Original link:tecdat.cn/?p=14887 

Generalized linear model (GLM)Is through theThe connection function, put the independent variablelinearThe combination is associated with a probability distribution of the dependent variable, which can be a Gaussian, binomial, polynomial, Poisson, gamma, or exponential distribution. The connection functions are:

  • Square root connection (for Poisson model)

Consider some random variable Y with mean μ and variance σ2. Taylor expansion

ifIf g (y) = SQRT {y} g (y) = y, then the second equation becomes

Therefore, we have variance stability through the square root transformation, which can be interpreted as a certain homology.

  • Log function of Bernoulli model

Suppose the variables are Poisson variables,

The previous model looked like Bernoulli regression analysis, with H as the link function, \ mathbb {P}

So now suppose that instead of observing N, we observe that Y = 1 (N> 0). In that case, running Bernoulli regressions with logarithmic linking functions, first with running Poisson regressions on the raw data, and then using them on our binary variables zero and non-zero. Let’s compare eλx and PX from standard logistic regression with some simulation data


regPois = glm(Y~.,data=base,family=poisson(link="log"))
regBinom = glm((Y==0)~.,data=base,family=binomial(link="probit"))
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What if px \ is obtained from Bernoulli’s regression and has a join function?


plot(prob,1-exp(-lambda),xlim=0:1,ylim=0:1)
abline(a=0,b=1,lty=2,col="red")
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The fit is good. Now, if we model the marital infidelity data set, published by Ray Fair in 1978 in the journal Political Economy (563 observations, nine variables) :


prob = predict(regBinom, type="response")
plot(prob,exp(-lambda),xlim=0:1,ylim=0:1)
abline(a=0,b=1,lty=2,col="red")
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In this case, the two models turn out to be very different. The second model is the same


plot(prob,1-exp(-lambda),xlim=0:1,ylim=0:1)
abline(a=0,b=1,lty=2,col="red")
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How do we explain this? Is it because the Poisson model is bad? We run the zero inflation model here to compare,


summary(regZIP)
 
Count model coefficients (poisson with log link):
             Estimate Std. Error z value Pr(>| z |) (Intercept) 0.002274 0.048413 0.047 0.963 1.019814 0.026186 38.945 X1<2e-16 ***
X2           1.004814   0.024172  41.570   <2e-16 *** 
Zero-inflation model coefficients (binomial with logit link): 
            Estimate Std. Error z value Pr(>| z |) (Intercept) 4.90190 2.07846 2.358 0.0184 2.00227 0.86897 2.304 0.0212 * * X1 - X2-0.01545-0.96121-0.016-0.9872 - Signif. Codes: 0 '* * *' 0.001 '* *' 0.01 '*' 0.05 '. '0.1 "' 1Copy the code

Because of the expansion of zero, we reject the assumption of the Poisson distribution here and can use logarithmic connections to check whether the Poisson distribution is a good model.

 

 

reference

1. Use SPSS to estimate the HLM hierarchical linear model

2. Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA) and Regular Discriminant Analysis (RDA) of R language

3. Lmer mixed linear regression model based on R language

4. Simple Bayesian linear regression simulation analysis of Gibbs sampling in R language

5. Use GAM (Generalized additive Model) to analyze power load time series in R language

6. Hierarchical linear model HLM using SAS, Stata, HLM, R, SPSS and Mplus

Ridge regression, lasso regression, principal component regression in R language: Linear model selection and regularization

8. Prediction of air quality ozone data by linear regression model in R language

9.R language hierarchical linear model case