# COMMA Notation Guide

## Kimberly Webb

### 2024-04-23

This guide is designed to summarize key notation and quantities used the COMMA R Package and associated publications.

Term | Definition | Description
--- | --- | ---
$X$ | -- | Predictor matrix for the true mediator and outcome.
$C$ | -- | Covariate matrix for the true mediator and outcome.
$Z$ | -- | Predictor matrix for the observed mediator, conditional on the true mediator
$Y$ | -- | Outcome variable.
M | $M \in \{1, 2\}$ | True binary mediator. Reference category is 2.
$m_{ij}$ | $\mathbb{I}\{M_i = j\}$ | Indicator for the true binary mediator.
$M^*$ | $M^* \in \{1, 2\}$ | Observed binary mediator. Reference category is 2.
$m^*_{i \ell}$ | $\mathbb{I}\{M^*_i = \ell \}$ | Indicator for the observed binary mediator.
True Mediator Mechanism | $\text{logit} \{ P(M = 1 | X, C ; \beta) \} = \beta_{0} + \beta_{X} X + \beta_{C} C$ | Relationship between $X$ and $C$ and the true mediator, $M$.
Observed Mediator Mechanism | $\text{logit}\{ P(M^* = 1 | M = m, Z ; \gamma) \} = \gamma_{1m0} + \gamma_{1mZ} Z$ | Relationship between $Z$ and the observed mediator, $M^*$, given the true mediator $M$.
Outcome Mechanism | $E(Y| X, C, M ; \theta) \} = \theta{0} + \theta_{X} X + \theta_{C} C \theta_{M}M + \theta_{XM}XM$ | Relationship between $X$, $C$, and $M$ and the outcome of interest $Y$.
$\pi_{ij}$ | $P(M_i = j | X, C ; \beta) = \frac{\text{exp}\{\beta_{j0} + \beta_{jX} X_i + \beta_{jC} C_i\}}{1 + \text{exp}\{\beta_{j0} + \beta_{jX} X_i + \beta_{jC} C_i\}}$ | Response probability for individual $i$'s true mediator category.
$\pi^*_{i \ell j}$ | $P(M^*_i = \ell | M_i = j, Z ; \gamma) = \frac{\text{exp}\{\gamma_{\ell j 0} + \gamma_{ \ell jZ} Z_i\}}{1 + \text{exp}\{\gamma_{\ell j0} + \gamma_{kjZ} Z_i\}}$ | Response probability for individual $i$'s observed mediator category, conditional on the true mediator.
$\pi^*_{i \ell}$ | $P(M^*_i = \ell | M_i, X, Z ; \gamma) = \sum_{j = 1}^2 \pi^*_{i \ell j} \pi_{ij}$ | Response probability for individual $i$'s observed mediator cateogry.
$\pi^*_{jj}$ | $P(M^* = j | M = j, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{ijj}$ | Average probability of correct classification for category $j$.
Sensitivity | $P(M^* = 1 | M = 1, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{i11}$ | True positive rate. Average probability of observing mediator $k = 1$, given the true mediator $j = 1$.
Specificity | $P(M^* = 2 | M = 2, Z ; \gamma) = \sum_{i = 1}^N \pi^*_{i22}$ | True negative rate. Average probability of observing mediator $k = 2$, given the true mediator $j = 2$.
$\beta_X$ | -- | Association parameter of interest in the true mediator mechanism.
$\gamma_{11Z}$ | -- | Association parameter of interest in the observed mediator mechanism, given $j=1$.
$\gamma_{12Z}$ | -- | Association parameter of interest in the observed mediator mechanism, given $j=2$.
$\theta_X$ | -- | Association parameter of interest in the outcome mechanism.
$\theta_M$ | -- | Association parameter relating the true mediator to the outcome.
$\theta_{XM}$ | -- | Association parameter for the interaction between $X$ and $M$ in the outcome mechanism.