Two Purposes, Two Questions
A regression model can be designed to describe an adjusted association or to forecast an outcome. These goals can call for different candidate terms, even when the response and data set are the same. A variable that helps forecast accurately may be a proxy, an administrative score, or a time index rather than a variable that answers the substantive question of interest.
The distinction matters because a model term is not self-explanatory. Including a variable changes what is held fixed when interpreting other coefficients. For forecasting, however, the central question is whether the model predicts well for the cases to which it will be applied. Neither purpose makes the other unnecessary; the important point is to choose terms and assess the model according to the stated goal.
What a Regression Coefficient Does—and Does Not—Say
For a quantitative response, a linear conditional-mean model can be written as
Here \(Y_i\) is the outcome for case \(i\), the \(X\)'s are the included predictors, and the \(\beta\)'s describe the model's conditional mean. Under this specification, \(\beta_j\) is the change in the modelled mean response for a one-unit increase in \(X_j\), holding the other included predictors fixed. This is an adjusted association defined by the model and its terms. It is not, by itself, a causal effect: regression adjustment alone does not establish that changing \(X_j\) would change \(Y\).
The model states a particular form for the conditional mean. Interpreting its coefficients as constant adjusted associations relies on that form being a reasonable description over the range being considered. The interpretation also depends on how variables are measured and coded. Normal errors are not required simply to state this conditional-mean interpretation. Additional assumptions are needed for particular inferential procedures, but they are not what makes a coefficient an explanation.
A predictive specification has a different success criterion. For numerical outcomes, one common criterion is mean squared prediction error: the average squared difference between an observed outcome and its forecast. A model that includes a hard-to-interpret proxy can be useful if it improves forecasts under the intended conditions. Its predictive usefulness does not turn the proxy into a meaningful explanatory variable.
Worked Example: An Adjusted Association With Treatment Dose
Suppose a hypothetical analysis models recovery time in days using treatment dose in milligrams and baseline severity. The fitted coefficient for dose is \(-0.30\) days per milligram. In this model, what adjusted association does a three-milligram dose difference correspond to, for patients with the same recorded baseline severity?
Calculation. The modelled difference is the dose coefficient multiplied by the dose difference:
Interpretation. The fitted model associates a dose of 5 mg, rather than 2 mg, with a mean recovery time 0.90 days shorter, conditional on the same baseline severity. This is an adjusted association in the specified model. It does not establish that raising a patient's dose would shorten recovery by 0.90 days; that would require a causal argument beyond the coefficient calculation.
How to Judge a Predictive Term
For a forecasting task, assess predictions on observations that were not used to fit the model. Otherwise, an apparent improvement can reflect how well a model matches the data it has already seen rather than how well it forecasts new cases. The comparison should use the same outcome definition and prediction-error measure for the candidate models. It should also reflect the conditions under which forecasts are intended to be used.
For numerical outcomes, the root mean squared error (RMSE) is the square root of the mean squared prediction error. It is in the same units as the outcome, and smaller RMSE means smaller average squared errors on that evaluation set. A lower RMSE on one small holdout set is evidence about that set, not a guarantee that a model will always forecast better.
Worked Example: A Risk Score That Improves Forecasts
Consider a hypothetical recovery-time forecasting task. Two models were fitted on training data. One uses the other candidate predictors; the second also includes a hospital risk score. The following four cases are a separate holdout set not used for fitting. Recovery time and predictions are in days.
| Case | Observed recovery time | Model without score | Model with score |
|---|---|---|---|
| A | 3 | 4 | 3 |
| B | 5 | 5 | 5 |
| C | 7 | 6 | 7 |
| D | 9 | 7 | 8 |
For the model without the score, the four prediction errors (observed minus predicted) are \(-1, 0, 1,\) and \(2\) days. Thus,
For the model with the score, the errors are \(0, 0, 0,\) and \(1\) day:
The score-containing model has the lower RMSE on this holdout set. That supports its use for forecasting in conditions like these, subject to further evaluation. The score may combine several inputs and may not represent a single, interpretable feature of recovery. Its predictive value does not show that changing the score would change recovery time, nor does it identify a mechanism.
