Know Why Generalized Linear Model is a Remarkable Synthesis Model!

# Know Why Generalized Linear Model is a Remarkable Synthesis Model!

Last updated:
15th Jun, 2023
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## Understanding the basics

GLM is very famous among individuals who deal with different regression models starting from Classical Linear Regression Models to Models for Survival Analysis. The term generalized linear model (GLIM or GLM) was coined and familiarized by McCullagh (1982) and Nelder (2nd edition 1989). GLM, in the simplest form as described in Rutherford 2001,  Data = Model + Error. It has a useful framework, which is the foundation of various statistical tests.

## What is a Generalized linear model?

Regression models in the generalized linear model (GLM) category can be used to simulate a variety of interactions between a response variable and one or more predictor variables. The generic linear model in R is a family of regression models that accommodates non-normal distributions and may be used in the software by using a function that accepts a number of parameters. A modification to the generalized linear model (GLM), known as a generalized linear mixed model (GLMM), includes random effects in addition to the typical fixed effects in the linear predictor.

The dependent variable is linearly associated with the components and covariates via a given link function in the generalized linear model, which extends the general linear model in R. The model also permits an abnormal distribution for the dependent variable. Through its extremely general model formulation, it covers a wide range of statistical models, including logistic models for binary data, log-linear models for count data, complementary log-log models for interval-censored survival data, and linear regression for normally distributed responses.

GLMs identify the equation that, given the values of the environmental factors, most accurately predicts the existence of a species. Three crucial elements make up the model:

• The response variable’s probability distribution.
• An overall score for the appropriateness of the environment is represented by the linear predictor (LP), which is a mixture of all predictor factors.
• The link function describes the relationship between the mean of the response and the linear predictor.

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## Revisiting the class of models

• Classical Linear Regression (CLR) Models, also referred to as Linear Regression models
• Analysis of Variance (ANOVA) models.
• Models which predict the odds of winning like the probability of machine failure
• Models used for explaining and predicting event counts
• Models for estimating lifespans of living and non-living things such as a processor or biological age of a plant etc.

Generalized Linear Model, as the name suggests, is like a canopy for all the above-given models with improved calculations and approximations.

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### SAS Laboratory

The next section takes into account two primary strategies. On the one hand, the evolution of the GLM scorecard is examined. But we are concentrating on machine learning. By concentrating on the primary phases of the modeling workflow, a general overview of SAS implementation is given. SAS makes it simple to recreate the details provided in the earlier parts.

SAS laboratory is GLM-focused.

One must first submit the same data file, after which the analysis is replicated using the next several stages in the SAS code below:

1. Establish a default flag,
2. Prepare and evaluate samples,
3. Explain binning and calculate WOE.
4. Conduct correlational research, and
5. Fit logit regression

## The Structure of Generalized Linear Models

A generalized linear model (or GLM1) consists of three major components:

1. Random Component: A random component known as a noise model or error model is the probability distribution of the response variable (Y).
2. Systematic Component : A linear predictor is a linear function of regressors, as mentioned below:

ηi = α + β1Xi1 + β2Xi2 +···+ βkXik

1. Link Function (denoted by η or g(μ) ): As the name suggests its the link between systematic and random components

Example : μi = E(Yi), to the linear predictor g(μi) = ηi = α + β1Xi1 + β2Xi2 +···+ βkXik

Generalized Linear Model applies to data by the process of maximum likelihood. This provides the estimates of the regression coefficients and estimated asymptotic standard errors of the coefficients.

The basic GLM for count data is the Poisson model with a log link. However, when the response variable is a count, its conditional variance increases more rapidly than its mean, producing a condition termed overdispersion and invalidating the use of the Poisson distribution. The quasi-Poisson GLM adds a dispersion parameter to handle overdispersed count data.

In general terms, quasi-likelihood estimation is one way of allowing for overdispersion, which is more significant variability in the data than expected from the statistical model used.

A similar model is based on the negative binomial distribution, which is not an exponential family. Negative-binomials in Generalized Linear Model cannot be determined by maximum likelihood. The zero-inflated Poisson regression model may be best suitable when there are more zeroes in the data than consistent with a Poisson distribution.

Features of Generalized linear model

The ability to scale- Large datasets and sophisticated models may be employed with GLMs because of their effective model fitting and prediction techniques.

Regularization- Using methods, GLMs may be regularized to decrease overfitting and enhance model performance.

The robustness- Due to the fact that they accept non-normal distributions of the response variable, GLMs can be resilient to outliers and other abnormalities in the data.

Simple to use- Particularly when contrasted to more intricate models like neural networks or decision trees, GLMs are comparatively simple to comprehend and apply.

Flexibility– Linear, logistic, Poisson, and exponential connections between the response and predictor variables can all be modeled using GLMs.

Interpreting a model- The link between the response and predictor variables, as well as each predictor’s impact on the response, are clearly interpreted by GLMs.

## Advantages of the Generalized Linear Model over traditional Ordinary Least Square (OLS) regression

There are many advantages of General Linear Models over the OLS regression, which can be summarised as below:

• Unlike OLS regression, the response Y is not required to be transformed every time to have a normal distribution.
• Modeling is more flexible as choosing a link is different from choosing a random component.
• A constant variance is NOT needed if the link gives additive effects.
• We have the optimal properties of the estimators as the models are attached via Maximum Likelihood estimation.
• All the inference tools and model checking for log-linear and logistic regression models apply for other GLMs too.
• There is usually only one process (procedure or function) in a software package to capture all the models listed in the table above; take, for instance, glm() (R Language) or PROC GENMOD (SAS).

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## Disadvantages of the Generalized Linear Model

Apart from the above-listed advantages, there are two major disadvantages which are important to know:

• Some restrictions like Linear function can have only a linear predictor in the systematic component.
• Responses cannot depend on each other.
• Makes strict assumptions on the distribution’s shape and the error terms’ unpredictability
• Prone to overfitting if the model is overly intricate or has an excessive number of predictor variables.
• Low predictability

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## Conclusion

If we summarize all the above information, we found that GLM is convenient with lower complexity. With GLM, response variables can have any form of exponential distribution type. Apart from this, it can deal with categorical predictors. The general linear model is relatability easy to interpret and allows a clear understanding of how each predictor is influencing the outcome.

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Director of Engineering @ upGrad. Motivated to leverage technology to solve problems. Seasoned leader for startups and fast moving orgs. Working on solving problems of scale and long term technology strategy.
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1What is a Poisson regression model?

2How is a general linear model different from a generalized linear model?

3What are some of the assumptions that a generalized linear model makes?

The majority of GLM assumptions are comparable to linear regression models, but some of the linear regression assumptions are changed. The data in a GLM is assumed to be independent and random. Errors are considered independent as well, although they don't have to be regularly distributed. While the response variable is not required to be independent, the distribution should belong to the exponential family.

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