Linear hybrid Model (5)--Generalized linear model

We know that the hybrid linear model is an extension of the general linear model, and the generalized linear model is further expanded on the basis of the mixed linear model, which makes the linear model more widely used. Each expansion is actually an extension of the applicable scope of the model, and the general linear model requires that the observed values be independent of each other, residuals (dependent variables) are subject to normal distribution, residual (dependent variable) variance homogeneity, and the mixed linear model cancels the mutual independence of the observed values and residuals (dependent variable) variance homogeneity of the requirements, Then the generalized linear model cancels the requirement that the residual (dependent variable) obey the normal distribution. Residuals do not have to obey the normal distribution, can obey two items, Poisson, minus two, normal, gamma, inverse Gaussian distribution, these distributions are collectively referred to as exponential distribution family, and introduced the connection function, according to the different dependent variable distribution, connection function and other combinations, can get a variety of different generalized linear models.

It is important to note that although generalized linear models do not require dependent variables to be normally distributed, they are required to be independent of each other, and if they do not conform to each other, the generalized estimation equations described later are required.

=================================================
A generalized linear model

The general form of a generalized linear model is:

There are several parts that make up the

1. Linear section

2. Random part Εi

3. Connection function

The connection function is a monotone (continuous and full smooth) function, the connection function plays "Y's estimated value μ" and "linear prediction of the independent variables" function, in the general linear model, the two is one thing, but when the value range of the independent variable is limited, it is necessary to expand the range of values through the connection function, so in the generalized linear The linear predictor of an independent variable is the function estimate of the dependent variable.

The generalized linear model sets the dependent variable to the exponential family probability distribution, so that the dependent variable can be not limited to a form of normal distribution, and the variance can be unstable.

The probability density function of the exponential distribution family is

where θ and φ are two parameters, θ is a natural parameter, φ is a discrete parameter, and a,b,c is a function

Parameter estimation of generalized linear models:

The parameter estimation of generalized linear model can not be used least squares, commonly weighted least squares or maximum likelihood method. The regression parameters need to be solved by iterative method.

Test and goodness of fit for generalized linear models:

The test of generalized linear model generally uses likelihood ratio test and wald test. Comparing the model with the likelihood ratio test, the regression coefficient is used wald test.

The likelihood ratio test is performed by comparing the logarithmic likelihood function of two nested models, such as model p nested within the model K, with a statistic of G:

g=-2* (LP-LK)

LP is the logarithmic likelihood function of model p, and LK is the logarithmic likelihood function of model K1

The independent variable in model p is part of the independent variable in the model k, and the other part is the variable to be tested, where G obeys the chi-square distribution of the k-p of freedom.

The goodness of fit for generalized linear models is typically measured using the following statistics:

Deviation statistics, Pearson chi-square statistic, aic,aicc,bic,caic criterion, the lower the value of the criterion the better

Linear hybrid Model (5)--Generalized linear model