Generalized linear modeling is the process of generalizing ordinary linear regressions by creating relationships between linear models and response variables through a link function. This technique was formulated by Robert Wedderburn and John Nelder to unify other statistical models like Poisson regression, logistic regression, and linear regression.
Linear predictor: This is a linear combination of the explanatory variable (x) and parameter (b)
Link function: The link function connects, or rather, links the parameter and the linear predictor for a probability distribution. When using Poisson regression the link function is typically the log function.
Probability distribution: This component generates the observed variable (y).
You do not necessarily have to transform the response (y) in order to have a normal distribution
There is more flexibility in modeling because the link choice is separate from the random component choice
If the link has additive effects, then a constant variance is not needed
The models are fitted using maximum likelihood estimation. This ensures the maximum properties of the estimators.
Disadvantages of generalized linear modeling
Does not allow the selection of features without step-by-step selection
There are strict assumptions around the randomness of error terms and the distribution shape
The predictor variables must be uncorrelated
Generalized linear models cannot detect non-linearity directly. However, this can be addressed manually through feature engineering
Generalized linear modeling is sensitive to outliers
Low predictive power
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Assumptions of generalized linear modeling
GLMs make some strict assumptions with regard to the data structure. These assumptions are based on:
Independence of each point of data
Proper distribution of the residuals
Proper specification of the structure of the variance
A linear relationship between the linear predictor and the response (y)
For general linear modeling to be effective, the residuals must be distributed normally and the variance should remain homogeneous across all the fitted model values. Also, the response (y) must have a linear relationship with the predictors. You should always double-check your model to make sure it does not violate this assumption. Generalized linear models that do not adhere to these assumptions can lead to the falsification of results.
Generalized linear model extensions
The standard generalized linear modeling assumes that all observations are not correlated. Over the years, developments have been made to generalized linear models to enable correlation between observations. These developments have resulted in the following extensions of generalized linear models:
Generalized estimation equations: This extension allows observations to be correlated without applying explicit probability models. They are commonly used when random effects and variances are not of integral interest because they allow correlation without defining its origin. The main purpose of a generalized estimation equation is to make an approximation of the average response (y) of a given population rather than defining regression parameters. For effective data analysis, generalized estimation equations are often used together with Huber-White standard errors.
Generalized linear mixed model: This extension incorporates effects in the linear predictor producing an explicit probability model that defines the origin or a correlation. Generalized linear mixed models are also referred to as mixed models or multilevel models. They are more computationally intensive than generalized estimation equations.
Generalized additive models: In this extension of generalized linear modeling, the linear predictor is not restricted to the covariates; it is the sum of the smoothing functions estimated from the set of data being analyzed.
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