Joint Level-and-Slope Links and Their Trade-Offs

Why Include Both Level and Slope?

A biomarker’s current value and its rate of change describe different features of a patient’s trajectory. A high value may be associated with event risk even when it is stable, while a rapid increase may carry information beyond the value reached so far. A joint level-and-slope link allows both features to contribute to the hazard at the same time. The central interpretive challenge is that they may give competing contributions, and the observed trajectories may not provide enough independent variation to estimate those contributions precisely.

Let $T_i$ denote the event time for subject $i$, and let $m_i(t)$ be that subject’s latent, or underlying, biomarker trajectory at time $t$. Its instantaneous slope is $m_i'(t)$, the derivative of the trajectory with respect to time. The hazard is conditional on the subject’s latent trajectory and included covariates; it is not a hazard conditional only on a noisy observed measurement.

Definition: A joint level-and-slope association link specifies the conditional event hazard as $$ h_i(t\mid b_i)=h_0(t)\exp\left\{\gamma^\mathsf{T}w_i+\alpha_L m_i(t)+\alpha_S m_i'(t)\right\}. $$ Here $h_0(t)$ is the baseline hazard, $w_i$ is a vector of event-model covariates, $\gamma$ is its coefficient vector, $b_i$ denotes the subject-specific random effects used to define the longitudinal trajectory, and $\alpha_L$ and $\alpha_S$ are the level and slope association parameters. The trajectory and its derivative are evaluated at the same time $t$.

A typical longitudinal submodel might be $Y_{ij}=m_i(t_{ij})+\epsilon_{ij}$, where $Y_{ij}$ is the measured biomarker at visit time $t_{ij}$, and $m_i(t)$ is modeled using fixed effects and subject-specific random effects. The joint model connects this submodel to the event process through the latent trajectory features in the hazard. Modeling the latent trajectory rather than substituting a noisy measurement is important: measurement error in a visit value should not be mistaken for true short-term movement.

The link assumes that the chosen trajectory is adequately described by the longitudinal model, that it is differentiable where the slope is used, and that the hazard form is appropriate. In the displayed specification, the association coefficients are constant over time, and the log hazard is linear in the current level and slope. Event and longitudinal outcomes are connected through the shared latent effects or trajectory; censoring is assumed non-informative conditional on the modeled information. These assumptions are substantive, not automatic consequences of fitting a joint model.

What the Two Coefficients Mean

For a fixed time $t$, compare two subjects with the same event covariates $w_i$, but different latent levels and slopes. Their conditional hazard ratio is

\frac{h_A(t\mid b_A)}{h_B(t\mid b_B)} = \exp\left\{\alpha_L[m_A(t)-m_B(t)] +\alpha_S[m_A'(t)-m_B'(t)]\right\}.

Thus $\alpha_L$ describes the change in log hazard associated with a one-unit higher current latent level, holding the current slope fixed. Likewise, $\alpha_S$ describes the change in log hazard associated with a one-unit higher slope, holding the current level fixed. These are conditional interpretations. They do not say that changing a biomarker by intervention would cause the hazard to change.

The two contributions can reinforce one another or compete. For a specified contrast, the level contribution is $\alpha_L\Delta m$, and the slope contribution is $\alpha_S\Delta m'$. Their sum determines the log hazard ratio. A positive coefficient alone does not guarantee a positive contribution: the subject’s difference in that trajectory feature may be negative. The earlier material on current value versus current slope and on opposing signs motivates keeping coefficients and trajectory contrasts distinct.

Worked Example: Combining Level and Slope Contributions

Suppose a fitted model has $\alpha_L=0.25$ per mmol/L and $\alpha_S=-0.40$ per (mmol/L per year). At a particular time, subject A’s latent level is 2 mmol/L higher than subject B’s, but A’s slope is 0.30 mmol/L per year lower. Assume their event covariates are otherwise equal. Calculate the conditional hazard ratio for A versus B.

The level contrast is $\Delta m=2$ mmol/L, and the slope contrast is $\Delta m'=-0.30$ mmol/L per year. Therefore,

\log(HR) =(0.25)(2)+(-0.40)(-0.30) =0.50+0.12 =0.62.

