Lesson 1: Observational and Explanatory Models

1 Modelling context

You are asked to "build a model" of something — electricity demand in a city, the spread of an infection in a dormitory, the price of a commodity, the wait time at a clinic. Before you choose any mathematical form, you must answer a prior question: what kind of claim will this model be expected to support?

Some questions are predictive. "Given next week's weather forecast, how much electricity will the city consume?" The answer can be produced by a model that has no idea why consumption depends on weather — it only needs to have captured the relationship accurately over the range of conditions you care about.

Other questions are interventional. "If we raise the price of electricity by 10 percent, how much will consumption fall?" This is a different kind of question. A model that merely describes the correlation between price and consumption cannot answer it, because the observed correlation may be driven by something else entirely — income, season, or the very weather you are also trying to control for.

The distinction is captured by two model types. Observational models describe patterns in data. Explanatory models propose a mechanism connecting inputs to outputs. The two can fit the same data equally well and still be useful for entirely different purposes.

The question determines the model, not the reverse

Choose the model type after you have stated the question. A common failure mode in modelling is to build a convenient model first and then use it to answer a question it was never designed to address.

2 Learning objectives

By the end of this lesson you will be able to

Define observational and explanatory models and give a concrete example of each.
Distinguish predictive questions from interventional questions.
Explain why a good statistical fit is not evidence of a mechanism.
Describe confounding and explain how it limits observational models.
State the claim-support scope of a model as part of its documentation.

3 Prerequisite review

This lesson assumes familiarity with the following ideas. If any is unfamiliar, review it before proceeding — the rest of the unit builds directly on them.

Function notation: y = f(x) means that y is determined by x through the rule f. Models extend this idea to several inputs and to a residual or noise term.
Conditional expectation: E[Y | X = x] is the average value of Y among all observations for which X equals x. It is the quantity an observational regression estimates.
Variance and residual: The residual is the difference between an observed value and the model's prediction. Its spread is the variance the model fails to explain.
Covariates: The input variables a model uses. Choosing which covariates to include is a modelling decision, not a preprocessing step.

4 Mathematical formulation

Observational models

An observational model asserts a relationship between observed quantities. It is typically written as

Y = f(X) + ε

where f is fitted from data and ε is a residual term capturing everything the model does not represent. The quantity estimated is the conditional expectation:

f(x) ≈ E[Y | X = x]

The model makes no claim about why f has the shape it has, or about what would happen if X were altered by intervention. It says: among the observations we have, Y tends to take this value when X takes this value.

Explanatory models

An explanatory model proposes a mechanism. It is typically written as a system of relations whose parameters have a structural interpretation — a rate of infection, a rate of cooling, a rate of adoption. For example, in continuous time:

dT/dt = −k (T − T_env)

where T is the temperature of an object, T_env is the ambient temperature, and k is a rate constant with a physical interpretation. The parameters are not fitted to reproduce an output pattern; they are estimated from data so that the mechanism reproduces the observed behaviour.

The interventional question

The key distinction between the two model types is captured by two different expectations. The observational model estimates:

E[Y | X = x]        (observed association)

An explanatory model, if its mechanism is correct, supports the interventional quantity:

E[Y | do(X = x)]    (effect of intervening on X)

The two coincide only when X is the sole cause of Y. When a third variable Z influences both X and Y, the observed association is confounded and the two expectations differ.

Confounding, formally

Suppose Z is a confounder: it affects both X and Y, and X may or may not affect Y. Then

P(X, Y, Z) = P(Z) · P(X | Z) · P(Y | X, Z)

and the observed association between X and Y mixes the direct effect of X with the effect of Z flowing along both paths. An observational model cannot disentangle these without additional assumptions or experimental variation. An explanatory model that explicitly represents Z and its relations can, at least in principle.

Figure 1.3. Confounding. Z influences both X and Y. The solid arrows are identifiable from observation; the dashed arrow is not, unless the model explicitly represents Z or the analysis uses experimental variation in X.

