Two Levels of Association
A biomarker can track clinical status among individual patients without reliably indicating whether a treatment will improve a clinical outcome. The distinction matters because surrogacy is used to support conclusions about treatment benefit, not merely to describe how two measurements vary together. Evidence must therefore be interpreted at the level of the question being asked: the relationship among patients, or the relationship between treatment effects across trials.
For clarity, suppose larger values of biomarker change \(S\) and clinical outcome change \(Y\) both represent improvement. In practice, signs and scales must be oriented consistently before associations are interpreted.
Individual-level analysis asks whether patients with greater biomarker improvement tend to have better clinical outcomes than other patients in the same treatment context. Trial-level analysis asks whether trials in which treatment produces a larger biomarker effect also tend to show a larger clinical treatment effect. The first compares people; the second compares randomized treatment contrasts. Neither association alone proves that changing the biomarker causes clinical improvement or that the biomarker can replace the clinical outcome.
Individual-Level Association
A simple individual-level summary is the within-treatment-group correlation between patients’ biomarker and clinical outcomes. Alternatively, a regression can adjust for randomized treatment assignment. For example, a linear model might include treatment assignment \(A_i\) and biomarker measurement \(S_i\) as predictors of the clinical outcome \(Y_i\):
Here, \(\beta_S\) describes the outcome difference associated with a one-unit difference in the biomarker, conditional on the variables in the model. It is an association parameter, not necessarily a causal effect. Its interpretation also depends on the model being appropriate, the biomarker and outcome being measured comparably, and relevant features such as treatment assignment and baseline status being handled appropriately. A single linear coefficient can be inadequate if the relationship is nonlinear or differs by treatment group.
Pooling treatment groups without adjustment can be misleading. If treatment changes both the biomarker and the clinical outcome, treated patients may have higher values of both even when there is no association between the two measurements within either group. That pooled pattern can reflect the treatment-group contrast rather than patient-level tracking.
Worked Example: A Strong Within-Group Association
In a hypothetical randomized trial, investigators examine biomarker improvement \(S\) and clinical improvement \(Y\) among patients in a treatment group. Suppose the sample variance of \(S\) is \(10.8\) biomarker-points squared, the sample variance of \(Y\) is \(10.0\) clinical-points squared, and their sample covariance is \(10.0\) biomarker-points times clinical-points. These are illustrative summary statistics, not results from a real study.
The sample correlation is the covariance divided by the product of the sample standard deviations:
The association is strong and positive in this group: patients with greater biomarker improvement tended to have greater clinical improvement. In a simple linear regression of \(Y\) on \(S\), the slope is \(10.0/10.8=0.9259\) clinical points per biomarker point. The fitted regression accounts for a sample variance of
Dividing that explained variance by the total sample variance of \(Y\) gives \(9.259/10.0=0.9259\), which also equals \(r_{S,Y}^2\) (up to rounding). This calculation establishes a strong linear association in the hypothetical patient data. It does not show that the same association holds in other treatment groups or that treatment effects on \(S\) predict treatment effects on \(Y\) across trials.
Trial-Level Association
Trial-level surrogacy is assessed using treatment-effect estimates, not patient measurements treated as if they were independent trials. For each trial \(j\), let \(\Delta S_j\) denote the estimated treatment effect on the biomarker and \(\Delta Y_j\) the estimated treatment effect on the clinical outcome, with both contrasts oriented so that positive values favor treatment. A simple across-trial regression is:
A positive \(\gamma\) indicates that trials with larger estimated biomarker effects tend to have larger estimated clinical effects in the data being considered. This relationship depends on the trials’ populations, interventions, endpoint definitions, measurement quality, and range of treatment effects. With few trials, an apparent correlation can be highly uncertain. The precision of each estimated treatment effect also matters; a raw correlation that ignores that precision can be a poor summary.
For a basic descriptive calculation, the across-trial sample correlation is the covariance of the trial-level effects divided by the product of their standard deviations. This requires a set of trials with comparable treatment contrasts and consistently oriented effects. It describes the observed association across those trials; it does not by itself establish that a biomarker can substitute for the clinical endpoint in a future trial.
Worked Example: Discordant Trial-Level Effects
Consider five hypothetical trials with estimated biomarker effects \(1,2,3,4,5\) biomarker units and clinical effects \(5,2,4,1,3\) clinical units, respectively. All effects are oriented so that positive values favor treatment. The trial-level means are both \(3\).
