Why Joint Diagnostics Need to Be Read Together
A joint model links repeated biomarker measurements to an event process, often through subject-specific random effects and a shared association structure. Its diagnostics therefore concern more than one outcome: a model may reproduce the biomarker trajectories while missing event timing, or may fit the overall event rate while misrepresenting how biomarker history relates to risk. A single residual plot cannot resolve all of these possibilities.
This workshop combines four diagnostic views: longitudinal residual plots, survival residuals, replicated-data checks, and influence analyses. The aim is not to count how many checks “pass.” It is to locate a pattern, assess whether it persists under diagnostics that account for uncertainty, and connect it to a specific model component that could plausibly produce it.
The particular joint model may use a different association structure, such as a slope or a cumulative feature of the trajectory. The diagnostic logic is the same, but a proposed revision must match the structure actually fitted. Interpret diagnostics under the model’s stated assumptions: the longitudinal mean and error structure are adequate, the random-effects distribution is suitable, the event hazard and association form are adequate, and censoring is conditionally independent of the event process given the modeled information. In practice, also check that estimation converged and that the time origin, event indicator, and measurement schedule were coded correctly. Diagnostic plots cannot repair an invalid data definition.
A Coordinated Reading Strategy
First describe the discrepancy rather than immediately naming a remedy. For example, “standardized residuals tend to be positive late in follow-up” is more useful than “the trajectory model is wrong.” Then ask which diagnostic views support that description, what competing explanations remain, and what focused revision would distinguish among them.
Record its outcome component, time range, subgroup, and direction. Check data coding and the measurement or censoring schedule before interpreting it as model failure.
Compare residual plots with replicated-data summaries and the other outcome component. Ask whether the same subjects or times are implicated, rather than assuming every warning has one cause.
Inspect influence results and uncertainty in the diagnostic reference, including simulation envelopes or bootstrap results when available. A pattern dominated by one subject calls for a different response from a broad population pattern.
Change the model feature suggested by the evidence, refit, and repeat the relevant checks. Retain a revision only if it addresses the discrepancy without creating important new ones.
For the longitudinal component, a basic residual is \(e_{ij}=Y_{ij}-\widehat m_i(t_{ij})\); a standardized version divides by an estimated residual standard deviation when that scaling is appropriate. Conditional residuals using estimated random effects are useful for locating subject-level patterns, but those random effects were estimated from the same measurements. Their residuals are not independent observations with a guaranteed standard normal distribution. Residual plots should therefore be read as structured evidence, not as a collection of independent tests.
For survival, define \(N_i(t)\) as the event-counting process, which is zero before an observed event and one afterward, and \(R_i(t)\) as the indicator that subject \(i\) is still under observation and at risk just before \(t\). The fitted cumulative hazard contribution through time \(u\) is \(\widehat A_i(u)=\int_0^u R_i(s)\widehat h_i(s)\,ds\). The martingale residual at the observed follow-up endpoint \(U_i=\min(T_i,C_i)\) is \(r_i^M=\delta_i-\widehat A_i(U_i)\), where \(C_i\) is censoring time. These residuals are not expected to be normally distributed. Their value is in detecting systematic structure across subjects or covariates, not in applying normal-theory cutoffs to individual values.
Why the Survival Residual Is Centered at Zero
The survival residual has a direct justification. Under a correctly specified hazard and the usual conditional independent-censoring assumptions, the event process minus its cumulative hazard compensator is a martingale. In counting-process notation, this is \(M_i(u)=N_i(u)-\int_0^u R_i(s)h_i(s)\,ds\). A martingale has mean-zero increments conditional on the information available before each increment. Evaluating at the observed endpoint gives the residual form above, with the fitted hazard substituted in practice.
This zero-mean property is a model-based reference, not a guarantee that fitted residuals average exactly zero in a finite sample. Parameters are estimated, random effects are latent, and censoring and follow-up determine what is observed. A residual pattern by time or covariate may indicate a hazard or association problem, but it can also reflect an inadequate baseline hazard, a subgroup difference, or a few influential events. The earlier material on survival residuals and joint checks develops the component-specific checks; here, the emphasis is on how those checks inform one another.
