# Advanced Causal Analysis

# Introduction

This vignette demonstrates advanced usage of the `causaldef` package, focusing on diagnostics for unobserved confounding and sensitivity analysis. We utilize real data incorporated in the package to showcase these features.

    library(causaldef)
    library(ggplot2)

    # DAG Helper
    plot_dag <- function(coords, edges, title = NULL) {
      edges_df <- merge(edges, coords, by.x = "from", by.y = "name")
      colnames(edges_df)[c(3,4)] <- c("x_start", "y_start")
      edges_df <- merge(edges_df, coords, by.x = "to", by.y = "name")
      colnames(edges_df)[c(5,6)] <- c("x_end", "y_end")

      ggplot2::ggplot(coords, ggplot2::aes(x = x, y = y)) +
        ggplot2::geom_segment(data = edges_df, ggplot2::aes(x = x_start, y = y_start, xend = x_end, yend = y_end),
                     arrow = ggplot2::arrow(length = ggplot2::unit(0.3, "cm"), type = "closed"),
                     color = "gray40", size = 1, alpha = 0.8) +
        ggplot2::geom_point(size = 14, color = "white", fill = "#E7B800", shape = 21, stroke = 1.5) + # different color
        ggplot2::geom_text(ggplot2::aes(label = name), fontface = "bold", size = 3, color = "black") +
        ggplot2::ggtitle(title) +
        ggplot2::theme_void(base_size = 14) +
        ggplot2::theme(plot.title = ggplot2::element_text(hjust = 0.5, face = "bold", margin = ggplot2::margin(b = 10))) +
        ggplot2::coord_fixed()
    }

# Unobserved Confounding and Negative Controls

Unobserved confounding is a major threat to causal inference. The `causaldef` package implements “Negative Control Outcomes” to detect residual confounding. A negative control outcome is a variable that is affected by unobserved confounders but is known not to be causally affected by the treatment.

We will use the `gene_perturbation` dataset provided in the package. This dataset comes from a CRISPR knockout experiment.

- Treatment (`knockout_status`): Whether a gene was knocked out (Treatment) or not (Control).
- Outcome (`target_expression`): Expression level of the target gene.
- Confounders: Technical factors like `library_size` and `batch`.
- Negative Control (`housekeeping_gene`): A housekeeping gene that should not be affected by the specific knockout but shares technical confounders.

## Loading the Data

    data("gene_perturbation")
    head(gene_perturbation)
    #>   sample_id batch library_size knockout_status target_expression
    #> 1        S1     3       948877        Knockout             8.643
    #> 2        S2     4       984900         Control            11.545
    #> 3        S3     4       868332         Control            12.325
    #> 4        S4     2      1052105         Control            12.941
    #> 5        S5     2       880271         Control             8.289
    #> 6        S6     2      1001246        Knockout             9.400
    #>   housekeeping_gene
    #> 1             6.594
    #> 2             9.424
    #> 3             8.568
    #> 4             9.224
    #> 5             6.380
    #> 6             6.706

## Running the Negative Control Diagnostic

We first specify the causal model, explicitly identifying the negative control outcome.

    spec_nc <- causal_spec(
      data = gene_perturbation,
      treatment = "knockout_status",
      outcome = "target_expression",
      covariates = c("library_size", "batch"),
      negative_control = "housekeeping_gene"
    )
    #> ✔ Created causal specification: n=500, 2 covariate(s)

    # Visualize the Structural Assumption
    coords <- data.frame(
      name = c("Unobserved", "Treatment", "Target", "Housekp"),
      x = c(0, -1.5, 1.5, 0),
      y = c(1, 0, 0, -1)
    )
    edges <- data.frame(
      from = c("Unobserved", "Unobserved", "Unobserved", "Treatment"),
      to =   c("Treatment", "Target", "Housekp", "Target")
    )
    plot_dag(coords, edges, title = "Why Negative Controls Work:\nShared Unobserved Confounding")

Now we run the diagnostic using `nc_diagnostic()`. We test if the Inverse Probability of Treatment Weighting (IPTW) method successfully removes confounding bias.

