Why a Regression Workflow Must Be Auditable
A regression result is not just a coefficient printed by software. It depends on which observations were used, how their values were prepared, which model was fitted, and what the diagnostics showed. If a reader cannot follow those connections, the result is difficult to verify, reproduce, or interpret correctly.
An auditable regression workflow makes those connections explicit. It provides a traceable route from the data as received to the conclusion as reported. Auditing does not mean that every analysis choice is automatically correct, nor does it prove that a model is appropriate. It means that a reader can see what was done, check whether the steps agree, and identify where a conclusion comes from.
The basic chain is:
Identify the original observations and the variables recorded for them.
Trace changes such as recoding, unit conversion, and decisions about which observations enter the analysis.
Identify the outcome, predictors, functional form, and any other features that define the fitted regression.
Check evidence relevant to the assumptions and limitations of the model, and record how it affects interpretation.
Link each estimate, interval, or conclusion to the model and analysis data that produced it.
Each arrow matters. A conclusion can be numerically copied correctly from a model output and still be misleading if the units were changed without notice, the analysis sample is unclear, or a diagnostic concern is omitted. An audit trail is therefore both a check on computation and a check on meaning.
Start With the Model and Its Assumptions
To see how a result can be traced, consider a simple linear regression with quantitative outcome \(Y\) and predictor \(x\):
Here \(i\) indexes observations, \(\beta_0\) is the population intercept, \(\beta_1\) is the population slope, and \(\varepsilon_i\) is the error for observation \(i\). For the usual small-sample \(t\)-based inference about the slope, assume the conditional mean is linear, the errors are independent with mean zero and constant variance, and the errors are normally distributed. These assumptions support the standard standard-error and \(t\)-interval calculations; they are not established merely because software returns output.
A diagnostic examines evidence about whether aspects of the model are plausible for the data. For example, a residual-versus-fitted plot can reveal patterns inconsistent with a simple linear mean or constant spread. A diagnostic is evidence to consider, not a mechanical certificate that assumptions hold. Its meaning depends on the model and the data being analyzed.
Worked Example: Trace a Slope From Data to a Conclusion
Suppose a hypothetical teaching data set records study hours \(x\) and quiz score \(Y\), in points, for five students. The data are not from a real study.
| Student | Study hours \(x\) | Quiz score \(Y\) (points) |
|---|---|---|
| A | 1 | 3 |
| B | 2 | 5 |
| C | 3 | 4 |
| D | 4 | 8 |
| E | 5 | 10 |
The fitted model is \(Y_i=\beta_0+\beta_1x_i+\varepsilon_i\). For the usual inference below, we would need a linear conditional mean, independent errors with mean zero and constant variance, and normally distributed errors. With only five observations, diagnostic checks are necessarily limited; the numerical calculation alone does not confirm these assumptions.
The means are \(\bar{x}=3\) hours and \(\bar{Y}=6\) points. The least-squares slope is \(b_1=S_{xy}/S_{xx}\), where \(S_{xy}=\sum (x_i-\bar{x})(Y_i-\bar{Y})\) and \(S_{xx}=\sum (x_i-\bar{x})^2\). Substituting the values gives:
The intercept is \(b_0=\bar{Y}-b_1\bar{x}=6-(1.7)(3)=0.9\) points. Thus the fitted line is \(\hat{Y}=0.9+1.7x\). Its fitted values at \(x=1,2,3,4,5\) are \(2.6,4.3,6.0,7.7,9.4\) points. Subtracting each fitted value from the observed score gives residuals \(0.4,0.7,-2.0,0.3,0.6\) points.
The residual sum of squares is \(0.4^2+0.7^2+(-2.0)^2+0.3^2+0.6^2=5.10\). For this simple regression, the residual standard deviation estimate is \(s=\sqrt{5.10/(5-2)}=\sqrt{1.70}=1.304\) points, and the standard error of the slope is \(SE(b_1)=s/\sqrt{S_{xx}}=1.304/\sqrt{10}=0.4123\) points per hour. With \(df=n-2=3\), the 95% \(t\)-interval uses \(t^*=3.182\):
The result to report is traceable: the slope comes from the five displayed pairs; the interval comes from that fitted model, its residual variation, and \(df=3\). Under the model assumptions, the interval procedure has 95% coverage in repeated sampling. In context, the fitted association is an increase of 1.7 quiz-score points per additional study hour. This observational illustration does not establish that studying caused the score difference.
Check the Links, Not Just the Output
A useful audit follows a result in both directions. Starting from a reported estimate, trace backward to the model output, the model specification, and the prepared observations. Starting from the raw data, trace forward to see whether every transformation and analysis decision is represented in the reported result. Both directions can catch errors that a single review misses.
