Why the Population Target Must Be Named
An outcome may be recorded only for units that satisfy an observation rule: a person responds to a survey, a firm reports its outcome, or a patient returns for follow-up. The mean calculated from recorded outcomes describes the units whose outcomes were observed. It does not automatically describe every unit in the population of interest. Before interpreting an outcome mean, specify which units define its target.
This distinction matters even when the available data are clean and the arithmetic is straightforward. A mean among observed units, a mean for the full population, and a mean for a covariate-defined subgroup are different quantities. Some can be calculated from the observed records; others may require information about outcomes that were not observed or additional assumptions.
The word “selected” refers to this defined indicator; it need not mean that units were sampled at random. Selection might depend on eligibility, a response decision, a threshold, or other features of the data process. This tutorial uses the observation rule to describe which outcomes are available, not to assert why selection occurs.
Three Distinct Outcome Targets
Suppose the target population consists of units drawn from a specified population distribution. The full-population outcome mean is the expected outcome for a unit drawn from that population, whether or not its outcome is observed. The selected-population outcome mean is the expected outcome among units in that population with \(S=1\). The latter is defined when \(P(S=1)>0\).
The two expectations refer to different groups. If \(Y\) is measured in dollars, both means are in dollars, but their populations differ. In observed data, the selected mean can often be summarized directly from recorded outcomes. The full-population mean includes outcomes for units with \(S=0\), so it is not directly calculated from those records alone.
Now let \(X\) denote covariates observed for units in the target population, and let \(A\) be a defined subgroup—for example, \(A\) might mean \(X\) falls in a specified age range. There are again two relevant targets: the mean for all units in subgroup \(A\), and the mean among selected units in that subgroup.
The subgroup mean \(\mu_A\) includes subgroup members whose outcomes are unobserved. The selected subgroup mean \(\mu_{A,S=1}\) does not. It is defined only if \(P(X\in A,S=1)>0\). A conditional target at a particular covariate value \(x\) is written \(E[Y\mid X=x]\); the corresponding selected conditional target is \(E[Y\mid X=x,S=1]\), when the conditioning events have positive probability. These targets should not be treated as interchangeable.
| Target | Units included | Directly summarized from observed outcomes? |
|---|---|---|
| \(E[Y]\) | All units in the target population | Not in general; outcomes for \(S=0\) are unobserved |
| \(E[Y\mid S=1]\) | Selected units | Yes, if the observed selected units represent the selected population |
| \(E[Y\mid X\in A]\) | All units in subgroup \(A\) | Not in general; some subgroup outcomes may be unobserved |
| \(E[Y\mid X\in A,S=1]\) | Selected units in subgroup \(A\) | Yes, if such units are observed and the data represent that selected subgroup |
What the Observed-Data Mean Calculates
For a sample of \(n\) units with selection indicators \(S_i\), the observed selected-outcome mean is the sum of outcomes for selected units divided by the number selected. Writing it with indicators makes clear which records enter the calculation.
Although \(Y_i\) is unobserved when \(S_i=0\), the product \(S_iY_i\) in this expression can be calculated from the observed data: it is the observed outcome when \(S_i=1\), and contributes zero when \(S_i=0\). This ratio is a sample summary of selected units. Under representative sampling from the target population and a positive probability of selection, it estimates the population selected mean \(E[Y\mid S=1]\). That interpretation does not turn it into an estimate of \(E[Y]\) without further information or assumptions.
The same construction applies within a covariate-defined subgroup. Let \(I(X_i\in A)\) equal 1 when unit \(i\) belongs to subgroup \(A\), and 0 otherwise. The selected subgroup sample mean is the total observed outcome for selected subgroup members divided by their count.
This calculation targets selected units in \(A\), not all units in \(A\). If no selected unit belongs to \(A\), the denominator is zero and the sample selected subgroup mean is undefined. Even when the denominator is positive, observed outcomes from selected subgroup members alone do not reveal the unobserved outcomes of the other subgroup members.
Worked Examples
Worked Example: Overall and Selected Means
Consider six workers in a hypothetical target population. Their weekly earnings, in hundreds of dollars, are 40, 50, 60, 70, 80, and 100. Suppose an outcome is observed for workers 1, 2, 4, and 6, so the selection indicators are \(1,1,0,1,0,1\). For this illustration, the complete earnings are shown so that both population targets can be calculated; an analyst who had only the selected records would not know the two unobserved earnings.
Full-population mean. The sum of all six earnings is \(40+50+60+70+80+100=400\). Thus the full-population mean is \(400/6=66.67\) hundreds of dollars, or \(\$6{,}667\) per week to the nearest dollar.
Selected-population mean. The selected workers have earnings \(40,50,70,100\). Their sum is \(260\), and four workers are selected, so their mean is \(260/4=65\) hundreds of dollars, or \(\$6{,}500\) per week.
