---
title: "Parameterization of the meta-d' model"
output: rmarkdown::html_vignette
bibliography: citations.bib
csl: apa.csl
link-citations: true
vignette: >
  %\VignetteIndexEntry{Parameterization of the meta-d' model}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  dpi = 150, fig.width = 8, fig.height = 6,
  fig.align = "center", out.width = "75%"
)
```

This vignette describes the parameterization of the meta-d' model used in the `hmetad` package. For an in-depth review, we refer readers to [@maniscalco2012; @maniscalco2014].


## Introduction
In metacognition research, decision-making is often categorized into two kinds. 

The *type 1* decision is the primary decision at hand, for example judging the orientation of a grating as left or right, or judging a word as either having been presented before (old) or not (new). In the context of the meta-d' model, the type 1 decision is a binary decision task (i.e., either a yes-no decision task or a two-alternative forced choice task).

In contrast, the *type 2* decision is the task of rating confidence in the type 1 decision. The meta-d' model is applicable where the type 2 decision is categorical (i.e., confidence is rated on an ordinal scale from `1` to `K`, where `1` indicates low confidence and `K` indicates high confidence).

In line with the distinction between type 1 and type 2 decisions, the meta-d' model is a conjunction of two models (one for each decision). Both models are based in Signal Detection Theory (SDT). However, the insight of the meta-d' model is that the information available for type 2 decisions might differ from the information available for type 1 decisions.



## Model for type 1 decisions
Under SDT, the observer is presented with a noisy signal dependent on the underlying stimulus $x_1 \sim \mathcal{D}_S(d')$ following a distribution $\mathcal{D}$ dependent on a stimulus $S \in \{0, 1\}$ and the observer's sensitivity $d'$. Typically, $\mathcal{D}$ is chosen to be an equal-variance normal distribution, such that 

$$\mathcal{D}_0(d') = \mathcal{N}\left(-\frac{d'}{2}, 1\right)$$ and $$\mathcal{D}_1(d') = \mathcal{N}\left(\frac{d'}{2}, 1\right)$$ 

However, this choice is arbitrary and other options are available.

Given the noisy encoding $x_1$, the observer is tasked with determining the true value of $S$. To do so, they simply threshold $x_1$ such that their response is 

$$R = [x_1 > c]$$

In this setup, the observer makes a correct response when $R = S$. More importantly, trials can be categorized into *hits* ($S = 1$ and $R = 1$), *misses* ($S = 1$ and $R = 0$), *false alarms* (FAs; $S = 0$ and $R = 1$), and *correct rejections* (CRs; $S = 0$ and $R = 0$). The generative model for type 1 decisions above implies the following response probabilities:

$$
P(R=r \;\vert\; S=s) = \begin{cases}
1 - F_{\mathcal{D}_s(d')}\left(c\right) & \textrm{if } r=1 \\
F_{\mathcal{D}_s(d')}\left(c\right) & \textrm{if } r=0
\end{cases}
$$


## Model for type 2 decisions
In classical SDT, type 2 decisions are treated just like type 1 decisions but with more stringent or more liberal response criteria -- when the signal passes a more stringent criterion, this is treated as a higher confidence choice. However, this assumes that observers have access to the same information when making type 1 and type 2 decisions. Relaxing this assumption, the meta-d' model assumes that type 2 decisions are derived from a separate decision variable: 

$$
x_2 \sim \begin{cases}
\mathcal{D}_S^{(-\infty, \textrm{meta-}c]}\left(\textrm{meta-}d'\right) & \textrm{if } R=0 \\
\mathcal{D}_S^{[\textrm{meta-}c, \infty)}\left(\textrm{meta-}d'\right) & \textrm{if } R=1
\end{cases}
$$

Importantly, this decision variable follows the same distribution as for the type 1 decision, with two differences. First, the distribution is truncated either above or below at the type 1 criterion $\textrm{meta-}c$ depending on the initial type 1 response. This is so that the type 2 decision cannot contradict the type 1 decision (i.e., the meta-d' model does not allow for changes of mind). Second, the sensitivity for the type 2 decision is $\textrm{meta-}d'$ rather than $d'$ to allow task-level sensitivity and metacognitive sensitivity to differ.

