A primer on causal graphs

Table of Contents

  1. Adjustment before effect estimation
    1. Principle
    2. Triplet structures
  2. Variable selection
  3. Expliciting assumptions
  4. Beyond the DAG

Causal graphs depict the assumed process underlying the generation of the analysed data. They explicitly represent the presence or absence of relationships between the variables, without making any assumptions about functional forms. Although different types of graph coexist (see Beyond the DAG section), the Directed Acyclic Graph (DAG), popularised by Judea Pearl, remains the cornerstone. This type of graph only allows oriented links between variables and prohibits cycles of influence. Working with graphs can support various research objectives, such as estimating unbiased causal effects and improving system comprehension by targeting efforts towards critical and unverified links. It can also enhance out-of-sample predictions and scenario projection, to name a few.

On this page, we briefly explain why DAGs are useful. Using nodes and arrows to sketch research assumptions is not only necessary for detection and attribution methods, but can also be highly beneficial for other research objectives. In fact, we are convinced it should become a reflex prior to any modelling exercise!

TL;DR - When should we draw DAGs?

Always, i.e. before any modelling exercise.

Adjustment before effect estimation

Principle

The availability of large amounts of observational data has paved the way for breakthrough statistical modelling approaches. However, it is also subject to many biases, such as observation effort, detection bias and any potential confounding variables affecting both the tested variable and the response. This is why, when it comes to estimating causal effects, the scientific community has adopted a set of rules and assumptions for causal inference that, when adhered to, enable unbiased estimation. Within the Structural Causal Model (SCM, see Causal paradigms), these rules belong to a general theory called do-Calculus and developed by Judea Pearl in 1995 (Pearl, 1995). See Entrop (2024) for an introductory blog post. The backdoor criterion is the most widespread identification tool derived from this theory. When targeting a total effect, it involves blocking all backdoor paths between the treatment and response variables.

Triplet structures

Confounders

AA path containing a confounder has to be included in the model, i.e. adjusted for. Controlling for an intermediary variable along the confounding path, e.g. B in T<--B<--C-->Y is not sufficient.

block-beta
    columns 5
    space:2 C((("Confounder"))) space:2
    
    space T C space
    T(("Exposure")) --> Y(("Outcome"))
    C --> T
    C --> Y

    style C fill:#e4686a,stroke:#333;
    style T fill:#6ae468;
    style Y fill:#47b8e4;

Real threats to correct effect estimation arise when confounding sources are known but unobserved due to a lack of data, or even unexpected.

  • How to properly deal with such unobserved confounding remains a major methodological challenge in causal inference (Byrnes & Dee, 2025).
  • Sensitivity analyses are designed to test the robustness of estimated effects against unobserved confounding factors (see section XXX).

Unobserved confounding is the Achilles heel of most nonexperimental studies

(Liu et al., 2013)

Colliders

Conversely, a path containing a collider should not be adjusted to avoid introducing a selection bias, see Munafò et al. (2018).

block-beta
    columns 5
    space:2 C((("Collider"))) space:2
    
    space T C space
    T(("Exposure")) ---> Y(("Outcome"))
    T --> C
    Y --> C

    style C fill:#dfb4f3,stroke:#333;
    style T fill:#6ae468;
    style Y fill:#47b8e4;

While colliders seem to be less frequent within ecological systems at first glance, controlling for the variables that actually influenced the selection of the study site can introduce such a bias.

Mediators

Finally, paths containing mediators (also called chains or pipes) should not be adjusted for when the total effect is the target estimand, but only when the direct effect is the target.

block-beta
    columns 5
    space:1 C((("Mediator"))):3 space:1 
    
    space T C space
    T(("Exposure")) -.-> Y(("Outcome"))
    T --> C
    C --> Y

style C fill:#fe85be,stroke:#333;
    style T fill:#6ae468;
    style Y fill:#47b8e4;
  • Mediation analyses aim to distinguish the direct effect of one variable on another from the indirect (mediated) effect, see section XXX.
  • The Frontdoor criterion exploits a special mediator configuration where:
    • The target estimand is the total effect of T on Y

    • An observed mediator captures the full effect of T on Y (no direct effect).

    • This enables the total effect estimation, even in the presence of unobserved confounders.

Resources

Table 2 fallacy

A now-famous pitfall in effect estimation is interpreting every coefficient from a fitted model as causal (Westreich & Greenland, 2013). The name comes from the condemned habit of reporting every variable coefficient in the second table of a research article, with the first table describing the dataset.

