FEV1 Decline and Cystic Fibrosis Progression

Why Link FEV1 to Progression?

In cystic fibrosis, forced expiratory volume in one second (FEV1) is repeatedly measured as lung function changes over time. A falling FEV1 may accompany worsening disease, but patients who become very ill may also miss measurements, undergo lung transplant, or die. An analysis that treats every recorded FEV1 as if it were observed independently of these outcomes can therefore give an incomplete picture of decline. A joint model connects the longitudinal trajectory to the time until a clinically important event.

Here the event is the first of lung transplant or death. This composite endpoint represents progression to either outcome; it does not assert that transplant and death have the same clinical meaning. The case study uses FEV1 as percent predicted, so a change is measured in percentage points. Baseline age and baseline disease severity are included to distinguish the FEV1 association from these important prognostic factors.

Definition: Let \(Y_{ij}\) be the FEV1 percent predicted for patient \(i\) at follow-up time \(t_{ij}\), and let \(T_i\) be time from study entry to the first lung transplant or death. Let \(\delta_i=1\) if that event is observed and \(\delta_i=0\) if follow-up is censored. Let \(A_i\) be baseline age, centered at 15 years, and let \(S_i=1\) indicate severe baseline disease (with \(S_i=0\) otherwise). A shared-parameter joint model describes the repeated measurements and event time together through patient-specific random effects.

A Model That Separates Age, Follow-Up, and Severity

One useful specification models the latent, error-free FEV1 trajectory as

Y_{ij}=m_i(t_{ij})+\epsilon_{ij}, \qquad m_i(t)=\beta_0+\beta_1t+\beta_2A_i+\beta_3S_i+\beta_4tS_i+b_{0i}+b_{1i}t.

Here \(\epsilon_{ij}\) is measurement error, commonly assumed independent with distribution \(N(0,\sigma^2)\), conditional on the random effects. The patient-specific random intercept \(b_{0i}\) and slope \(b_{1i}\) are commonly modeled jointly as mean-zero normally distributed variables with an estimated covariance matrix. The fixed coefficient \(\beta_1\) describes the average annual change for the reference age and non-severe group; \(\beta_4\) allows the annual change to differ for severe disease. The age term \(\beta_2\) represents a baseline difference at entry, not an effect of one more year of follow-up.

A current-value association model for the event hazard is

h_i(t\mid b_i)=h_0(t)\exp\{\gamma_A A_i+\gamma_S S_i+\alpha[m_i(t)-70]\}.

The centering at 70 percentage points makes the association term refer to the latent FEV1 level relative to 70. The baseline hazard \(h_0(t)\) describes the event rate over follow-up time for a patient at the reference covariate values and latent FEV1 of 70. The coefficient \(\alpha\) is the log hazard ratio for a one-percentage-point higher latent FEV1, conditional on age and severity. A negative \(\alpha\) means higher FEV1 is associated with a lower event hazard, after adjustment. It is an association, not evidence that changing FEV1 itself would cause the hazard to change.

The model relies on assumptions that should be made explicit. The mean trajectory and hazard covariate forms must be adequate for the data; the random-effects and residual distributions must be reasonable; and, conditional on the modeled trajectory and covariates, the event process must be represented by the specified hazard. Censoring is assumed non-informative conditional on modeled information. The model also treats the transplant-or-death composite as the event of interest rather than modeling its components separately. Its shared-parameter structure links outcomes through the latent trajectory and random effects, not by treating an observed FEV1 value as error-free.

Why the Joint Likelihood Uses the Latent Trajectory

For a given patient and random effects \(b_i=(b_{0i},b_{1i})\), the longitudinal contribution is the product of the measurement densities. The event-time contribution is the hazard at the observed event time, raised to the event indicator, multiplied by the survival probability through the observed follow-up. The patient’s likelihood contribution integrates the product over the random-effects distribution:

L_i=\int \left\{\prod_j f(Y_{ij}\mid b_i)\right\} h_i(T_i\mid b_i)^{\delta_i} \exp\!\left[-\int_0^{T_i}h_i(u\mid b_i)\,du\right] f(b_i)\,db_i.

