# Simulating Data from Multilevel Logistic Regression

I want to simulate data from multilevel logistic regression .

I focus on the following multilevel logistic model with one explanatory variable at level 1 (individual level) and one explanatory variable at level 2 (group level) :

$$\text{logit}(p_{ij})=\pi_{0j}+\pi_{1j}x_{ij}\ldots (1)$$ $$\pi_{0j}=\gamma_{00}+\gamma_{01}z_j+u_{0j}\ldots (2)$$ $$\pi_{1j}=\gamma_{10}+\gamma_{11}z_j+u_{1j}\ldots (3)$$

where , $u_{0j}\sim N(0,\sigma_0^2)$ , $u_{1j}\sim N(0,\sigma_1^2)$ , $\text{cov}(u_{0j},u_{1j})=\sigma_{01}$

I prefer R and started to write codes for the simulation as :

 set.seed(36)

x <- rnorm(1000)  ### individual level variable , x_ij

z <- rnorm(1000)  ### group level variable , z_j


If I have initial value for $\gamma_{00}=-1 , \gamma_{01}=0.3,\gamma_{10}=0.3,\gamma_{11}=0.3$ , how can I generate $\pi_{0j},\pi_{1j}$ since there is $u_{0j},u_{1j}$ in equation (2) and (3) ?

In your case, basically $\boldsymbol u = (u_0, u_1)$ is distributed as bi-variate normal. You can use R package mvtnorm to generate $u_{0j}, u_{1j}$. Then you can get $\pi_{0j}, \pi_{1j}$ easily. (You would wish to do all those in terms of random vector manipulation.)

The following code generates $\boldsymbol u = (u_0, u_1)$ for you.

require(mvtnorm)
set.seed(1234)
## need variance values as input
s2_0 <- 2
s2_1 <- 3
s01 <- 0.5

## generate bi-variate normal rv for u0, u1
mean <- c(0, 0)
sigma <- matrix(c(s2_0, s01, s01, s2_1), ncol = 2)

u <- rmvnorm(1000, mean = mean, sigma = sigma, method = "chol")


Here I assume you know the values of the hyperparameters. And you get something like following.

> dim(u)
 1000    2
[,1]       [,2]
[1,]  0.03563353 -2.0906992
[2,] -2.93442022  1.8783072
[3,]  0.82448029  0.7432658
[4,] -0.92269067 -0.9954788
[5,] -1.39514484 -0.9776595
[6,] -1.51995711 -0.8265220
[7,] -0.13484601 -1.3445112
[8,]  0.12429872  1.6618925
[9,] -1.40900867 -0.8850944
[10,]  3.10290171 -1.4500239

• Could you please tell me what is hyperparameter ? – ABC Jun 1 '15 at 4:16
• @ABC, I mean $\sigma_0^2, \sigma_1^2$ and $\sigma_{01}$ in your case. Basically a hypterparameter is the one for your prior distribution, not the one you wish to estimate and make inference about. That is mainly a term in Bayesian statistics. I might misuse it here. Apologize for misleading. – SixSigma Jun 1 '15 at 4:27