M-Step Expected Log-Likelihood with respect to Gamma

Objective function of the form: \(Q_{\gamma} = \sum_{i = 1}^N \Bigl[\sum_{j = 1}^2 \sum_{k = 1}^2 w_{ij} y^*_{ik} \text{log} \{ \pi^*_{ikj} \}\Bigr]\). Used to obtain estimates of \(\gamma\) parameters.

Usage

q_gamma_f(gamma_v, Z, obs_Y_matrix, w_mat, sample_size, n_cat)

Arguments

gamma_v

A numeric vector of regression parameters for the observed outcome mechanism, Y* | Y (observed outcome, given the true outcome) ~ Z (misclassification predictor matrix). In matrix form, the gamma parameter matrix rows correspond to parameters for the Y* = 0 observed outcome, with the dimensions of Z. In matrix form, the gamma parameter matrix columns correspond to the true outcome categories \(j = 1, \dots,\) n_cat. The numeric vector gamma_v is obtained by concatenating the gamma matrix, i.e. gamma_v <- c(gamma_matrix).

Z

A numeric design matrix.

obs_Y_matrix

A numeric matrix of indicator variables (0, 1) for the observed outcome Y*. Rows of the matrix correspond to each subject. Columns of the matrix correspond to each observed outcome category. Each row should contain exactly one 0 entry and exactly one 1 entry.

w_mat

Matrix of E-step weights obtained from w_j.

sample_size

An integer value specifying the number of observations in the sample. This value should be equal to the number of rows of the design matrix, Z.

n_cat

The number of categorical values that the true outcome, Y, and the observed outcome, Y* can take.

Value

q_beta_f returns the negative value of the expected log-likelihood function, \(Q_{\gamma} = \sum_{i = 1}^N \Bigl[\sum_{j = 1}^2 \sum_{k = 1}^2 w_{ij} y^*_{ik} \text{log} \{ \pi^*_{ikj} \}\Bigr]\), at the provided inputs.