Comparison of Models for Analyzing Seasonal Activity using

Comparison of Models for Analyzing Seasonal Activity using Longitudinal Count Data
Daniel J. Hocking and Kimberly J. Babbitt
University of New Hampshire
Introduction
Activity patterns of most animals are influenced by environmental
conditions. A clear understanding of how organisms respond to
environmental and climatic conditions is important for biological
assessment surveys, management plans, and monitoring of
populations. It is also critical for understanding animal responses to
climate change. However, challenges arise when taking repeated
counts of animals on the same sites. The potential correlation of the
data at a given site must be accounted for to avoid
pseudoreplication.
Generalized linear mixed models (GLMM) are most frequently used
to account for correlation through random effects when interested in
count or binomial response variables. The expected count (Y) at site i
on occasion j given the independent variables (X) and the random
effect of site (bi) are related exponentially.
The regression model is linearized assuming a Poisson error
distribution and a log link function. GLMMs rely on maximum
likelihood estimation for calculating parameter estimates. Because
the counts are dependent on the random effects, GLMM estimates
are considered subject-specific (conditional). This means that the
fixed effects are interpreted as the effect of one unit change in X on
Y at a given site (on a log scale).
An alternative method of accounting for correlation within sites is to
use generalized estimating equations (GEE). For count data, GEE
models also assume a Poisson distribution and log link, but estimates
are averaged over all sites (subjects) to produced populationaveraged (marginal) coefficient estimates using a quasi-likelihood
estimator.
Additionally, the variance structure of GEE models can be explicitly
modeled and always includes an overdispersion term (), making
negative binomial and Poisson log-normal distributions unnecessary.
The ability to specify the variance structure of the model and the
overdispersion term allow for great flexibility in GEE models.
Additionally, the population-averaged estimation changes the
inference to more closely match the interest of most ecologists. The
coefficients are interpreted as the effect of one unit change in X on Y
on average across sites (on a log scale).
Objectives: To compare coefficients and model predictions using
GLMM and GEE models of red-backed salamander (Plethodon
cinereus) seasonal surface activity

Methods

We conducted nighttime visual encounter surveys on five sites in
a New Hampshire forest dominated by American beech (Fagus
grandifolia). Sites were 20-m diameter circular plots (314 m2)
We surveyed each site 91 times over four years from 2008-2011
We obtained meteorological data from nearby weather stations to
include air temperature, rainfall in the previous 24 hours, relative
humidity, number of days since previous rain (>0.1 cm), and wind
speed in our models
To account for complex phenology and responses that differ across
the year, we used a harmonic sine-cosine function of day of the
year and interactions terms with climatic conditions
We started with a beyond optimal GLMM and selected the best
nested model using AIC. Because over overdispersion in the
Poisson GLMM, we used site and observation as random effects in
all GLMM for a Poisson-lognormal model
We used the same predictor variables in the GEE model but did
not include the observation-level effect since there is an
overdispersion term
We also used mean daily conditions over the past 20 years to

Results
All models are wrong, but some are useful George E. P. Box
We observed 4,622 red-backed salamanders (10 0.6 per plot-night)
Greatest number of salamanders per site-night was 100
We observed zero salamanders on 100 of 455 site-nights

Variable
(Intercept)
airT
airT2
RainAmt24
RainAmt242
RH
windspeed
droughtdays
sin(0.0172 * DOY)
cos(0.0172 * DOY)
airT*RainAmt24
airT*windspeed
RH*windspeed
airT*sin(0.0172 * DOY)
airT*cos(0.0172 * DOY)
RainAmt24*sin(0.0172 *
DOY)
RainAmt24*cos(0.0172 *
DOY)
airT*RH
RainAmt24*droughtdays
airT*RainAmt24*sin(0.0172
* DOY)
Figures:
airT*RainAmt24*cos(0.0172
Red line = predicted
(mean) count from the GEE;
* DOY)

GLMM
Estimat GLMM
e
SE
11.028 2.239
1.416
1.679
0.641
0.307
0.947
0.350
-0.123
0.022
12.284 2.497
2.014
0.448
0.095
0.036
-1.354
0.753
-4.921
0.969
-0.267
0.281
-0.212
0.133
-1.678
0.463
1.236
0.494
3.981
0.628

GEE
Estimat
e
-9.669
4.018
0.035
0.504
-0.090
11.363
0.955
0.086
-0.333
-2.918
-0.014
-0.036
-0.931
0.479
2.379

GEE
SE
0.894
0.640
0.105
0.131
0.009
0.973
0.183
0.010
0.252
0.320
0.101
0.045
0.187
0.165
0.202

-0.642

0.324

-0.725

0.116

1.457
-1.320
-0.051

0.414
1.668
0.017

1.056
-3.228
-0.035

0.153
0.633
0.006

0.493

0.266

0.602

0.093

grey area = 95%
CI for GEE
-1.104Dark0.312
-0.707
0.113

Blue line = predicted (mean, bi=0) count from GLMM;
GLMM

Light grey area = 95% CI for

Discussion
Coefficient estimates for GLMM and GEE models were considerably
different but agreed in direction and generally in magnitude
except the intercept
Coefficients are not independently interpretable because of
potential of harmonic functions to be out of phase; therefore
predictions are needed for model comparison
GLMM and GEE models suggest very similar patterns, although
GLMM models predict slightly fewer surface active animals on
average
On the natural log scale GLMM 95% CI are uniform around the
mean estimate but on the response scale the CI increase as the
predicted values increase owing to the exponential nature of the
equation
Despite smaller coefficient SE, greater overall uncertainty in GLMM
than in GEE models
Even when conditions are favorable in the summer, few
salamanders are expected to be surface active
Red-backed salamander surface activity shows a bimodal
distribution with peak activity in mid-May and mid-October
Salamander activity in response to temperature is dependent on
season, consistent with acclimation models
Likely that salamanders have a peak activity associated with
temperature but the effects were confounded with day of the year
in these models

Recommendations

Use GEE models for count and binomial data when populationaveraged inference is of interest but data insufficient for
hierarchical detection models
Use GEE when additional variance-covariance structures need to
be specified
Plot fitted or predicted values when using GLMM to show full level
of uncertainty in estimates

Future Directions
Validate GLMM and GEE models to determine the accuracy of
predictions
Compare model selection for GLMM and GEE models using AIC and
QIC, respectively
Use simulations to evaluate the effects of spatial and temporal
replication on GLMM and GEE models
Examine how well post hoc marginalized GLMMs compare with
GEE predictions

Acknowledgments
We would like to thank J. Veysey and M. Ducey for extended discussion of
mixed models and S. Wile, E. Willey, J. Bartolotta, and M. deBethune for help
in the field. This work was funded through the UNH Agricultural Field Station
and DJH received support from the UNH COLSA, the UNH Graduate School,
and the Department of NR&E.

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