Description

This function fits the aggregated source data using mixed models. The fitting procedures are done by the package spaMM which we use to jointly fit the mean isotopic values and their associated residual dispersion variance in a spatially explicit manner.

Usage

Arguments

Details

The detailed statistical definition of the isoscape model is described in Courtiol & Rousset 2017 and summarized in Courtiol et al. 2019.

Briefly, the fitting procedure of the isoscape model is divided into two fits: mean_fit and disp_fit. mean_fit corresponds to the fit of the "mean model", which we will use to predict the mean isotopic values at any location in other functions of the package. disp_fit corresponds to the fit of the "residual dispersion model", which we will use to predict the residual dispersion variance associated to the mean predictions. mean_fit is a linear mixed-effects model (LMM) with fixed effects, an optional spatial random effect with a Matérn correlation structure and an optional uncorrelated random effect accounting for variation between sources unrelated to their location. disp_fit is a Gamma Generalized LMM (Gamma GLMM) that also has fixed effects, an optional spatial random effect with a Matérn correlation structure and an optional uncorrelated random effect. For the GLMM the residual variance is fixed to its theoretical expectation.

The dataframe data must contain a single row per source location with the following columns: mean_source_value (the mean isotopic value), var_source_value (the unbiased variance estimate of the isotopic value at the location), n_source_value (the number of measurements performed at the location, could be 1) and source_ID (a factor defining the identity of the sources at a given location).

The arguments mean_model_fix and disp_model_fix allow the user to choose among different fixed-effect structures for each model. These arguments are lists of booleans (TRUE or FALSE), which define which of the following fixed effects must be considered: the elevation (elev), the absolute value of the latitude (lat_abs), the squared latitude (lat_2), the longitude (long) and the squared longitude (long_2). An intercept is always considered in both models.

In the models, the mean (for the mean model) or the log residual variance (for the residual dispersion model) follow a Gaussian distribution around a constant value. The arguments mean_model_rand and disp_model_rand allow to choose among different random effects for each model influencing the realizations of these Gaussian random processes. For each model one can choose not to include random effects or to include an uncorrelated random effect, a spatial random effect, or both (default). Setting "uncorr" = TRUE implies that the realizations of the random effect differ between sources for reasons that have nothing to do with the relative geographic location (e.g. some micro-climate or some measurement errors trigger a shift in all measurements (mean model) or a shift in the variance between measurements (residual dispersion model) performed at a given source by the same amount). Setting "spatial" = TRUE (default) implies that the random realizations of the Gaussian process follow a Matérn correlation structure. Put simply, this implies that the closer two locations are, the more similar the means (or the log residual variance) in isotopic values are (e.g. because they are likely to be traversed by the same air masses).

The arguments uncorr_terms allow the choice between two alternative parametrizations for the uncorrelated random effect in the fits: "lambda" or "nugget" for each model. When using "lambda", the variance of the uncorrelated random terms is classically modelled by a variance. When a spatial random effect is considered, one can alternatively choose "nugget", which modifies the Matérn correlation value when distance between location tends to zero. If no random effect is considered, one should stick to the default setting and it will be ignored by the function. The choice of the parametrization is a matter of personal preferences and it does not change the underlying models, so the estimations for all the other parameters of the models should not be impacted by whether one chooses "lambda" or "nugget". However, only uncertainty in the estimation of "lambda" can be accounted for while computing prediction variances, which is why we chose this alternative as the default. Depending on the data one parametrization may lead to faster fit than the other.

The argument spaMM_method is also a list of two strings where the first element defines the spaMM functions used for fitting the mean model and the second element defines the spaMM method used for fitting the residual dispersion model. The possible options are "HLfit", "corrHLfit" and "fitme". Note that "HLfit" shall only be used in the absence of a Matérn correlation structure and "corrHLfit" shall only be used in the presence of it. In contrast, "fitme" should work in all situations. Which method is best remains to be determined and it is good practice to try different methods (if applicable) to check for the robustness of the results. If all is well one should obtain very similar results with the different methods. If this is not the case, carefully check the model output to see if one model fit did not get stuck at a local minimum during optimization (which would translate in a lower likelihood, or weird isoscapes looking flat with high peaks at very localised locations).

The argument dist_method allows modifying how the distance between locations is computed to estimate the spatial correlation structure. By default, we consider the so-called "Earth" distances which are technically called orthodromic distances. They account for earth curvature. The alternative "Euclidean" distances do not. For studies performed on a small geographic scale, both distance methods should lead to similar results.

The arguments control_mean and control_dist are lists that are transmitted to the spaMM fitting functions (defined by spaMM_method). These lists can be used to finely control the fitting procedure, so advanced knowledge of the package spaMM is required before messing around with these inputs.

We highly recommend users to examine the output produced by isofit. Sometimes, poor fit may occur and such models should therefore not be used for building isoscapes or performing assignments.

Value

This function returns a list of class ISOFIT containing two inter-related fits: mean_fit and disp_fit. The returned list also contains the object info_fit that contains all the call arguments.

Note

There is no reason to restrict mean_fit and disp_fit to using the same parametrization for fixed and random effects.

Never use a mean_fit object to draw predictions without considering a disp_fit object: mean_fit is not fitted independently from disp_fit.

For all methods, fixed effects are being estimated by Maximum Likelihood (ML) and dispersion parameters (i.e. random effects and Matern correlation parameters) are estimated by Restricted Maximum Likelihood (REML). Using REML provides more accurate prediction intervals but impedes the accuracy of Likelihood Ratio Tests (LRT). Our choice for REML was motivated by the fact that our package is more likely to be used for drawing inferences than null hypothesis testing. Users interested in model comparisons may rely on the conditional AIC values that can be extracted from fitted models using the function AIC from spaMM.

Variable names for data must be respected to ensure a correct utilization of this package. Alteration to the fixed effect structure is not implemented so far (beyond the different options proposed) to avoid misuse of the package. Users that would require more flexibility should consider using spaMM directly (see Courtiol & Rousset 2017) or let us know which other covariates would be useful to add in IsoriX.

Source

https://kimura.univ-montp2.fr/~rousset/spaMM.htm

References

Courtiol, A., Rousset, F. (2017). Modelling isoscapes using mixed models. https://www.biorxiv.org/content/10.1101/207662v1

Courtiol A, Rousset F, Rohwäder M, Soto DX, Lehnert L, Voigt CC, Hobson KA, Wassenaar LI, Kramer-Schadt S (2019). Isoscape computation and inference of spatial origins with mixed models using the R package IsoriX. In Hobson KA, Wassenaar LI (eds.), Tracking Animal Migration with Stable Isotopes, second edition. Academic Press, London.

Rousset, F., Ferdy, J. B. (2014). Testing environmental and genetic effects in the presence of spatial autocorrelation. Ecography, 37(8):781-790.

Bowen, G. J., Wassenaar, L. I., Hobson, K. A. (2005). Global application of stable hydrogen and oxygen isotopes to wildlife forensics. Oecologia, 143(3):337-348.

See Also

spaMM for an overview of the spaMM package

fitme and corrHLfit for information about the two possible fitting procedures that can be used here

Matern.corr for information about the Matérn correlation structure

prepsources for the function preparing the data for isofit

Examples