Connecting the Layers in the Models

How the Different Layers in the Formal Models Interact

Note: For context see here:

Introduction

In Layered Explanations for Statistical and Other Formal Models, I introduced a layered framework for understanding statistical inference and formal modeling. The goal was to separate concerns so that different aspects of statistical reasoning—mathematical, conceptual, application, interpretation, and validation—could be analyzed independently. However, while treating these layers as distinct allows for clarity in analysis, in practice, they interact dynamically. Understanding these interactions is just as important as understanding the layers themselves, as errors at one level can propagate through the system in complex ways.

This essay explores how the different layers interact, focusing primarily on the conceptual layer, which serves as a bridge between the mathematical foundation of statistics and its real-world applications. Misunderstandings or weaknesses in the conceptual layer can lead to misapplications of statistical methods, misinterpretations of results, and ultimately flawed validation processes. The interplay between these layers helps explain why many statistical practices in science lead to misleading conclusions and why the replication crisis has become a widespread issue.

Discussion: Interactions Between Layers

The layers do not exist in isolation. Instead, they depend on and influence each other, forming a web of interactions that shape how statistical inference is conducted, interpreted, and validated. The most important interactions revolve around the conceptual layer, which informs how statistical techniques are applied, how results are interpreted, and how methods are validated. Below, I examine these key interactions.

1. Conceptual Layer and Mathematical Layer

The mathematical layer provides the formal structure of statistical models, but it does not dictate how these models should be used—this is the role of the conceptual layer.

• Mathematics depends on conceptual clarity – The development of statistical methods requires at least a basic conceptual understanding of the problem being modeled. If this conceptual understanding is weak or flawed, the resulting mathematics may be inadequate for real-world inference.

• Conceptual frameworks shape mathematical choices – The same problem can be approached with different mathematical tools, depending on conceptual preferences. For example, frequentist statistics, Bayesian inference, and signal detection methods all offer different ways to think about probability, but their choice depends on conceptual assumptions about uncertainty and evidence.

• Mathematics can expose conceptual errors – If a statistical model produces nonsensical or contradictory results, this often signals a flaw in the conceptual framework. Historically, probability paradoxes and contradictions in early statistical theories led to refinements in both mathematical and conceptual thinking.

Thus, while the mathematical layer provides the formal rigor, it is ultimately the conceptual layer that determines its meaningful application.

2. Conceptual Layer and Application Layer

The application layer consists of the actual use of statistical methods in research. The way statistical techniques are applied depends entirely on the conceptual framework guiding their use.

• Proper conceptual understanding leads to correct application – If researchers correctly understand the meaning of p-values, confidence intervals, and regression assumptions, they are more likely to apply them correctly in empirical studies.

• Conceptual misunderstandings lead to misapplication – Many researchers misapply statistical techniques because they misunderstand their assumptions.

  • Example: A criminology professor asks a research assistant to compute a massive matrix of correlations without correcting for multiple comparisons, leading to meaningless results.

  • Example: A psychologist runs an ANOVA without checking whether assumptions of normality and homogeneity of variance hold, leading to unreliable p-values.

• Institutional norms and rote application – Many statistical methods are applied mechanically, following academic conventions rather than conceptual reasoning. Null Hypothesis Significance Testing (NHST) is a prime example, as many researchers conduct p-value-driven research without understanding its limitations.

Thus, a weak conceptual foundation leads directly to poor application, reinforcing flawed research practices.

3. Conceptual Layer and Interpretive Layer

The interpretation layer is where statistical results are given meaning. However, statistical output does not interpret itself—researchers must contextualize results within a conceptual framework.

• Conceptual clarity leads to meaningful interpretation – If researchers correctly understand what statistical results represent, they can make appropriate inferences from them.

• Conceptual confusion leads to misinterpretation – Many misinterpretations arise because researchers impose incorrect meanings onto statistical measures.

  • Example: Treating p-values as the probability that the null hypothesis is true, rather than as a measure of consistency between data and a null model.

  • Example: Interpreting regression coefficients as direct causal effects, even when confounding variables are unaccounted for.

• Over-reliance on statistical significance – Many researchers ignore effect sizes and practical relevance, interpreting statistically significant but trivial results as meaningful findings. This leads to misleading conclusions in psychology, nutrition, and medicine.

Thus, errors in the conceptual layer not only lead to misapplication but also to misleading interpretations of results, contributing to unreliable science.

4. Conceptual Layer and Validation/Verification Layer

The validation and verification layer assesses whether statistical methods and their interpretations are actually fit for purpose. However, validation efforts themselves are filtered through the conceptual framework, making conceptual clarity crucial.

• Proper validation depends on conceptual soundness – If researchers understand the purpose of statistical methods, they can evaluate whether they achieve their intended goals.

• Conceptual errors lead to flawed validation – If researchers assume flawed statistical concepts are valid, they may validate methods that should be rejected.

  • Example: The continued reliance on NHST in many fields, despite its conceptual weaknesses, perpetuates a flawed validation framework.

  • Example: The replication crisis has revealed that many published findings do not hold up under reanalysis—yet, without conceptual scrutiny, replication efforts may fail to address the root problems.

• Circular validation problems – If the conceptual layer is flawed, replication studies and validation efforts can become self-referential rather than corrective. Replicating a flawed methodology does not make it more valid—it only reinforces the original problem.

Thus, if the conceptual layer is weak, validation efforts may be meaningless, leading to false confidence in flawed statistical practices.

Summary

1. The Conceptual Layer as a Linchpin

The conceptual layer is the bridge between mathematics, application, interpretation, and validation. Its clarity—or lack thereof—determines the integrity of statistical inference.

2. How Conceptual Weaknesses Corrupt the Process

A flawed conceptual layer does not necessarily affect mathematical correctness, but it can distort how methods are applied, how results are interpreted, and how findings are validated. This explains why:

• Many researchers misapply statistical methods without realizing it.

• Interpretations of results are often shaped by institutional conventions rather than scientific reasoning.

• Validation efforts may reinforce flawed methodologies rather than correct them.

3. The Risk of Rote Science

While statistical methods can theoretically be applied and interpreted by rote, this would be intellectually hollow and lead to meaningless results. Yet, many researchers do not depart too far from this model, using statistics mechanically without engaging deeply with the conceptual layer.

4. The Need for Deeper Conceptual Engagement

To improve scientific research, we must:

• Strengthen conceptual understanding among researchers.

• Challenge conventional but flawed statistical practices.

• Ensure validation efforts are not circular but genuinely corrective.

By examining statistical inference through layered interactions, we can diagnose the weaknesses in current practices more precisely than conventional critiques. If the conceptual layer is unclear, the entire framework collapses, making conceptual clarity the foundation of meaningful scientific inference.