---
title: "2. Wide Correlation Workflows"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{2. Wide Correlation Workflows}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

## Scope

This vignette covers the basic wide-data correlation estimators. These methods
start from one numeric matrix or data frame, treat columns as variables, and
return a square matrix indexed by those columns.

The main functions in this group are:

- `pearson_corr()`
- `spearman_rho()`
- `kendall_tau()`
- `dcor()`

They answer related but not identical questions, so method choice should be
driven by the structure of the data rather than by habit alone.

## A common input pattern

```{r}
library(matrixCorr)

set.seed(10)
z <- rnorm(80)
u <- rnorm(80)
X <- data.frame(
  x1 = z + rnorm(80, sd = 0.35),
  x2 = 0.85 * z + rnorm(80, sd = 0.45),
  x3 = 0.25 * z + 0.70 * u + rnorm(80, sd = 0.45),
  x4 = rnorm(80)
)
```

All four estimators accept this same wide input format.

```{r}
R_pear <- pearson_corr(X)
R_spr  <- spearman_rho(X)
R_ken  <- kendall_tau(X)
R_dcor <- dcor(X)

print(R_pear, digits = 2)
summary(R_spr)
```

This toy dataset is intentionally structured so that `x1` and `x2` form a
clear linear pair, `x3` is only moderately related to that first block, and
`x4` is close to null. That makes the shared output structure easier to
interpret than a pure-noise example.

## Pearson correlation

Pearson correlation targets linear association on the original measurement
scale. It is the natural first choice when variables are continuous, the
relationship is approximately linear, and there is no strong concern about
outlier sensitivity.

```{r}
plot(R_pear)
```

If confidence intervals are required, they can be requested directly.

```{r}
R_pear_ci <- pearson_corr(X, ci = TRUE)
summary(R_pear_ci)
estimate(R_pear_ci)
confint(R_pear_ci)
ci(R_pear_ci)
tidy(R_pear_ci)
```

## Spearman and Kendall

Spearman's rho and Kendall's tau are rank-based estimators. They are useful
when monotone association is of interest and the analysis should be less
sensitive to departures from strict linearity. Both functions also support
optional large-sample confidence intervals through `ci = TRUE`.

```{r}
set.seed(11)
x <- sort(rnorm(60))
y <- x^3 + rnorm(60, sd = 0.5)
dat_mon <- data.frame(x = x, y = y)

pearson_corr(dat_mon)
spearman_rho(dat_mon)
kendall_tau(dat_mon)
```

In this setting the relationship is monotone but not linear, so a rank-based
summary is often the clearer first description.

When interval estimation is required, the same matrix-style interface is kept.

```{r}
fit_spr_ci <- spearman_rho(X, ci = TRUE)
fit_ken_ci <- kendall_tau(X, ci = TRUE)

summary(fit_spr_ci)
summary(fit_ken_ci)
```

## Distance correlation

Distance correlation addresses a broader target. It is designed to detect
general dependence rather than only linear or monotone structure. The function
also supports optional hypothesis testing through `p_value = TRUE`.

```{r}
set.seed(12)
x <- runif(100, -2, 2)
y <- x^2 + rnorm(100, sd = 0.2)
dat_nonlin <- data.frame(x = x, y = y)

pearson_corr(dat_nonlin)
dcor(dat_nonlin)
```

This is a typical situation where Pearson correlation can be close to zero even
though the variables are clearly dependent.

If a formal inferential summary is needed, p-values can be requested directly.

```{r}
fit_dcor_p <- dcor(dat_nonlin, p_value = TRUE)
summary(fit_dcor_p)
```

## Missing values

The default wide-data behaviour is strict validation. Missing values are
rejected unless the function explicitly supports a relaxed mode through
`na_method = "pairwise"`.

```{r}
X_miss <- X
X_miss$x2[c(3, 7)] <- NA

try(pearson_corr(X_miss))
pearson_corr(X_miss, na_method = "pairwise")
```

When `na_method = "pairwise"`, the package uses pairwise complete observations
for the affected estimator. That is convenient, but it also means different
pairs may be based on different effective sample sizes.

## Practical guidance

In ordinary wide-data work, the following sequence is usually defensible.

- Start with `pearson_corr()` when the variables are continuous and linear
  association is the scientific target.
- Use `spearman_rho()` or `kendall_tau()` when a monotone summary is preferred.
- Use `dcor()` when non-linear dependence is plausible and a zero Pearson
  correlation would be misleading.

The next vignette addresses settings where this basic family is still not
sufficient because the data are contaminated by outliers or the number of
variables is large relative to sample size.
