Build Core and Extended Specifications

Why Build a Core Model and Extensions?

A regression specification is a statement of which outcome and predictors a model will use. A research question may make some terms essential: the outcome being studied, the focal predictor whose association is of interest, and any adjustment variables needed to define that question. Other variables may be scientifically plausible additions, but their inclusion should not quietly replace the original question.

A useful strategy is to start with a core specification containing the essential terms, then write a small set of extended specifications that add plausible terms for a stated reason. This makes the comparison interpretable. It also prevents a long list of predictors from obscuring which association the analysis was designed to examine.

Definition: A core specification is a regression model containing the outcome, focal predictor, and other terms essential to the stated research question. An extended specification retains the core terms and adds one or more plausible optional terms. The primary target is the scientific quantity the question asks about, described in terms of the outcome, focal predictor, population, scale, and any essential conditioning or adjustment.

The target is not simply a formula, and model specifications do not guarantee that they estimate exactly the same numerical quantity. Adding a predictor can change the meaning or value of a fitted coefficient. The practical goal is to keep the scientific question fixed, make each model’s terms explicit, and be honest about how the added terms affect interpretation.

State the Target Before Listing Terms

For a continuous outcome, a basic additive regression model can be written as

Y_i = \beta_0 + \beta_A A_i + \sum_{j=1}^{q}\beta_j Z_{ij} + \varepsilon_i .

Here \(Y_i\) is the outcome for observational unit \(i\), \(A_i\) is the focal predictor, and \(Z_{ij}\) are other included predictors. The \(\beta\) terms are model coefficients, and \(\varepsilon_i\) represents the part of the outcome not described by the included terms. This is a model form, not a claim that the relationship is known to be linear or that all relevant predictors have been measured.

For the usual conditional-mean interpretation of this model, one assumes that the mean of the error is zero at each included predictor combination: \(E(\varepsilon_i\mid A_i,Z_{i1},\ldots,Z_{iq})=0\). Usual regression inference also relies on appropriate sampling and error assumptions for its particular procedure. Merely writing a core model does not establish that those assumptions hold; they need to be considered when analyzing data.

Before choosing extensions, write a plain-language target that names:

the outcome and its measurement scale;
the focal predictor and the contrast or change of interest;
the population or setting to which the question refers; and
any essential conditioning terms that are part of the intended question.

This statement acts as a reference point. An extension should add a term for a clear reason while leaving those target-defining choices in place. If a proposed addition changes the outcome, replaces the focal predictor, or changes an essential conditioning definition, it is not just an extension of the same target.

From Essential Terms to Plausible Additions

A term belongs in the core when omitting it would leave the stated question inadequately represented. For example, the focal predictor cannot be omitted if the question concerns its association with the outcome. A term may also be essential because the question explicitly asks for an association conditional on that variable, or because it is needed to represent the population comparison being described.

An optional term is not an arbitrary extra. It should have a subject-matter or measurement rationale, such as representing a plausible additional source of outcome variation or an important recorded characteristic. Its status as optional means that the primary question does not require it in the core—not that the term is unimportant or should be included automatically.

For each proposed term, ask two separate questions: Why might this term belong in a model? and Would including it change what the focal coefficient means? These questions help distinguish a reasonable extension from a model that silently answers a different question. A term that occurs after the focal exposure in a scientific process, for example, may not be an innocuous adjustment when the target concerns the overall association. The role of a term depends on the question, not only on its availability in the data.

Worked Example: A Core Model for Follow-Up Blood Pressure

Suppose a hypothetical clinic dataset records follow-up systolic blood pressure \(Y\), an indicator \(A\) for receiving a particular care program, age \(G\) in years, baseline systolic blood pressure \(B\), and smoking status \(S\) coded 1 for current smoker and 0 otherwise. The research question concerns the association between program participation and follow-up pressure, comparing participants with nonparticipants of the same age and baseline pressure.

Core terms. The outcome is follow-up pressure, \(Y\). The focal predictor is program participation, \(A\). Age and baseline pressure are essential because the question explicitly defines the comparison at the same values of those variables. The core model is

Y_i = \beta_0 + \beta_A A_i + \beta_G G_i + \beta_B B_i + \varepsilon_i .

