Why Calculate the Shift?
When participants with more severe health are less likely to return for a planned biomarker assessment, the people measured at that visit may not represent the original cohort. A simple comparison of their biomarker average with the cohort average makes this selection visible. It also forces an important bookkeeping question: do we know the biomarker values, or at least the mean values, for the people who were actually retained?
This tutorial uses hypothetical numbers to calculate that shift. The calculations describe a fixed assessment time and a specified original cohort. They do not, by themselves, show that retention caused a change in anyone’s biomarker, or that the observed retained-sample mean estimates a biological trajectory.
In a real follow-up study, \(Y\) is generally unobserved for people who were not retained. The original-cohort mean at follow-up therefore may not be directly available. In a numerical illustration, however, we can specify all the relevant values or summaries so that both means can be calculated and compared.
A Grouped Calculation
Suppose the original cohort is divided into severity groups. For group \(g\), let \(n_g\) be the original group size, \(\bar{Y}_g\) its original-cohort mean biomarker value, and \(m_g\) the number retained. The original-cohort mean is the size-weighted average of the group means:
To calculate the retained-sample mean, we need the mean among the retained participants in each group, denoted \(\bar{Y}_{g,R}\):
These formulas use different weights and, potentially, different group means. Knowing \(\bar{Y}_g\) does not generally tell us \(\bar{Y}_{g,R}\). They are equal if values are constant within each group, or if the retained members happen to have the same mean as the full group; otherwise, the retained-group means or totals must be supplied.
Worked Example: A Lower Mean After Selective Retention
Consider a hypothetical cohort assessed at one follow-up visit. Biomarker values are in mg/L. Each participant’s value is constant within their severity group for this example, so the group mean also applies to the retained members of that group.
| Severity group | Original count | Group mean (mg/L) | Retained count | Retained-group mean (mg/L) |
|---|---|---|---|---|
| Low | 50 | 8 | 45 | 8 |
| Moderate | 30 | 16 | 21 | 16 |
| High | 20 | 30 | 5 | 30 |
Original-cohort mean. The group totals are \(50(8)=400\), \(30(16)=480\), and \(20(30)=600\) mg/L. Thus,
Retained-sample mean. There are \(45+21+5=71\) retained participants. Since this example explicitly assumes each group’s values are constant, the retained group totals are \(45(8)=360\), \(21(16)=336\), and \(5(30)=150\) mg/L. Therefore,
The shift is \(11.92-14.80=-2.88\) mg/L (using unrounded values, approximately \(-2.885\) mg/L). In this hypothetical cohort, the retained-sample mean is about 2.88 mg/L lower than the original-cohort mean. Retention rates also differ across groups: \(45/50=0.90\) for low severity, \(21/30=0.70\) for moderate severity, and \(5/20=0.25\) for high severity. The lower retained mean is consistent with the much lower retention of the group with the highest biomarker values.
Why the Retained-Group Means Matter
The previous calculation works because the example gave enough information to calculate the retained totals. If biomarker values vary within severity groups, the original group mean need not equal the mean among the group’s retained members. Retention may select people with comparatively high or low values even within the same group.
Worked Example: Using Explicit Retained-Group Means
In a second hypothetical cohort, 40 participants are in a lower-severity group and 40 are in a higher-severity group. Their original-cohort biomarker means are 6 mg/L and 18 mg/L, respectively. Among the lower-severity group, 32 participants are retained and their mean is 6.5 mg/L. Among the higher-severity group, 12 are retained and their mean is 20 mg/L. These retained-group means are given directly, so the retained totals can be calculated without assuming that retained members have the original group means.
The original-cohort total is \(40(6)+40(18)=240+720=960\) mg/L, giving
The retained total is \(32(6.5)+12(20)=208+240=448\) mg/L, and the retained count is \(32+12=44\). Therefore,
The shift is \(10.18-12.00=-1.82\) mg/L. As a consistency check, the nonretained lower-severity participants have total \(40(6)-208=32\) mg/L, or mean \(32/8=4\) mg/L. The nonretained higher-severity participants have total \(40(18)-240=480\) mg/L, or mean \(480/28\approx17.14\) mg/L. These values are compatible with the stated cohort and retained totals. Retention rates are \(32/40=0.80\) and \(12/40=0.30\); both the lower retention of the higher-severity group and the higher retained-group means within groups contribute to the difference.
A Short Derivation of the Mean Shift
The shift has a concise expression in terms of the association between the biomarker and retention. In a finite cohort, expectations can be read as averages over participants. Since \(R\) is 1 for retained participants and 0 otherwise, \(E[R]=P(R=1)\), and \(E[YR]/P(R=1)\) is the retained-sample mean. Provided at least one participant is retained,
Thus, when retained participants tend to have lower \(Y\) values than the full cohort, \(\operatorname{Cov}(Y,R)\) is negative and the retained mean is lower. This identity describes a difference in averages; it does not establish why retention and biomarker values are associated. It also does not supply the missing biomarker values for people who were not retained.