Exponentiating gives $HR=\exp(0.62)\approx1.859$. Equivalently, the two factors are $\exp(0.50)\approx1.649$ for the level contrast and $\exp(0.12)\approx1.128$ for the slope contrast; their product is approximately $1.859$. Under this model, at the specified time and conditional on the modeled trajectory and covariates, A’s hazard is about 1.86 times B’s. Both contrasts contribute in the same direction to this comparison, even though the slope coefficient is negative.

When the Terms Compete

The combined link makes it possible for the level and slope terms to point in opposite directions for a particular comparison. A profile may have a higher current level, which raises its modeled hazard if $\alpha_L>0$, but a falling slope, which lowers its modeled hazard if $\alpha_S>0$. The net association depends on the size of both weighted differences, not on which feature sounds more clinically important.

Worked Example: A Higher Level Offset by a Slower Increase

Suppose $\alpha_L=0.30$ per mmol/L and $\alpha_S=0.60$ per (mmol/L per year). At a given time, profile A is 1 mmol/L above profile B but is increasing 0.50 mmol/L per year more slowly. The subjects have equal event covariates. Find the hazard ratio for A versus B and identify the competing contributions.

Here $\Delta m=1$ mmol/L and $\Delta m'=-0.50$ mmol/L per year. The level contribution is $0.30(1)=0.30$, while the slope contribution is $0.60(-0.50)=-0.30$. Consequently,

\log(HR)=0.30-0.30=0, \qquad HR=\exp(0)=1.

The fitted link assigns equal conditional hazards to these two profiles at this time, because the specified contributions exactly cancel. This does not establish that the profiles have identical risk in all respects, nor that level and slope are unimportant. It means only that their modeled association contributions sum to zero for this particular contrast.

More generally, comparisons with equal modeled association contribution satisfy $\alpha_L\Delta m+\alpha_S\Delta m'=0$. When $\alpha_S\ne0$, this can be rearranged as $\Delta m'=-(\alpha_L/\alpha_S)\Delta m$. The equation describes a trade-off in the fitted linear predictor, not a biological law. It also shows why reporting only one coefficient can misrepresent a comparison: the other term may offset or amplify it.

Assessing Distinct Contributions

The fitted coefficients are conditional on the other link term. To ask whether level contributes beyond slope, test $H_0:\alpha_L=0$ while retaining the slope term. To ask whether slope contributes beyond level, test $H_0:\alpha_S=0$ while retaining the level term. A joint test of $H_0:\alpha_L=\alpha_S=0$ asks whether the two association terms, considered together, improve on a model without either term. These are different hypotheses; a joint test does not identify which term is responsible for evidence against the null.

For an estimated coefficient vector $\hat{\boldsymbol{\alpha}}=(\hat{\alpha}_L,\hat{\alpha}_S)^\mathsf{T}$ with estimated covariance matrix $\widehat V$, a Wald statistic for the joint null is

W=\hat{\boldsymbol{\alpha}}^\mathsf{T}\widehat V^{-1}\hat{\boldsymbol{\alpha}}, \qquad W\ \dot{\sim}\ \chi^2_2 \quad\text{under }H_0.

The chi-square reference distribution is an approximation whose quality depends on adequate information and the usual regularity conditions for the fitted model. For a one-coefficient null, the corresponding Wald statistic is the squared estimate divided by its estimated variance and is compared approximately with a chi-square distribution with one degree of freedom. Likelihood-based comparisons of nested models can also assess removal of one term or both, provided the models are fitted comparably and the required regularity conditions hold.

Worked Example: A Joint Wald Test for Level and Slope

Suppose a joint model estimates $\hat{\alpha}_L=0.35$ and $\hat{\alpha}_S=-0.50$, with estimated covariance matrix

\widehat V= \begin{pmatrix} 0.0324 & -0.025\\ -0.025 & 0.0484 \end{pmatrix}.

