Why the Target Matters
When an omics assay is repeated over time, “the effect of time” or “the association with outcome” is not a complete statistical question. A study might ask how a biomarker changes within a particular subject, how its mean changes in a population, or whether a recent measurement predicts an upcoming event. These are different targets. They can have different answers even when they use the same assay results.
An estimand is the precisely defined quantity a study aims to learn. Defining it requires specifying the population, the outcome or measurement, the time points or contrast, and how events such as death or missing measurement are handled. The estimand is not the fitted model or its coefficient; those are tools that may be used to estimate it.
Three Distinct Targets
A subject-specific trajectory describes how the measurement changes over time for an individual, or for an individual characterized by specified subject-level features. It may refer to an underlying expected trajectory rather than the noisy assay reading at each visit. For example, one can describe a person’s expected biomarker change per month. Such a trajectory is individual-level: averaging the slopes across people produces a different target.
A population-average change compares means across a defined population at specified times. If the target population is all enrolled subjects and the measurement is defined at both times, one possible contrast is
This contrast is measured in the units of \(Y\). It is not generally the same as the average of individual slopes unless the slope definition, time interval, and population align with that contrast. A mean contrast also does not imply that every subject changed by that amount.
A time-updated prognostic association asks how a measurement available by a given time is associated with an outcome occurring afterward. For a time-to-event outcome, a standard conditional association can be represented by a proportional hazards model:
where \(\lambda_i(t\mid \mathcal{H}_i(t))\) is the event hazard at time \(t\), conditional on the subject’s observed history \(\mathcal{H}_i(t)\), \(Z_i(t)\) is a specified current or most recent omics value known before the risk interval, and \(\lambda_0(t)\) is the baseline hazard. Under this model, \(\exp(\beta)\) is the conditional hazard ratio for a one-unit increase in \(Z_i(t)\), holding the other included history fixed. This is a prognostic association, not automatically a causal effect.
For that interpretation, the proportional hazards relationship must be suitable over the time range of interest, the covariate scale and history must be defined, and the measurement used to predict an event must precede that event. The model also relies on appropriate handling of censoring and the risk sets. A measurement taken after an event cannot be used as a predictor of that event.
Worked Example: Comparing Individual Trajectories
Suppose two hypothetical subjects have an assay reported in normalized units at months 0, 6, and 12. For a simple descriptive comparison, define each subject’s endpoint slope as the change from month 0 to month 12 divided by 12 months. This calculation treats the recorded endpoints as the quantities of interest; it does not assume a linear trajectory between visits.
Subject A has values 2, 5, and 8. The endpoint slope is
Subject B has values 8, 7, and 6. The endpoint slope is
The two subjects therefore have different endpoint slopes: the assay rises for A and falls for B over the specified interval. The month-6 readings show that endpoint slopes do not summarize all within-person behavior: A’s increments are 3 and 3, while B’s are \(-1\) and \(-1\). With other data, intermediate values could depart substantially from the endpoint trend. The target here is each subject’s endpoint change rate, not a population mean or a survival association.
Worked Example: A Population-Average Change
In a hypothetical study, the same four subjects have observed, assay-defined measurements at baseline and month 12. Their baseline values are 2, 4, 6, and 8 normalized units; their month-12 values are 3, 5, 7, and 9. Define the target as the month-12 mean minus the baseline mean in these four subjects.
The baseline mean is \((2+4+6+8)/4=20/4=5.00\) normalized units. The month-12 mean is \((3+5+7+9)/4=24/4=6.00\) normalized units. Thus,
This is a descriptive estimate of a population-average contrast for the stated four-subject target. Each subject’s paired change is 1.00 unit, so the average of the paired changes is also 1.00 unit. That equality follows from using the same complete subjects at both times: the mean of the differences equals the difference of the means. If different subjects contributed at each time, the difference of observed means would not in general describe the average within-subject change for a common group.
If the intended target were a broader population, these four values alone would not establish that population’s mean change. A sampling design and suitable analysis would be needed to quantify uncertainty and support generalization.
Deriving the Paired Mean Contrast
For \(n\) subjects with measurements at both specified times, define each paired change by \(D_i=Y_i(t_1)-Y_i(t_0)\). The sample mean change is
The last equality follows by distributing the sum over subtraction. It shows why, for the same complete subjects and equal weights, the average paired change equals the difference between the two sample means. The identity is arithmetic, not an assumption about the data-generating process. It does not make the contrast causal, correct bias from missing measurements, or make the observed subjects representative of a larger population.
A common target instead conditions on survival. For example, \(E\{Y_i(t)\mid T_i>t\}\) is the mean among subjects alive and event-free at time \(t\), when the measurement is defined for them. Comparing such means across times compares populations that may differ because some subjects have experienced the event. It is not the same as the mean change in a fixed group of subjects.