The sums of squared deviations for each set of effects are \(10\). The sum of cross-products of deviations is:
Each sample variance is \(10/(5-1)=2.5\), and the sample covariance is \(-5/(5-1)=-1.25\). Thus the trial-level correlation is:
In this small hypothetical set, trials with larger estimated biomarker effects do not have larger estimated clinical effects; the observed association is negative. This does not prove a negative relationship in a broader trial population, particularly with only five trials, but it clearly does not support a positive trial-level relationship in these data. A strong patient-level association in a separate study would not override this discordance.
Why the Two Levels Can Disagree
The unit of analysis changes the question. Within a trial, patients differ in disease severity, adherence, exposure, biology, and measurement error. Across trials, the relevant variation is in treatment contrasts and in trial characteristics. A biomarker can therefore track outcome differences between patients while treatment effects on that biomarker fail to track treatment effects on the clinical endpoint.
A related problem arises when individual data are pooled across randomized groups. The following hypothetical data illustrate how treatment assignment alone can induce a positive pooled correlation, even when the within-group association is zero.
Worked Example: A Pooled Correlation Created by Treatment Groups
Suppose a hypothetical trial has four control patients with biomarker values \(1,1,3,3\) and clinical values \(2,0,2,0\), and four treated patients with biomarker values \(7,7,9,9\) and clinical values \(6,4,6,4\). Within each arm, the deviations in the two measurements have a cross-product sum of zero, so there is no linear within-arm association in these data.
Across all eight patients, the means are \(5\) biomarker units and \(3\) clinical units. The sum of cross-products around these overall means is \(48\); the sums of squared deviations are \(80\) for the biomarker and \(40\) for the clinical outcome. The pooled correlation is therefore:
The pooled association is strongly positive, although neither arm has a within-arm linear association. The group means differ in both outcomes, so the treatment-group contrast creates the pooled pattern. Consequently, a pooled patient-level correlation should not be interpreted as evidence of individual-level tracking unless the analysis accounts for treatment assignment and other relevant design features.
Interpretation, Diagnostics, and Limits
A sound interpretation keeps the level of evidence explicit. A positive individual-level association supports the statement that the biomarker and clinical outcome tend to vary together among the patients analyzed, under the specified adjustment and model. A positive trial-level association supports the statement that estimated treatment effects on the two endpoints tend to vary together across the trials analyzed. Neither statement alone establishes that treatment effects on the biomarker predict clinical benefit in new settings or that the biomarker is a valid replacement endpoint.
Before interpreting either association, check that the endpoint directions and scales are consistent, that the treatment contrasts are comparable, and that the analysis respects the randomization and clustering structure. Inspect plots rather than relying only on a correlation: a single influential patient or trial, a nonlinear relationship, or a restricted range can make a numerical summary misleading. For individual-level analyses, compare within-arm patterns with pooled patterns and consider whether baseline status or treatment assignment accounts for the apparent relationship. For trial-level analyses, inspect each trial’s effects and their uncertainty; a correlation based on a handful of trials can change substantially when one trial is removed.
The main alternatives answer different questions. Separate within-arm analyses or treatment-adjusted patient-level regression help describe individual-level association. An across-trial regression or correlation of estimated randomized effects addresses trial-level association. Neither can be substituted for the other, and neither alone answers whether the biomarker captures the treatment’s effect on the clinical outcome. Detailed evaluation of surrogate performance requires further evidence and methods beyond these descriptive associations.
- Pitfall: “Patients with larger biomarker changes do better, so treatment effects on the biomarker must predict clinical benefit across trials.” The first clause concerns patient-level association; the conclusion concerns trial-level association. It does not follow.
- Correction: Estimate and evaluate the two associations at their respective levels. Report them separately, with their uncertainty and the limits of the trials or patients analyzed.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- What is the difference between individual-level and trial-level surrogacy?
- A pooled correlation is positive, but the association is near zero within each randomized arm. Give one explanation for this pattern.
- Across four trials, the estimated treatment effects on a biomarker are consistently larger in trials with larger clinical effects. What level of association does this describe, and what does it not establish by itself?
- Why should the five-trial correlation in the worked example be interpreted cautiously?
Solutions. (1) Individual-level surrogacy concerns association among patients; trial-level surrogacy concerns association across trials between estimated treatment effects. (2) Treatment assignment may shift both outcomes, producing a pooled association even when within-arm association is absent. (3) This describes a trial-level association. It does not prove causation, guarantee prediction in new trials, or establish that the biomarker can replace the clinical endpoint. (4) Five trials provide little information about the across-trial relationship, and one influential trial or uncertainty in the effect estimates could substantially change the observed correlation.