Then, to determine the confidence level $C \in \{ 1 \ldots K\}$, the observer rates confidence using one of two sets of $K-1$ ordered confidence criteria ($\textrm{meta-}c_2^0$ or $\textrm{meta-}c_2^1$):

$$
\begin{align*}
    C &= \begin{cases}
    1+\Sigma_{k=1}^{K-1}[x_2 < \textrm{meta-}c_{2,k}^0] & \textrm{if } R=0 \\
    1+\Sigma_{k=1}^{K-1} [x_2 > \textrm{meta-}c_{2,k}^1] & \textrm{if } R=1
    \end{cases}
\end{align*}
$$

This generative model implies that, conditional on the stimulus and the type 1 response, the type 2 response probabilities are

\begin{align*}
    P(C=c \;\vert\; R=r,S=s) &= \begin{cases}
    \frac{F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c\right) -
    F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c_{2,1}^0\right)}
    {F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c\right)}
    & \textrm{if } r=0, c=1 \\
    
    \frac{F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c_{2,k}^0\right) -
    F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c_{2,k+1}^0\right)}
    {F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c\right)}  
    & \textrm{if } r=0, 1 \le c \le K \\
    
    \frac{F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c_{2,K}^0\right)}
    {F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c\right)}
    & \textrm{if } r=0, c = K \\
    
    \frac{F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c_{2,1}^1\right) -
    F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c \right)}
    {1 - F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c\right)}
    & \textrm{if } r=1, c=1 \\
    
    \frac{F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c_{2,k+1}^1\right) -
    F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c_{2,k}^1\right)}
    {1 - F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c\right)}
    & \textrm{if } r=1, 1 \le c \le K \\
    
    \frac{1 - F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c_{2,K}^1 \right)}
    {1 - F_{\mathcal{D}_s(\textrm{meta-}d')}\left(\textrm{meta-}c\right)}
    & \textrm{if } r=1, c = K
    \end{cases}
\end{align*}

In these formulas, the numerator is the probability that $x_2$ lies between successive confidence thresholds and the denominator is the probability of a type 1 response $r$ given the type 2 parameters $\textrm{meta-}d'$ and $\textrm{meta-}c$ to account for the truncation of the type 2 signal distributions at $\textrm{meta-}c$.


## Joint model for type 1 and type 2 decisions
Ultimately, we are interested in the joint type 1 and type 2 response probabilities for each stimulus. This can be conveniently decomposed into the type 1 response probability and the conditional type 2 response probability as follows:

$$
P(R=r, C=c \;\vert\; S=s) = 
P(R=r \;\vert\; S=s) \; P(C=c \;\vert\; R=r, S=s) 
$$

Given the joint response probabilities for each stimulus, we can formulate the log likelihood of the meta-d' in one of two ways. If trial-level effects are of interest, one can model individual trials using a categorical likelihood:

$$
LL \;=\; \sum_n \textrm{categorical}\_\textrm{lpmf}\left(\textrm{joint}\_\textrm{response}(r_n,c_n) \;\vert\; P(R=r_n, C=c_n \;\vert\; S=s_n)\right)
$$

However, this formulation requires the likelihood to be evaluated once per trial, which for well-powered experiments can take a long time. So, by default the `hmetad` package uses a multinomial likelihood over the aggregated data. Where $N_{s,r,c}$ is the number of trials with $S=s$, $R=r$, and $C=c$:

$$
\begin{align*}
LL \;&=\; \textrm{multinomial}\_\textrm{lpmf}\left(N_{0,r,c} \;\vert\; P(R=r, C=c \;\vert\; S=0)\right) \\
&\;\quad+\; \textrm{multinomial}\_\textrm{lpmf}\left(N_{1,r,c} \;\vert\; P(R=r, C=c \;\vert\; S=1)\right)
\end{align*}
$$

This formulation only requires the model likelihood to be evaluated twice (once per stimulus), dramatically increasing the efficiency of model fitting. Because of the increase in efficiency, the multinomial likelihood is the default in the `hmetad` package. If a categorical likelihood is desired (e.g., for estimating trial-level effects or crossed random effects), it can be used with the argument `categorical=TRUE`.