In this model, \(\beta_A\) describes the difference in model-mean follow-up pressure associated with \(A=1\) rather than \(A=0\), at fixed age and baseline pressure. Its units are mmHg per program indicator contrast. That is a conditional association, not a claim that participation caused a change.

Possible extension. Smoking status is recorded and could plausibly be related to follow-up pressure. An extended model adds it while retaining all core terms:

Y_i = \beta_0 + \beta_A A_i + \beta_G G_i + \beta_B B_i + \beta_S S_i + \varepsilon_i .

The core has four coefficients including the intercept: \(1+1+1+1=4\). The extended model has five: \(4+1=5\). The extension still asks about program participation and follow-up pressure, but now the coefficient for \(A\) compares at fixed smoking status as well as age and baseline pressure. That difference in conditioning should be stated; the two values of \(\beta_A\) need not be equal.

For clarity, suppose a purely illustrative set of coefficients for the extended model were \(\beta_0=90\), \(\beta_A=-3\), \(\beta_G=0.2\), \(\beta_B=0.4\), and \(\beta_S=2\). For a hypothetical participant aged 50, with baseline pressure 130 mmHg and \(S=1\), the model-mean calculation is \(90+(-3)(1)+(0.2)(50)+(0.4)(130)+(2)(1)=90-3+10+52+2=151\) mmHg. These illustrative values are not fitted estimates; the arithmetic only shows how the terms enter the written specification.

Write Extensions That Answer a Clear Secondary Question

A small set of extensions is often more informative than a single expanded model containing every available variable. Each extension can add a term or a compact group of related terms for a stated reason. For example, one extension might add a measure of household resources, while another might add a measure of neighborhood context. Keeping these additions distinguishable makes it easier to see which modeling choice is being considered.

The core terms should remain in each extension if the intent is to examine how additions affect the original question. Otherwise, a change in the focal coefficient could reflect removal of a core term rather than the effect of the proposed addition. Maintaining the outcome definition is equally important: changing from follow-up pressure to change in pressure, for instance, changes the outcome and therefore the target.

Worked Example: Tutoring and a Standardized Test Score

In a hypothetical school dataset, \(Y\) is a mathematics test score in points, \(A\) indicates participation in a tutoring program, and \(P\) is the student’s score on a comparable test taken before the program. The question is about the association between tutoring and the later score among students with the same prior score. The core model is

Y_i = \beta_0 + \beta_A A_i + \beta_P P_i + \varepsilon_i .

The intended focal comparison is the difference in model-mean later scores for \(A=1\) versus \(A=0\) at the same prior score. The model contains three coefficients: intercept, tutoring, and prior score, so \(1+1+1=3\).

Suppose the dataset also records attendance \(D\), measured as the number of school days attended before the later test. If attendance is considered a plausible additional predictor of the later score, one extension is

Y_i = \beta_0 + \beta_A A_i + \beta_P P_i + \beta_D D_i + \varepsilon_i .

This extension has four coefficients, \(3+1=4\). It retains the same outcome, tutoring indicator, and prior-score comparison. However, the coefficient for tutoring is now conditional on attendance as well as prior score. If attendance is itself affected by tutoring, that conditioning may no longer represent the same kind of association as the core comparison. The extension is therefore a candidate to consider only if its role is compatible with the question; recording a variable does not by itself make it an appropriate addition.

To illustrate the arithmetic without suggesting an actual analysis, take \(\beta_0=40\), \(\beta_A=2\), \(\beta_P=0.5\), and \(\beta_D=0.1\). For a hypothetical student with \(A=1\), \(P=60\), and \(D=170\), the model-mean score is \(40+2(1)+0.5(60)+0.1(170)=40+2+30+17=89\) points. The numerical prediction follows from the specified model; it does not establish that tutoring caused a two-point increase.

A Short Justification: What an Extension Changes

The relationship between a core model and an extension can be seen directly from their equations. Suppose a core model has mean expression \(m_{\text{core}}(A,Z)\), and an extension adds an optional predictor \(W\) with coefficient \(\gamma\). The extended mean expression is

m_{\text{extended}}(A,Z,W) = m_{\text{core}}(A,Z) + \gamma W .