The observed records give the selected mean, not the full-population mean. The latter uses the earnings of workers 3 and 5, whose outcomes are unobserved under the stated rule. The fact that this hypothetical example reveals those values for calculation does not make them available in a dataset that contains only selected outcomes.
Worked Example: Full and Selected Means Within Covariate Groups
In a hypothetical study, \(X\) records whether a person lives in Area A or Area B, and \(Y\) is weekly spending in dollars. In Area A, four people have outcomes \(\$20,\$30,\$40,\$50\); the outcomes are observed for the first and third people. In Area B, four people have outcomes \(\$10,\$30,\$50,\$70\); the outcomes are observed for the second and fourth people. The complete values are given here only to illustrate how subgroup targets differ.
Area A, full subgroup target. The mean for all Area A residents is \((20+30+40+50)/4=\$35\) per week.
Area A, selected subgroup target. The observed Area A outcomes are \(\$20\) and \(\$40\). Their mean is \((20+40)/2=\$30\) per week. This is \(E[Y\mid X\in A,S=1]\) as a population target, and its sample analogue here; it is not the full Area A mean.
Area B targets. The full Area B mean is \((10+30+50+70)/4=\$40\) per week. The selected Area B outcomes are \(\$30\) and \(\$70\), giving \((30+70)/2=\$50\) per week.
Thus the four relevant quantities are \(\$35\) and \(\$30\) for Area A, and \(\$40\) and \(\$50\) for Area B, where each pair gives the full subgroup mean and selected subgroup mean, respectively. Knowing the area label for every person identifies subgroup membership, but does not by itself supply the unobserved spending outcomes needed for the full subgroup targets.
Worked Example: A Subgroup With Few Observed Outcomes
Suppose a hypothetical population has five younger and five older adults, and \(Y\) is a health score known to lie between 0 and 20 points. The outcome is observed for four younger adults, with scores \(2,4,6,8\), and one older adult, with score \(10\). The age group is known for everyone, but the other five scores are unobserved.
Observed selected mean. The observed score sum is \(2+4+6+8+10=30\), and five outcomes are observed. The mean among selected adults is \(30/5=6\) points. This describes the observed, selected units in these data.
Younger subgroup. The four observed younger scores sum to \(20\), and one younger score is missing. Since the missing score is between 0 and 20, the full younger-group mean could be as low as \((20+0)/5=4\) or as high as \((20+20)/5=8\) points. The selected younger mean is \(20/4=5\) points.
Older subgroup. The one observed older score is \(10\); four scores are missing. The full older-group mean could be as low as \((10+4\cdot0)/5=2\) or as high as \((10+4\cdot20)/5=18\) points. The selected older mean is \(10/1=10\) points.
The outcome range gives bounds, not point values, for the two full subgroup means. Knowing how many adults belong to each group, and observing at least one outcome in each group, does not determine the missing scores. The selected subgroup means are calculable from the observed records, but they answer different questions from the full subgroup means.
Observation Rules and the Scope of the Target
A selection indicator is meaningful only after its rule and reference population are specified. For example, “\(S=1\) for people who completed the follow-up questionnaire” defines a different selected group from “\(S=1\) for people eligible for follow-up.” If the definition changes, the selected-population target changes too. State what \(S=1\) means in the application, which units are in the target population, and which covariates are available for classifying subgroups.
It is also important to distinguish an outcome that is unobserved from an outcome that is observed at a restricted value or recorded under a censoring rule. This tutorial defines the target for a setting where \(Y\) is recorded for \(S=1\) and unavailable for \(S=0\). The meaning of a recorded limited outcome depends on its own measurement rule; it should not be silently re-labelled as an unobserved outcome.
Common Mistake: Renaming the Observed Mean
A common mistake is to calculate the average of recorded outcomes and call it “the population mean.” That wording is wrong unless the target population is specifically the selected population, or unless a separate argument establishes how the observed outcomes relate to the full-population target. The arithmetic alone establishes only which observed records were averaged.
The correction is to name the target before reporting the number. For example: “The mean weekly earnings among workers with observed earnings was \(\$6{,}500\)” identifies the selected group and the units. It does not claim that the mean for all workers was \(\$6{,}500\). Similarly, say “the observed mean among selected older adults” rather than “the mean for older adults” when older adults with unobserved outcomes are part of the intended subgroup.
Check Your Understanding
For each question, be explicit about which units define the target.
- A survey records annual income only for respondents. State the selection indicator and distinguish the respondent mean from the mean for the full target population.
- Write the population target for the mean outcome among selected units in a subgroup defined by \(X\in A\).
- In a sample of 80 units, 50 have observed outcomes. What does the ratio of the sum of those 50 outcomes to 50 calculate directly?
- Outcomes are observed for some, but not all, members of subgroup \(A\). Does knowing subgroup membership for everyone identify \(E[Y\mid X\in A]\)? Explain.
- What happens to the sample selected subgroup mean if no selected unit belongs to subgroup \(A\)?