Model specification, log-likelihood, scores and second derivatives

Notation

β — vertical coefficient vector.
X — Covariate matrix with one row per observation.
X_i — i’th row from X
Y — Vertical binary outcome vector.
k — number of covariates.
n — number of observations.
i — observation index.
j — covariate index.

Model

$$ P(Y_i = 1) = \frac{\lambda}{1 + \text{exp}(\boldsymbol{X_i}\boldsymbol{\beta})} = \frac{\text{exp}(\theta)}{(1 + \text{exp}(\theta))(1 + \text{exp}(\boldsymbol{X_i}\boldsymbol{\beta}))} $$

θ is the logit transformation of λ: $\theta = \text{log}(\frac{\lambda}{1-\lambda})$

Optimisation is done using the θ parameterisation because it does not constrain the likelihood.

Log likelihood

l(θ, β) = ∑_i y_i θ − log(1 + exp(X_iβ)) − log(1 + exp(θ)) + (1 − y_i)log(1 + exp(X_iβ)(1 + exp(θ)))

Scores

Second derivatives

References

Dunning AJ (2006). “A model for immunological correlates of protection.” Statistics in Medicine, 25(9), 1485-1497. https://doi.org/10.1002/sim.2282.