Generalized Additive Mixed Modelsesapubs.org/archive/ecol/E094/238/appendix-C.pdf · Generalized...
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Generalized Additive Mixed Models
Initial data-exploratory analysis using scatter plots indicated a non linear dependence of
the response on predictor variables. To overcome these difficulties, Hastie and Tibshirani (1990)
proposed generalized additive models (GAMs). GAMs are extensions of generalized linear
models (GLMs) in which a link function describing the total explained variance is modeled as a
sum of the covariates. The terms of the model can in this case be local smoothers or simple
transformations with fixed degrees of freedom (e.g. Maunder and Punt 2004). In general the
model has a structure of:
Where and has an exponential family distribution. is a response variable, is
a row for the model matrix for any strictly parametric model component, is the corresponding
parameter vector, and the are smooth functions of the covariates, .
In regression studies, the coefficients tend to be considered fixed. However, there are
cases in which it makes sense to assume some random coefficients. These cases typically occur
in situations where the main interest is to make inferences on the entire population, from which
some levels are randomly sampled. Consequently, a model with both fixed and random effects
(so called mixed effects models) would be more appropriate. In the present study, observations
were collected from the same individuals over time. It is reasonable to assume that correlations
exist among the observations from the same individual, so we utilized generalized additive
mixed models (GAMM) to investigate the effects of covariates on movement probabilities. All
the models had the probability of inter-island movement obtained from the BBMM as the
dependent term, various covariates (SST, Month, Chlorophyll concentration, maturity stage, and
wave energy) as fixed effects, and individual tagged sharks as the random effect. The GAMM
used in this study had Gaussian error, identity link function and is given as:
Where k = 1, …q is an unknown centered smooth function of the kth covariate and
is a vector of random effects following All models were implemented using the
mgcv (GAM) and the nlme (GAMM) packages in R (Wood 2006, R Development Core Team
2011).
Spatially dependent or environmental data may be auto-correlated and using models that
ignore this dependence can lead to inaccurate parameter estimates and inadequate quantification
of uncertainty (Latimer et al., 2006). In the present GAMM models, we examined spatial
autocorrelation among the chosen predictors by regressing the consecutive residuals against each
other and testing for a significant slope. If there was auto-correlation, then there should be a
linear relationship between consecutive residuals. The results of these regressions showed no
auto-correlation among the predictors.
Predictor terms used in GAMMs
Predictor
variable
Type Description Values
Sea surface
temperature
(SST)
Continuous Monthly aver. SST on each of the grid cells 20.7° - 27.5°C
Chlorophyll a
concentration
(Chlo)
Continuous Monthly aver. Chlo each of grid cells 0.01 – 0.18 mg m-3
Wave energy Continuous Monthly aver. W. energy on each of grid cells 0.01 – 1051.2 kW m-1
Month Categorical Month the Utilization Distribution
was generated
January to December (1-
12)
Maturity stage Categorical Maturity stage of shark Mature male TL> 290cm
Mature female TL > 330
cm
Distribution of residual and model diagnostics
The process of statistical modeling involves three distinct stages: formulating a model, fitting the
model to data, and checking the model. The relative effect of each xj variable over the dependent
variable of interest was assessed using the distribution of partial residuals. The relative influence
of each factor was then assessed based on the values normalized with respect to the standard
deviation of the partial residuals. The partial residual plots also contain 95% confidence
intervals. In the present study we used the distribution of residuals and the quantile-quantile (Q-
Q) plots, to assess the model fits. The residual distributions from the GAMM analyses appeared
normal for both males and females.
Males
Residuals distribution
Residuals
Fre
quen
cy
-2 0 2 4
020
0400
600
800
1000
12
00
-4 -2 0 2 4
-20
24
Q-Q plot
Theorethical quantiles
Sam
ple
quantile
s
Females
Hastie, T.J., and R.J. Tibshirani. 1990. Generalized Additive Models. CRC press, Boca Raton,
FL.
Latimer, A. M., Wu, S., Gelfand, A. E., and Silander, J. A. 2006. Building statistical models to
analyze species distributions. Ecological Applications, 16: 33–50.
Maunder, M.N., and A.E. Punt. 2004. Standardizing catch and effort: a review of recent
approaches. Fisheries Research 70: 141-159.
Wood, S.N. 2006. Generalized Additive Models: an introduction with R. Boca Raton, CRC
Press.