COMMA Notation Guide
Kimberly Webb
2024-04-23
COMMA_Notation_Guide.RmdThis 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. |