Why Reading the Table Carefully Matters
Regression software can display several numbers beside each model term: an estimate, a standard error, a test statistic, a p-value, and possibly confidence limits. These numbers answer different questions. A coefficient estimate describes the fitted association on a particular scale; its standard error describes the estimate’s sampling variability on that same scale. Confusing either with the model’s residual standard deviation can lead to incorrect conclusions about precision.
This tutorial focuses on locating and checking those quantities in a regression output table. Before interpreting them, confirm which outcome and predictors were modeled, how each variable was measured, and which kind of standard error the software reports.
The Model and the Main Output Fields
For a multiple linear regression with \(n\) observations and \(p\) estimated coefficients, including the intercept, write the model as
Here \(Y_i\) is the outcome for observation \(i\), the \(x\)'s are predictors, \(\beta_j\) are population coefficients, and \(\varepsilon_i\) is the error. The fitted coefficient estimate for \(\beta_j\) is commonly written \(b_j\). Software may label it “Estimate” or “Coefficient.”
Conventional model-based standard errors and the familiar residual degrees of freedom rely on the usual linear regression setup: the conditional mean of \(Y_i\) is represented by the specified linear predictor, errors have mean zero, observations’ errors are independent, and errors have a common variance. Normal errors support exact small-sample \(t\)-based inference. If software instead reports robust or clustered standard errors, those are a different calculation; read the label and documentation rather than assuming the conventional version.
A coefficient’s standard error is not the residual standard deviation. The residual standard deviation describes the typical size of residuals in outcome units. A coefficient standard error describes uncertainty in one estimated coefficient, so its units are outcome units per unit of that predictor (for a slope), or outcome units (for an intercept). In a table, residual standard deviation may appear as “Residual standard error,” “Residual SD,” or a similar model-level summary, not in the coefficient row.
Finding and Checking the Relevant Columns
A coefficient table often has one row per model term. The row name identifies the coefficient; columns commonly include its estimate and standard error. Other columns may report a test statistic, p-value, or lower and upper confidence limits. The exact labels and ordering vary across software, so use the column headings rather than relying on position.
| Output field | What it refers to | Basic check |
|---|---|---|
| Estimate or coefficient | Fitted value \(b_j\) for the named term | Check the sign, predictor, and measurement units |
| Std. Error | Estimated sampling standard deviation of \(b_j\) | It should use the same coefficient scale and units |
| Residual df | Degrees of freedom left for estimating residual variation | For usual full-rank least squares, check \(n-p\) |
| Lower and upper limits | Endpoints of a reported interval for a coefficient | They should be on the coefficient’s scale |
| Residual standard error | Summary of residual spread around the fitted model | It is in outcome units, not slope units |
The coefficient’s meaning depends on the predictor’s coding and units. A slope for a numeric predictor is the fitted change in the mean outcome associated with a one-unit increase in that predictor, holding the other included predictors fixed. This is an adjusted association, not by itself evidence that changing the predictor causes the outcome to change.
Worked Example: Locating a Slope Estimate and Its Standard Error
Suppose a hypothetical analysis relates systolic blood pressure, measured in mmHg, to weekly exercise hours, age, and smoking status. The analysis includes 48 observations and estimates an intercept plus three predictor coefficients. A portion of the software output is:
| Term | Estimate | Std. Error | t value | Residual df |
|---|---|---|---|---|
| Exercise (hours/week) | -1.84 | 0.62 | -2.97 | 44 |
The coefficient of interest is in the “Estimate” column: \(b_{\text{exercise}}=-1.84\). In this model, the fitted mean systolic blood pressure is 1.84 mmHg lower for each additional weekly exercise hour, comparing observations with the same modeled age and smoking status. This is an adjusted association.
The coefficient’s standard error is \(0.62\), in mmHg per exercise hour. It does not mean that an individual’s blood pressure typically differs from the fitted value by 0.62 mmHg; that would confuse coefficient uncertainty with residual variation.
The degrees-of-freedom check gives \(n-p=48-4=44\), because the model estimates four coefficients: one intercept and three slopes. The reported residual df of 44 is therefore consistent with the stated usual least-squares setup. This check would need reconsideration if the model used a different fitting or inference method.
Check Whether the Scales Match
A useful reading habit is to write the units beside each number. For a slope, the estimate, its standard error, and its confidence limits should all have units of outcome per unit of predictor. For an intercept, those quantities use the outcome’s units. Residual degrees of freedom are a count and have no measurement units.
Software can also show standardized coefficients, which express changes in standard-deviation units, or it may show coefficients after a predictor has been recoded. Such entries should not be compared directly with unstandardized values as if they had the same scale. Check labels such as “standardized,” “scaled,” or units embedded in the term name.
