| Term | Definition | Description |
|---|---|---|
| X | – | Predictor matrix for the true outcome. |
| Z(1) | – | Predictor matrix for the first-stage observed outcome, conditional on the true outcome. |
| Z(2) | – | Predictor matrix for the second-stage observed outcome, conditional on the true outcome and first-stage observed outcome. |
| Y | Y ∈ {1, 2} | True binary outcome. Reference category is 2. |
| yij | 𝕀{Yi = j} | Indicator for the true binary outcome. |
| Y*(1) | Y*(1) ∈ {1, 2} | First-stage observed binary outcome. Reference category is 2. |
| yik*(1) | 𝕀{Yi*(1) = k} | Indicator for the first-stage observed binary outcome. |
| Y*(2) | Y*(2) ∈ {1, 2} | Second-stage observed binary outcome. Reference category is 2. |
| yiℓ*(2) | 𝕀{Yi*(2) = ℓ} | Indicator for the second-stage observed binary outcome. |
| True Outcome Mechanism | logit{P(Y = j|X; β)} = βj0 + βjXX | Relationship between X and the true outcome, Y. |
| First-Stage Observation Mechanism | logit{P(Y*(1) = k|Y = j, Z(1); γ(1))} = γkj0(1) + γkjZ(1)(1)Z(1) | Relationship between Z(1) and the first-stage observed outcome, Y*(1), given the true outcome Y. |
| Second-Stage Observation Mechanism | logit{P(Y*(2) = ℓ|Y*(1) = k, Y = j, Z(2); γ(2))} = γℓkj0(2) + γℓkjZ(2)(2)Z(2) | Relationship between Z(2) and the second-stage observed outcome, Y*(2), given the first-stage observed outcome, Y*(1), and 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*(1) | $P(Y^{*(1)}_i = k | Y = j, Z^{(1)} ; \gamma^{(1)}) = \frac{\text{exp}\{\gamma^{(1)}_{kj0} + \gamma^{(1)}_{kjZ^{(1)}} Z_i^{(1)}\}}{1 + \text{exp}\{\gamma^{(1)}_{kj0} + \gamma^{(1)}_{kjZ^{(1)}} Z_i^{(1)}\}}$ | Response probability for individual i’s first-stage observed outcome category, conditional on the true outcome. |
| πiℓkj*(2) | $P(Y^{*(2)}_i = \ell | Y^{*(1)} = k, Y = j, Z^{(2)} ; \gamma^{(2)}) = \frac{\text{exp}\{\gamma^{(2)}_{\ell kj0} + \gamma^{(2)}_{\ell kjZ^{(2)}} Z_i^{(2)}\}}{1 + \text{exp}\{\gamma^{(2)}_{\ell kj0} + \gamma^{(2)}_{\ell kjZ^{(2)}} Z_i^{(2)}\}}$ | Response probability for individual i’s second-stage observed outcome category, conditional on the first-stage observed outcome and the true outcome. |
| πik*(1) | $P(Y^{*(1)}_i = k | X, Z^{(1)} ; \gamma^{(1)}) = \sum_{j = 1}^2 \pi^{*(1)}_{ikj} \pi_{ij}$ | Response probability for individual i’s first-stage observed outcome cateogry. |
| πjj*(1) | $P(Y^{*(1)} = j | Y = j, Z^{(1)} ; \gamma^{(1)}) = \sum_{i = 1}^N \pi^{*(1)}_{ijj}$ | Average probability of first-stage correct classification for category j. |
| πjjj*(2) | $P(Y^{*(2)} = j | Y^{*(1)}_i = j, Y = j, Z^{(2)} ; \gamma^{(2)}) = \sum_{i = 1}^N \pi^{*(2)}_{ijjj}$ | Average probability of first-stage and second-stage correct classification for category j. |
| First-Stage Sensitivity | $P(Y^{*(1)} = 1 | Y = 1, Z^{(1)} ; \gamma^{(1)}) = \sum_{i = 1}^N \pi^{*(1)}_{i11}$ | True positive rate. Average probability of observing first-stage outcome k = 1, given the true outcome j = 1. |
| First-Stage Specificity | $P(Y^{*(1)} = 2 | Y = 2, Z^{(1)} ; \gamma^{(1)}) = \sum_{i = 1}^N \pi^{*(1)}_{i22}$ | True negative rate. Average probability of observing first-stage outcome k = 2, given the true outcome j = 2. |
| βX | – | Association parameter of interest in the true outcome mechanism. |
| γ11Z(1)(1) | – | Association parameter of interest in the first-stage observation mechanism, given j = 1. |
| γ12Z(1)(1) | – | Association parameter of interest in the first-stage observation mechanism, given j = 2. |
| γ111Z(2)(2) | – | Association parameter of interest in the second-stage observation mechanism, given k = 1 and j = 1. |
| γ121Z(2)(2) | – | Association parameter of interest in the second-stage observation mechanism, given k = 2 and j = 1. |
| γ112Z(2)(2) | – | Association parameter of interest in the second-stage observation mechanism, given k = 1 and j = 2. |
| γ122Z(2)(2) | – | Association parameter of interest in the second-stage observation mechanism, given k = 2 and j = 2. |