r - 用固定的 r2 模拟逻辑回归的数据

Question

我想模拟逻辑回归的数据，我可以在其中预先指定其解释方差。看看下面的代码。我模拟了四个自变量，并指定每个 logit 系数的大小应为 log(2)=0.69。这很有效，解释方差(我报告 Cox & Snell 的 r2)为 0.34。

但是，我需要以一种预先指定的 r2 将从回归中产生的方式指定回归系数。因此，如果我想生成一个 r2，我们可以说正好是 0.1。需要如何指定系数？我有点挣扎于此..

# Create independent variables
sigma.1 <- matrix(c(1,0.25,0.25,0.25,   
                0.25,1,0.25,0.25,   
                0.25,0.25,1,0.25,    
                0.25,0.25,0.25,1),nrow=4,ncol=4)
mu.1 <- rep(0,4) 
n.obs <- 500000 

library(MASS)
sample1 <- as.data.frame(mvrnorm(n = n.obs, mu.1, sigma.1, empirical=FALSE))

# Create latent continuous response variable 
sample1$ystar <- 0 + log(2)*sample1$V1 + log(2)*sample1$V2 + log(2)*sample1$V3 + log(2)*sample1$V4

# Construct binary response variable
sample1$prob <- exp(sample1$ystar) / (1 + exp(sample1$ystar))
sample1$y <- rbinom(n.obs,size=1,prob=sample1$prob)

# Logistic regression
logreg <- glm(y ~ V1 + V2 + V3 + V4, data=sample1, family=binomial)
summary(logreg)

输出是:

Call:
glm(formula = y ~ V1 + V2 + V3 + V4, family = binomial, data = sample1)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.7536  -0.7795  -0.0755   0.7813   3.3382  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept) -0.002098   0.003544  -0.592    0.554    
V1           0.691034   0.004089 169.014   <2e-16 ***
V2           0.694052   0.004088 169.776   <2e-16 ***
V3           0.693222   0.004079 169.940   <2e-16 ***
V4           0.699091   0.004081 171.310   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 693146  on 499999  degrees of freedom
Residual deviance: 482506  on 499995  degrees of freedom
AIC: 482516

Number of Fisher Scoring iterations: 5

Cox 和 Snell 的 r2 给出:

library(pscl)
pR2(logreg)["r2ML"]

> pR2(logreg)["r2ML"]
 r2ML 
0.3436523 

Answer 1

如果您向 ystar 变量添加一个随机误差项，生成 ystat.r，然后使用它，您可以调整标准偏差，直到它符合您的规范。

sample1$ystar.r <- sample1$ystar+rnorm(n.obs, 0, 3.8)  # tried a few values
sample1$prob <- exp(sample1$ystar.r) / (1 + exp(sample1$ystar.r))
sample1$y <- rbinom(n.obs,size=1,prob=sample1$prob)
logreg <- glm(y ~ V1 + V2 + V3 + V4, data=sample1, family=binomial)
summary(logreg)  # the estimates "shrink"
pR2(logreg)["r2ML"]
#-------
     r2ML 
0.1014792