## Fixing the type 1 threshold for type 2 responses
The meta-d' model requires the parameter $\textrm{meta-}c$ to be fixed to be equal with respect to the type 1 criterion $c$. As discussed by [@maniscalco2014], there are multiple ways of fixing $\textrm{meta-}c$. The `hmetad` package implements two:

Under the fixed parameterization, $\textrm{meta-}c = c$. This parameterization is used as the default, since it was also used in previous versions of the [Hmeta-d toolbox](https://github.com/metacoglab/HMeta-d) (see also @fleming2017).

Alternatively, under the relative parameterization, $\frac{\textrm{meta-}c}{\textrm{meta-}d'} = \frac{c}{d'}$, which is achieved by setting $\textrm{meta-}c = M c$. This parameterization was used in [@maniscalco2012; @maniscalco2014].

To switch between these two parameterizations, the `fit_metad` and `sim_metad` functions have an argument `metac_absolute` which is `TRUE` by default. To use the relative parameterization, simply set `metac_absolute=FALSE` in your call to each function.

## Model parameterization

To further increase the efficiency model fitting and help with convergence, the `hmetad` package parameterizes the meta-d' model so that all parameters are unconstrained variables (i.e., they fall in the range $(-\infty, \infty)$). The parameters for type 1 responses ($d'$ and $c$) are already unconstrained, so they are estimated normally. However, the parameters for the type 2 parameters are bounded.

First, instead of fitting $\textrm{meta-}d'$ directly, the `hmetad` package models the $M$-ratio $M = \frac{\textrm{meta-}d'}{d'}$. While this parameterization helps regularize against strong differences between $\textrm{meta-}d'$ and $d'$, the $M$-ratio is still bounded by zero. So, the `hmetad` package models the $M$-ratio on the logarithmic scale, i.e., $\textrm{log }M = \textrm{log}\frac{\textrm{meta-}d'}{d'}$. In this parameterization, one can compute $\textrm{meta-}d'$ as $\textrm{meta-}d' = e^{\textrm{log }M}d'$.

Second, the confidence criteria $\textrm{meta-}c_{2,1:K}^0$ and $\textrm{meta-}c_{2,1:K}^1$ each have two constraints. Namely, $\textrm{meta-}c_{2,1:K}^0$ must be strictly decreasing and less than $\textrm{meta-}c$, whereas $\textrm{meta-}c_{2,1:K}^1$ must be strictly increasing and greater than $\textrm{meta-}c$. To deal with these constraints, the `hmetad` package estimates the differences between successive confidence criteria:

$$
\textrm{dmeta-}c_{2,k}^0 = \begin{cases}
  \textrm{meta-}c - \textrm{meta-}c_{2,1}^0 & \textrm{if } k = 1 \\
  \textrm{meta-}c_{2,k-1}^0  - \textrm{meta-}c_{2,k}^0 & \textrm{if } 2 \le k \le K
\end{cases}
$$
$$
\textrm{dmeta-}c_{2,k}^1 = \begin{cases}
  \textrm{meta-}c_{2,1}^1 - \textrm{meta-}c & \textrm{if } k = 1 \\
  \textrm{meta-}c_{2,k}^1  - \textrm{meta-}c_{2,k-1}^1 & \textrm{if } 2 \le k \le K
\end{cases}
$$

Like for $M$-ratio, the differences between successive confidence criteria are also modeled on the logarithmic scale (parameters are named `metac2zero<k>diff` and `metac2one<k>diff`). Under this parameterization, the confidence criteria can be computed as follows:

$$
\begin{align*}
\textrm{meta-}c_2^0 &= \textrm{meta-}c - \textrm{cumulative}\_\textrm{sum}\left(e^{\textrm{dmeta-}c_2^0}\right) \\
\textrm{meta-}c_2^1 &= \textrm{meta-}c + \textrm{cumulative}\_\textrm{sum}\left(e^{\textrm{dmeta-}c_2^1}\right)
\end{align*}
$$

## References
