COMBO Notation Guide - Two-stage Misclassification Model

Kim Hochstedler

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 $j = 2$.