Linear Regression and Generalized Linear Models (GLM) are both statistical methods used for understanding the relationship between variables. Understanding the difference between GLM and Linear Regression is essential for accurate model selection, tailored to data types and research questions. It’s crucial for predicting diverse outcomes, ensuring valid statistical inference, and is vital in interdisciplinary research. In this blog, we will learn about the differences between Linear Regression and GLM by delving into their distinct characteristics, suitable applications, and the importance of choosing the right model based on data type and research objective.
What’s the difference between GLM & Linear Regression?
Linear Regression and Generalized Linear Models (GLM) are two closely related statistical methods used for modeling the relationship between a dependent variable (response) and one or more independent variables (predictors). Here are the definitions for both GLM and Linear Regression:
- Linear Regression:
- Linear Regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables, assuming that the relationship is linear. Check out this paper to learn more about linear regression – All of Linear Regression.
- The response variable is continuous and is usually assumed to be normally distributed. This means the dependent variable can take any value along a continuum, such as height, weight, temperature, or prices.
- Generalized Linear Models (GLM):
- GLM is a flexible generalization of ordinary linear regression that allows for the dependent variable to have a distribution other than a normal distribution.
- GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function. The link function is a function of the expected value of the response variable. This link function describes how the mean of the response variable depends on the linear predictor of independent variables. This allows modeling of non-linear relationships.
- The response variable in GLM can be of various types, including continuous, binary, count, or categorical. This includes distributions like binomial (for binary data), Poisson (for count data), and others.
The following is the list of key differences between GLM and Linear Regression:
| Aspect | Linear Regression | Generalized Linear Models (GLM) |
|---|---|---|
| Response Variable Type | Continuous and normally distributed. | Can be various types: continuous, binary, count, etc. |
| Relationship with Independent Variables | Linear relationship assumed. | Relationship defined by a link function, can be non-linear. |
| Error Distribution | Errors are normally distributed with constant variance (homoscedasticity). | Distribution of errors can vary, not restricted to normal. Includes Poisson, binomial, etc. |
| Model Flexibility | Less flexible, suitable for datasets where the response variable has a linear relationship with predictors. | More flexible, can model a wide range of data types and relationships. |
| Use Cases | Suitable for predicting values where the response is a continuous measure (e.g., house prices). | Suitable for cases like binary outcomes (logistic regression), count data (Poisson regression), etc. |
| Assumptions | Assumes linearity, homoscedasticity, and independence of errors. | More general, does not assume normal distribution of errors, and can handle heteroscedasticity. |
| Link Function | No link function (identity link is implied). | Uses a link function to relate the mean of the response variable to the linear predictor (e.g., logit link for logistic regression). |
| Python code | sklearn.linear_model.LinearRegression, statsmodels.api.OLS | sklearn.linear_model.LogisticRegression (for GLM with logistic link), statsmodels.api.GLM, statsmodels.api.Logit |
| R code | lm() | glm() |
When to use GLM vs Linear Regression?
When deciding between Generalized Linear Models (GLM) and Linear Regression, consider the following three key points:
- Type of Response Variable:
- Use Linear Regression when the response variable is continuous and approximately normally distributed.
- Choose GLM if the response variable is not continuous or normally distributed, such as binary (e.g., yes/no), count (e.g., number of events), or categorical data.
- Relationship Between Variables:
- Opt for Linear Regression if there is a linear relationship between the independent and dependent variables.
- Use GLM when the relationship is not linear or when it needs a special form of modeling (e.g., logit function for binary outcomes in logistic regression).
- Error Distribution and Assumptions:
- Linear Regression assumes homoscedasticity (constant variance of errors) and normally distributed errors.
- GLM is more flexible and can accommodate various types of error distributions (e.g., Poisson, binomial) and does not require the assumption of homoscedasticity.
These criteria are fundamental in guiding the choice between GLM and Linear Regression, ensuring the selection of the most appropriate model for your data analysis needs.
GLM vs Linear Regression: Examples
Here are two unique examples for each, where GLM and Linear Regression would be most appropriate. These examples illustrate situations where the inherent characteristics of the data and the nature of the relationship between variables make either Linear Regression or GLM the more suitable choice for analysis.
Linear Regression Examples:
- Real Estate Pricing: Predicting the selling price of houses based on features like square footage, number of bedrooms, location, and age of the property. In this scenario, the response variable (house price) is continuous, and a linear relationship is typically assumed between the house prices and the features.
- Academic Performance: Estimating a student’s final grade based on continuous predictors such as hours spent studying, attendance rate, and scores in previous exams. The final grade is a continuous outcome expected to have a linear relationship with these predictors.
Generalized Linear Models (GLM) Examples:
- Disease Diagnosis: Predicting the probability of a patient having a particular disease (say, diabetes) based on various factors like age, body mass index, family history, and blood pressure. This is a binary outcome (disease: yes/no), making logistic regression, a type of GLM, appropriate.
- Traffic Accident Count Analysis: Modeling the number of traffic accidents occurring at an intersection based on factors like traffic volume, day of the week, and weather conditions. Since the response variable (number of accidents) is a count, a Poisson regression, which is a GLM suitable for count data, would be appropriate.
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