Why Units and Reference Groups Belong With Estimates
A regression table can display precise numerical estimates and still leave readers unsure what they describe. A row labelled Age: 0.42 does not say whether age was measured in years or decades, what outcome scale the estimate uses, or whether the value is an unstandardized or transformed coefficient. Likewise, a row labelled Hospital B: 3.1 does not identify the group against which Hospital B is compared. These details are part of the estimate’s meaning, not decoration.
This tutorial focuses on how to make those details visible in a table or its notes. It does not replace a model specification: the report should still identify the fitted model and analysis variables. Instead, it ensures that a reader can connect each reported estimate to the measurement scales and comparison represented by that model.
Match Coefficient Units to the Model Term
For an unstandardized additive linear regression model without interactions, a coefficient for a quantitative predictor has units of outcome units per predictor unit. A coefficient for an indicator variable has units of the outcome and corresponds to the indicator’s coded contrast with its reference category, holding other predictors fixed. These descriptions require the stated model conditions: interactions can make a categorical contrast depend on another predictor, and transformations change the scale to which a coefficient applies.
The arithmetic behind the units is straightforward. If an outcome is measured in millimetres of mercury (mmHg) and a predictor in years, then the slope’s units are mmHg per year. Changing the predictor’s reporting unit changes the numerical coefficient. For example, one decade is ten years, so the slope per decade is ten times the slope per year. The fitted relationship has not changed; its coefficient is expressed on a different predictor scale.
A table can put units directly in row labels, such as Age (per 10 years), or provide them in a clearly linked note. A note is especially useful when many rows share the same outcome scale. Avoid a bare label such as Age if the data dictionary or model specification does not make the unit easy to find.
Worked Example: Reporting a Slope on the Recorded Scale
Suppose a hypothetical additive linear model, with no interactions, predicts systolic blood pressure in mmHg from age recorded in years and an indicator for a program group. The fitted age coefficient is \(0.28\), and the program coefficient is \(-2.4\). The comparison group for the program indicator is explicitly coded as usual care. Show how the age row should identify its units, and calculate the corresponding estimate per decade.
The outcome is measured in mmHg and age in years, so the age coefficient’s units are mmHg per year. A suitable row label is Age (per year), with the outcome’s scale stated in the table heading or note.
One decade contains 10 years. Therefore, the estimate per decade is
The table could instead label this row Age (per 10 years) and report \(2.8\). The program row should identify Program group: program vs usual care; its estimate is in mmHg because the indicator is a categorical contrast in this additive model. The labels convey the units and comparison without requiring readers to infer them from abbreviations.
Make Categorical Reference Levels Explicit
A categorical predictor with several levels is commonly represented by indicator terms, with one level serving as the reference. Each displayed category coefficient is tied to that coding choice. Thus, a table should name the reference level and, if possible, write the comparison direction in each coefficient row. Writing Rural vs urban (reference) is clearer than writing only Rural.
For an additive model without interactions, an indicator coefficient is the fitted-value difference between its category and the reference category when the other predictors are held fixed. This wording should not be transferred unqualified to a model with interactions. With an interaction, the difference may depend on the value of another predictor. A categorical coefficient by itself may then describe the comparison only at that predictor’s reference or zero value.
Worked Example: A Three-Level Predictor
Consider a hypothetical additive linear model for clinic waiting time in minutes. The predictor clinic type has three levels: community, regional, and teaching. Community is the reference level. The two fitted indicator coefficients are \(4.6\) for regional and \(9.2\) for teaching. Show appropriate table rows and identify what the numbers are measured in.
Because the outcome is waiting time in minutes and the clinic terms are category indicators, both estimates are in minutes. Since this is an additive model with no interactions, each coefficient is a fitted-value difference from community clinics at fixed values of the other predictors.
A clear table could use these rows:
| Model term | Estimate | Units and comparison |
|---|---|---|
| Clinic type: regional vs community (reference) | 4.6 | Minutes |
| Clinic type: teaching vs community (reference) | 9.2 | Minutes |
The estimates are not comparisons between regional and teaching clinics. The labels show that each is compared with community clinics. The table note should also state that waiting time is measured in minutes and that other included predictors are held fixed for these additive-model contrasts.
When an Interaction Changes the Comparison
A row label must reflect the model actually fitted. For example, suppose a model contains an indicator \(I\), a quantitative predictor \(x\), and their product \(Ix\). Let \(I=0\) denote the reference group and \(I=1\) the other group. The model terms involving these variables are \(\beta_1 I+\beta_2x+\beta_3Ix\). At a fixed value \(x\), the fitted-value difference between the \(I=1\) group and the \(I=0\) group is not generally just \(\beta_1\).
To see the comparison, subtract the fitted value for \(I=0\) from the fitted value for \(I=1\). For \(I=1\), the terms are \(\beta_1+\beta_2x+\beta_3x\); for \(I=0\), they are \(\beta_2x\). The difference is \(\beta_1+\beta_3x\). Consequently, \(\beta_1\) represents the group contrast at \(x=0\), not at every value of \(x\). If \(x\) has been centered, zero refers to its centering value, which should be documented.
