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.