose being felled by only a few hunters, which means that the variation within HMPs could
be high.

Although the covariate model ranked highest for rock ptarmigan, there was no apparent effect
of covariates for that species. A possible explanation may be that rock ptarmigan is felled in
only three counties, all located close to mountain chains. For this study, counties with no
hunting for each species were not included, and if it had, the result for rock ptarmigan would
probably have turned out differently, and the results would have shown an effect of several
factors. However, when only including the actual area where the species was felled, it seems to
have a more negligible effect.

Other countries in Europe (e.g., Cyprus and Finland) are experiencing similar problems with
voluntary hunting report systems, as in Sweden, and experiences that harvest estimates are
sensitive to deviating values at low reporting levels (Åhl et al., 2020). In addition, in Europe,
there is an ongoing problem: where hunting estimates do not take into account any
measurements of uncertainty and that the hunters who report may not represent all hunters. This
suggests that there is a need to develop better methods for harvest reporting and harvest
estimations. By including valuable information about land cover, geographical distribution,
wildlife accidents, and the impact of climate, other countries other than Sweden can benefit
from this kind of harvest estimation model, where covariates are taking into account.

There is no method for monitoring population size and distribution for many of the game
species in Sweden, even if harvest is high or low. Therefore, harvest estimates are vital for
wildlife management since it could indicate changes in population size (monitoring long-term
survival of wildlife species) (Bergqvist et al., 2016). However, if harvest estimates should
work as an indicator of population size, the estimates should be as accurate as possible. Even
if the null model, created by Lindström and Bergqvist, improved the former harvest
estimation model, the uncertainties were still too large for most game species. The covariate
model reduced the large uncertainties and estimated a more precise harvest estimate.
Furthermore, a model with a lower credible interval can detect trends in hunting patterns over
the years, which also is important for species distribution and hunters game preference. This
implies that the covariate model can be used as a tool for population size and distribution

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changes, which is vital for wildlife management, especially since there is no reliable
methodology for inventorying many of the game species.

There are still some limits to the covariate model. During posterior prediction for unobserved
hunting, cross-validation with training and validation data was not applied due to the time limit,
limiting the model. Without cross-validation with training and validation data, the performed
prediction can still be uncertain, even if the performed prediction were reliable. Another aspect
necessary to consider was that the covariate model may be slightly over-fitted and may not
work so well on other datasets. Further analysis should therefore include cross-validation with
training and validation data to reduce any bias in the result. However, the results still show that
the covariate model results in a smaller credible interval in posterior prediction than the null
model, and LOO-CV was lower for all species, although there was not always a significant
enough difference. Lastly, there is still limited knowledge of which hunting teams that report
their harvest. Even if SAHWM continuously informs and encourages hunting teams to report
their total harvest, low reporting remains a problem since there are uncertainties about hunting
and reporting preferences. Therefore, if harvest reporting should increase for all hunting teams
throughout Sweden, and if the covariate model would be implemented as a harvest estimation
model for all game species in Sweden, harvest estimates would work as a strong indicator of
population size and distribution changes.

4.1 Conclusion
Since reporting is voluntary for most common felled game, the number of reports can vary
between hunting teams, HMP, and counties. There is therefore a demand for statistical methods
to estimate harvest in Sweden. In this study, I assessed and expanded a model for estimating
the relevance of covariates for harvest estimation of felled game.

This study aimed to improve the current estimation method, and by including covariates at the
HMP level, this study has developed a framework for studying the effect of climate, land cover,
and geographical distribution on harvest estimation. Second, this study suggests that harvest
estimates can be improved by taking these factors (covariates) into account. Based on LOO-
CV and the 95% credibility interval, which did not include zero for the remaining covariates, it
can be said that the covariate model is a better model than the null model for nine out of ten
focal species. To conclude, by taking covariates into account as the finest spatial resolution that
can be considered from the available data, instead of only using HMP as the finest scale, the
covariate model reduced the predictive variation. Using covariates at the precinct level, models

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