This expression gives the essential justification for joint modeling: each plausible latent trajectory contributes to both the repeated-measurement likelihood and the event-time likelihood. A patient’s event information can therefore inform the distribution of their latent FEV1 path, while the repeated measurements inform the hazard association. This dependence is particularly relevant when the chance of observing another measurement changes as a patient approaches transplant or death.

Worked Example: Constructing an Age- and Severity-Specific Trajectory

Suppose a hypothetical fitted longitudinal model has coefficients \(\beta_0=82\), \(\beta_1=-3.2\), \(\beta_2=-0.6\), \(\beta_3=-8.0\), and \(\beta_4=-1.1\). The units of \(\beta_1\) and \(\beta_4\) are percentage points per follow-up year. Consider a patient aged 17 at entry, with severe baseline disease, at two years of follow-up. Suppose the estimated random effects for this patient are \(b_{0i}=-3.0\) and \(b_{1i}=-0.5\).

The centered age is \(A_i=17-15=2\), and \(S_i=1\). Substituting into the trajectory gives

\begin{aligned} m_i(2) &=82+(-3.2)(2)+(-0.6)(2)+(-8.0)(1)+(-1.1)(2)(1)+(-3.0)+(-0.5)(2)\\ &=82-6.4-1.2-8.0-2.2-3.0-1.0\\ &=60.2. \end{aligned}

The model’s estimated latent FEV1 at two years is 60.2 percent predicted. It is not the same as a single noisy observed measurement. The decline terms are also distinct: the severe-disease interaction adds 1.1 percentage points of annual decline relative to the non-severe group, while the random slope makes this particular patient’s trajectory steeper still.

Interpreting the FEV1 Association

The association coefficient is easiest to communicate over a clinically interpretable contrast. If two patients have the same age and severity, but their latent FEV1 values differ by \(d\) percentage points at time \(t\), the ratio of their instantaneous hazards is \(\exp(\alpha d)\). The comparison concerns their latent current values at that time; it is not a comparison of observed measurements with their measurement error.

Worked Example: Hazard Ratio for a Ten-Point FEV1 Difference

Suppose a hypothetical analysis estimates \(\alpha=-0.045\) per percentage point, with standard error \(0.018\). Compare patients with latent FEV1 values 10 percentage points apart, at the same age and severity.

The estimated hazard ratio for the patient with the higher FEV1 is

\exp(10\hat{\alpha})=\exp\{10(-0.045)\}=\exp(-0.450)=0.638.

Thus, at a given time and conditional on the model, the estimated instantaneous hazard of transplant or death for a patient with 10 percentage points higher latent FEV1 is 0.638 times the hazard for an otherwise comparable patient. This is about a 36.2% lower hazard, since \(1-0.638=0.362\); it is not a 36.2% lower probability of an event over a fixed period.

A Wald 95% interval for \(\alpha\) is \(-0.045\pm1.96(0.018)\), or approximately \((-0.0803,-0.00972)\). For the 10-point contrast, exponentiating the endpoints multiplied by 10 gives

\left(\exp\{10(-0.0803)\},\exp\{10(-0.00972)\}\right) =(0.448,\ 0.907).

Under the model and the Wald approximation, this interval corresponds to hazard ratios from 0.448 to 0.907 for a 10-percentage-point higher latent FEV1. It excludes 1, indicating evidence of an adjusted association at the 5% level under a two-sided Wald test. The interval describes uncertainty in the model-based contrast; it is not a probability statement about the fixed parameter.

From a Hazard Contrast to a Conditional Risk

An association parameter alone does not give an absolute event probability. Absolute risk also depends on the baseline hazard, age, severity, the trajectory, and the time interval. The following calculation illustrates this distinction under a deliberately simple piecewise-constant baseline hazard.

Worked Example: A One-Year Conditional Event Risk

Suppose, for a hypothetical patient at two years of follow-up, the baseline hazard is \(0.080\) per year for the interval from year 2 to year 3. Let \(\gamma_A=0.035\) per year of baseline age, \(\gamma_S=0.55\), and \(\alpha=-0.045\) per FEV1 percentage point. Consider a 20-year-old patient with severe disease and latent FEV1 \(m_i(2)=55\). Assume the latent value and covariates remain fixed over this one-year interval for this illustration.

