---
title: "5. Case study: confounder audit"
author: "Abdelkarim Hani Ghrieb"
date: "`r Sys.Date()`"
output:
  BiocStyle::html_document:
    toc: true
    toc_float: true
vignette: >
  %\VignetteIndexEntry{5. Case study: confounder audit}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

# Status note

This is a synthetic-data walkthrough used to verify `design_audit()`
behavior on a deliberately confounded cohort. It is a vignette, not a study.
The narrative follows the *three-tier* structure used across the case
studies: a **naive analysis** first, then **rnaSentry standing guard**, then
the **counterfactual**.

# Setup

We build a cohort of 180 samples with two design variables that are almost
interchangeable:

- `region` is a 3-level factor derived by cutting a latent normal variable;
- `batch` is a *noisy copy* of `region` (80% of samples keep their `region`
  label; the rest are shuffled).

We also plant a real batch effect in expression: six genes are multiplied by
a batch-dependent factor. Finally, survival depends on the latent variable
(so `region`, and through its overlap `batch`, are genuine survival
correlates) -- a realistic setting in which an unadjusted screen can mistake
batch-driven genes for prognostic ones.

```{r build-cohort}
library(rnaSentry)
library(SummarizedExperiment)
library(survival)

set.seed(12)
n <- 180
latent <- rnorm(n)
region <- factor(cut(latent, breaks = 3, labels = c("R1", "R2", "R3")))
batch <- as.character(region)
flip <- runif(n) < 0.2
batch[flip] <- sample(c("R1", "R2", "R3"), sum(flip), replace = TRUE)
batch <- factor(batch)

counts <- matrix(rpois(50 * n, lambda = 200), nrow = 50, ncol = n,
                 dimnames = list(paste0("g", seq_len(50)), paste0("S", seq_len(n))))
counts[1:6, ] <- counts[1:6, ] *
  matrix(1 + 0.6 * as.integer(batch), nrow = 6, ncol = n, byrow = TRUE)

event_time <- rexp(n, rate = 0.05 * exp(0.8 * latent))
censor_time <- rexp(n, rate = 0.03)
cd <- S4Vectors::DataFrame(batch = batch, region = region,
                           time = pmin(event_time, censor_time),
                           event = as.integer(event_time < censor_time),
                           row.names = colnames(counts))
se <- SummarizedExperiment(assays = list(counts = counts), colData = cd)
se
```

# Tier 1: the naive analysis

A naive analyst runs univariate Cox regressions for every gene against
survival and takes the top hits -- without checking whether the candidate
genes are driven by batch, and without auditing the design.

```{r naive-screen, warning = FALSE}
lg <- log2(counts + 1)
stat <- do.call(rbind, lapply(rownames(lg), function(g) {
    s <- summary(coxph(Surv(cd$time, cd$event) ~ lg[g, ]))
    c(beta = s$coefficients[1, 1], p = s$coefficients[1, 5])
}))
rownames(stat) <- rownames(lg)
naive_top <- head(stat[order(stat[, "p"]), , drop = FALSE], 6)
data.frame(gene = rownames(naive_top),
           planted_batch_effect = rownames(naive_top) %in% paste0("g", 1:6),
           naive_top)
```

The six genes whose expression we multiplied by the batch factor rank as the
most "prognostic" genes in the cohort. An unguarded screen would put exactly
these batch artifacts into a signature. A naive model that then includes both
of the near-identical design variables compounds the problem by
double-counting collinear terms:

```{r double-count}
summary(coxph(Surv(time, event) ~ batch + region, data = as.data.frame(cd)))
```

`batch` and `region` carry almost the same information; fitting both inflates
standard errors and produces unstable coefficients that look like two
"independent" effects when there is really one.

# Tier 2: rnaSentry standing guard

## Design audit

```{r design-audit}
da <- design_audit(se, design_vars = c("batch", "region"),
                   redundant_effect_size = 0.5)
```

### Pairwise redundancy

```{r pairwise-table}
da$pairwise_table
```

`batch` and `region` are almost interchangeable (Cramer's V = 0.82), and the
pair is flagged `redundant`. Including both in a model would double-count
nearly the same information.

### Confounder scan

```{r confounder-table}
da$confounder_table
```

Both variables are flagged as associated with the expression surrogate:
`batch` with eta-squared 0.97, `region` with epsilon-squared 0.55 (it leaks
the same signal through its overlap with `batch`).

