The difference between generalized linear models and ordinary linear models

Generalized linear models and general linear models are commonly used regression analysis methods in statistics. Although the two terms are similar, they differ in some ways. Generalized linear models allow the dependent variable to follow a non-normal distribution by linking the predictor variables to the dependent variable through a link function. The general linear model assumes that the dependent variable obeys a normal distribution and uses linear relationships for modeling. Therefore, generalized linear models are more flexible and have wider applicability.

1. Definition and scope

The general linear model is a regression analysis method that is suitable for linear relationships between dependent variables and independent variables. relationship situation. It assumes that the dependent variable follows a normal distribution.

Generalized linear model is a regression analysis method suitable for dependent variables that do not necessarily obey a normal distribution. It can describe the relationship between dependent variables and independent variables by introducing link functions and distribution families.

2. Distribution Assumption

General linear model: The general linear model assumes that the dependent variable follows a normal distribution, which means that it is suitable for continuous type, symmetrically distributed dependent variable.

Generalized linear model: The generalized linear model does not make specific assumptions about the distribution of the dependent variable, and can be applied to many types of dependent variables, such as binomial distribution, Poisson distribution, etc. .

3. Link function

General linear model: The link function used in the general linear model is the identity function, which will The linear combination maps directly to the dependent variable.

Generalized linear model: The generalized linear model maps the linear combination of independent variables to an appropriate range by introducing a link function. For example, for the binomial distribution, you can use the logit function as a link function to map linear combinations of independent variables to probabilities between 0 and 1.

4. Distribution family

General linear model: The dependent variable in the general linear model obeys the normal distribution, so the distribution family is normal distribution family.

Generalized linear model: The dependent variable in the generalized linear model can obey a variety of distributions, so there are many distribution families to choose from, such as the binomial distribution family, the Poisson distribution family, etc. .

5. Parameter estimation

General linear model: The general linear model uses the least squares method for parameter estimation.

Generalized linear model: Generalized linear model uses the maximum likelihood method for parameter estimation.

6. Model optimization

General linear model: A variety of methods can be used for model optimization in general linear models, such as stepwise regression, crossover Verification etc.

Generalized linear model: There are relatively few optimization methods in generalized linear models, and the maximum likelihood method is generally used for model optimization.

To sum up, the generalized linear model is a broader regression analysis method suitable for situations where the dependent variable does not necessarily follow a normal distribution. It introduces link functions and distribution families to describe the relationship between dependent variables and independent variables. In contrast, the general linear model assumes that the dependent variable obeys a normal distribution, uses the identity function as the link function, and is suitable for symmetrically distributed dependent variables. In practical applications, appropriate models need to be selected based on specific problems.

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