用固定R2模拟多元回归数据：如何合并相关变量？

Question

我想有四个预测我在哪里可以自由指定

• 整体解释模型
• 所有标准化回归系数的大小的变化来模拟数据进行多元线性回归
• 的预测变量彼​​此相关的程度

我到达了满足前两点的解决方案，但是基于所有ind自变量彼此无关（请参阅下面的代码）。为了得到标准化的回归系数，我从平均值= 0和方差= 1的总体变量中抽样。

# Specify population variance/covariance of four predictor variables that is sampled from 
sigma.1 <- matrix(c(1,0,0,0, 
        0,1,0,0, 
        0,0,1,0,  
        0,0,0,1),nrow=4,ncol=4) 
# Specify population means of four predictor varialbes that is sampled from 
mu.1 <- rep(0,4) 

# Specify sample size, true regression coefficients, and explained variance 
n.obs <- 50000 # to avoid sampling error problems 
intercept <- 0.5 
beta <- c(0.4, 0.3, 0.25, 0.25) 
r2 <- 0.30 

# Create sample with four predictor variables 
library(MASS) 
sample1 <- as.data.frame(mvrnorm(n = n.obs, mu.1, sigma.1, empirical=FALSE)) 

# Add error variable based on desired r2 
var.epsilon <- (beta[1]^2+beta[2]^2+beta[3]^2+beta[4]^2)*((1 - r2)/r2) 
sample1$epsilon <- rnorm(n.obs, sd=sqrt(var.epsilon)) 

# Add y variable based on true coefficients and desired r2 
sample1$y <- intercept + beta[1]*sample1$V1 + beta[2]*sample1$V2 + 
beta[3]*sample1$V3 + beta[4]*sample1$V4 + sample1$epsilon 

# Inspect model 
summary(lm(y~V1+V2+V3+V4, data=sample1)) 

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

Residuals: 
    Min  1Q Median  3Q  Max 
-4.0564 -0.6310 -0.0048 0.6339 3.7119 

Coefficients: 
      Estimate Std. Error t value Pr(>|t|)  
(Intercept) 0.496063 0.004175 118.82 <2e-16 *** 
V1   0.402588 0.004189 96.11 <2e-16 *** 
V2   0.291636 0.004178 69.81 <2e-16 *** 
V3   0.247347 0.004171 59.30 <2e-16 *** 
V4   0.253810 0.004175 60.79 <2e-16 *** 
--- 
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.9335 on 49995 degrees of freedom 
Multiple R-squared: 0.299, Adjusted R-squared: 0.299 
F-statistic: 5332 on 4 and 49995 DF, p-value: < 2.2e-16 

问题：如果我的预测变量是相关的，因此，如果没有非对角线元素为0指定其方差/协方差矩阵中，R2和回归系数主要来自我多么希望他们能有所不同，例如通过使用

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) 

有什么建议吗？ 谢谢！

Answer 1

在考虑了我的问题之后，我找到了答案。

上面的代码首先对预测变量进行采样，并给定相互之间的相关度。然后根据r2的期望值添加错误列。然后将所有这些加在一起，为y添加一列。

到目前为止，创建错误的行仅仅是

var.epsilon <- (beta[1]^2+beta[2]^2+beta[3]^2+beta[4]^2)*((1 - r2)/r2) 
sample1$epsilon <- rnorm(n.obs, sd=sqrt(var.epsilon)) 

所以它假定每贝塔系数贡献100％至y的解释（=没有独立变量的相互关系）。但是如果x变量是相关的，那么每个beta都不是（！）贡献100％。这意味着误差的方差必须更大，因为这些变量需要相互之间的一些变化。

多大？刚刚适应误差项的创建类似如下：

var.epsilon <- (beta[1]^2+beta[2]^2+beta[3]^2+beta[4]^2+cor(sample1$V1, sample1$V2))*((1 - r2)/r2) 

所以在何种程度上独立的变量加入cor(sample1$V1, sample1$V2)相关的只是添加到误差方差。在相互关系为0.25的情况下，例如，通过使用

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) 

cor(sample1$V1, sample1$V2)类似于0.25并且这个值被加到误差项的方差。

假设所有的相互关系相同，像这样，可以指定独立变量之间的任何程度的相互关系，连同真正的标准化回归系数和所需的R2。

证明：

sigma.1 <- matrix(c(1,0.35,0.35,0.35, 
        0.35,1,0.35,0.35, 
        0.35,0.35,1,0.35,  
        0.35,0.35,0.35,1),nrow=4,ncol=4) 
# Specify population means of four predictor varialbes that is sampled from 
mu.1 <- rep(0,4) 

# Specify sample size, true regression coefficients, and explained variance 
n.obs <- 500000 # to avoid sampling error problems 
intercept <- 0.5 
beta <- c(0.4, 0.3, 0.25, 0.25) 
r2 <- 0.15 

# Create sample with four predictor variables 
library(MASS) 
sample1 <- as.data.frame(mvrnorm(n = n.obs, mu.1, sigma.1, empirical=FALSE)) 

# Add error variable based on desired r2 
var.epsilon <- (beta[1]^2+beta[2]^2+beta[3]^2+beta[4]^2+cor(sample1$V1, sample1$V2))*((1 - r2)/r2) 
sample1$epsilon <- rnorm(n.obs, sd=sqrt(var.epsilon)) 

# Add y variable based on true coefficients and desired r2 
sample1$y <- intercept + beta[1]*sample1$V1 + beta[2]*sample1$V2 + 
    beta[3]*sample1$V3 + beta[4]*sample1$V4 + sample1$epsilon 

# Inspect model 
summary(lm(y~V1+V2+V3+V4, data=sample1)) 

> summary(lm(y~V1+V2+V3+V4, data=sample1)) 

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

Residuals: 
    Min  1Q Median  3Q  Max 
-10.7250 -1.3696 0.0017 1.3650 9.0460 

Coefficients: 
      Estimate Std. Error t value Pr(>|t|)  
(Intercept) 0.499554 0.002869 174.14 <2e-16 *** 
V1   0.406360 0.003236 125.56 <2e-16 *** 
V2   0.298892 0.003233 92.45 <2e-16 *** 
V3   0.247581 0.003240 76.42 <2e-16 *** 
V4   0.253510 0.003241 78.23 <2e-16 *** 
--- 
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 2.028 on 499995 degrees of freedom 
Multiple R-squared: 0.1558, Adjusted R-squared: 0.1557 
F-statistic: 2.306e+04 on 4 and 499995 DF, p-value: < 2.2e-16