### What the surrogate gradient looks like

The surrogate the confounder scan tests against is the leading principal
component of the expression data. `pca_audit()` performs the same scan
against a single batch variable, one PC at a time:

```{r pca-batch-tests}
pa <- pca_audit(se, batch_col = "batch")
pa$batch_tests
```

`batch` is flagged as associated with the leading PCs. Plotting the scores
makes the structure visible: the three batches separate along PC1, which is
exactly the surrogate gradient `design_audit()` measures:

```{r pca-batch-plot, fig.width = 7, fig.height = 5, fig.align = "center", fig.cap = "PC1-PC2 scatter of the batch-confounded cohort, colored by batch. The three batch clusters separate along PC1 (the surrogate gradient); this is the structure the confounder scan detects."}
plot_pca_audit(pa, color_by = "batch")
```

Coloring the same scores by `region` gives a nearly identical picture,
because `batch` is a noisy copy of `region`:

```{r pca-region-plot, fig.width = 7, fig.height = 5, fig.align = "center", fig.cap = "The same PC1-PC2 scatter colored by region. The near-identical structure is why the redundancy scan reports Cramer's V = 0.82 and the recommended formula drops `region`."}
plot_pca_audit(pa, color_by = "region")
```

The two candidate variables track the same expression structure, so the
redundancy scan collapses them: keeping only `batch` in the design captures
the shared confound without double-counting collinear terms.

### Recovered design

```{r recovered-design}
da$formula_text
```

The audit therefore recommends the minimal formula `~ batch`, dropping the
redundant `region` term. This is the behavior verified by the test suite
(`confounder-detection` and `design-audit` blocks): a redundant design term
is reported, not silently kept.

## Batch-adjusted screening

Armed with the recommended design, the guarded discovery run adjusts its
univariate screening for `batch`. The batch-driven genes are no longer the
top hits:

```{r adj-screen, warning = FALSE}
adj <- do.call(rbind, lapply(rownames(lg), function(g) {
    s <- summary(coxph(Surv(cd$time, cd$event) ~ lg[g, ] + batch,
                       data = as.data.frame(cd)))
    c(beta = s$coefficients[1, 1], p = s$coefficients[1, 5])
}))
rownames(adj) <- rownames(lg)
guarded_top <- head(adj[order(adj[, "p"]), , drop = FALSE], 6)
data.frame(gene = rownames(guarded_top),
           planted_batch_effect = rownames(guarded_top) %in% paste0("g", 1:6),
           guarded_top)
```

Once `batch` is in the model, the planted genes lose their survival
association. The pipeline records this decision in the flag ledger:

```{r guarded-sig, warning = TRUE}
sig_guarded <- build_signature(se, "time", "event", top_n = 6,
                               repeats = 3, folds = 3,
                               design_terms = "batch", seed = 12)
sig_guarded$genes
sig_guarded$flags
```

# Tier 3: the counterfactual

| | Naive analysis (Tier 1) | rnaSentry (Tier 2) |
|---|---|---|
| Screening | unadjusted; the 6 batch-driven genes rank as most "prognostic" | adjusted for `batch`; batch-driven genes no longer top hits |
| Design | `batch + region` fitted together; collinear terms double-counted | `redundant` flag on the pair (Cramer's V = 0.82); minimal `~ batch` recommended |
| Surrogate association | never checked | `batch` eta-squared 0.97, `region` epsilon-squared 0.55 reported |
| Ledger | none | `redundant_variable`, `adjusted_screening` recorded |

**What a guarded analysis can claim:** without `design_audit()`, this cohort
would produce a six-gene "prognostic" signature that is entirely a batch
artifact, plus a model reporting two collinear "independent" risk factors.
With it, the batch confounder is surfaced with its effect size, the redundant
term is dropped, and screening is adjusted so the discovered signature is not
a batch echo.

# Reading

The lesson is that the recommended formula is only as good as the design
variables you supply. `design_audit()` will not invent a missing confounder;
it audits what it is given. Supply every variable that could plausibly
confound expression (batch, plate, sex, subtype), then trust the recommended
formula over an ad-hoc pick.

# Session info

```{r session-info}
sessionInfo()
```