EDSS Progression and Disability Milestones in Multiple Sclerosis

Why the Event Definition Matters

In multiple sclerosis, the Expanded Disability Status Scale (EDSS) provides repeated assessments of disability. A joint model can relate a participant’s evolving EDSS profile to the time until a prespecified disability milestone. The modeling target depends on both parts of that description: how the repeated scores are represented and exactly what counts as reaching the milestone.

EDSS is an ordered scale: larger scores indicate greater disability, but the distance between adjacent scores need not represent an equal change in disability. Scores also have a bounded range, and the scale’s interpretation is not uniform across its range. Treating the score as a normally distributed continuous measurement can therefore impose questionable assumptions. An ordinal longitudinal model preserves the ordering without assuming equal spacing.

A milestone must be defined before the analysis. For example, an investigator might define progression as a specified increase from baseline EDSS that is confirmed at a later assessment. The threshold, confirmation period, rules for participants with different baseline scores, and event-date convention should be stated explicitly. Such rules are study definitions, not universal properties of EDSS.

Definition: In this case study, the longitudinal outcome is the repeated ordered EDSS score \(Y_{ij}\) for participant \(i\) at assessment time \(t_{ij}\). The event time \(T_i\) is the time to a milestone defined in advance, such as a specified, confirmed increase from baseline. A joint model links the participant’s latent disability trajectory to the hazard of that event while accounting for dependence between repeated scores and event times.

An Ordinal Longitudinal Model Linked to the Event

One way to represent the repeated scores is a cumulative logit mixed model. Let \(b_{0i}\) and \(b_{1i}\) be participant-specific random intercept and slope, and let \(X_i\) denote baseline covariates. For ordered EDSS cutpoints \(c\), the model is

\operatorname{logit}\{P(Y_{ij}\le c\mid b_i)\} =\kappa_c-\eta_i(t_{ij}), \qquad \eta_i(t)=\beta_0+\beta_1t+\beta_2^\top X_i+b_{0i}+b_{1i}t.

The cutpoints \(\kappa_c\) are ordered, and the latent predictor \(\eta_i(t)\) is oriented so that a larger value corresponds to a lower probability of being at or below a given EDSS cutpoint—that is, a tendency toward higher, worse scores. The model estimates probabilities for the ordered categories; \(\eta_i(t)\) is not itself an observed EDSS score or a number of EDSS points.

A proportional-hazards event model can link the current latent disability level to milestone risk:

h_i(t\mid b_i)=h_0(t)\exp\{\gamma^\top X_i+\alpha\eta_i(t)\}.

Here \(h_i(t\mid b_i)\) is the event hazard at time \(t\), conditional on the participant’s random effects and covariates; \(h_0(t)\) is the baseline hazard; and \(\alpha\) is the association parameter. With this link, a one-unit increase in the latent predictor corresponds to a hazard ratio of \(\exp(\alpha)\), conditional on the model’s covariates and random effects. It does not mean a one-point increase in EDSS.

A different specification may link the hazard to the latent slope \(\eta_i'(t)\), or to both current level and slope. A current-value link addresses whether participants with greater current latent disability have higher event hazards. A slope link asks whether the rate of latent change adds information beyond current level. The choice must match the scientific question and be supported by the data; adding links does not automatically improve the model.

Assumptions to make explicit: The ordinal model’s cutpoints and link adequately describe the category probabilities; the random-effects distribution and trajectory form are reasonable; and the event model’s hazard structure, including proportional hazards when used, is adequate. The shared random effects capture the modeled dependence between repeated EDSS scores and event time. Inference also relies on suitable handling of censoring and assessment times. If death prevents the disability milestone from occurring, treating death as ordinary independent censoring requires justification; a competing-event formulation may be needed.

Reading the Model in a Hypothetical Case

Worked Example: How a Latent Trajectory Changes Category Probabilities

Suppose a hypothetical fitted ordinal model gives a cutpoint \(\kappa_c=0.80\) for the category boundary EDSS 4.0. For a participant, the estimated latent predictor is \(0.20\) at one assessment and \(0.70\) at a later assessment. Calculate the model probabilities of EDSS at or below 4.0 and above 4.0 at each time.

