# 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.
$$Z$$ Predictor matrix for the first-stage observed outcome, conditional on the true outcome.
V 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.
$$Y^*$$ $$Y^* \in \{1, 2\}$$ First-stage observed binary outcome. Reference category is 2.
$$y^*_{ik}$$ $$\mathbb{I}\{Y^*_i = k\}$$ Indicator for the first-stage observed binary outcome.
$$\tilde{Y}$$ $$\tilde{Y} \in \{1, 2\}$$ Second-stage observed binary outcome. Reference category is 2.
$$\tilde{y}_{i \ell}$$ $$\mathbb{I}\{\tilde{Y}_i = \ell \}$$ 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 $$\text{logit}\{ P(Y^* = k | Y = j, Z ; \gamma) \} = \gamma_{kj0} + \gamma_{kjZ} Z$$ Relationship between $$Z$$ and the first-stage observed outcome, $$Y^*$$, given the true outcome $$Y$$.
Second-Stage Observation Mechanism $$\text{logit}\{ P(\tilde{Y} = \ell | Y^* = k, Y = j, V ; \delta) \} = \delta_{\ell kj0} + \delta_{\ell kjV} V$$ Relationship between $$V$$ and the second-stage observed outcome, $$\tilde{Y}$$, given the first-stage observed outcome, $$Y^*$$, 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.
$$\pi^*_{ikj}$$ $$P(Y^*_i = k | Y_i = j, Z ; \gamma) = \frac{\text{exp}\{\gamma_{kj0} + \gamma_{kjZ} Z_i\}}{1 + \text{exp}\{\gamma_{kj0} + \gamma_{kjZ} Z_i\}}$$ Response probability for individual $$i$$’s first-stage observed outcome category, conditional on the true outcome.
$$\tilde{\pi}_{i \ell kj}$$ $$P(\tilde{Y}_i = \ell | Y^*_i = k, Y_i = j, Z ; \delta) = \frac{\text{exp}\{\delta_{\ell kj0} + \delta_{\ell kjV} V_i\}}{1 + \text{exp}\{\delta_{\ell kj0} + \delta_{\ell kjV} V_i\}}$$ Response probability for individual $$i$$’s second-stage observed outcome category, conditional on the first-stage observed outcome and 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 first-stage observed outcome cateogry.
$$\pi^*_{jj}$$ $$P(Y^* = j | Y = j, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{ijj}$$ Average probability of first-stage correct classification for category $$j$$.
$$\tilde{\pi}_{jjj}$$ $$P(\tilde{Y} = j | Y^* = j, Y = j, Z ; \delta) = \sum_{i = 1}^N \tilde{\pi}_{ijjj}$$ Average probability of first-stage and second-stage correct classification for category $$j$$.
First-Stage 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 $$j = 1$$.
Second-Stage 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 first-stage observation mechanism, given $$j=1$$.
$$\gamma_{12Z}$$ Association parameter of interest in the first-stage observation mechanism, given $$j=2$$.
$$\delta_{111Z}$$ Association parameter of interest in the second-stage observation mechanism, given $$k = 1$$ and $$j = 1$$.
$$\delta_{121Z}$$ Association parameter of interest in the second-stage observation mechanism, given $$k = 2$$ and $$j = 1$$.
$$\delta_{112Z}$$ Association parameter of interest in the second-stage observation mechanism, given $$k = 1$$ and $$j = 2$$.
$$\delta_{122Z}$$ Association parameter of interest in the second-stage observation mechanism, given $$k = 2$$ and $$j = 2$$.