When Retention Does Not Shift the Mean
A contrast helps show what is distinctive about selective retention. If participants are independently retained with the same probability \(p>0\), then, conditional on at least one participant being retained, the expected retained-sample mean equals the fixed cohort mean. In a particular finite sample, random variation can still make the two means differ. If retention rates are equal across severity groups, the groups’ shares among retained participants are preserved, but that alone does not guarantee equality of means if retention selects biomarker values within groups.
Worked Example: Equal Retention Rates Across Groups
Suppose a hypothetical cohort has 40 participants with values of 5 mg/L and 60 participants with values of 15 mg/L. The original mean is \(\{40(5)+60(15)\}/100=(200+900)/100=11.00\) mg/L. Now suppose 24 participants from the first group and 36 from the second are retained. Retention is \(24/40=0.60\) and \(36/60=0.60\); because each group’s values are constant, the retained group means remain 5 and 15 mg/L.
The retained mean is \(\{24(5)+36(15)\}/(24+36)=(120+540)/60=11.00\) mg/L, so the shift is \(11.00-11.00=0.00\) mg/L. In this constructed example, equal retention rates preserve the group composition exactly, and there is no within-group selection. It illustrates a benchmark, not a guarantee that every realized study with similar retention rates will have exactly zero shift.
Assumptions, Diagnostics, and Limits
Before calculating or interpreting a shift, specify the target cohort, the follow-up time, the biomarker unit, and what counts as retained. The cohort mean and the retained mean must refer to the same assessment time and biomarker definition. The calculations above are exact for the stated hypothetical counts and summaries; no sampling model or significance test is needed to obtain these descriptive differences.
For grouped data, check that the group counts sum to the stated cohort and retained totals, that group means are weighted by the correct counts, and that every retained total uses a retained-group mean. If only original group means and retained counts are available, the retained mean is generally not calculable. It is calculable from those quantities only under an additional condition, such as constant biomarker values within each group, or an explicit assumption that the retained mean equals the original group mean.
A useful diagnostic is to compare retention rates across severity groups and, when available, compare retained and nonretained biomarker summaries within groups. Large differences in retention rates show that the group composition can change. They do not, by themselves, determine the exact size or direction of the overall mean shift: within-group values and selection also matter. Conversely, similar group retention rates do not rule out selection within groups.
The retained-sample average answers a descriptive question about people measured at that visit. The original-cohort average answers a different question about everyone enrolled. With incomplete follow-up, the latter is usually not observed directly. Weighting, imputation, and joint modeling are alternative analytical approaches that require additional assumptions and are not interchangeable with simply reporting the retained mean. A toy calculation where all cohort values are supplied demonstrates the arithmetic; it does not validate any particular method for recovering unobserved values in an actual study.
Common Mistakes
- Using original group means for retained totals without justification. Multiplying a retained count by the original group mean silently assumes that the retained members have that same mean. State this assumption, establish constant values within groups, or provide retained-group means.
- Comparing counts instead of rates. A group with fewer retained participants may simply have started smaller. Compare \(m_g/n_g\), not just \(m_g\).
- Calling a mean difference an individual change. A retained mean below the cohort mean does not mean each retained participant’s biomarker fell. It is a comparison between two sets of people.
- Attributing the shift to a cause based on arithmetic alone. A numerical difference describes selection in the stated data; it does not establish a biological mechanism or a causal effect.
Check Your Understanding
Answer each question, then compare your work with the solutions below.
- A cohort has 20 participants with a group mean of 4 mg/L and 30 with a group mean of 10 mg/L. What is the cohort mean?
- In the first group from question 1, 10 participants are retained. The original group mean is 4 mg/L, but no retained-group mean or within-group values are given. Can you calculate the retained total for that group? Explain.
- A cohort’s mean is 12 mg/L and its retained-sample mean is 9 mg/L. What is the shift, and how should it be interpreted?
- Two groups each have 50 participants. Retention is 40 in the first group and 10 in the second. Calculate each retention rate. Does this information alone determine the retained-sample mean?
Solutions. (1) The cohort total is \(20(4)+30(10)=80+300=380\) mg/L, so the mean is \(380/50=7.60\) mg/L. (2) No. The 10 retained participants’ mean is not given, and it need not equal 4 mg/L; the retained total requires their mean or their individual values. (3) The shift is \(9-12=-3\) mg/L. The measured participants’ average is 3 mg/L lower than the original-cohort average; this is a difference between group averages, not a claim that individuals’ values fell. (4) The rates are \(40/50=0.80\) and \(10/50=0.20\). These rates indicate different retention across groups, but without group biomarker values and retained-group means they do not determine the retained-sample mean.