Introduction

Generalized Linear Models (GLM) are an extension of ‘simple’ linear regression models, which predict the response variable as a function of multiple predictor variables. Linear regression models work on a few assumptions, such as the assumption that we can use a straight line to describe the relationship between the response and the predictor variables. This implies that a constant change in a predictor leads to a continuous change in the response variable. This assumption is often violated in ecological data; therefore, these models are extended into GLMs to deal with non-normal distributed data.

GLMs find the equation that best predicts a species’ occurrence for the values of the environmental variables. The model has three critical parts

1. The probability distribution of the response variable.
2. The linear predictor (LP), a combination of all predictor variables representing an overall score for the environmental suitability.
3. The link function, describing how the mean of the response relates to the linear predictor.

Thus, the relationship between the response and the predictors is not linear, but the link function provides a transformation of the response so that the transformed response is linearly related to the predictors.

A GLM with binomial data, such as the presence/absence of a species, is commonly called “logistic regression”. In this case, the link function is a logit function, which is the log of the odds ratio (probability of presence/probability of absence) (Figure 1). Figure 1. Plot showing how the relationship between a binary response variable and predictors can be made linear through transformation.

The coefficient of a predictor variable (the number used to multiply a variable) in a logistic regression model can be easily interpreted, as in the following hypothetical example. Suppose a predictor, such as average annual temperature, has a positive coefficient of 0.3 in an estimated model of the species occurrence. In that case, this implies that a one-unit increase in temperature results in a rise in exp (0.3) = 1.35 (the log-odds ratio), or 35%, in the probability of species presence.

The  values of the variable coefficients are estimated by maximum likelihood estimation (MLE), which maximizes the "agreement" of the predicted species occurrences with the observed data. In other words, MLE finds the values of the coefficients that result in a model under which you would be most likely to get the observed results. Most GLM models, including the GLM provided in EcoCommons, use the iteratively reweighted least squares (IWLS) method for MLE.

• The response variable can have any form of exponential distribution type
• Able to deal with categorical predictors
• Relatively easy to interpret and allows a clear understanding of how each of the predictors are influencing the outcome
• Less susceptible to overfitting than for example CTA or MARS algorithms

Limitations

• Needs relatively large datasets. The more predictor variables, the larger the sample size (N) required. As a rule of thumb, the number of predictor variables should be less than N/10.
• Sensitive to outliers

Assumptions

No assumptions are made about the distributions of the environmental variables. However, they should not be highly correlated with one another because this could cause problems with the estimation.

Requires absence data

Yes.

Configuration options

EcoCommons allows the user to set model arguments as specified below.

 random_seed Seed used for generating random values. Using the same seed value, i.e. 123, ensures that running the same model, with the same data and settings generates the same result, despite stochastic processes such as machine learning or cross-validation. Number of repetitions (nb_run_eval) Integer value, corresponding to the number of repetitions to be done for calibration/validation splitting. (default = 10) Data split percentage (data_split) Numeric value between 0 and 100, corresponding to the percentage of data used to calibrate the models (calibration/validation splitting). (default = 100) prevalence Allows to give more or less weight to particular observations; default = NULL: each observation (presence or absence) has the same weight; if value < 0.5: absences are given more weight; if value > 0.5: presences are given more weight. (algorithm parameter) Variable importance (var_import) Integer value, corresponding to the number of permutations to be done for each variable to estimate variable importance. (default = 0) Scale models (rescale_all_models) A logical value defining whether all models predictions should be scaled with a binomial GLM or not. (default = FALSE) Evaluate all models (do_full_models) A logical value defining whether models calibrated and evaluated over the whole dataset should be computed or not. (default = TRUE) Regression type (type) Type of regression to model: linear ("simple"), quadratic or polynomial. (default =  quadratic) Variable interaction (interaction_level) Number of interactions between predictor variables that need to be considered. (default = 0 ) Test fit model (test) Criteria that should be used to test the fit of the model in stepwise predictor selection; if 'none' the stepwise procedure will be switched off. (default = AIC ) Family (family) Description of the error distribution of the response variable and the link function used in the model. (default = binomial) Mean start value (mustart) Starting values for the vector of means. (default = 0.5 ) Tolerance (control_epsilon) Positive convergence tolerance. (default =  1e-8) Maximum interactions (control_maxit) The maximum number of IWLS iterations to find maximum likelihood estimates. (default = 25 ) Interaction output (control_trace) If output should be produced for each IWLS iteration. (default = FALSE )

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