The GLMM for binary data using probit link function is 1 [P(Y it = 1 |

The GLMM for binary data using probit link function is

Φ^–1[P(Y_it = 1 | u_i)] = x’_it β + z’_it u_i,

where Φ is the N(0, 1) cdf and u_i has N(0, ∑) pdf, f(u_i; ∑).

a. Show that the marginal mean is

P(Y_t = 1) = ʃ P(Z – z’_it u_i ≤ x’_it β) f(u_i; ∑) d u_i,

where Z is a standard normal variate that is independent of u_i.

b. Since Z – _z’it u_i has a N(0, 1 + z’_it ∑z_it) distribution, deduce that

Φ^–1[P(Y_t = 1)] = x’_it β[1 + z’_it ∑z_it]^–1/2.

Hence, the marginal model is a probit model with attenuated effect. In the univariate random intercept case, show the marginal effect equals that from the GLMM divided by √1 + σ².