Variable selection

Graphs are a key tool in strategies for identifying causal effects. While it is their main function, it is not the only one. They can also assist the choice of variable selection in a predictive task. In their recent study, Pichler & Hartig (2023) illustrate an expected but key result: relying only on causal drivers — instead of all collected data including a collider — significantly improves out-of-distribution prediction at a marginal cost for in-distribution predictions. Adapting modern deep learning (DL) techniques to this result is a key element of success for better generalisation.

The adjustment sets obtained by applying the backdoor criterion to DAGs are minimal; they only include the variables that need to be accounted for to achieve unbiased effect estimates. However, they are not the only neither necessarily the best option, depedending on the aim: on top of adjustement, efficiency, precision and robustness may also be priorities. Different strategies are positioned in Witte & Didelez (2019). For instance, the outcome approach leads to an increased precision for many estimation models, but may miss confounders that are weakly associated with the outcome but strongly associated with the treatment. Variable selection in high-dimensional settings targeting improved effect estimation rather than purely predictive modelling is an active research area (Tang et al., 2023).

Expliciting assumptions

Rigor & communication

Another key advantage of representing assumptions about the studied system with causal graphs is rendering them explicit. While it may seem straightforward at first, it actually proves to be an incredibly useful exercise in improving communication and justifying results.

In research articles, inputs are often fed to models based on ecological knowledge (typically bioclimatic variables), or on justifications specific to the problem at hand (e.g. human footprint considered as the inverse of habitat connectivity). However, the links between these inputs are rarely considered, particularly in prediction tasks. This can impact the measures of variable contribution and the conclusions drawn. By requiring consideration of the interdependencies between variables and their potential consequences, causal graphs improve the rigour and transparency of modelling in ecological science (Borger & Ramesh, 2024). Overall, they increase reproducibility and confidence in results.

DAGs should be read not only in terms of the represented nodes and arrows, but also in terms of the absence of arrows between nodes, which is equally an important assumption.

Discussion & iterative science

By explicitly setting out research hypotheses, DAGs facilitate constructive discussion of results in relation to ecological assumptions. Alternative assumptions and subsequent results can be used to challenge conclusions in a reasoned debate based on graph structures. Such a practice has the potential to greatly benefit iterative science (Brodie et al., 2025).

Beyond the DAG

DAGs are neither the only option for representing causal relationships nor the most flexible. In fact, many other options exist, some of which are specifically designed to extend DAGs or relax constraints. See Vowels, Camgoz, & Bowden (2022) section 2 for a survey.

Firstly, when a large number of unobserved confounders hinder the readability of a DAG, switching to an Acyclic Directed Mixed Graph (ADMG) can help researchers recover a clear scheme that supports identification strategies (Richardson, 2003). In ADMGs , a unidirected edge implies a direct or indirect link, and a bidirected edge represent a link including an unboserved confounder -or latente variable- hidden in the middle. Causal search algorithms commonly converge on Markov equivalence classes rather than a unique DAG, i.e. a set of distinct graphs that satisfy the same conditional independence relationships found in the data. These sets are typically represented using Completed Partially Directed Acyclic Graphs (CPDAGs), in which an edge is directed only if it is present in all graphs and is otherwise left undirected.

When dealing with time-series under the stationarity hypothesis, DAGs can be unfolded with various time steps for each variable, using window causal graphs (Runge et al., 2023). Causal precedence (only events that occurred before or at the same time can cause a variable) enables some of the many possible links to be pruned. Assaad, Devijver, & Gaussier (2022) suggest a trade-off between the full window causal graph and a summary causal graph that aggregates all time steps. The extended summary causal graph represents all past time steps together, while keeping them distinct from the present state variables (see their Figure 1).

Finally, recent developments offer causal structures that can be used to leverage specific organisational structures of information, enabling better causal identification:

  • Hierarchical graph structures (Weinstein & Blei, 2024) model unit-level variables with nested subunit-level variables and potential interactions in both directions.
  • Cluster DAGs (C-DAGs) allow you to define relationships between groups of variables without requiring you to specify within-group relationships (Anand et al., 2023).

These graph structures, which are designed to capture the interdependencies of complex systems, have great potential for application across scales in ecology.