5 Worked example

Two models of a cooling cup of coffee

A cup of coffee is placed on a desk. Its temperature is recorded once per minute for 30 minutes. You are asked to model the cooling process.

Model A — observational.

Fit an exponential decay to the observed temperature: T(t) = a + b · e^−ct, with a, b, c estimated by least squares.
The fit is excellent: residuals are small and show no obvious structure.
The model predicts T(t) accurately for any t in the observed window and a little beyond it.
The parameter c has no interpretation beyond "the rate at which this particular dataset decayed."

Model B — explanatory.

Propose Newton's law of cooling: dT/dt = −k (T − T_env), where T_env is the ambient temperature and k is a rate constant with units of inverse time.
Solve analytically: T(t) = T_env + (T₀ − T_env) · e^−kt.
Fit k to the same data. The fit is also excellent — in fact, for this dataset, Model A and Model B are nearly indistinguishable on residuals.
The parameter k has a physical meaning and can be compared with values measured in other experiments.

Where they differ.

Ask: "What if the room temperature were raised by 5°C?"

Model A cannot answer this. It contains no variable for room temperature; the observation window contains only one room temperature.
Model B can answer it, because T_env appears explicitly in the mechanism. Substituting a new value gives a new predicted trajectory. The answer is only as good as the assumption that Newton's law holds over the new range.

The two models have the same residuals and different claim-support scopes. The question determines which is appropriate.

6 Verification check

Verification asks whether the implementation matches the specification. For an observational model, the checks are numerical; for an explanatory model, they also include limiting behaviour of the mechanism.

Checks to perform

Residual structure. Plot the residuals against time and against each covariate. Structure indicates the model form is wrong, not that the fit needs improvement.
Range check. Confirm the model was fitted on the range of X you intend to apply it to. Extrapolation is not verified by a good in-sample fit.
Unit consistency. If k has units of inverse minutes, the time input must be in minutes, not seconds.
Limiting behaviour (explanatory). As t → ∞, T(t) → T_env. Confirm the solution has this limit before fitting.
Dimensional analysis (explanatory). The argument of the exponential must be dimensionless. k · t must have no units, so k must carry inverse units of t.

7 Interpretation

The crucial interpretive point of this lesson is that model quality is not a property of the model alone. It is a relationship between three things: the model, the data, and the question the model is meant to answer.

A model that fits the data beautifully but answers the wrong question is not a good model. It is a well-fitted answer to a question nobody asked. Conversely, a model with a larger residual that correctly represents the mechanism and supports the intended intervention may be far more useful — even if it would "lose" a competition based on in-sample fit.

Statement of claim-support scope

Every model in this course must be accompanied by a sentence stating what it can and cannot be used to claim. For an observational model, that sentence will typically be of the form: "This model describes the association between X and Y within the observed range of X, and does not support claims about interventions on X." For an explanatory model: "This model proposes a mechanism that reproduces the observed behaviour under the stated assumptions, and supports conditional predictions within the range of conditions for which the mechanism is assumed to hold."

8 Uncertainty and limitations

Both model types carry uncertainty, but of different kinds.

Sources of uncertainty by model type
Source	Observational	Explanatory
Sampling variability	Yes — estimates depend on the sample	Yes — parameter estimates depend on the sample
Model-form uncertainty	Yes — the functional form is a choice	Yes — the mechanism is an assumption
Confounding	Structural, and often unresolvable from observation alone	Reduced if the confounder is modelled explicitly
Extrapolation risk	High — the model has no mechanism to guide it	Lower, but depends on the mechanism holding
Intervention validity	Not supported	Supported to the extent the mechanism is correct

The important asymmetry is in the last two rows. An observational model cannot support interventional claims, no matter how good the fit. An explanatory model can, but only as far as its mechanism is correct — and correctness of a mechanism is not established by fit alone.

