---
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}
utils::data("dyadic_univariate_example", package = "dyadicMarkov")

head(dyadic_univariate_example)
dim(dyadic_univariate_example)
```

The states are integer-coded. In this example, both members can take the values 1 or 2.

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

## Empirical transition counts

The first step is to compute empirical transition counts with `countEmp()`. For `states = 2`, the rows represent the four possible previous dyadic states `(FM_t, SM_t)`, and the columns represent the state of the first member at the next time point, `FM_{t+1}`.

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

emp_uni
class(emp_uni)
```

The resulting object keeps ordinary matrix behavior.

```{r univariate-counts-matrix}
dim(emp_uni)
rowSums(emp_uni)
```

## Maximum-likelihood transition probabilities

The empirical counts can be converted into transition probabilities using `mleEstimation()`. Rows with positive totals are normalized independently. For an unobserved previous-state combination, the transition probabilities are unidentified; the package returns a uniform row as an implementation convention.

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

round(fit_uni, 3)
rowSums(fit_uni)
class(fit_uni)
```

The estimated transition matrix summarizes the empirical transition structure of the observed dyadic sequence.

## 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.

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

pat_uni
```

For this example, the selected pattern is stored in the `pattern` component.

```{r univariate-pattern-details}
pat_uni$pattern
pat_uni$TEST.AM
pat_uni$TEST.PM
summary(pat_uni)
```

The returned object is a list with an additional S3 class. It can therefore be printed and summarized, while still allowing direct access to its components.

## 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.

## References