Why Small Reporting Choices Matter
A regression report can contain a correctly fitted model and still mislead its readers. A coefficient without units is hard to interpret; a category effect without its reference group is ambiguous; and a small p-value can be made to sound like proof of a claim the analysis did not establish. These problems are often not errors in arithmetic. They are errors in how results are presented and described.
A useful audit asks whether a reader can tell what was estimated, on what scale, relative to which group, and with what uncertainty. It also checks that the written conclusion stays within what the fitted model and study design support. The examples below use hypothetical analyses to show how to make that audit concrete.
A Practical Audit
Before interpreting an estimate, identify what the coefficient represents. In a linear regression, the outcome is modeled using predictors; a coefficient describes a change in the fitted mean outcome associated with a one-unit change in a quantitative predictor, holding the other included predictors fixed. For a categorical predictor, the coefficient describes a difference from its reference category, again holding the other included predictors fixed. The words “one unit” and “reference category” must be made explicit for the report to be interpretable.
For conventional coefficient confidence intervals and t-tests, the model’s inferential conditions also matter. The usual framework assumes independent observations, a suitable mean model, constant error variance for conventional standard errors, and approximately normal errors for exact small-sample t inference. A report should not imply that a p-value or interval repairs a model or design that does not support those conditions. Earlier material on model specification and diagnostic findings addresses how those issues are assessed; this tutorial focuses on how to avoid misleading presentation and interpretation.
A short audit can proceed in this order:
Check that the outcome units and predictor units, including any transformations or rescaling, are stated.
For categorical predictors, identify the reference group and name the comparison group.
Round sensibly and present intervals and p-values with enough context to interpret them.
Make sure the conclusion describes the fitted association or prediction, and does not claim causation or generality without justification.
Worked Examples of Common Reporting Errors
Worked Example: Recovering Units and the Comparison
Suppose a hypothetical observational analysis models fasting glucose in milligrams per deciliter (mg/dL) using weekly activity, age, and current smoking status. Activity is recorded in 10-minute increments per week. Smoking status is coded as current smoker or not a current smoker, with not a current smoker as the reference group. The fitted activity coefficient is \(-0.38\), with a 95% confidence interval from \(-0.71\) to \(-0.05\) mg/dL per 10-minute weekly increment and \(p=0.024\).
Assume each participant contributes one observation, the observations can reasonably be treated as independent, and checks of the fitted model found no serious departure from the linear mean and constant-variance assumptions. The conventional t-based interval and test are being used with those conditions in mind.
A weak report might say, “Activity was significant, coefficient \(-0.38\).” This omits both scales: the reader cannot tell what outcome is changing or what a one-unit increase in activity means. A better report names both units and makes clear that this is an adjusted association:
“In this fitted model, each additional 10 minutes of weekly activity was associated with an estimated 0.38 mg/dL lower mean fasting glucose, conditional on age and smoking status (95% CI: \(-0.71\) to \(-0.05\) mg/dL; \(p=0.024\)).”
The interval is in ascending numerical order on the glucose scale. It can also be expressed as an estimated decrease of 0.05 to 0.71 mg/dL per 10-minute increment. Those endpoints must be ordered from the smaller decrease to the larger decrease; writing “0.71 to 0.05 mg/dL lower” reverses them.
The smoking coefficient needs its comparison stated too. If the estimate were \(3.2\) mg/dL, for example, its interpretation would be an estimated 3.2 mg/dL higher mean fasting glucose for current smokers than for those who were not current smokers, conditional on activity and age. Saying only “smoking status: 3.2” leaves the comparison direction unclear. Since this is an observational analysis, neither coefficient alone establishes that changing activity or smoking status would change glucose.
Worked Example: Rounding Without Hiding the Result
In a hypothetical study of 40 independent participants, a linear regression models a test score in points from weekly study hours and two other predictors. The study-hour coefficient is estimated as 2.35 score points per additional weekly hour, with standard error \(0.86\). Assume the mean model is suitable, the residual checks show no serious pattern or marked variance change, and the residual distribution is adequate for the conventional t-based inference used here. With three predictors and an intercept, the residual degrees of freedom are \(40-4=36\).
The test statistic for \(H_0:\beta=0\) is the estimate divided by its standard error:
For \(df=36\), the two-sided p-value is approximately \(0.010\). Using the 95% t critical value \(t^*=2.028\), the interval calculation is:
A sensible report rounds these results to, for example, an estimate of 2.35 points per weekly hour, a 95% confidence interval of 0.61 to 4.09 points, and \(p=0.010\). Displaying an estimate such as 2.347281, a standard error of 0.864119, and a p-value of 0.009647 would imply more practical precision than the data and measurement scale warrant. Rounding should not change the substantive conclusion, and the interval and estimate should use compatible units.
