广义线性模型

• 线性模型的扩展，通过联结函数建立响应变量的数学期望值与线性组合的预测变量之间的关系。其特点是不强行改变数据的自然度量，数据可以具有非线性和非恒定方差结构。是线性模型在研究响应值的非正态分布以及非线性模型简洁直接的线性转化时的一种发展。
• R中实现广义线性模型：
  glm()函数—glm(formula,family=family(link=function),data=)

Logistic回归分析

• 逻辑回归（Logistic Regression）是一种用于解决二分类问题的机器学习方法，是一种广义的线性回归分析模型。用于估计某种事物的可能性。比如某用户购买某商品的可能性，某病人患有某种疾病的可能性，以及某广告被用户点击的可能性等。
• 逻辑回归（Logistic Regression）与线性回归（Linear Regression）都是一种广义线性模型（generalized linear model）。逻辑回归假设因变量 y 服从伯努利分布，而线性回归假设因变量 y 服从高斯分布。因此与线性回归有很多相同之处，去除Sigmoid映射函数的话，逻辑回归算法就是一个线性回归。可以说，逻辑回归是以线性回归为理论支持的，但是逻辑回归通过Sigmoid函数引入了非线性因素，因此可以轻松处理0/1分类问题。
• Logistic回归分类：

1. 条件Logistic回归：配对或配伍设计资料
2. 非条件Logistic回归: 适用于成组设计的统计资料
3. 因变量可以是：两项分类，无序多项分类，有序多项分类等

• logistic示例：

1. 数据准备

data(Affairs, package="AER")
summary(Affairs)
table(Affairs$affairs)

# create binary outcome variable
Affairs$ynaffair[Affairs$affairs > 0] <- 1
Affairs$ynaffair[Affairs$affairs == 0] <- 0
Affairs$ynaffair <- factor(Affairs$ynaffair, 
                           levels=c(0,1),
                           labels=c("No","Yes"))
table(Affairs$ynaffair)

2. 筛选变量

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

3. 查看结果

summary(fit.full)
# Call:
#   glm(formula = ynaffair ~ gender + age + yearsmarried + children + 
#         religiousness + education + occupation + rating, family = binomial(), 
#       data = Affairs)
# 
# Deviance Residuals: 
#   Min       1Q   Median       3Q      Max  
# -1.5713  -0.7499  -0.5690  -0.2539   2.5191  
# 
# Coefficients:
#   Estimate Std. Error z value Pr(>|z|)    
# (Intercept)    1.37726    0.88776   1.551 0.120807    
# gendermale     0.28029    0.23909   1.172 0.241083    
# age           -0.04426    0.01825  -2.425 0.015301 *  
#   yearsmarried   0.09477    0.03221   2.942 0.003262 ** 
#   childrenyes    0.39767    0.29151   1.364 0.172508    
# religiousness -0.32472    0.08975  -3.618 0.000297 ***
#   education      0.02105    0.05051   0.417 0.676851    
# occupation     0.03092    0.07178   0.431 0.666630    
# rating        -0.46845    0.09091  -5.153 2.56e-07 ***
#   ---
#   Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# (Dispersion parameter for binomial family taken to be 1)
# 
# Null deviance: 675.38  on 600  degrees of freedom
# Residual deviance: 609.51  on 592  degrees of freedom
# AIC: 627.51
# 
# Number of Fisher Scoring iterations: 4

4. 有统计学意义的变量有以下4个：age + yearsmarried + religiousness + rating 挑选这四个变量继续做logistic回归分析：

fit.reduced <- glm(ynaffair ~ age + yearsmarried + religiousness + 
                     rating, data=Affairs, family=binomial())
summary(fit.reduced)

5. 两个模型比较：结果显示P值不显著，没有统计学差异，说明两个模型评价效果差异不大

# compare models
anova(fit.reduced, fit.full, test="Chisq")
# Analysis of Deviance Table
# 
# Model 1: ynaffair ~ age + yearsmarried + religiousness + rating
# Model 2: ynaffair ~ gender + age + yearsmarried + children + religiousness + 
#   education + occupation + rating
# Resid. Df Resid. Dev Df Deviance Pr(>Chi)
# 1       596     615.36                     
# 2       592     609.51  4   5.8474   0.2108

6. 亚组分析：查看不同评分的风险分析，相当于亚组分析。其他因素取均数，只计算评分的风险分别情况

# calculate probability of extramariatal affair by marital ratings
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")
testdata
# rating      age yearsmarried religiousness      prob
# 1      1 32.48752     8.177696      3.116473 0.5302296
# 2      2 32.48752     8.177696      3.116473 0.4157377
# 3      3 32.48752     8.177696      3.116473 0.3096712
# 4      4 32.48752     8.177696      3.116473 0.2204547
# 5      5 32.48752     8.177696      3.116473 0.1513079

7. 过度离势分析

# evaluate overdispersion
fit <- glm(ynaffair ~ age + yearsmarried + religiousness +
             rating, family = binomial(), data = Affairs)
fit.od <- glm(ynaffair ~ age + yearsmarried + religiousness +
                rating, family = quasibinomial(), data = Affairs)
pchisq(summary(fit.od)$dispersion * fit$df.residual,  
       fit$df.residual, lower = F)
# 0.340122

两模型比较后计算P值>0.05,说明不存在过度离势。