If \(\gamma=0\), the added term contributes nothing for any value of \(W\), and the extended mean expression reduces to the core expression. This shows why the core is contained within the extension as a special case. But it does not show that fitting the two models to data must give the same focal coefficient. When the extension is fitted, the model allocates variation among the included predictors under a different specification. The estimated focal coefficient can therefore change.

This is why “same primary target” should be treated as a discipline for designing and describing the candidate models, not as a promise that every model produces the same estimate. State what association is primary, keep its outcome and focal predictor fixed, and describe how additional conditioning terms alter the coefficient’s interpretation. If a proposed extension changes the target itself, identify it as a different question rather than presenting it as a routine robustness check.

Worked Example: A Crop Yield Extension

Suppose a hypothetical agricultural dataset records crop yield \(Y\) in kilograms per plot, fertilizer amount \(F\) in kilograms per plot, and soil organic matter \(O\) as a percentage. The primary question is the association between fertilizer amount and yield at the same measured soil organic matter. The core specification is

Y_i = \beta_0 + \beta_F F_i + \beta_O O_i + \varepsilon_i .

This model has three coefficients: \(1+1+1=3\). A possible extension adds irrigation amount \(I\), measured in millimeters:

Y_i = \beta_0 + \beta_F F_i + \beta_O O_i + \beta_I I_i + \varepsilon_i .

It has four coefficients, \(3+1=4\), and keeps yield as the outcome and fertilizer as the focal predictor. The coefficient \(\beta_F\) is now interpreted at fixed soil organic matter and irrigation amount. That may be a useful extension if irrigation is relevant to the intended comparison. It is not automatically equivalent to the core target, because the core does not condition on irrigation.

For an arithmetic illustration, let the extended model have illustrative coefficients \(\beta_0=10\), \(\beta_F=1.5\), \(\beta_O=2\), and \(\beta_I=0.3\). For a hypothetical plot with \(F=4\), \(O=3\), and \(I=20\), the model-mean yield is \(10+1.5(4)+2(3)+0.3(20)=10+6+6+6=28\) kg per plot. These numbers demonstrate substitution into the model only; they are not observations or fitted results.

Common Mistakes and Why Specification Matters

Adding every available variable. Availability is not a scientific rationale. Include an optional term only when its role is plausible and compatible with the target.
Calling the focal coefficient unchanged because the question is unchanged. The scientific question can remain the focus while the coefficient’s conditioning set—and therefore its interpretation or value—changes.
Dropping a core term in an extension. If the goal is to examine an addition to the same core, retain the core terms. Removing one creates a different comparison.
Changing the outcome definition mid-comparison. A model for a later measurement and a model for change from baseline address different outcomes. They should not be described as simple extensions of one unchanged outcome specification.
Reading a coefficient as causal by default. A regression coefficient describes a model-based association under the stated specification and assumptions. A causal interpretation requires additional design and assumptions; the equation alone does not provide them.

Core and extended specifications matter because they make the purpose of each model visible. A reader can see which terms define the primary question, which terms are optional, and what conditioning changes from one model to another. This clarity is more useful than presenting a long formula without explaining why its terms are there.

Key takeaway: Define the outcome, focal predictor, population, and essential conditioning first. Put those terms in the core model, then add only plausible optional terms in clearly described extensions. Keep the research question in view, but do not assume that adding a term leaves the focal coefficient numerically or conceptually unchanged.

Check Your Understanding

Try these questions without looking back.

A study asks about the association between a program indicator and a later outcome at the same baseline outcome level. Which variables belong in the core, and what does the focal coefficient compare?
Why can adding a plausible predictor change the interpretation or value of the focal coefficient even when the outcome and focal predictor stay the same?
A proposed extension replaces the measured outcome with change from baseline. Is this an extension of the same target? Explain.
A dataset contains ten additional variables. Does that alone justify adding all ten to the core model? Give a reasoned answer.
What does it mean for the core model to be a special case of an extension, and what does that fact not guarantee about fitted coefficients?