---
title: "Univariate dyadic workflow"
bibliography: references.bib
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Univariate dyadic workflow}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

## Overview

This vignette shows the univariate workflow implemented in `dyadicMarkov`. In the univariate setting, one categorical variable is observed repeatedly for the two members of a dyad. The workflow follows the single-case method described in @bollen2023: empirical transition counts are computed from the dyadic sequence, transition probabilities are estimated by maximum likelihood, and restricted actor/partner structures are compared to identify the pattern of interaction.

The example uses the data set `dyadic_univariate_example` included in the package. The data are synthetic and are used only to illustrate the required input structure and the package workflow.

Although the data are synthetic, the two columns can be read like real ordered observations from a dyad. For example, `FM` and `SM` could represent two partners, a parent and child, a therapist and client, or any two interacting members observed at repeated occasions. The integer states represent coded categories of a behavior or response. In a binary application, `state 1` and `state 2` could represent absence and presence of a coded behavior, two interaction states, or two response categories defined by the researcher.

## Data

The univariate example contains one categorical variable for the first member (`FM`) and the second member (`SM`) of a dyad. Each row corresponds to one measurement occasion. In the first analysis, the sequence of `FM`, the first member, is analyzed; `SM` is the second member or partner.

```{r univariate-data}
# Load the example data included with dyadicMarkov
utils::data("dyadic_univariate_example", package = "dyadicMarkov")

head(dyadic_univariate_example)
dim(dyadic_univariate_example)
```

In this example, the categorical states are coded as 1 and 2 for both members.

```{r univariate-states}
table(dyadic_univariate_example$FM)
table(dyadic_univariate_example$SM)
```

## Empirical transition counts

`countEmp()` computes the empirical transition counts for the first member from the two observed dyadic sequences. For `states = 2`, the resulting matrix has four rows corresponding to the possible previous dyadic states $(FM_t, SM_t)$ and two columns corresponding to the possible next states of the first member, $FM_{t+1}$. The returned `dyadic_counts` object retains ordinary matrix behavior and provides `print()`, `summary()`, and `utils::toLatex()` methods. The `summary()` method reports information including the matrix dimensions, total count, and row sums, while `utils::toLatex(emp_uni)` produces a LaTeX representation for reports or manuscripts.

```{r univariate-counts}
emp_uni <- dyadicMarkov::countEmp(
  chainFM = dyadic_univariate_example$FM,
  chainSM = dyadic_univariate_example$SM,
  states = 2L
)

print(emp_uni)
summary(emp_uni)
```

## Maximum-likelihood transition probabilities

The empirical counts are converted into estimated transition probabilities with `mleEstimation()`. For each previous-state combination that is observed in the data, the corresponding row of counts is divided by its row total so that the estimated transition probabilities sum to one.

If a previous-state combination is never observed, its row total is zero and there is therefore no information in the data from which to estimate its transition probabilities. In this case, `dyadicMarkov` returns equal probabilities for all possible next states as an implementation convention.

The returned `dyadic_mle` object retains ordinary matrix behavior and provides `print()`, `summary()`, and `utils::toLatex()` methods. The `summary()` method reports the matrix structure and row sums, while `utils::toLatex(fit_uni)` produces a LaTeX representation of the estimated transition matrix.

```{r univariate-mle}
fit_uni <- dyadicMarkov::mleEstimation(emp_uni)

print(fit_uni)
summary(fit_uni)
```

## Univariate pattern identification

The univariate method uses the A-family matrix codes for its dependence patterns: A1 denotes actor-partner, A2 actor only, and A3 partner only. The pattern nomenclature is summarized in Table 2 of @bollen2026.

The function `univariatePattern()` implements the univariate Likelihood-Ratio Test (LRT) procedure. It compares the unrestricted actor-partner structure with actor-only and partner-only restricted structures. `dyadicMarkov` evaluates these comparisons using Pearson's chi-squared statistic, $X^2 = \sum (O-E)^2/E$. The two test outcomes are then combined to classify the sequence as actor-partner, actor only, partner only, or independence.

The returned `dyadic_pattern` object contains the selected interaction pattern and the corresponding test results and provides `print()`, `summary()`, and `plot()` methods. The printed output gives the selected pattern, `summary()` reports the two restriction tests and their decisions at the specified significance level, and `plot()` displays the observed categorical sequences of the two members. Individual components can also be accessed directly, including `pat_uni$pattern`, `pat_uni$TEST.AM`, and `pat_uni$TEST.PM`.

```{r pattern-methods, fig.width=8.5, fig.height=4.1, out.width="99%", fig.align="center"}
pat_uni <- dyadicMarkov::univariatePattern(
  chainFM = dyadic_univariate_example$FM,
  chainSM = dyadic_univariate_example$SM,
  states = 2L,
  alpha = 0.05
)

print(pat_uni)
summary(pat_uni)
plot(pat_uni)
```

## Interpretation

In this example, the selected pattern is `PM (A3)`. This indicates that the previous state of the second member is retained in the restricted structure, whereas the previous state of the first member is not retained. In the terminology of the univariate method, this corresponds to a partner-only pattern.

The result should be interpreted as a pattern description for the analyzed sequence. The function call above analyzes the sequence of `FM`, the first member, with `SM` as the second member or partner. Reversing the two arguments analyzes the sequence from the perspective of the second member. Thus, describing both members of a dyad requires two calls, and each returned pattern is specific to the analyzed sequence.

```{r reverse-perspective}
pat_uni_reverse <- dyadicMarkov::univariatePattern(
  chainFM = dyadic_univariate_example$SM,
  chainSM = dyadic_univariate_example$FM,
  states = 2L,
  alpha = 0.05
)

pat_uni_reverse
```

In this example, reversing the two members also returns `PM (A3)`, but this does not occur in general: each call describes the pattern of the sequence supplied as the first member, conditional on the sequence supplied as the second member. For a complementary approach to visualization and clustering of dyadic longitudinal sequences, see @bollen2024.

## References