Test the joint null that neither level nor slope is associated with the hazard. For a two-dimensional vector and covariance matrix, the quadratic form can be written as

W= \frac{\hat{\alpha}_L^2\widehat{\operatorname{Var}}(\hat{\alpha}_S) -2\hat{\alpha}_L\hat{\alpha}_S\widehat{\operatorname{Cov}}(\hat{\alpha}_L,\hat{\alpha}_S) +\hat{\alpha}_S^2\widehat{\operatorname{Var}}(\hat{\alpha}_L)} {\widehat{\operatorname{Var}}(\hat{\alpha}_L)\widehat{\operatorname{Var}}(\hat{\alpha}_S) -\widehat{\operatorname{Cov}}(\hat{\alpha}_L,\hat{\alpha}_S)^2}.

Substituting the values,

W= \frac{(0.35)^2(0.0484)-2(0.35)(-0.50)(-0.025)+(-0.50)^2(0.0324)} {(0.0324)(0.0484)-(-0.025)^2} = \frac{0.005279}{0.00094316} \approx 5.59714.

With $df=2$, the approximate upper-tail p-value is $P(\chi^2_2\ge 5.59714)=\exp(-5.59714/2)\approx0.0609$. At $\alpha=0.05$, this joint test does not reject the null. The result is not evidence that the parameters are zero; it indicates that this test does not provide sufficient evidence against their joint null at the chosen level. Separate one-parameter tests answer different questions and should be reported as such.

Why Separate Estimation Can Be Difficult

The longitudinal model generates a level and a slope for each subject and time. If subjects with high levels tend also to have high slopes, the two features are correlated in the event-risk information. The model may then have difficulty distinguishing their conditional associations: many combinations of $\alpha_L$ and $\alpha_S$ can yield similar linear predictors over the observed trajectories. This is a precision and interpretation problem, even when the software returns a converged fit.

A practical assessment should look beyond whether each coefficient has a small p-value. Examine the coefficient standard errors and covariance, the estimated correlation between coefficients, and whether conclusions about predicted hazard contrasts change materially when each term is removed. Inspect the observed and fitted trajectories to understand whether the data contain subjects with different levels at similar slopes, or different slopes at similar levels. If such contrasts are scarce, the data may support the combined prediction better than it supports attributing that prediction separately to level and slope.

The level-only alternative sets $\alpha_S=0$, and the slope-only alternative sets $\alpha_L=0$. Each is simpler, but each imposes that the omitted feature carries no additional association after accounting for the retained one. A model containing both is more flexible in this respect, but does not guarantee that the two effects are separately well estimated. Compare these specifications using the scientific question and appropriate model-based evidence; do not choose a link solely because one coefficient happens to cross a significance threshold.

Common warning signs include very large standard errors, strong coefficient covariance, estimates that change substantially when one term is removed, and fitted hazard contrasts that rely on combinations of level and slope rarely represented in the data. These may reflect limited independent trajectory variation, a poorly supported linear association form, or shortcomings in the longitudinal or event model. They are reasons to investigate the assumptions and data support, not automatic proof that the joint link is wrong.

Common pitfall: It is incorrect to interpret $\alpha_L$ as the overall association of biomarker level with the hazard when the slope term is in the model. It is the association for a one-unit level difference at the same slope. Likewise, $\alpha_S$ is the slope association at the same level. The correction is to state the conditioning explicitly and calculate a hazard ratio for the joint level-and-slope contrast of interest.

Check Your Understanding

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

If $\alpha_L=0.20$ and $\alpha_S=0.50$, what is the log hazard ratio for a contrast with $\Delta m=-1.5$ and $\Delta m'=0.40$? What is the hazard ratio?
In a model containing both terms, what does the null hypothesis $\alpha_S=0$ test?
Why can a joint test of $\alpha_L=\alpha_S=0$ fail to reject even when one separate coefficient test rejects?
Name two signs that the data may not distinguish the level and slope associations precisely.

Solutions. (1) The log hazard ratio is $0.20(-1.5)+0.50(0.40)=-0.30+0.20=-0.10$, so $HR=\exp(-0.10)\approx0.9048$. (2) It tests whether slope has an association conditional on the current level term and other modeled covariates. (3) The tests use different null hypotheses and account for the joint covariance; the joint evidence can be insufficient even when a component-specific test provides evidence against its own null. (4) Examples include large standard errors, strong covariance between coefficient estimates, and substantial changes in either estimate when the other term is removed.