    set.seed(123)
    # Check if IPTW approach effectively removes confounding using the negative control
    nc_res <- nc_diagnostic(spec_nc, method = "iptw", n_boot = 100)
    #> ℹ Using kappa = 1 (conservative). Consider domain-specific estimation or sensitivity analysis via kappa_range.
    #> ✔ No evidence against causal assumptions (p = 0.14851 )
    print(nc_res)
    #>
    #> -- Negative Control Diagnostic ----------------------------------------
    #>
    #> * screening statistic (weighted corr): 0.0821
    #> * delta_NC (association proxy): 0.0821
    #> * delta bound (under kappa alignment): 0.0821 (kappa = 1 )
    #> * screening p-value: 0.14851
    #> * screening method: weighted_permutation_correlation
    #>
    #> RESULT: NOT REJECTED. This is a screening result, not proof that confounding is absent.
    #> NOTE: Your effect estimate must exceed the Noise Floor (delta_bound) to be meaningful.

Interpreting Negative Control Diagnostics:

This diagnostic tests a crucial assumption: “Did my adjustment remove ALL confounding?”

Key Results to Check:

1. p_value:

   - > 0.05: [PASS] – No evidence that treatment affects the negative control after adjustment

   - \(\leq\) 0.05: [FAIL] – Treatment still associated with negative control -> residual confounding detected
2. falsified:
   - FALSE: Assumption not falsified -> Adjustment appears adequate
   - TRUE: Assumption falsified -> Likely unobserved confounding or model misspecification
3. delta_nc:
   - Magnitude of residual association with negative control
   - Smaller values indicate better performance
   - Should be close to zero if adjustment is successful
4. delta_bound:
   - Upper bound on your true deficiency: \(\delta\) \(\leq\) \(\kappa\) \(\times\) \(\delta\)_NC
   - Provides a worst-case estimate even when you can’t directly measure \(\delta\)

Scientific Implications:

The negative control is powerful because:

- It doesn’t rely on assumptions about what confounders exist
- It provides empirical evidence (not just theoretical arguments)
- Falsification is strong evidence – if it fails, your conclusions are suspect

If falsified:

- Do NOT ignore this result
- Investigate: batch effects, technical artifacts, unmeasured biology
- Consider: additional covariates, different methods, or acknowledging the limitation

## Addressing the Falsification

If the diagnostic fails (as it might with unobserved batch effects not fully captured), it indicates that your current adjustment set is insufficient to block all back-door paths to the negative control (and likely the outcome).

Recommended Actions:

1. Enrich the Confounder Set: If possible, measure and include additional potential confounders (e.g., more granular technical variables).
2. Sensitivity Analysis: Since you have evidence of unobserved confounding, use `confounding_frontier()` to determine how strong this confounding would need to be to invalidate your results.
3. Alternative Designs: Consider methods that rely on different assumptions, such as Instrumental Variables (if available) or Difference-in-Differences.
4. Transparent Reporting: If you cannot fix the issue, you must report the failure. “We detected residual confounding (\(\delta_{nc} = ...\)) using negative controls. Consequently, causal estimates interpretability is limited.”

# Sensitivity Analysis: Confounding Frontiers

When we suspect unobserved confounding, we can quantify how sensitive our causal conclusions are to potential unmeasured confounders \(U\). The `confounding_frontier()` function maps the “deficiency” (information loss) as a function of the strength of confounding paths \(U \to A\) (\(\alpha\)) and \(U \to Y\) (\(\gamma\)).