The interpretation is that, in this fitted model, one additional weekly study hour is associated with an estimated 2.35-point higher mean test score, conditional on the other included predictors. The interval describes values compatible with the data and model under the confidence-interval procedure; it is not a statement that there is a 95% probability the fixed coefficient lies inside this particular interval.
Worked Example: Making Significance Stars and Claims Informative
Suppose a hypothetical observational study models a follow-up health measure in mg/dL using program participation and measured baseline characteristics. Program participants and nonparticipants are separate individuals; the model uses independent observations, and residual checks do not show major violations of the conditions for the conventional t-based test. The reported adjusted coefficient for participation is \(-4.2\) mg/dL, with standard error \(1.8\), \(df=96\), and a two-sided p-value of about \(0.022\).
The test statistic is:
Using \(t^*=1.985\) for a 95% interval with 96 degrees of freedom gives:
A table that prints only “\(-4.2^{*}\)” does not say what the star means, which test was used, or which other variables were controlled. If stars are used, their thresholds must be defined, for example, “\(*p<0.05,\ **p<0.01\), two-sided tests.” Even with that legend, the p-value and interval provide more useful context than the star alone. A small p-value is not the probability that the null hypothesis is true, and it does not measure the size or importance of the association.
A defensible summary is: “In the fitted model, program participation was associated with an estimated 4.2 mg/dL lower mean follow-up measure than nonparticipation, conditional on the included baseline characteristics (95% CI: \(-7.77\) to \(-0.63\) mg/dL; \(p=0.022\)).” This is an adjusted association. The observational comparison does not, by itself, establish that program participation caused the difference. A report claiming “the program reduced the measure by 4.2 mg/dL” would go beyond the analysis unless the design and assumptions support a causal interpretation.
A Short Justification: Why Units Must Follow the Coefficient
The units of a regression coefficient follow directly from the fitted model. If the outcome \(Y\) is measured in mg/dL and predictor \(x\) is measured in 10-minute increments per week, then a slope \(\beta_1\) multiplies a change in \(x\) to produce a change in the fitted mean of \(Y\). Thus its units are outcome units divided by predictor units:
For a change of \(\Delta x\) increments, the fitted mean changes by \(\beta_1\Delta x\) mg/dL, with other predictors held fixed. This is why a coefficient is not fully described by its number alone. Reporting “\(-0.38\)” without naming the scales loses the information needed to interpret \(\beta_1\Delta x\) in context.
Common Mistakes and Corrections
| Misleading choice | Why it is a problem | Correction |
|---|---|---|
| Reporting “estimate = 4.1” | The outcome units and predictor increment are unknown. | Name the outcome scale and the change in predictor represented by the coefficient. |
| Reporting a category coefficient without a reference group | The direction and meaning of the comparison are ambiguous. | Name both the comparison category and reference category. |
| Printing many decimal places | It suggests unjustified precision and obscures the main result. | Round estimates, uncertainty, and p-values consistently to practical precision. |
| Using stars without a definition or other context | Readers cannot tell the threshold or test, and stars conceal magnitude and uncertainty. | Define any star notation and report estimates, intervals, and p-values as appropriate. |
| Writing a causal conclusion from an adjusted observational association | Adjustment alone does not establish causation. | Describe the conditional association and state limitations imposed by the design. |
The central pitfall is to treat a regression table as self-explanatory. A coefficient may be numerically correct while its scale, comparison, or interpretation is missing. Similarly, a small p-value does not rescue a vague report or authorize a broader claim. Read each sentence alongside the model specification and ask whether every comparison and conclusion can be traced to what was actually fitted.
Check Your Understanding
For each question, focus on what a reader needs in order to interpret the report accurately.
- A report gives a slope of \(-1.6\) but does not state the units of the outcome or predictor. What information should be added?
- A categorical predictor is coded as “Group B versus Group A,” but the report gives only the coefficient. What comparison should the interpretation name?
- Why can reporting a coefficient to six decimal places be misleading, even when that is what the software prints?
- A table marks a coefficient with two stars but gives no legend. What is missing, and why should the stars not replace an interval or p-value?
- An observational model finds a negative adjusted coefficient for program participation. What is a suitably cautious way to describe the result, and what claim would require additional justification?