Turn a Research Question Into Candidate Model Terms

From a Question to a Model

A regression model is useful only when it represents a question that can be answered with the available measurements. A broad prompt such as “Does exercise affect health?” does not yet say what outcome to analyze, how exercise is measured, or what pattern of association is of interest. A focused question makes those choices explicit enough to propose candidate model terms.

This tutorial uses a question about exercise and blood pressure to illustrate the translation. The aim is not to calculate fitted coefficients or decide whether an association is statistically significant. It is to make the research question and the preliminary model correspond clearly, before using data to estimate anything.

Definition: The response variable is the measured outcome whose variation the regression model describes. The primary predictor is the explanatory measurement whose relationship with the response is the main subject of the research question. A model term is a component of the model’s mean structure, such as a predictor, a squared predictor, or a product of two predictors. A preliminary specification is a proposed set of terms, not a conclusion about their estimated values or importance.

State the Question Precisely

Consider this hypothetical research question: Among adults aged 30 to 65, is the number of hours of moderate-to-vigorous exercise per week associated with resting systolic blood pressure, comparing adults of the same age and recorded sex? This wording identifies a population, an outcome, a principal measurement, and the comparison intended by the question.

For adult \(i\), let \(Y_i\) be resting systolic blood pressure in millimeters of mercury (mmHg), \(X_i\) be usual exercise hours per week, \(A_i\) be age in years, and \(S_i\) be a recorded-sex indicator. For illustration, set \(S_i=0\) for one recorded category and \(S_i=1\) for the other; the categories and coding would need to be defined from the actual measurement process. Here \(Y_i\) is the response, \(X_i\) is the primary predictor, and age and recorded sex are included because the question explicitly asks for a comparison at the same values of those measurements.

A preliminary additive linear model for this question is

Y_i=\beta_0+\beta_1X_i+\beta_2(A_i-50)+\beta_3S_i+\epsilon_i.

The \(i\) subscript identifies the adult. The \(\beta\) values are coefficients to be estimated, and \(\epsilon_i\) represents the part of the observed response not represented by the specified mean structure. Age is centered at 50 years so that the intercept refers to an age within the stated range; centering changes the intercept’s reference point, not the question’s response or primary predictor. This model proposes a straight-line association with exercise and additive age and recorded-sex terms. It does not establish that this shape is correct.

Working model statement: A regression specification describes the conditional mean of the response as a function of the stated predictors and terms. In the example, the working statement is \(E(Y_i\mid X_i,A_i,S_i)=\beta_0+\beta_1X_i+\beta_2(A_i-50)+\beta_3S_i\), with errors defined as the differences between observations and that conditional mean. This mean representation alone does not assert a particular error variance or prove that observations are independent; those assumptions must be considered for later estimation and inference.

Worked Example: Naming the Response and Primary Predictor

Suppose a hypothetical data set records exercise and blood pressure for five adults. The entries below are illustrative, not results from a real study.

Adult	Resting systolic pressure (mmHg)	Exercise (hours/week)	Age (years)	Recorded-sex indicator
1	132	1	40	0
2	126	3	50	1
3	129	2	55	0
4	121	5	45	1
5	135	0	60	0

Response. Resting systolic pressure is \(Y_i\), measured in mmHg, because the question asks how this outcome varies with exercise. The observed value for adult 1 is \(Y_1=132\) mmHg.

Primary predictor. Exercise is \(X_i\), measured in hours per week. For adult 4, \(X_4=5\) hours per week. It is the primary predictor because its relationship with blood pressure is the focus of the question.

Other stated measurements. Age is recorded in years, and the indicator distinguishes the two recorded categories under the stated coding. The question asks for a comparison at the same age and recorded sex, so these measurements appear in the preliminary mean structure as age and indicator terms. The table does not, by itself, determine whether a linear association is an adequate description or whether any coefficient differs from zero.

Translate the Question’s Pattern Into Terms

The wording of a question can indicate what kind of relationship its preliminary model should be able to represent. A question about a constant average change per additional exercise hour suggests a linear term \(X_i\). A question about a changing rate of association may call for a curvature term such as \(X_i^2\). A question about whether the exercise association differs between recorded groups may call for a product term. These are different mathematical statements of different research questions; they are not interchangeable decorations.

For the additive model above, the comparison that corresponds to one more exercise hour, holding age and recorded sex fixed, follows directly from its conditional mean:

\begin{aligned} &E(Y_i\mid X_i+1,A_i,S_i)-E(Y_i\mid X_i,A_i,S_i)\\ &=\{\beta_0+\beta_1(X_i+1)+\beta_2(A_i-50)+\beta_3S_i\}\\ &\quad-\{\beta_0+\beta_1X_i+\beta_2(A_i-50)+\beta_3S_i\}\\ &=\beta_1. \end{aligned}

The intercept, age contribution, and recorded-sex contribution cancel, leaving \(\beta_1\). Thus, under this specified linear and additive mean structure, \(\beta_1\) represents the modelled difference in mean systolic pressure, in mmHg, for one additional exercise hour per week at fixed age and recorded sex. This is a model-based association, not evidence that changing exercise would cause that difference.

Worked Example: A Question About Curvature

Suppose the research question is revised: Among adults of the stated age range, is the exercise–blood-pressure association curved rather than adequately summarized by a constant change per additional exercise hour? The response remains systolic pressure \(Y_i\), and exercise \(X_i\) remains the primary predictor. The question now explicitly asks about curvature, so a candidate quadratic term can be added to the mean structure:

E(Y_i\mid X_i)=\beta_0+\beta_1X_i+\beta_2X_i^2.