The effect of changing a predictor’s units can be derived directly. Suppose \(x^*=x/10\), so that \(x^*\) counts tens of the original predictor units. Since \(x=10x^*\), a term \(b x\) in the fitted model becomes \(10b x^*\). The fitted values do not change, but the coefficient per new unit is multiplied by 10. Its standard error and any confidence limits for that coefficient are multiplied by 10 as well, because they are all expressed on the coefficient’s scale.
Worked Example: Checking a Change in Predictor Units
A hypothetical model predicts annual service cost in dollars from equipment age. When age is measured in years, the output reports an age coefficient of \(6.2\) dollars per year and a standard error of \(2.1\) dollars per year. The same predictor is then recorded in decades, with \(x^*=x/10\).
One decade represents ten years, so the estimated association per decade is \(10(6.2)=62\) dollars per decade. The standard error on that scale is \(10(2.1)=21\) dollars per decade. If the original output also reported limits of 2.0 and 10.4 dollars per year, the corresponding limits on the decade scale would be \(10(2.0)=20\) and \(10(10.4)=104\) dollars per decade.
The coefficient is numerically larger after the change, but this does not mean the fitted relationship or its precision has changed. Both estimate and standard error changed units by the same factor. Comparing 62 dollars per decade with 2.1 dollars per year would be a scale mismatch.
Reading Reported Confidence Limits
When a software table provides lower and upper confidence limits, they are endpoints for the coefficient named in that row. Check that the stated confidence level is known, that both endpoints use the coefficient’s units, and that the estimate falls between them. The midpoint and half-width offer a quick arithmetic check, subject to rounding. If the interval is symmetric, its midpoint should be close to the estimate; its half-width should be consistent with the standard error times the method’s critical multiplier.
This check does not establish that an interval is valid. The calculation also depends on the interval method, confidence level, degrees of freedom where relevant, and assumptions behind the reported standard error. If these details are absent, avoid treating the limits as independently verified.
Worked Example: Checking the Coefficient Limits in a Table
A hypothetical least-squares analysis uses 35 observations to model a laboratory measurement, in units, from three predictors and an intercept. The output gives the following row for one predictor:
| Term | Estimate | Std. Error | 95% lower | 95% upper |
|---|---|---|---|---|
| Predictor A (per unit) | 0.48 | 0.15 | 0.17 | 0.79 |
The estimate and standard error are in measurement units per unit of Predictor A. The limits should have those same units. Their midpoint is \((0.17+0.79)/2=0.48\), matching the estimate. Their half-width is \((0.79-0.17)/2=0.31\), and \(0.31/0.15\approx2.07\). That ratio is a plausible multiplier for a conventional 95% interval with residual \(df=35-4=31\), allowing for rounding. It is a consistency check, not a reconstruction of the interval method.
The limits bracket the estimate, as a conventional symmetric interval should. If the table’s limits had instead been labelled in standardized units while the estimate remained unstandardized, the midpoint check would not be meaningful until the scales were reconciled.
Common Mistakes and a Reliable Reading Routine
- Using residual standard error as a coefficient standard error. They refer to different quantities and generally have different units. Find the coefficient-specific “Std. Error” in the row of interest.
- Ignoring the term label. A table can contain several slopes. Confirm that the row refers to the predictor, interaction, or category contrast you intend to discuss.
- Comparing numbers without checking units. A slope per year and a slope per decade are not on the same scale. Convert one before comparing them.
- Assuming every “Std. Error” uses the same method. Conventional, robust, and clustered standard errors can differ. Read the output description to identify the method.
- Treating residual df as the sample size. In usual full-rank least squares it is \(n-p\), not \(n\). Count the estimated coefficients, including the intercept, when checking it.
- Reading a coefficient as a causal effect. Regression adjustment describes a conditional association under the model; a causal interpretation requires additional design and assumptions.
A practical sequence is: identify the outcome and model terms; select the intended coefficient row; record its estimate and standard error with units; locate the residual df and interval limits if reported; then check that each quantity belongs to the same coefficient scale. This short audit catches many misreadings before any further inference is attempted.
Check Your Understanding
Use the definitions and scale checks above to answer each question.
- A model has 60 observations and estimates an intercept plus five slopes. What residual degrees of freedom are expected under the usual full-rank least-squares calculation?
- A slope is reported as 3.4 mmHg per year, with standard error 1.1 mmHg per year. What units should its confidence limits use?
- Explain why a residual standard error of 5 outcome units is not the standard error of a slope measured in outcome units per hour.
- If a predictor is changed from years to decades by dividing its values by 10, how do the slope and its standard error change numerically?
- A reported interval has endpoints 1.2 and 2.0, while the coefficient estimate is 2.4. What basic consistency check fails, and what should you inspect in the output?