At the first assessment,

P(Y\le4.0)=\frac{1}{1+\exp\{-(0.80-0.20)\}} =\frac{1}{1+\exp(-0.60)} =0.6457.

Thus the probability of a score above 4.0 is \(1-0.6457=0.3543\). At the later assessment,

P(Y\le4.0)=\frac{1}{1+\exp\{-(0.80-0.70)\}} =\frac{1}{1+\exp(-0.10)} =0.5250, \qquad P(Y>4.0)=1-0.5250=0.4750.

The fitted trajectory is associated with a larger modeled probability of being above 4.0 at the later assessment. This is a model-based probability comparison, not a claim that the participant’s observed score must increase or that a change of \(0.50\) in \(\eta_i(t)\) is a particular number of EDSS points.

Worked Example: Interpreting the Current-Value Association

Suppose the hypothetical event-model estimate is \(\hat{\alpha}=0.70\). Compare two participants at the same time and with the same modeled covariates, whose latent predictors differ by \(0.60\). Under the current-value link, the hazard ratio for the participant with the higher predictor is

\exp\{\hat{\alpha}(0.60)\} =\exp(0.70\times0.60) =\exp(0.42) =1.522.

The higher latent disability level is associated with an estimated conditional instantaneous hazard about 52.2% higher at that time, under this model. This is not a 52.2% increase in the probability of reaching the milestone over a specified period, and it does not establish that disability causes progression. The association is on the model’s latent ordinal scale, not directly per EDSS point.

For illustration, if the baseline hazard at that time were \(0.0800\) events per year and the comparison participant had \(\eta_i(t)=0\) with no covariate contribution, the hazard at \(\eta_i(t)=0.60\) would be

0.0800\exp(0.70\times0.60)=0.1218 \quad\text{events per year}.

This calculation illustrates a hazard at a particular time under the stated model; it is not a cumulative event probability.

Milestone Timing, Confirmation, and Censoring

A confirmed milestone creates a timing question as well as a classification rule. Consider a hypothetical participant with baseline EDSS 3.0. Suppose the study defines progression as an increase of at least 1.0 point that persists for at least six months. The participant has EDSS 3.5 at month 6, EDSS 4.0 at month 12, and EDSS 4.0 again at month 18.

Worked Example: Dating a Confirmed Progression Event

The milestone threshold is \(3.0+1.0=4.0\). The month-6 score of 3.5 does not meet it. The month-12 score of 4.0 meets the threshold, and the month-18 score of 4.0 confirms that the increase persisted for six months. If the estimand is time of first threshold attainment, the event occurred sometime after the last assessment below threshold and no later than month 12, given these assessment data. If the study instead defines event time as confirmation, the recorded event time is month 18.

These conventions represent different event-time definitions. Assigning month 18 as the event time measures time to confirmation, not necessarily time of onset. Assigning month 12 as the onset time uses retrospective confirmation, while the assessment schedule only locates threshold attainment within an interval. The analysis should use a convention consistent with the scientific question and the available event-time model; when onset is only known to lie in an interval, an interval-censored approach may be more faithful than pretending the exact onset time was observed.

The event rule can depend on the same EDSS assessments used in the longitudinal component. This is clinically natural, but it means the two outcomes are closely connected by construction. A strong estimated association may partly reflect how the milestone is defined, not an independent predictive signal. The joint model accounts for modeled dependence; it does not make the event definition independent of the scores.

Right-censoring also needs a clear interpretation. A participant who has not met the milestone by last follow-up is typically censored then, provided the censoring process is non-informative after conditioning on the modeled information. Loss to follow-up related to unobserved worsening can violate that condition. Death before progression is different: it can prevent the milestone from occurring, so treating it as ordinary censoring may misrepresent the event process.

Justification for the Hazard-Ratio Interpretation

The current-value link gives the hazard-ratio interpretation directly. For two otherwise comparable participants at time \(t\), let their latent predictors be \(\eta_A(t)\) and \(\eta_B(t)\), with the same covariate contribution. Dividing their hazards cancels the shared baseline hazard and covariate term:

\frac{h_A(t\mid b_A)}{h_B(t\mid b_B)} = \frac{h_0(t)\exp\{\gamma^\top X+\alpha\eta_A(t)\}} {h_0(t)\exp\{\gamma^\top X+\alpha\eta_B(t)\}} = \exp\{\alpha[\eta_A(t)-\eta_B(t)]\}.

