| Term | Definition | Description |
|---|---|---|
| \(X\) | – | Predictor matrix for the true outcome. |
| \(Z\) | – | Predictor matrix for the observed outcome, conditional on the true outcome. |
| \(Y\) | \(Y \in \{1, 2\}\) | True binary outcome. Reference category is 2. |
| \(y_{ij}\) | \(\mathbb{I}\{Y_i = j\}\) | Indicator for the true binary outcome. |
| \(Y^*\) | \(Y^* \in \{1, 2\}\) | Observed binary outcome. Reference category is 2. |
| \(y^*_{ik}\) | \(\mathbb{I}\{Y^*_i = k\}\) | Indicator for the observed binary outcome. |
| True Outcome Mechanism | \(\text{logit} \{ P(Y = j | X ; \beta) \} = \beta_{j0} + \beta_{jX} X\) | Relationship between \(X\) and the true outcome, \(Y\). |
| Observation Mechanism | \(\text{logit}\{ P(Y^* = k | Y = j, Z ; \gamma) \} = \gamma_{kj0} + \gamma_{kjZ} Z\) | Relationship between \(Z\) and the observed outcome, \(Y^*\), given the true outcome \(Y\). |
| \(\pi_{ij}\) | \(P(Y_i = j | X ; \beta) = \frac{\text{exp}\{\beta_{j0} + \beta_{jX} X_i\}}{1 + \text{exp}\{\beta_{j0} + \beta_{jX} X_i\}}\) | Response probability for individual \(i\)’s true outcome category. |
| \(\pi^*_{ikj}\) | \(P(Y^*_i = k | Y_i = j, Z ; \gamma) = \frac{\text{exp}\{\gamma_{kj0} + \gamma_{kjZ} Z_i\}}{1 + \text{exp}\{\gamma_{kj0} + \gamma_{kjZ} Z_i\}}\) | Response probability for individual \(i\)’s observed outcome category, conditional on the true outcome. |
| \(\pi^*_{ik}\) | \(P(Y^*_i = k | Y_i, X, Z ; \gamma) = \sum_{j = 1}^2 \pi^*_{ikj} \pi_{ij}\) | Response probability for individual \(i\)’s observed outcome cateogry. |
| \(\pi^*_{jj}\) | \(P(Y^* = j | Y = j, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{ijj}\) | Average probability of correct classification for category \(j\). |
| Sensitivity | \(P(Y^* = 1 | Y = 1, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{i11}\) | True positive rate. Average probability of observing outcome \(k = 1\), given the true outcome \(j = 1\). |
| Specificity | \(P(Y^* = 2 | Y = 2, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{i22}\) | True negative rate. Average probability of observing outcome \(k = 2\), given the true outcome \(j = 2\). |
| \(\beta_X\) | – | Association parameter of interest in the true outcome mechanism. |
| \(\gamma_{11Z}\) | – | Association parameter of interest in the observation mechanism, given \(j=1\). |
| \(\gamma_{12Z}\) | – | Association parameter of interest in the observation mechanism, given \(j=2\). |