Worked Example: A Time-Updated Prognostic Association
Consider a hypothetical proportional hazards model in which \(Z_i(t)\) is a most-recent assay value, in normalized units, recorded before the start of the relevant risk interval. Suppose the fitted coefficient is \(\hat{\beta}=0.40\) per normalized unit. Under the model’s proportional-hazards assumption, the estimated hazard ratio for a one-unit higher assay value is
In this hypothetical model, among subjects with the same included covariate history at a given time, a one-unit higher current assay value is associated with an estimated event hazard 1.492 times as large, or about 49.2% higher, during the modeled risk comparison. This is not a 49.2% increase in event probability, and it does not show that increasing the assay value would cause the hazard to rise. A confidence interval and a test would be needed to describe statistical uncertainty; neither is supplied by this coefficient alone.
The timing is part of the target: the assay must be available before the interval whose event hazard it is used to describe. If a subject’s event occurs before an assay visit, that future assay cannot legitimately serve as a predictor for the earlier risk time.
Population, Survival, and Missing Measurements
A population-average estimand needs a clearly stated target population and a measurement definition at each time. In survival settings, death can make a later assay undefined rather than merely unobserved. Three questions must not be conflated:
- Mean among survivors: What is the mean assay value among those alive at each specified time? The population can change over time.
- Change among observed pairs: What is the mean change among subjects with measurements at both times? This describes a selected complete-case group unless the target is explicitly that group.
- Change in a fixed original population: What would the contrast be for all initially eligible subjects? If the assay is undefined after death, this requires an explicit strategy for defining the target; a mean assay value after death cannot simply be presumed to exist.
For missing assays among living subjects, the target can still be well-defined, but observed-data estimates may be biased if measurement availability depends on the unobserved assay or related history. Generalizing from measured subjects requires assumptions about the visit and missingness process, as well as an analysis suited to those assumptions. Irregular visit times also matter: an unweighted mean of recorded values can give more influence to subjects with more visits, and it need not estimate a mean at a fixed time.
Diagnostics, Alternatives, and Limitations
Before estimation, inspect measurement times, numbers at risk, assay availability by visit, and the distribution of follow-up. Plot individual profiles alongside population summaries: a population mean can hide sharply different subject trajectories. For a time-updated prognostic target, verify that covariate values precede the risk intervals, check whether the proportional-hazards assumption is plausible, and assess whether conclusions depend on a small number of subjects or sparse event times. These checks can reveal problems; they cannot prove that missingness or censoring assumptions hold.
| Target | Question answered | What it does not answer |
|---|---|---|
| Subject-specific trajectory | How does the assay change for an individual over time? | How much the population mean changes or whether the assay predicts an event. |
| Population-average change | How do means differ at defined times in a specified population? | Whether every subject changes similarly or whether the change causes an outcome. |
| Time-updated prognostic association | How is an available assay value associated with subsequent event hazard? | Whether intervening to change the assay would change event risk. |
These targets are alternatives in the sense that they answer different questions, not competing estimators of one universal quantity. A flexible model for individual profiles may be useful when individual trajectories are central. A population-average analysis targets mean behavior. A survival model with correctly timed covariates targets prognostic association. The mixed-model and time-dependent survival methods developed later provide modeling tools for some of these questions; choosing a method does not remove the need to state the estimand first.
Common Mistakes
- Calling a population mean an individual trajectory. A mean increase of 1 unit does not imply that every subject increased by 1 unit.
- Comparing means from changing survivor groups as if they were paired changes. The survivor mean at a later time may describe a different set of people.
- Using a future assay to predict an earlier event. This introduces information unavailable at the prediction time and can distort the prognostic association.
- Interpreting a prognostic association causally. A hazard ratio for an observed assay value does not, by itself, estimate the effect of intervening on that value.
- Leaving the assay scale unspecified. A one-unit contrast is interpretable only after defining the scale, such as raw concentration, log concentration, or normalized units.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A mean assay value is 4 normalized units at month 0 and 5 at month 6. What is the population-average contrast, and does it imply every subject increased by 1 unit?
- At month 12, an assay mean is calculated only among subjects alive at that time. What population does this mean describe?
- A proportional hazards model gives \(\hat{\beta}=\log(0.80)\) for a one-unit higher assay value. State the estimated hazard ratio and interpret it without making a causal claim.
- Why is an assay collected after an event not a valid time-updated predictor of that event?
Solutions. (1) The contrast is \(5-4=1\) normalized unit. It does not imply that every subject increased by that amount; opposite individual changes can average to 1. (2) It describes the subjects alive at month 12 with values contributing to the mean, not necessarily the original enrolled population. (3) The estimated hazard ratio is \(\exp\{\log(0.80)\}=0.80\). Under the model and conditional on included covariates, a one-unit higher assay value is associated with an estimated 20% lower hazard; this does not establish that changing the assay causes lower hazard. (4) The post-event assay was not available before the event and cannot represent information known at the prediction time.