General Linear Regression Model may be considered because there are circumstances where the linear regression model may not be suitable

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Generalized Linear Regression

It is an advanced statistical modeling technique formulated by John Nelder and Robert Wedderburn in 1972. It encompasses many other models, which allows the response variable y to have an error distribution other than a normal distribution. The models include Linear Regression, Logistic Regression, and Poisson Regression. In a Linear Regression Model, the dependant variable ‘y’ is expressed as a linear function of all the independent variables 'x .'The underlying relationship between the response and the predictors is linear; we can visualize the relationship in a straight line.

Generalized linear regression models allow us to build a linear relationship between the response and predictors, even though their underlying relationship is not linear. We can use a link function, which links the response variable to a linear model. Unlike Linear Regression models, the error distribution of the response variable need not be normally distributed. The errors in the response variable are assumed to follow an exponential family of distribution (i.e., normal, binomial, Poisson, or gamma distributions) since we are trying to generalize a linear regression model that can also be applied in these cases, the name Generalized Linear Models.

General Linear Regression Model may be considered because there are circumstances where the linear regression model may not be suitable. Such cases are like when the relationship between X and y is not linear. There exists some non-linear relationship between them. For example, y increases exponentially as X increases. Also, when the variance of errors in y (commonly called Homoscedasticity in Linear Regression) is not constant and varies with X.

Components of Generalized Linear Regression Model

A GLM consists of three components: 

·         A random component,

·         A systematic component, and

·         A link function.

Systematic Component/Linear Predictor:

It is just the linear combination of the Predictors and the regression coefficients.

β0+β1X1+β2X2

Link Function:

Represented as η or g(μ), it specifies the link between a random and systematic components. It indicates how the expected/predicted value of the response relates to the linear combination of predictor variables.

Random Component:

It refers to the probability distribution, from the family of distributions, of the response variable.

The family of distributions, called an exponential family, includes normal distribution, binomial distribution, or poisson distribution. Normal distribution falls under the identity function, binomial function falls under logit or sigmoid function. Poisson distribution falls under log function.

Assumptions of Generalized Linear Regression Model

There are some basic assumptions for Generalized Linear Models as well. Most of the premises are similar to Linear Regression models, while some of the beliefs of Linear Regression are modified. The premises are as follows;

·         The original response variable need not have a linear relationship with the independent variables, but the transformed response variable (through the link function) is linearly dependent on the independent variables.

·         Homoscedasticity (that is, constant variance) need not be satisfied. Response variable Error variance can increase or decrease with the independent variables.

·         The response variable y does not need to be normally distributed, but the distribution is from an exponential family (example; binomial, Poisson, multinomial, normal)

 

·         Feature engineering on the Independent variable can be applied, i.e., instead of taking the original raw independent variables, variable transformation can be done, and the transformed independent variables, such as taking a log transformation, squaring the variables, reciprocal of the variables,  can also be used to build the GLM model.

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