We will demonstrate this using the `hct_outcomes` dataset (Hematopoietic Cell Transplantation).

    data("hct_outcomes")

    # Convert treatment to numeric for the Gaussian model approximation
    hct_outcomes$conditioning_numeric <- as.numeric(hct_outcomes$conditioning_intensity) - 1

    # Simplified specification focusing on conditioning intensity
    spec_sens <- causal_spec(
      data = hct_outcomes,
      treatment = "conditioning_numeric",
      outcome = "time_to_event", # Simplified for this example
      covariates = c("age", "disease_status")
    )
    #> ✔ Created causal specification: n=800, 2 covariate(s)

We compute the confounding frontier. This does not require observing \(U\), but rather exploring the space of hypothetical confounding strengths.

    frontier <- confounding_frontier(
      spec_sens,
      alpha_range = c(-1, 1),
      gamma_range = c(-1, 1),
      grid_size = 40
    )
    #> ℹ Computing benchmarks for observed covariates...
    #> ✔ Computed confounding frontier: 40x40 grid

    # The result contains a grid we can visualize
    head(frontier$grid)
    #>        alpha gamma      delta
    #> 1 -1.0000000    -1 0.01464513
    #> 2 -0.9487179    -1 0.01529749
    #> 3 -0.8974359    -1 0.01600208
    #> 4 -0.8461538    -1 0.01676353
    #> 5 -0.7948718    -1 0.01758630
    #> 6 -0.7435897    -1 0.01847429

## Visualizing the Frontier

We can visualize the regions where deficiency is high (problematic) vs low.

    ggplot(frontier$grid, aes(x = alpha, y = gamma, fill = delta)) +
      geom_tile() +
      scale_fill_viridis_c(option = "magma", direction = -1) +
      labs(
        title = "Confounding Frontier",
        x = "Confounding Strength U -> A (alpha)",
        y = "Confounding Strength U -> Y (gamma)",
        fill = "Deficiency\n(Delta)"
      ) +
      theme_minimal() +
      geom_contour(aes(z = delta), color = "white", breaks = c(0.01, 0.05, 0.1))

Interpreting the Confounding Frontier:

This visualization maps the “danger zones” of unobserved confounding:

The Axes:

- x-axis (\(\alpha\)): Strength of confounding U -> A (unobserved confounder affects treatment)
- y-axis (\(\gamma\)): Strength of confounding U -> Y (unobserved confounder affects outcome)
- Color/fill (\(\delta\)): Resulting deficiency (information loss)

Reading the Plot:

1. Dark regions (low \(\delta\) \(\approx\) 0):

   - “Safe zones” where causal identification is robust
   - Small amounts of unobserved confounding don’t destroy identification
   - You can proceed with confidence in these scenarios

2. Bright regions (high \(\delta\) > 0.15):

   - “Danger zones” where unobserved confounding severely compromises identification
   - Large distributional distance from the ideal experiment (TV distance > 0.15)
   - Causal conclusions are unreliable without additional assumptions

3. Contour lines (white lines at \(\delta\) = 0.01, 0.05, 0.1):

   - Boundaries between different risk levels
   - The “frontier” where identification breaks down

Scientific Decision-Making:

Use domain knowledge to assess plausibility:

- If you believe |\(\alpha\)| < 0.3 and |\(\gamma\)| < 0.3 (weak confounding), and the plot shows low \(\delta\) in that region -> Proceed with confidence
- If plausible scenarios put you in high-\(\delta\) regions -> Your conclusions are fragile, acknowledge this limitation
- If uncertain about confounding strength -> Report results across the frontier to show sensitivity

Example Interpretation:
“Our sensitivity analysis shows that if unobserved confounding U -> A and U -> Y are both weak (|\(\alpha\)|, |\(\gamma\)| < 0.2), the deficiency remains below 0.05, suggesting robust identification. However, if either confounding strength exceeds 0.5, identification breaks down (\(\delta\) > 0.1). Given the biological plausibility of weak confounding in this system, we conclude our estimates are reasonably robust.”

The confounding frontier shows the boundary: if we are outside the “safe” zone defined by the frontier, we cannot claim causal identification without further assumptions (see Akdemir 2026, DOI: 10.5281/zenodo.18367347 for theoretical details).

# Conclusion

The `causaldef` package provides tools not just for estimation, but for falsification and sensitivity analysis.

1. Negative Controls allow us to empirically test our “no unobserved confounding” assumption using auxiliary data.
2. Confounding Frontiers allow us to theoretically explore how robust our study is to potential unobserved confounders.