A Useful Prediction Term Is Not Necessarily an Explanation
A term can summarize patterns that are useful for forecasting without corresponding to a well-defined substantive contrast. A risk score, a coded category, or a time index may carry information about the outcome. Its coefficient can still be mathematically defined in a fitted model, but a numerical coefficient is not automatically a scientifically informative explanation.
Worked Example: Forecasting With a Week Index
A hypothetical laboratory records weekly test volume over four weeks: 102, 109, 121, and 128 tests. For illustration, fit the simple linear model \(\hat{Y}_t=b_0+b_1t\), where \(t\) is the sequential week number. For these data, \(\bar{t}=2.5\) and \(\bar{Y}=115\).
The fitted slope is the sum of cross-products divided by the sum of squared deviations in week number:
The intercept is
The fitted line is therefore \(\hat{Y}_t=92.5+9t\). At week 5, it forecasts
The week index captures a trend in these four observations and may help forecast the next week if the pattern continues. But the coefficient of 9 does not mean that the passage of one week causes nine more tests. The index is a way to represent the observed ordering and trend, not an intervention or a mechanism. Four weeks also provide limited evidence about whether the trend will continue, so this calculation alone is not enough to establish forecasting performance.
Why the Goals Can Lead to Different Terms
Suppose an analyst wants to describe the adjusted association between a treatment and an outcome. The terms should be chosen in light of that association and the variables that define the intended adjustment. A proxy that is highly predictive may alter the conditional association being described when it is added. The resulting treatment coefficient answers a question conditional on the proxy as well as the other included terms; it is not simply a more accurate version of the original coefficient.
For prediction, the question is instead whether the term improves forecasts for new cases. A variable need not be a cause or a readily interpretable construct to be informative. However, a term that improves fit to the training data is not automatically helpful: evaluate its prediction errors on data not used to fit the model. Also, a forecast can only use information available when that forecast is made; a strong retrospective association is not enough to establish practical predictive value.
A short justification explains why additional information can help prediction in principle. Let \(X\) denote information already used to forecast \(Y\), and let \(W\) be an additional variable. Under squared-error loss, the best forecast given \(X\) is the conditional mean \(E(Y\mid X)\). For any proposed forecast \(a\), write \(m=E(Y\mid X)\). Since \(E(Y-m\mid X)=0\),
The cross-product vanishes because its conditional expectation is \(E[(Y-m)\mid X](m-a)=0\). The remaining term \((m-a)^2\) is nonnegative, so choosing \(a=m\) minimizes conditional mean squared error. With more information, the best possible forecast can therefore do no worse on average:
This is a statement about the best possible forecasts when the conditional means are known. In practice, they are estimated from finite data. Adding a term can make a fitted model more variable or less reliable, so a larger candidate model is not guaranteed to perform better on new observations. That is why predictive value must be evaluated, not assumed from a training fit.
A Practical Decision Rule
Specify the adjusted association to describe, or specify the outcome to forecast and how prediction errors will be assessed.
For an explanatory specification, ask how including it changes the conditional association of interest. For a predictive specification, ask whether it improves forecasts for new cases.
Describe explanatory coefficients as adjusted associations unless a separate causal justification is available. Describe predictive terms by their forecasting performance, not as causes simply because they help predict.
Common Mistake
Wrong reasoning: “This variable is strongly associated with the outcome, so it belongs in the explanatory model and explains the outcome.” Strong prediction alone does not establish that conclusion. The variable may be a proxy, its inclusion may change the adjusted association being studied, and regression coefficients do not automatically identify causes.
Correction: First decide whether the model is for an adjusted association or for forecasting. For the former, justify the term by the association the model is meant to describe and interpret the resulting coefficient conditionally. For the latter, test whether it improves prediction on suitable data not used for fitting. Keep those claims separate.
Check Your Understanding
Use the distinction between adjusted association and forecasting to answer these questions.
- What does a regression coefficient describe when interpreted as an adjusted association, and why is that not automatically a causal effect?
- A composite risk score lowers RMSE on a separate holdout set. What does that support, and what does it not establish about the score?
- Why should predictive performance be assessed on observations not used to fit the model?
- A sequential week number helps forecast laboratory volume. Why does its fitted slope not show that a week of time causes the change in volume?
- How can adding a useful predictive term change the adjusted association described by another coefficient?