Worked Example: Labeling a Group Coefficient With an Interaction
Suppose a hypothetical additive-scale outcome \(Y\), measured in points, is modelled using a group indicator \(I\), age \(x\) measured in years, and their interaction. Group A is the reference (\(I=0\)); Group B has \(I=1\). The fitted terms are \(1.5I+0.20x+0.08Ix\). Show the group contrast at age 0 and age 10, and explain how to label the coefficient for \(I\).
At age \(x\), the fitted Group B minus Group A contrast is \(1.5+0.08x\) points. At age 0, it is
At age 10 years, it is
Thus, a table row labelled only Group B would conceal the relevant condition. A more informative label is Group B vs Group A at age 0 years (reference value) for the \(I\) coefficient, accompanied by a separate row for the interaction term, labelled Group B vs Group A contrast change per year of age. If age had been centered at 50 years instead, the main group coefficient would describe the contrast at age 50, and the note would need to say so.
Document Transformations and Table Notes
A coefficient can be labelled correctly only if the reporting makes clear what variable entered the model. Common cases include predictors recorded in different units, centered or rescaled predictors, and transformed outcomes or predictors. The table should not imply that a coefficient is per year if the fitted predictor was age divided by ten, or that an outcome is in its original units if the model used a transformed outcome. Give enough information for the reader to identify the model scale; do not rely on a software variable name alone.
A concise table note can report shared information once: the outcome and its units, the unit or coding of quantitative predictors, category reference levels, and any centering or transformation that affects the meaning of a term. For example: “Outcome: systolic blood pressure (mmHg). Age entered per 10 years. Program group coded program versus usual care; usual care is the reference. Age centered at 50 years.” The note should be consistent with the model terms and the row labels.
The units of an interaction term follow from the product of the interacting variables. If \(I\) is a dimensionless indicator and \(x\) is measured in years, the coefficient of \(Ix\) has outcome units per year. If both interacting predictors are quantitative, the product term’s units are the product of their units, and the coefficient is outcome units per such product unit. State the actual scales in the table or note rather than expecting readers to reconstruct them.
Worked Example: Checking a Rescaled Predictor and Its Label
A hypothetical model predicts recovery score in points using a variable \(z\) defined as age in years divided by 5. Its fitted coefficient is \(-0.60\). The table currently labels the row “Age.” Determine the coefficient’s units and revise the label. Then calculate the coefficient for age expressed per year.
The model predictor \(z\) increases by 1 when age increases by 5 years. Therefore, \(-0.60\) has units of points per 5 years of age. A clear row label is Age (per 5 years), or the table note must define \(z=\text{age in years}/5\).
For a one-year increase, the corresponding estimate is the per-five-year estimate divided by 5:
The row should not be called simply “Age” with no note, because the coefficient would then appear to be per year even though the model was fitted using five-year increments. The numerical value and units must describe the same predictor scale.
A Practical Reporting Check
Before sharing a regression table, inspect each estimate from the reader’s perspective. Can the reader identify the outcome scale, the predictor’s unit or coding, and the relevant reference group? For an interaction, can the reader tell which contrast varies and at what value a main-effect term applies? If any answer depends on guessing from a variable name, revise the row label or note.
Use the actual model variable, including any rescaling, centering, transformation, or product term.
Give the outcome units and the predictor units or category coding needed to understand the estimate.
For a categorical term, identify the displayed category and its reference; for interactions, specify the relevant predictor value or contrast dependence.
Ensure they describe the same model and do not imply a different unit, reference category, or interaction condition.
Common Mistake
A common mistake is to report a categorical coefficient as a general group difference when the fitted model includes an interaction. For example, saying “Group B is 1.5 points higher than Group A” from a model containing \(I\), \(x\), and \(Ix\) ignores the interaction. The group difference is \(1.5+0.08x\) in the worked example, so it depends on \(x\); the coefficient \(1.5\) alone is the contrast only at \(x=0\). Correct the label by naming that reference value, or report a contrast at a specified value of \(x\). A second frequent problem is omitting a predictor’s unit after rescaling: a coefficient per five years is not a coefficient per year.
Check Your Understanding
Use the model terms and table-label principles above to answer these questions.
- A model uses weight measured in kilograms to predict an outcome measured in centimetres. What units should be associated with the unstandardized weight coefficient?
- A categorical predictor has levels A, B, and C, with A as the reference. What comparison should be stated for the coefficient row for C?
- In a model containing \(I\), \(x\), and \(Ix\), why can the coefficient of \(I\) not generally be labelled as the group difference at every value of \(x\)?
- A predictor entered in a model is age in years divided by 10. What must a table label or note communicate about its coefficient?
- If a quantitative predictor is centered at 40 years, at what age does the main categorical coefficient in a model with a group-by-centered-age interaction describe the group contrast?