References

  1. Pearl, J. (1995). Causal Diagrams for Empirical Research. Biometrika, 82, 669–688. https://doi.org/10.2307/2337329
  2. Entrop, J. P. (2024). The 3 Rules of Do-Calculus – Joshua Entrop 👋. In Quarto. https://www.joshua-entrop.com/post/the_3_rules_of_do_calculus.html
  3. Byrnes, J. E. K., & Dee, L. E. (2025). Causal Inference With Observational Data and Unobserved Confounding Variables. Ecology Letters, 28. https://doi.org/10.1111/ele.70023
  4. Liu, W., Kuramoto, S. J., & Stuart, E. A. (2013). An Introduction to Sensitivity Analysis for Unobserved Confounding in Nonexperimental Prevention Research. Prevention Science, 14, 570–580. https://doi.org/10.1007/s11121-012-0339-5
  5. Munafò, M. R., Tilling, K., Taylor, A. E., Evans, D. M., & Davey Smith, G. (2018). Collider Scope: When Selection Bias Can Substantially Influence Observed Associations. International Journal of Epidemiology, 47, 226–235. https://doi.org/10.1093/ije/dyx206
  6. Ankan, A., Wortel, I. M. N., & Textor, J. (2021). Testing Graphical Causal Models Using the R Package “Dagitty.” Current Protocols, 1, e45. https://doi.org/10.1002/cpz1.45
  7. Cinelli, C., Forney, A., & Pearl, J. (2020). A Crash Course in Good and Bad Controls [SSRN Scholarly Paper]. https://doi.org/10.2139/ssrn.3689437
  8. Westreich, D., & Greenland, S. (2013). The Table 2 Fallacy: Presenting and Interpreting Confounder and Modifier Coefficients. American Journal of Epidemiology, 177, 292–298. https://doi.org/10.1093/aje/kws412
  9. Pichler, M., & Hartig, F. (2023). Can Predictive Models Be Used for Causal Inference? arXiv. https://doi.org/10.48550/arXiv.2306.10551
  10. Witte, J., & Didelez, V. (2019). Covariate Selection Strategies for Causal Inference: Classification and Comparison. Biometrical Journal, 61, 1270–1289. https://doi.org/10.1002/bimj.201700294
  11. Tang, D., Kong, D., Pan, W., & Wang, L. (2023). Ultra-High Dimensional Variable Selection for Doubly Robust Causal Inference. Biometrics, 79, 903–914. https://doi.org/10.1111/biom.13625
  12. Borger, M., & Ramesh, A. (2024). Let’s DAG in – How DAGs Can Help Behavioural Ecology Be More Transparent. https://ecoevorxiv.org/repository/view/7606/
  13. Brodie, J. F., Mohd-Azlan, J., Chen, C., Wearn, O. R., Deith, M. C. M., Ball, J. G. C., Slade, E. M., Burslem, D. F. R. P., Teoh, S. W., Williams, P. J., Nguyen, A., Moore, J. H., Goetz, S. J., Burns, P., Jantz, P., Hakkenberg, C. R., Kaszta, Z., Cushman, S., Coomes, D., … Luskin, M. S. (2025). Reply to: Causal Claims, Causal Assumptions and Protected Area Impact. Nature, 638, E42–E44. https://doi.org/10.1038/s41586-024-08513-7
  14. Vowels, M. J., Camgoz, N. C., & Bowden, R. (2022). D’ya Like DAGs? A Survey on Structure Learning and Causal Discovery. ACM Computing Surveys, 55, 82:1–82:36. https://doi.org/10.1145/3527154
  15. Richardson, T. (2003). Markov Properties for Acyclic Directed Mixed Graphs. Scandinavian Journal of Statistics, 30, 145–157. https://doi.org/10.1111/1467-9469.00323
  16. Runge, J., Gerhardus, A., Varando, G., Eyring, V., & Camps-Valls, G. (2023). Causal Inference for Time Series. Nature Reviews Earth & Environment, 4, 487–505. https://doi.org/10.1038/s43017-023-00431-y
  17. Assaad, C. K., Devijver, E., & Gaussier, E. (2022). Discovery of Extended Summary Graphs in Time Series. Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence, 96–106. https://proceedings.mlr.press/v180/assaad22a.html
  18. Weinstein, E. N., & Blei, D. M. (2024). Hierarchical Causal Models. arXiv. https://doi.org/10.48550/arXiv.2401.05330
  19. Anand, T. V., Ribeiro, A. H., Tian, J., & Bareinboim, E. (2023). Causal Effect Identification in Cluster DAGs. arXiv. https://doi.org/10.48550/arXiv.2202.12263