Thus a latent-predictor difference of one gives \(\exp(\alpha)\), and a difference of \(d\) gives \(\exp(\alpha d)\). This comparison is conditional on the model and applies to instantaneous hazards at the same time. It is neither a comparison of cumulative risks nor a causal effect.

Diagnostics, Alternatives, and Limitations

Diagnostics should assess whether the chosen model represents the data and event process well, not merely whether the software converged. Useful checks include:

Compare observed category frequencies over time with model-based predicted probabilities. Systematic discrepancies can indicate an inadequate trajectory, cutpoint structure, or proportional-odds assumption.
Inspect observed and predicted participant trajectories, including whether the random-effects distribution and time trend capture the range of repeated scores. Sparse follow-up may make random slopes or nonlinear time trends difficult to estimate.
Check the event-model fit, including whether proportional hazards is plausible and whether the chosen current-value or slope link is supported. Examine uncertainty in the association estimate, not just its point estimate.
Review the milestone algorithm, confirmation records, censoring, missed visits, and event dates. Inconsistent confirmation or assessment schedules can distort event timing even when the statistical model converges.

A standard Cox model with baseline EDSS is simpler, but it does not use the evolving score trajectory. A Cox model with the most recent observed EDSS can use updated measurements, but the observed score is measured with error, visits may be irregular, and the last measurement may be stale. Separate ordinal longitudinal and event analyses do not, by themselves, represent their shared participant-level dependence. A joint model is useful when repeated scores and event time are related and this dependence matters for the scientific target or prediction.

These alternatives are not interchangeable in every setting. A joint model is more demanding: it depends on the specified longitudinal and event submodels, may be sensitive to the assumed random-effects distribution, and can be difficult to estimate with few events or sparse repeated measurements. A latent ordinal scale can be less immediately interpretable than raw EDSS points. The model also cannot correct a poorly defined milestone, eliminate bias from unmeasured factors, or establish a causal effect of disability on progression.

Common Pitfalls

Wrong reasoning: “EDSS is numeric, so a one-point change is a constant amount of disability and a normal model is automatically appropriate.” Correction: EDSS is ordered and bounded, and adjacent values need not be equally spaced. An ordinal model is one way to respect that structure.
Wrong reasoning: “A hazard ratio of 1.52 means a 52% higher chance of progression.” Correction: It describes a conditional instantaneous hazard ratio at a given time, not a probability or cumulative risk ratio.
Wrong reasoning: “A strong link proves the EDSS trajectory causes the milestone.” Correction: The association is conditional on the model and may be strengthened because EDSS helps define the milestone.
Wrong reasoning: “The event date is obvious once confirmation occurs.” Correction: Threshold attainment, confirmation, and recorded event time can differ. Define which time the analysis targets.

Key takeaway: A defensible EDSS joint analysis begins with a clinically explicit, prespecified milestone and a longitudinal model appropriate for ordered scores. Interpret the event association on the model’s latent scale, report the event-time convention, and treat the result as a model-based association rather than a causal effect.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

If \(\hat{\alpha}=0.40\), what hazard ratio corresponds to a latent-predictor difference of \(0.50\)?
Why might an EDSS milestone confirmed six months after threshold attainment have an event time different from the date of confirmation?
For a cutpoint \(\kappa_c=1.10\) and latent predictor \(\eta_i(t)=0.60\), calculate the model probability of being at or below that cutpoint.
Give one reason a joint model may be preferable to a Cox model using only baseline EDSS, and one limitation that remains.

Solutions. (1) \(\exp(0.40\times0.50)=\exp(0.20)=1.221\), so the estimated conditional hazard is about 22.1% higher for that latent-predictor difference. (2) Confirmation establishes persistence retrospectively; threshold attainment occurred earlier, and with scheduled assessments its exact time may only be known to lie within an interval. (3) \(P(Y\le c)=1/[1+\exp\{-(1.10-0.60)\}]=1/[1+\exp(-0.50)]=0.6225\), so the probability above the cutpoint is \(0.3775\). (4) A joint model can use the evolving repeated EDSS profile and represent its dependence with event time, unlike a baseline-only score. It still relies on its longitudinal, event, and censoring assumptions and does not establish causality.