# How can I make simulated logistic regression model more noisy?

When I want to simulate Y coming from the linear regression model, $$Y_i = X_i ^T \beta + \epsilon_i,$$

I can use code like:

x = rnorm(100); beta = 1
y = x %*% beta + rnorm(100, sd = 3)


If I want to make the estimate of $$\beta$$ "noiser" (increase standard error), I can increase the standard deviation in the second line above.

How can I increase the standard error of a logistic regression coefficient? I can simulate from a logistic regression model with:

x = rnorm(100); beta = 1
nu = x %*% beta        # linear predictor
pr = 1/(1+exp(-nu_1))         # pass through an inv-logit function
y = rbinom(100,1,pr)      # bernoulli response variable

• nu = x %*% beta + rnorm(100,0,1) would do it, altering the standard deviation as required. – jbowman Jan 19 at 0:11
• @jbowman's approach works well, but--unlike the ordinary least squares case--it renders the model invalid by introducing "overdispersion." Whether that's a problem depends on what you're trying to accomplish with your simulation. – whuber Jan 19 at 1:48
• I think @whuber's comment holds the key to getting a good answer - what is your ultimate objective? Why are you trying to make the estimate of $\beta$ "noisier"? Does bias matter? ... – jbowman Jan 19 at 17:38

You could make the $$x \times \beta$$ values be on average substantially below (or above) zero instead of approx. zero (e.g. subtract 2 from all x values). That results in a more unbalanced data set.