q_delta_f {COMBO} R Documentation

## M-Step Expected Log-Likelihood with respect to Delta

### Description

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

### Usage

q_delta_f(
delta_v,
V,
obs_Ystar_matrix,
obs_Ytilde_matrix,
w_mat,
sample_size,
n_cat
)


### Arguments

 delta_v A numeric array of regression parameters for the second-stage observed outcome mechanism, \tilde{Y} | Y^*, Y (second-stage observed outcome, given the first-stage observed outcome and the true outcome) ~ V (misclassification predictor matrix). The \delta vector is obtained from the array form. In array form, the first dimension (matrix rows) of delta corresponds to parameters for the \tilde{Y} = 1 second-stage observed outcome, with the dimensions of the V The second dimension (matrix columns) correspond to the first-stage observed outcome categories Y^* \in \{1, 2\}. The third dimension of delta_start corresponds to to the true outcome categories Y \in \{1, 2\}. The numeric vector \delta is obtained by concatenating the delta array, i.e. delta_v <- c(delta_array). V A numeric design matrix. obs_Ystar_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. obs_Ytilde_matrix A numeric matrix of indicator variables (0, 1) for the observed outcome \tilde{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_2stage. 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, V. n_cat The number of categorical values that the true outcome, Y, and the observed outcomes can take.

### Value

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

[Package COMBO version 1.1.0 Index]