第13章 广义线性模型

注：R语言的再复习之路
 
 

1. 广义线性模型和glm()函数

广义线性模型的拟合形式为：
$$g({\mu }_Y)={\beta }_0+\sum _{j=1}^p{\beta }_j{X}_j,$$
其中$g({\mu }_Y)$是条件均值函数（称为连接函数），这里$Y$服从指数组分布中的一种即可。

1.1 glm()函数

基本形式：glm(formula, family = family(link = function), data = )

分布族	默认的连接函数
binomial	link = 'logit'
gaussian	link = 'identity'
gamma	link = 'inverse'
poisson	link = 'log'
quasibinomial	link = 'logit'
quasipoisson	link = 'log'

1.2 连用的函数

函数	展示拟合模型的细节
summary()	展示拟合模型的细节
coefficients(), coef()	列出拟合模型的参数
confint()	给出模型参数的置信区间
residuals()	列出拟合模型的残差值
anova()	生成两个拟合模型的方差分析表
plot()	生成评价拟合模型的诊断图
predict()	用拟合模型对新数据集进行预测
deviance()	拟合模型的偏差
df.residual()	拟合模型的残差自由度

1.3 模型拟合和回归诊断

### 绘制初始变量的预测值与残差的图形
plot(predict(model, type = 'response'), residuals(model, type = 'deviance'))

### 识别离群点、高杠杆值点、强影响点
plot(hatvalues(model))
plot(rstudent(model))
plot(cooks.distance(model))

### 综合性的诊断图
library(car)
influencePlot(model)

 
 

2. Logistic回归

### 基本的数据整理，将出轨次数转化成二值型因子
data(Affairs, package = 'AER')
Affairs <- within(Affairs, {
    ynaffair <- NA
    ynaffair[affairs > 0] <- 1
    ynaffair[affairs == 0] <- 0
    })
Affairs$ynaffair <- factor(Affairs$ynaffair, levels = c(0, 1), labels = c('No', 'Yes'))

### 构建Logistic回归
fit.full <- glm(ynaffair ~ gender + age + yearsmarried + children + religiousness + education + occupation + rating, data = Affairs, family = binomial())
summary(fit.full)

### 把不显著的变量删除，再次构建Logistic回归
fit.reduced <- glm(ynaffair ~ age + yearsmmaried + religiousness + rating, data = Affairs, family = binomial())
summary(fit.reduced)

### 对两个拟合模型进行方差分析
anova(fit.reduced, fit.full, test = 'Chisq')  ## 若检验表明两个拟合模型没有显著差别，则删除以上的变量是合理的

2.1 解释模型参数

coef(fit.reduced)
exp(coef(fit.reduced))

将拟合参数指数化后，参数的数值表明该参数增加一个单位后，响应变量优势比$\frac{Y}{1-Y}$变化的幅度。

2.2 评价预测变量对结果概率的影响

### 评价婚姻评分对婚外情概率的影响
testdata <- data.frame(rating = c(1, 2, 3, 4, 5), age = mean(Affairs$age), yearsmarried = mean(Affairs$yearsmarried), religiousness = mean(Affairs$religiousness))
testdata$prob <- predict(fit.reduced, newdata = testdata, type = 'response')

### 评价年龄对婚外情概率的影响
testdata <- data.frame(rating = mean(Affairs$rating), age = seq(17, 57, 10), yearsmarried = mean(Affairs$yearsmarried), religiousness = mean(Affairs$religiousness))
testdata$prob <- predict(fit.reduced, newdata = testdata, type = 'response')

2.3 过度离势

过度离势：观测到的响应变量的方差大于期望的二项分布的方差

• 方法1
  比较二项分布模型的残差偏差与残差自由度，即
  $$\varphi =\text{残差偏差}/\text{残差自由度}。$$
  若$\varphi$比1大很多，便可以认为存在过度离势。

deviance(fit.reduced)/df.residual(fit.reduced)

• 方法2

fit <- glm(ynaffair ~ age + yearsmarried + religiousness + rating, data = Affairs, family = binomial())
fit.od <- glm(ynaffair ~ age + yearsmarried + religiousness + rating, data = Affairs, family = quasibinomial())
pchisq(summary(fit.od)$dispersion * fit$df.residual, fit$df.residual, lower = F)

 
 

3. 泊松回归

data(breslow.dat, package = 'robust')
fit <- glm(sumY ~ Base + Age + Trt, data = breslow.dat, family = poisson())
summary(fit)

3.1 解释模型参数

coef(fit)
exp(coef(fit))

注：这里将参数指数化并不是一成不变的，要根据具体的模型而来，观察怎样变换能够更好地解释预测变量对于响应变量的影响。

3.2 过度离势

• 方法1

deviance(fit) / df.residual(fit)

• 方法2

fit <- glm(sumY ~ Base + Age + Trt, data = breslow.dat, family = poisson())
fit.od <- glm(sumY ~ Base + Age + Trt, data = breslow.dat, family = quasipoisson())
pchisq(summary(fit.od)$dispersion * fit$df.residual, fit$df.residual, lower = F)

• 方法3

library(qcc)
qcc.overdispersion.test(breslow.dat$sumY, type = 'poisson')