A short justification for checking counts is straightforward: a fitted model uses a particular set of rows. If the documented sequence of inclusion and missing-value decisions does not produce the number of rows in the model output, then at least one step is unaccounted for or incorrectly described. A numerical result may still be computable, but its analysis sample is not yet auditable.
Worked Example: Find a Break in the Sample-Count Chain
In a hypothetical observational analysis, the raw file contains 120 records. Eight are ineligible under the stated study criteria. Among the remaining records, five have a missing outcome and three have a missing predictor. Assume those two sets of missing records do not overlap, and the analysis uses only records with both variables observed.
The count after applying the stated decisions should be \(120-8-5-3=104\) observations. If the model output says \(n=102\), the reported steps do not account for two additional records. That mismatch is a warning to trace the analysis data and the model input before interpreting the coefficients.
The audit does not justify silently changing the reported count to 104 or assuming the software is wrong. It calls for locating the discrepancy: perhaps another exclusion was applied but not described, or perhaps the stated counts do not match the actual data. Once the difference is resolved, the account of the analysis sample and the model output should agree. The conclusion is not yet fully auditable while the two-record gap remains unexplained.
Diagnostics also belong in the chain. If a plot shows a strong pattern in residuals, a report that presents an unqualified linear-model interpretation omits evidence relevant to that interpretation. Conversely, mentioning that a diagnostic was examined is not enough: the reader needs to know what it showed and whether it limits the conclusion. The goal is not to list every possible check, but to connect relevant checks to the model and the claims made from it.
Worked Example: Trace a Unit Conversion Into a Reported Slope
Consider a second hypothetical analysis relating weekly exercise time to a numerical wellness score. The raw variable is recorded in minutes per week, but the preparation step expresses it in hours per week by dividing each value by 60. A linear regression is then fitted using hours as the predictor. Assume the linear-model conditions stated above are suitable for the intended inference; the unit check itself cannot establish those conditions.
Suppose the fitted slope is \(0.42\) score points per hour. To verify the units, let \(x_{\text{hours}}=x_{\text{minutes}}/60\). The fitted contribution from the predictor is:
Thus the same fitted association is \(0.007\) score points per minute, or \(0.42\) points per 60-minute increase. If a report instead labels \(0.42\) as points per minute, the numerical estimate has been copied but its unit has not followed the preparation step. Tracing the reported coefficient back to the transformed predictor reveals the error. Neither unit expression implies that exercise causes a change in the score.
What a Reader Should Be Able to Verify
An auditable report need not reproduce every line of computation. It should, however, give enough information for a reader to answer a consistent set of questions:
- What observations and variables are the analysis based on?
- What preparation or selection steps connect those observations to the data used by the model?
- What model was fitted, and which output supports the reported estimate?
- What relevant diagnostic evidence was examined, and how does it qualify the interpretation?
- Do the sample size, coefficient units, interval, and written conclusion agree with one another?
The chain also clarifies the boundary between a computational result and a scientific conclusion. A regression coefficient summarizes a specified model applied to a specified analysis data set. It does not by itself establish that the data represent a broader population, that the model captures every important feature, or that an association is causal. Those claims require justification beyond a successful model fit.
Common Mistakes
- Treating a software output as a complete record. Output may show estimates but not explain how the analysis observations were obtained. Connect it to the data and preparation steps.
- Assuming matching sample sizes prove the workflow is correct. Matching counts are a useful consistency check, not evidence that variables were coded correctly or the model assumptions are reasonable.
- Reporting a coefficient without tracing its units. The predictor may have been rescaled or transformed. Follow its definition into the fitted model before interpreting the slope.
- Presenting diagnostics as a pass-or-fail stamp. A diagnostic can reveal concerns but cannot prove assumptions true. Describe the evidence and its implications with appropriate caution.
- Turning association into causation in the final sentence. A regression result alone does not establish a causal effect. Phrase the conclusion at the level supported by the design and analysis.
Check Your Understanding
Use the workflow chain to answer each question.
- A model output has 86 observations, but the documented inclusion and missing-data steps account for 90. What should be checked before interpreting the estimates?
- Why does a residual diagnostic provide evidence about a model assumption rather than proof that the assumption is true?
- A predictor is converted from minutes to hours before fitting the model. What must be checked before reporting the slope in context?
- What is the difference between tracing a reported coefficient backward and tracing the workflow forward from raw data?
- A fitted regression shows an association between treatment and outcome in observational data. Why is that association alone insufficient to claim that treatment caused the outcome difference?