---
title: "COMMA Notation Guide"
author: "Kimberly Webb"
date: "2024-04-23"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteEngine{knitr::knitr}
  %\VignetteIndexEntry{COMMA Notation Guide}
  %\usepackage[UTF-8]{inputenc}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = FALSE)

library(kableExtra)
```

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

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


notation_table <- data.frame(Term, Definition, Description)
```

```{r}
kableExtra::kbl(notation_table, escape = FALSE, booktabs = TRUE) %>%
  kableExtra::kable_styling(latex_options = "HOLD_position")
```