Simulate data for logistic regression with fixed r2

Question

I would like to simulate data for a logistic regression where I can specify its explained variance beforehand. Have a look at the code below. I simulate four independent variables and specify that each logit coefficient should be of size log(2)=0.69. This works nicely, the explained variance (I report Cox & Snell's r2) is 0.34.

However, I need to specify the regression coefficients in such a way that a pre-specified r2 will result from the regression. So if I would like to produce an r2 of let's say exactly 0.1. How do the coefficients need to be specified? I am kind of struggling with this..

# 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)

The output is:

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

And Cox and Snell's r2 gives:

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

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

Answer 1

If you add a random error term to the ystar variable making ystat.r and then work with that, you can tweek the standard deviation until it meets you specifications.

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

Answer 2

R-squared (and its variations) is a random variable, as it depends on your simulated data. If you simulate data with the exact same parameters multiple times, you'll most likely get different values for R-squared each time. Therefore, you cannot produce a simulation where the R-squared will be exactly 0.1 just by controlling the parameters.

On the other hand, since it's a random variable, you could potentially simulate your data from a conditional distribution (conditioning on a fixed value of R-squared), but you would need to find out what these distributions look like (math might get really ugly here, cross validated is more appropriate for this part).