Statistical analyses
All statistical analyses were
performed with R 3.5.3
(R
Core Team, 2014).
Variations with
age
To test whether the leukocyte concentration and counts varied with age,
we used the leukocyte concentration (log-transformed) as dependent
variable and the age as an explanatory variable in a LMM and the counts
of lymphocytes, neutrophils, monocytes and eosinophils as dependent
variables and the age as an explanatory variable in four GLMMs with a
Poisson distribution (appropriate for the observed distribution of count
data). Body mass at capture, sex, capture date, year of capture and the
interactions between capture date and year of capture and between age
and sex, were further included as fixed effects. Because individuals
were sampled several times over the years, we included individual’s
identity as random intercepts (Table 1A). The functions “lmer” and
“glmer” in the package “lme4” (Bates et al., 2015) were used to fit
Linear Mixed Models (LMMs) and Generalized Linear Mixed Models (GLMMs)
(Bolker et al., 2009). Models including linear, quadratic or no effect
of age were considered and compared using Akaike’s Information Criterion
(AICc). The age effect models with the lowest AICc (Table S2) were
selected. Afterwards, exploratory variables were removed following a
backwards elimination procedure. The results obtained with the backwards
elimination procedure were confirmed using an AICc selection (Table S3).
We measured zero-inflation and variance inflation factors (VIFs) in all
our models using the R package “performance” (Lüdecke et al., 2020).
The only correlations between fixed effects we observed were between the
body mass at capture and the year or the capture date as expected from
the huge annual variability in body mass and the increase in body mass
due to fat accumulation during the active season. For all models, we
checked a posteriori the distribution of the residuals to assess
the fit of the models to the observed data. Since we observed moderate
overdispersion (all dispersion ratios < 2.58) in some of our
models (models for lymphocytes and neutrophils), we estimated all
models’ parameters using a Bayesian approach. From the final models, we
used the “sim” function from the R-package “arm” to simulate values
from the posterior distributions of the model parameters (Gelman &
Yu-Sung, 2020). The 95% credible intervals (CI) around the mean were
obtained after 5000 simulations. Assessment of statistical support was
obtained from the posterior distribution of each parameter. We
considered a fixed effect to be important if zero was not included
within the 95% CI.
Partitioned age
effect
To separate within- from
between-individual variation with age, we tested for between- and
within-individual age effect, using the same models as above but
partitioning the age of each individual into ‘average age’ and ‘delta
age’ (following
van
de Pol & Wright, 2009)) (Table 1B). ‘Average age’ corresponds to the
average of all ages at which an individual was sampled, and ‘delta age’
to the difference between its age at sampling and its ‘average age’. The
‘average age’ represents the between-individual age effect, which
corrects for the potential selective disappearance of individuals, while
‘delta age’ represents the within-individual age effect
(van
de Pol & Wright, 2009). Models were selected using both a backwards
elimination procedure and an information theoretic approach (see results
in Table S4) and statistical support for parameters were estimated as
above.
Finally, to test if the between-
and within-individual age effects were significantly different, which
would indicate selective (dis)appearance, we ran the five selected final
models including both age and ‘average age’ as explanatory variables
(Table 1C). In these models, ‘age’ represents the within-individual
effect and ‘average age’ the difference between within- and
between-individual effects
(van
de Pol & Wright, 2009).
Immune phenotype
and survival probability
We tested whether the death
probability depended on leukocyte characteristics with mixed-effects Cox
right-censored regression models
(Nenko
et al., 2018; Ripatti & Palmgren, 2000; Therneau et al., 2003). These
models included leukocyte concentration or counts as time-dependent
covariates and survival as response variable using the “coxme”
function in the “coxme” R package
(Therneau,
2018). The age at first capture and the sex were also included as fixed
effects. Individual identity and year of birth were added as random
effects to take into account repeated measurements and cohort effects
(Table 2). The data were encoded with a zero as starting point for all
individuals and with the years to death, to the end of the study, or to
the next capture (for individuals with repeated data) as stop
(Therneau,
2018). For the repeated data, the next interval started with the end of
the previous interval. A ‘1’ was assigned to the event variable, if the
individual died during the interval. We assumed that an individual died
if it was neither captured nor observed the following spring (monitored
until 2018). A hazard ratio higher than one indicates that the
corresponding explanatory variable is associated with an increased risk
to die. All individuals were followed until death (n = 27 for leukocyte
concentration and n = 43 for leukocyte counts) or still alive in 2018 (n
= 4 for leukocyte concentration and n = 6 for leukocyte counts). Three
individuals were excluded from this analysis because their fate (alive
or dead) was uncertain, due to capture permit forbidding to monitor
their families in 2017 and 2018.