Worked Example: Reading a Longitudinal Residual Pattern
Suppose a hypothetical fitted model has estimated longitudinal residual standard deviation \(\widehat\sigma=4.0\) mg/L. For one subject, the observed measurements at months 2, 6, and 10 are 50, 55, and 65 mg/L, while the fitted latent means are 52, 57, and 58 mg/L. Calculate standardized residuals and describe the pattern.
Using \(r_{ij}=(Y_{ij}-\widehat m_i(t_{ij}))/\widehat\sigma\), the three residuals are \( (50-52)/4.0=-0.50\), \( (55-57)/4.0=-0.50\), and \( (65-58)/4.0=1.75\). The first two observations are modestly below the fitted mean; the last is 1.75 estimated residual standard deviations above it. The positive late residual suggests the fitted trajectory may rise too slowly for this subject, but one subject is not enough to establish a population-wide time trend.
Suppose the same upward late pattern appears across many subjects and in replicated-data checks of late biomarker values. That convergence supports examining the trajectory’s time form, such as whether its mean structure allows sufficient curvature. If instead the pattern is limited to this subject and the subject-deletion fit changes little, the evidence is weaker for a general trajectory revision. Before either conclusion, check the recorded month and units, and examine whether irregular visit timing or a subgroup characteristic accounts for the pattern.
Worked Example: Interpreting Two Survival Residuals
For a hypothetical subject with an observed event, suppose the fitted cumulative hazard through the event time is \(0.80\). For a second subject censored at the end of follow-up, suppose the fitted cumulative hazard through censoring is \(0.35\). Calculate each martingale residual.
For the observed event, \(\delta_i=1\), so \(r_i^M=1-0.80=0.20\). For the censored subject, \(\delta_i=0\), so \(r_i^M=0-0.35=-0.35\). These values are not probabilities and should not be read as individual goodness-of-fit scores with a universal cutoff.
Now suppose the event residuals tend to be positive among subjects with high fitted biomarker levels, while replicated event histories also show too few events in that group. The joint evidence motivates checking the association component and the event model in that region. It does not, by itself, prove that \(\alpha\) is too small: baseline-hazard misspecification, sparse follow-up, or a subgroup effect could also contribute. Compare the finding with longitudinal fit and with the association form used in the fitted model.
Replication and Influence Put Patterns in Context
A replicated-data check asks whether data generated from the fitted model reproduce features that matter for the scientific use of the model. Replicates should reflect the fitted joint structure, including random effects, longitudinal measurements, event process, and the observation or censoring scheme relevant to the check. Compare prespecified summaries, such as late measurements, event counts by follow-up interval, or joint patterns of biomarker history and event occurrence. If the observed summary is unusual relative to the replicated summaries, that is evidence of a discrepancy; it is not a calibrated hypothesis test unless the procedure was specifically constructed as one.
Influence analysis asks whether conclusions depend strongly on particular subjects or events. A simple coefficient sensitivity measure for parameter \(\theta\) is \(D_i=(\widehat\theta_{(-i)}-\widehat\theta)/SE(\widehat\theta)\), where \(\widehat\theta_{(-i)}\) is the estimate after deleting subject \(i\). This scaled change describes sensitivity relative to the original standard error; it is not a universal significance threshold. In joint models, deletion can affect both components and the association estimate, so inspect what changes and whether the deletion removes a clinically unusual but valid observation.
Worked Example: Combining Influence and Replicated-Data Evidence
In a hypothetical joint-model fit, the association estimate is \(\widehat\alpha=0.65\) per mg/L, with \(SE(\widehat\alpha)=0.20\) per mg/L. Deleting one subject gives \(\widehat\alpha_{(-i)}=0.38\) per mg/L. Calculate the scaled change and decide what it says.
The change is \(D_i=(0.38-0.65)/0.20=-0.27/0.20=-1.35\). Thus, deleting this subject lowers the estimated association by 1.35 original standard errors. This signals sensitivity worth investigating; it does not show that the subject is erroneous or that the model should automatically exclude the subject.
