What Is the Future Outcome?
A prediction interval is about a future observation, but “a future observation” is not specific enough to define its target. We need to say which response is being predicted and at what predictor values. A prediction interval for one patient with a specified baseline score, for example, has a different target from an interval for the average response of all patients with that score.
The distinction matters because a single person's outcome can vary around the average outcome for people with the same predictors. A prediction interval is meant to cover one future response from that conditional group—not the group average, and not an outcome forecast that averages across cases with different predictor values.
Modeling a Conditional Outcome
To make the distinction precise, write the response for a case with predictor values \(x\) as a conditional mean plus that case's deviation from the mean:
Here \(\mu(x)=E(Y\mid X=x)\) is the population mean response among cases with predictor values \(x\), and \(\varepsilon=Y-\mu(x)\) is the individual case's deviation from that mean. For this decomposition, the defining condition is \(E(\varepsilon\mid X=x)=0\). Regression models may add further assumptions for estimation and interval construction; those assumptions are not needed to identify which outcome is the target.
For a new case at \(x^*\), its response is \(Y_{\mathrm{new}}=\mu(x^*)+\varepsilon_{\mathrm{new}}\). The value \(\mu(x^*)\) describes the center of the conditional response distribution. The future response also includes the new case's individual deviation. Thus, even if the conditional mean were known exactly, it would not determine the new case's outcome.
This is a distinction between what the interval is about, not merely between two ways to report uncertainty. Asking “What is the average response at this profile?” calls for a mean target. Asking “What response might this next case have?” calls for an individual-outcome target. The same regression model and predictor profile can be involved in both questions, but the targets are not interchangeable.
Specify the Case Before Naming the Target
A predictor profile is the full set of predictor values that defines the case of interest. If the model includes treatment group, baseline score, and age, it is not enough to specify only the baseline score when the target also depends on treatment and age. A clear target statement names the response, gives the predictor profile, and identifies the case as new or future.
The phrase “a new case” does not necessarily mean a person who has never appeared in any data collection. It means the future response being considered is an outcome not used as the response in fitting the model. If a new response is measured later for someone whose earlier information is known, the target still has to specify which response occasion and which predictor values are being conditioned on.
Worked Example: One Future Recovery Time at a Fixed Profile
Suppose a hypothetical regression relates recovery time, measured in days, to treatment group and a baseline symptom score. Consider a new patient assigned to Treatment A with a baseline score of 12. The intended target is the patient's recovery time.
The specified profile is Treatment A and baseline score 12. The conditional mean \(\mu(\text{Treatment A},12)\) is the average recovery time for the population of patients represented by that profile. It is not the target here. The target is one future response, \(Y_{\mathrm{new}}\): the recovery time, in days, of one new patient with those specified values.
For illustration, suppose the conditional mean at this profile is 18 days. The decomposition gives \(Y_{\mathrm{new}}=18+\varepsilon_{\mathrm{new}}\) days, where the deviation is in days and has conditional mean zero under the stated model. The arithmetic identifies 18 days as the conditional mean component; it does not assert that this patient's recovery time will be 18 days. The individual response remains the target.
Three Targets That Sound Similar
A prediction question can be ambiguous when it does not say whether the predictors are fixed or uncertain, or whether the requested quantity is an individual response or a mean. These distinctions can be organized by asking two questions: Are predictor values specified? and Is the target one case or an average?
| Question | Target | What varies across cases? |
|---|---|---|
| What is the mean response for cases at \(x^*\)? | \(\mu(x^*)\), a conditional population mean | Responses among cases at the specified profile, summarized by their mean |
| What response will one new case at \(x^*\) have? | \(Y_{\mathrm{new}}\mid X_{\mathrm{new}}=x^*\), an individual conditional outcome | The individual response among cases at that profile |
| What outcome should be forecast when the future case's predictors are not specified? | An outcome forecast that accounts for the distribution of predictor values | Both which predictor profile occurs and the response at that profile |
The third question is not simply the second question with the profile left blank. If future predictor values are unknown, a forecast must state how those values are represented—for example, by a specified distribution of future cases. Without that information, “the future outcome” does not identify a unique target. A conditional prediction interval at one \(x^*\) remains conditional on that profile; it does not automatically cover outcomes for a mix of profiles.
Worked Example: Mean Response or One New Patient?
A hypothetical clinic model relates a patient's follow-up score, measured in points, to a baseline score. At a baseline score of 40, suppose the model's conditional mean follow-up score is 52 points.
If the question is “What is the average follow-up score among patients with baseline score 40?”, the target is \(\mu(40)=52\) points. It refers to a population mean, not to one selected patient's score.