The centered age is \(A_i=20-15=5\), and the centered latent FEV1 is \(55-70=-15\). Therefore, the conditional hazard is

\begin{aligned} h_i &=0.080\exp\{(0.035)(5)+(0.55)(1)+(-0.045)(-15)\}\\ &=0.080\exp(0.175+0.550+0.675)\\ &=0.080\exp(1.400)=0.324\ \text{per year}. \end{aligned}

For a constant hazard over one year, the event probability is \(1-\exp(-h_i\times1)\). Substitution gives \(1-\exp(-0.324)=0.277\), or about 27.7%. This is a model-based risk conditional on the stated covariates and latent trajectory assumption. A full prediction would account for uncertainty in the fitted parameters and random effects, rather than treating their estimates as known.

Diagnostics, Alternatives, and Limitations

In a cystic fibrosis application, check first whether the longitudinal model reproduces the observed distribution of FEV1 over follow-up, including differences by age and baseline severity. Plot residuals against fitted values and time; examine whether a straight-line trajectory is plausible or whether curvature, changing variability, or a different random-effects structure is needed. Inspect whether residual patterns remain after accounting for the patient-specific intercepts and slopes. For the event model, assess whether the chosen baseline hazard is flexible enough and whether age, severity, or the latent FEV1 association shows evidence of misspecified form.

Also inspect the joint fit itself: convergence warnings, unstable estimates, implausibly large standard errors, or strong sensitivity to quadrature settings can indicate numerical or specification problems. Compare fitted and observed event patterns over follow-up and across clinically relevant age and severity groups. The association estimate should be checked against alternative plausible trajectory forms, because a misspecified longitudinal path can distort the inferred current value and hence the hazard association.

The main alternatives answer different questions. A separate mixed model for FEV1 and a separate survival model are simpler, but do not directly account for the shared latent trajectory when event occurrence and measurement availability are linked. A survival model using the most recent observed FEV1 as a time-dependent covariate can be easier to fit, but that value is noisy and may be measured at irregular times; it is not the model’s error-corrected latent current value. The joint model estimates the association through the latent trajectory, at the cost of stronger distributional and functional-form assumptions.

The endpoint also limits interpretation. Combining transplant and death is useful when the scientific question is time to either progression outcome, but it can conceal different associations for the two components. If the scientific question concerns their separate risks, a model that distinguishes event types is needed. Further limitations include measurement error not captured by the assumed residual model, unmeasured prognostic factors, informative censoring not explained by modeled data, and extrapolation beyond the observed follow-up range.

A frequent mistake is to interpret \(\exp(10\alpha)\) as a 10-point FEV1 difference causing a change in event probability. The correction is to call it an adjusted hazard ratio for the latent FEV1 contrast, conditional on the model’s covariates and assumptions. Likewise, age and follow-up time must not be conflated: baseline age adjusts for differences between patients at entry, while \(t\) indexes change since entry. Including both allows those roles to be represented separately.

Key takeaway: In this case study, the joint model links the latent FEV1 trajectory to time until transplant or death while adjusting for baseline age and disease severity. Interpret the association on a specified FEV1 scale and contrast, and distinguish a hazard ratio from an absolute risk or a causal effect.

Check Your Understanding

Answer each question, then compare with the solutions below.

If \(\hat{\alpha}=-0.030\) per percentage point, what is the hazard ratio for a 5-point higher latent FEV1, holding age and severity fixed?
Why does the trajectory model include both baseline age and follow-up time?
For a constant hazard of \(0.20\) per year over six months, calculate the event probability over that interval.
Give one reason a joint model can be preferable to using the last observed FEV1 in a survival model, and one important limitation of the joint model.

Solutions. (1) The contrast is \(\exp\{5(-0.030)\}=\exp(-0.150)=0.861\). The estimated instantaneous hazard is about 0.861 times as large, or 13.9% lower, for the higher-FEV1 patient, conditional on the model. (2) Baseline age represents an entry-age difference between patients; follow-up time represents within-patient change after entry. They are different quantities and can have different relationships with FEV1. (3) Six months is \(0.5\) years, so \(1-\exp\{-(0.20)(0.5)\}=1-\exp(-0.10)=0.0952\), or 9.52%. (4) A joint model uses a latent trajectory that accounts for measurement error and links repeated measurements with the event process; the last observed value is noisy and may be irregularly timed. A limitation is reliance on assumptions about the trajectory, random effects, hazard form, and censoring.