9 Common implementation errors

Watch for these

Treating a good fit as evidence of a mechanism. Any sufficiently flexible function can fit a finite dataset. Fit is necessary for a useful model, not sufficient for a correct one.
Using an observational model to answer an interventional question. This is the single most common misuse of models in applied work, and it produces confident, wrong answers.
Extrapolating beyond the observed range. An observational model has no basis for prediction outside the range of inputs it has seen. The functional form it happened to fit may be wildly wrong just outside the data.
Forgetting to state the claim-support scope. Without a stated scope, a downstream user will assume the model supports whatever claim they happen to need.
Mistaking a confounder for a mediator. A variable that lies on the causal path from X to Y is a mediator and should be handled differently from a confounder.

10 Independent modelling task

Your turn

Choose a system you know something about — a workplace process, a household system, a public service.
Write down one predictive question about the system and one interventional question. Be specific about the quantity and the range.
For each question, state which model type is required and why. If you believe both can be answered by the same model, justify the assumption under which that is true.
Identify at least one plausible confounder for the interventional question. Explain what it would do to the observed association.
Write one sentence stating the claim-support scope of each model.

Keep this task in your working notes. It will feed directly into the design and audit component of Assignment 1.

11 Reflection prompt

Reflect before continuing

Think of a model you have encountered — a forecast, a risk score, a recommendation system, a business projection. Was it observational or explanatory? What was it being used to claim? Was that claim consistent with what the model could actually support?

Write two or three sentences. The point is not to produce a polished answer but to notice how often the two categories are conflated in practice.

12 Lesson summary

Key takeaways

Observational models describe. They estimate E[Y | X = x] and support prediction within the observed range.
Explanatory models propose a mechanism. They can support interventional claims, but only to the extent the mechanism is correct.
A good fit does not establish a mechanism. Both model types can fit the same data equally well.
Confounding breaks the link between observation and intervention. A third variable that influences both X and Y makes the observed association misleading for interventional purposes.
Every model must state its claim-support scope. This is the sentence that tells a downstream user what the model can and cannot be asked to do.

Coming next

Lesson 2 addresses the second orthogonal axis of classification: whether the model's output is a single number or a distribution.

§ Self-quiz, practice, and homework

Practice exercises

Classify each as observational or explanatory, and as deterministic or stochastic. Justify in one sentence each: (a) a weather forecast, (b) Newton's law of cooling, (c) a spam filter, (d) an epidemiological SIR model, (e) a queueing model of a call centre.
For the coffee-cooling example in the lesson, write two additional questions that Model A cannot answer but Model B can.
Describe a situation from your own experience where a predictive claim was used as if it were interventional. What would have been required to justify the intervention claim?
Identify a plausible confounder for each of: (a) the association between ice-cream sales and drowning incidents, (b) the association between hospital size and patient survival rates.
Write one sentence stating the claim-support scope of a simple linear regression fitted to (X, Y) data.

Homework problems

Model classification portfolio. Choose five published or widely-used models from different fields (at least one from each of: physical sciences, social sciences, business). For each, write a 100-word profile that states: the model type, the mechanism (if any), the data used, and what the model can and cannot be used to claim. Submit as a single document.
Confounding analysis. Choose a documented observational study where the researchers identified a confounder. Describe the study, the confounder, and how the researchers handled it. State whether you find the handling convincing and why. 500 words.
Claim-support statement. For a model you have built or encountered in your own work, write a one-paragraph claim-support statement. Include: the model type, the data range, what the model can be used to claim, and what it cannot. Then describe one specific decision that would require a different model type.

Mini project

Choose a real system you have access to — a household process, a small business, a public service. Write a two-page modelling brief that answers: (i) What is the predictive question? (ii) What is the interventional question? (iii) For each, would you build an observational or explanatory model, and why? (iv) Identify at least two confounders for the interventional question. (v) State the claim-support scope for each model. This brief will feed directly into Assignment 1.