COMBO Notation Guide - Two-stage Misclassification Model

Notation

This guide is designed to summarize key notation and quantities used the COMBO R Package and associated publications.
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.
$$Y$$ $$Y \in \{1, 2\}$$ True binary outcome. Reference category is 2.
$$y_{ij}$$ $$\mathbb{I}\{Y_i = j\}$$ Indicator for the true binary outcome.
$$Y^{*(1)}$$ $$Y^{*(1)} \in \{1, 2\}$$ First-stage observed binary outcome. Reference category is 2.
$$y^{*(1)}_{ik}$$ $$\mathbb{I}\{Y^{*(1)}_i = k\}$$ Indicator for the first-stage observed binary outcome.
$$Y^{*(2)}$$ $$Y^{*(2)} \in \{1, 2\}$$ Second-stage observed binary outcome. Reference category is 2.
$$y^{*(2)}_{i \ell}$$ $$\mathbb{I}\{Y^{*(2)}_i = \ell \}$$ Indicator for the second-stage 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, $$Y$$.
First-Stage Observation Mechanism $$\text{logit}\{ P(Y^{*(1)} = k | Y = j, Z^{(1)} ; \gamma^{(1)}) \} = \gamma^{(1)}_{kj0} + \gamma^{(1)}_{kjZ^{(1)}} Z^{(1)}$$ Relationship between $$Z^{(1)}$$ and the first-stage observed outcome, $$Y^{*(1)}$$, given the true outcome $$Y$$.
Second-Stage Observation Mechanism $$\text{logit}\{ P(Y^{*(2)} = \ell | Y^{*(1)} = k, Y = j, Z^{(2)} ; \gamma^{(2)}) \} = \gamma^{(2)}_{\ell kj0} + \gamma^{(2)}_{\ell kjZ^{(2)}} Z^{(2)}$$ Relationship between $$Z^{(2)}$$ and the second-stage observed outcome, $$Y^{*(2)}$$, given the first-stage observed outcome, $$Y^{*(1)}$$, and the true outcome $$Y$$.
$$\pi_{ij}$$ $$P(Y_i = j | X ; \beta) = \frac{\text{exp}\{\beta_{j0} + \beta_{jX} X_i\}}{1 + \text{exp}\{\beta_{j0} + \beta_{jX} X_i\}}$$ Response probability for individual $$i$$’s true outcome category.
$$\pi^{*(1)}_{ikj}$$ $$P(Y^{*(1)}_i = k | Y = j, Z^{(1)} ; \gamma^{(1)}) = \frac{\text{exp}\{\gamma^{(1)}_{kj0} + \gamma^{(1)}_{kjZ^{(1)}} Z_i^{(1)}\}}{1 + \text{exp}\{\gamma^{(1)}_{kj0} + \gamma^{(1)}_{kjZ^{(1)}} Z_i^{(1)}\}}$$ Response probability for individual $$i$$’s first-stage observed outcome category, conditional on the true outcome.
$$\pi^{*(2)}_{i \ell kj}$$ $$P(Y^{*(2)}_i = \ell | Y^{*(1)} = k, Y = j, Z^{(2)} ; \gamma^{(2)}) = \frac{\text{exp}\{\gamma^{(2)}_{\ell kj0} + \gamma^{(2)}_{\ell kjZ^{(2)}} Z_i^{(2)}\}}{1 + \text{exp}\{\gamma^{(2)}_{\ell kj0} + \gamma^{(2)}_{\ell kjZ^{(2)}} Z_i^{(2)}\}}$$ Response probability for individual $$i$$’s second-stage observed outcome category, conditional on the first-stage observed outcome and the true outcome.
$$\pi^{*(1)}_{ik}$$ $$P(Y^{*(1)}_i = k | X, Z^{(1)} ; \gamma^{(1)}) = \sum_{j = 1}^2 \pi^{*(1)}_{ikj} \pi_{ij}$$ Response probability for individual $$i$$’s first-stage observed outcome cateogry.
$$\pi^{*(1)}_{jj}$$ $$P(Y^{*(1)} = j | Y = j, Z^{(1)} ; \gamma^{(1)}) = \sum_{i = 1}^N \pi^{*(1)}_{ijj}$$ Average probability of first-stage correct classification for category $$j$$.
$$\pi^{*(2)}_{jjj}$$ $$P(Y^{*(2)} = j | Y^{*(1)}_i = j, Y = j, Z^{(2)} ; \gamma^{(2)}) = \sum_{i = 1}^N \pi^{*(2)}_{ijjj}$$ 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 $$j = 2$$.
$$\beta_X$$ Association parameter of interest in the true outcome mechanism.
$$\gamma^{(1)}_{11Z^{(1)}}$$ Association parameter of interest in the first-stage observation mechanism, given $$j=1$$.
$$\gamma^{(1)}_{12Z^{(1)}}$$ Association parameter of interest in the first-stage observation mechanism, given $$j=2$$.
$$\gamma^{(2)}_{111Z^{(2)}}$$ Association parameter of interest in the second-stage observation mechanism, given $$k = 1$$ and $$j = 1$$.
$$\gamma^{(2)}_{121Z^{(2)}}$$ Association parameter of interest in the second-stage observation mechanism, given $$k = 2$$ and $$j = 1$$.
$$\gamma^{(2)}_{112Z^{(2)}}$$ Association parameter of interest in the second-stage observation mechanism, given $$k = 1$$ and $$j = 2$$.
$$\gamma^{(2)}_{122Z^{(2)}}$$ Association parameter of interest in the second-stage observation mechanism, given $$k = 2$$ and $$j = 2$$.