Suppose further that this subject has a late biomarker value poorly reproduced by the model, but the overall replicated event summaries are adequate and the other subjects show no comparable residual trend. The focused next step is to verify this subject’s records and assess whether the longitudinal trajectory or residual variance is inadequate for this type of trajectory. A broad change to the association structure would be less directly supported. If many subjects instead show the same longitudinal and event-related discrepancies, the evidence for a structural model revision is stronger than a single deletion result.
This integrated reading differs from relying on one diagnostic family alone. Residual plots are efficient for locating patterns but can be sensitive to estimated latent effects and visit schedules. Replicated-data checks assess whether chosen data features are reproduced, but conclusions depend on the summaries selected. Influence analyses reveal sensitivity, not the source of misspecification. No one approach replaces the others; their value comes from different weaknesses and from agreement about a specific pattern.
From Diagnostic Pattern to Targeted Revision
| Recurring evidence | Possible component to examine | Important qualification |
|---|---|---|
| Time-ordered longitudinal residual trend and poor replicated trajectory summaries | Longitudinal mean form or residual structure | Check units, visit timing, and subgroup composition first |
| Survival residual pattern by time with replicated event-count discrepancy | Baseline hazard or time representation | Assess whether the pattern is driven by a few events |
| Event-related discrepancy associated with biomarker history, alongside adequate separate component checks | Association structure | Alternative association forms require scientific justification |
| Residual pattern concentrated in a subgroup and echoed by subgroup replication checks | Subgroup effects or relevant interactions | Do not add interactions solely to chase small-sample noise |
| Large deletion sensitivity with otherwise isolated evidence | Data verification and subject-level fit | Influence is not a rule for excluding valid observations |
After a revision, refit the model and revisit the diagnostics that motivated it as well as the other outcome component. A change that improves late biomarker fit but worsens event-history replication has not automatically improved the joint model. Report persistent discrepancies, uncertainty, and sensitivity rather than presenting a preferred fit as if all uncertainty had disappeared.
Common Pitfall and Limits
A common mistake is to treat a conspicuous residual or a large deletion change as a diagnosis: “This subject has a large residual, so the model is wrong for everyone,” or “Deleting this subject improves the estimate, so the subject should be removed.” The correction is to separate three questions: Is the observation correctly recorded? Is its pattern repeated across subjects or replicated-data summaries? Does a targeted, scientifically defensible revision improve the relevant checks without damaging others? A single unusual case can be influential and valid at the same time.
Diagnostics also have limited power when events are few, follow-up is short, biomarker measurements are sparse, or the relevant subgroup is small. Conditional residuals may look reassuring because random effects adapt to each subject; conversely, noisy subject-level residuals may look alarming by chance. Replicated-data conclusions depend on the summaries and observation process represented. Thus, absence of a visible discrepancy is not proof that the model is correct, and a visible discrepancy does not uniquely identify its cause.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A subject has an observed event and fitted cumulative hazard \(1.10\) through that event. What is the martingale residual, and what should you avoid inferring from it alone?
- In a joint model, late longitudinal residuals are positive across many subjects, and replicated late values are systematically lower than observed. Which component is most directly implicated, and what should be checked before revising it?
- An association estimate changes from \(0.50\) to \(0.30\), with original \(SE=0.10\), after one subject is deleted. Calculate the scaled change and interpret it.
- Why is a replicated-data discrepancy not automatically a hypothesis test with a calibrated p-value?
Solutions. (1) \(r^M=1-1.10=-0.10\). It is not a probability or a stand-alone verdict about an individual or the model. (2) The longitudinal trajectory component is the most direct target. Check measurement units, visit-time coding, schedules, and subgroup composition; then consider whether the time form is inadequate. (3) \(D=(0.30-0.50)/0.10=-2.00\), a decrease of two original standard errors. This indicates sensitivity, not grounds by itself for deletion. (4) Its interpretation depends on the chosen summaries and replication procedure; calibration as a formal test requires a procedure designed to provide that test’s reference distribution and error properties.