The term \(X_i^2\) is formed by squaring exercise hours per week, so its units are \((\text{hours/week})^2\). For an adult exercising 3 hours per week, \(X_i^2=3^2=9\) \((\text{hours/week})^2\). For 5 hours per week, \(X_i^2=5^2=25\) \((\text{hours/week})^2\). The corresponding model mean at 3 hours is \(\beta_0+3\beta_1+9\beta_2\); at 5 hours it is \(\beta_0+5\beta_1+25\beta_2\).

The squared term gives the proposed mean curve the capacity to bend; the question does not guarantee that the estimated bend will be substantial or that this is the best model. If the question instead specifies only a constant average change across the range, the linear term alone is the direct preliminary translation. In either case, the response and primary predictor remain unchanged.

Worked Example: A Question About Different Group Associations

Now consider a separate, hypothetical version of the question: Does the linear exercise–blood-pressure association differ between the two recorded-sex categories? Let \(S_i=0\) denote the first category and \(S_i=1\) the second. The question asks whether the exercise slope varies with the group, so the candidate mean structure includes an exercise-by-group product:

E(Y_i\mid X_i,S_i)=\beta_0+\beta_1X_i+\beta_2S_i+\beta_3X_iS_i.

When \(S_i=0\), the mean is \(\beta_0+\beta_1X_i\), so its slope with respect to exercise is \(\beta_1\). When \(S_i=1\), the mean is \((\beta_0+\beta_2)+(\beta_1+\beta_3)X_i\), so its slope is \(\beta_1+\beta_3\). The term \(X_iS_i\) therefore allows the two group-specific linear associations to differ. For example, at \(X_i=4\) hours per week and \(S_i=1\), the modelled mean is \(\beta_0+4\beta_1+\beta_2+4\beta_3\).

The product is included here because the research question asks about differing associations, not because an interaction should automatically be added whenever a grouping variable is present. The expression specifies a question the model can represent; it does not test or answer that question on its own.

A Practical Translation Sequence

A careful specification begins with the question rather than with a list of available columns. The following sequence keeps the model’s response, principal relationship, and proposed structure tied to the intended comparison.

Define the outcome in measurable terms.
State what is recorded for each observational unit and its units. In the running example, the outcome is resting systolic blood pressure in mmHg, not general cardiovascular health.

Name the main predictor and its measurement.
Specify the quantity or category whose relationship with the outcome is central. Here it is usual exercise hours per week, not an unspecified notion of fitness.

Make the intended comparison explicit.
Say whether the question concerns an average linear change, possible curvature, or a difference in association across groups. Include additional measurements only when the question’s wording or stated comparison calls for them.

Write candidate terms and check their meanings.
Use notation that makes the proposed mean structure visible. Check that each term represents the stated question and that its units and coding are interpretable.

Worked Example: Translate a Different Focused Question

A hypothetical clinic asks: Among patients receiving a specified treatment, is the average change in symptom score associated with treatment duration, allowing the average change to differ by clinic? The question defines the response as change in symptom score, the primary predictor as treatment duration, and a group comparison by clinic. Assume for this illustration that there are two clinics and \(C_i=0\) or \(1\) denotes the clinic.

Let \(Y_i\) be the patient’s change in symptom score, measured in score points, and let \(T_i\) be treatment duration in days. Because the question asks whether the duration association differs by clinic, a preliminary candidate structure is

E(Y_i\mid T_i,C_i)=\beta_0+\beta_1T_i+\beta_2C_i+\beta_3T_iC_i.

For clinic \(C_i=0\), the model mean is \(\beta_0+\beta_1T_i\); for clinic \(C_i=1\), it is \((\beta_0+\beta_2)+(\beta_1+\beta_3)T_i\). Thus the proposed terms allow clinic-specific intercepts and duration slopes, matching the question’s stated comparison. The response is not treatment duration, and the clinic label is not the primary predictor: each has a different place in the translated question. This specification says nothing by itself about whether the treatment caused a symptom-score change.

Common Mistakes and Why the Translation Matters

Starting from the columns instead of the question. A data set may contain many measurements, but their presence alone does not make each one part of the preliminary model. First define the outcome and the specific relationship the question asks about.
Confusing a predictor with a model term. Exercise is a measured variable; \(X_i^2\) and \(X_iS_i\) are constructed terms involving that variable. A question about curvature or group differences may require terms beyond the original measurement itself.
Reading an association question as a causal conclusion. “Associated with” describes a relationship in the model or data. It does not mean that an intervention on the predictor would cause the observed response difference.
Treating a preliminary model as already validated. A clear translation makes the intended structure explicit, but it does not show that the form is adequate for the data or that the assumptions needed for later inference hold.

This translation matters because an unclear question can produce a model whose coefficients answer something different from the investigator’s intended question. Stating the response, primary predictor, comparison, and candidate terms before estimation makes that mismatch easier to detect. It also keeps scientific interpretation attached to measurements and units rather than to symbols alone.

Key takeaway: A focused research question should identify a measurable response, a primary predictor, and the pattern or comparison of interest. Candidate regression terms are the mathematical translation of that wording; they are proposals to examine, not automatic evidence of an association or a causal effect.

Check Your Understanding

For each question, identify the response, primary predictor, and the main feature of a preliminary mean structure that the wording calls for.

Among adults, is weekly exercise duration associated with resting systolic blood pressure through a straight-line average relationship?
A question asks whether the exercise–blood-pressure association differs between two recorded groups. What kind of term represents that requested difference?
In the clinic example, why is change in symptom score the response rather than treatment duration?
What does the exercise coefficient represent in the additive linear model when age and recorded sex are held fixed? Include its units and the model condition needed for that interpretation.
Why does including a candidate term in a preliminary specification not establish that the corresponding association is causal?