| Term | Definition | Description |
|---|---|---|
| X | – | Predictor matrix for the true outcome. |
| Z | – | Predictor matrix for the observed outcome, conditional on the true outcome. |
| Y | Y ∈ {1, 2} | True binary outcome. Reference category is 2. |
| yij | 𝕀{Yi = j} | Indicator for the true binary outcome. |
| Y* | Y* ∈ {1, 2} | Observed binary outcome. Reference category is 2. |
| yik* | 𝕀{Yi* = k} | Indicator for the observed binary outcome. |
| True Outcome Mechanism | logit{P(Y = j|X; β)} = βj0 + βjXX | Relationship between X and the true outcome, Y. |
| Observation Mechanism | logit{P(Y* = k|Y = j, Z; γ)} = γkj0 + γkjZZ | Relationship between Z and the observed outcome, Y*, given the true outcome Y. |
| πij | $P(Y_i = j | X ; \beta) = \frac{\text{exp}\{\beta_{j0} + \beta_{jX} X_i\}}{1 + \text{exp}\{\beta_{j0} + \beta_{jX} X_i\}}$ | Response probability for individual i’s true outcome category. |
| πikj* | $P(Y^*_i = k | Y_i = j, Z ; \gamma) = \frac{\text{exp}\{\gamma_{kj0} + \gamma_{kjZ} Z_i\}}{1 + \text{exp}\{\gamma_{kj0} + \gamma_{kjZ} Z_i\}}$ | Response probability for individual i’s observed outcome category, conditional on the true outcome. |
| πik* | $P(Y^*_i = k | Y_i, X, Z ; \gamma) = \sum_{j = 1}^2 \pi^*_{ikj} \pi_{ij}$ | Response probability for individual i’s observed outcome cateogry. |
| πjj* | $P(Y^* = j | Y = j, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{ijj}$ | 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. |
| βX | – | Association parameter of interest in the true outcome mechanism. |
| γ11Z | – | Association parameter of interest in the observation mechanism, given j = 1. |
| γ12Z | – | Association parameter of interest in the observation mechanism, given j = 2. |