If the question is “What follow-up score will one new patient with baseline score 40 have?”, the target is \(Y_{\mathrm{new}}\mid X_{\mathrm{new}}=40\). Its decomposition is \(Y_{\mathrm{new}}=52+\varepsilon_{\mathrm{new}}\) points. For instance, a hypothetical deviation of \(-3\) points would yield \(52+(-3)=49\) points for that case. The value 49 is an illustration of an individual outcome, not a replacement for the mean target.
The model and specified baseline value are the same in both questions. The wording changes the target: 52 points is a mean, while 49 points in the illustration is one possible individual response. A confidence interval for the mean and a prediction interval for the individual response therefore answer different questions.
When the Predictors Are Not Fixed
Sometimes the future case has not yet been selected, so its predictor values are unknown. Then an analyst might want a forecast averaged over a defined population of future cases. This is a different target from the response for a case at a specified profile. To see the distinction, suppose a hypothetical future service will be used by patients with baseline scores of 30 or 50, with equal probability. Suppose their conditional mean outcomes are 40 and 60 points, respectively.
Worked Example: A Forecast Across Two Possible Profiles
For a patient known to have baseline score 30, the conditional mean is 40 points; for a patient with baseline score 50, it is 60 points. If the future patient's score is not known, but the specified forecast population has equal proportions at these two profiles, its average conditional mean is:
The 50-point result is an average across the stated mix of predictor profiles. It is not the conditional mean at baseline score 30 or 50, and it is not an individual outcome guaranteed for the next patient. The actual future patient's profile could be either one, and that patient's response may also differ from the conditional mean at that profile.
The equal weights are part of the hypothetical forecast definition. If the future population had a different mix, the average would change; for example, weights of 0.75 and 0.25 would give \(0.75(40)+0.25(60)=30+15=45\) points. This calculation illustrates why an unconditional forecast needs a defined distribution of predictor values. It does not turn a conditional prediction interval into an interval for that forecast.
Why the Target Definition Matters
The target determines what an interval's endpoints are supposed to contain. For a conditional prediction interval, the question is whether the observed response of one future case with the specified profile falls between the endpoints. The relevant response has the same units as the outcome—days, points, dollars, or another response scale. Predictor values specify which conditional group is relevant; they are not themselves the outcome being covered.
The individual-outcome target also explains why one should not describe a prediction interval as a range for the “true mean.” A population mean is a fixed feature of the specified population and profile, while a future response is one realized outcome. Conversely, a confidence interval for a mean should not be described as a range expected to contain most individual responses. “Mean” and “individual outcome” refer to different quantities.
The target must also be plausible in context. A prediction for a profile far outside the predictor values represented by the data may rely on an unsupported extrapolation. Defining the target does not establish that the model is suitable for that profile; it makes the question explicit so that model relevance can be assessed.
A Short Justification: Why a Mean Is Not the Future Response
The difference follows directly from the definition of the conditional mean. For a specified profile \(x^*\), the conditional mean is the expected response across cases at that profile:
An expectation summarizes a distribution; it does not require every value in that distribution to equal the expectation. In the model decomposition, subtracting the mean gives the case-specific deviation:
Even though \(E(\varepsilon_{\mathrm{new}}\mid X_{\mathrm{new}}=x^*)=0\), an individual \(\varepsilon_{\mathrm{new}}\) need not be zero. Therefore, \(\mu(x^*)\) is not generally the value of \(Y_{\mathrm{new}}\). This is why the interval target must be named as either the conditional mean or the individual response before an interval is interpreted.
Common Mistakes
- Calling the conditional mean “the predicted patient's outcome.” The mean is an average for cases with the specified profile. A prediction interval targets one future response, which can differ from that average.
- Leaving the profile unstated. “A new patient's score” is incomplete if the model uses several predictors. State the predictor values, or explain how their distribution is defined for an unconditional forecast.
- Treating a conditional interval as unconditional. An interval defined for a case with baseline score 40 does not automatically describe a future patient whose baseline score could be 30, 40, or 50.
- Confusing the predictor with the response. A future case's predictor values locate the conditional target. The quantity to be covered is the future response, in the response's units.
- Interpreting an individual-outcome interval as a range for most cases. A prediction interval for one future case is not, by itself, a claim about the fraction of all cases in a population that fall in its range.
Check Your Understanding
For each question, identify whether the target is a conditional mean, one conditional future response, or an unconditional forecast.
- A model relates salary in dollars to years of experience. What is the target in the question, “What salary might one new employee with 5 years of experience receive?”
- At a specified treatment and baseline profile, explain the difference between the population mean response and one new patient's response.
- A forecast concerns future customers whose age distribution is unknown. What additional information is needed to define an unconditional forecast target?
- Why does a conditional prediction interval for a baseline score of 40 not automatically describe outcomes for patients with other baseline scores?
- Suppose the conditional mean response is 25 units and a new case's deviation is 4 units. What individual response follows from the decomposition, and why is it not the conditional mean?