Nonlinear Transformations of the Current Marker Value

Why transform the current marker value?

A linear current-value association assigns the same change in log hazard to the same absolute increase in the latent marker, wherever that increase occurs. That may be unsuitable when the association bends with marker level: an increase from a low value might matter differently from the same increase at a high value. A nonlinear transformation lets the hazard depend on the marker's current level in a specified curved way.

The key is to transform the latent marker trajectory, not to replace it with a noisy observed measurement. At time \(t\), write \(m_i(t)\) for the underlying marker value for subject \(i\). The longitudinal model describes this trajectory; the hazard model can then use a function of it. The association remains conditional on the model's other covariates and random effects.

Model and hazard-ratio contrast

A general conditional hazard specification for one such link is

Here \(h_0(t)\) is the baseline hazard, \(\mathbf w_i\) is a vector of other covariates, \(\boldsymbol\gamma\) their coefficient vector, and \(\mathbf b_i\) denotes the subject-specific random effects that help determine the latent trajectory. This is a conditional hazard: its interpretation holds given the random effects and covariates in the model. The hazard has units of inverse time; the exponential term is dimensionless.

For a marker contrast, compare two model-implied hazards at the same time, with the same other covariates and other terms held fixed. If the marker values are \(m_A\) and \(m_B\), the hazard ratio for \(B\) relative to \(A\) is

The baseline hazard and shared covariate terms cancel. This is a conditional model contrast, not automatically a population-averaged comparison and not a causal effect. If the comparison also changes covariates, their linear-predictor differences must be included in the exponent.

Square transformation: curvature and symmetry

For \(f(m)=m^2\), the hazard is proportional to \(\exp\{\alpha m_i(t)^2\}\), apart from the baseline and other covariates. The finite contrast is

Unlike a linear current-value link, a square link does not generally give the same hazard ratio for equal marker increments at different starting levels. Its shape is also symmetric around zero: \(m\) and \(-m\) have the same transformed value. Thus, for a marker that can be negative, the model associates the same hazard with equal-magnitude positive and negative values, all else equal. If the marker is strictly positive, the square is increasing in the marker, but the change in its value grows as the marker increases.

The parameter \(\alpha\) has units reciprocal to the squared marker units, so \(\alpha m^2\) is dimensionless. Its sign determines whether larger squared values raise or lower the conditional hazard. For a local change around marker value \(m\), the derivative of the log hazard with respect to the marker is \(2\alpha m\); the direction of association can therefore vary across zero.

Log transformation: relative contrasts and domain

A log link is \(f(m)=\log(m/c)\), where \(m>0\) and \(c>0\) is a fixed reference value in the same units as \(m\). The ratio for two positive marker values is

The reference value makes the logarithm's argument dimensionless. It cancels from a contrast, so changing \(c\) does not change this hazard ratio when the same \(c\) is used for both values. The log link gives a relative, rather than absolute, marker contrast: doubling the marker produces the same ratio at different starting values, provided both are in the model's valid range. The parameter \(\alpha\) is dimensionless.

Positivity is a substantive requirement, not a computational detail. If the latent marker can be zero or negative, \(\log(m/c)\) is undefined there. Adding an arbitrary constant to force positivity changes the model and its interpretation; it is not a harmless repair. The transformation must be scientifically defensible over the trajectory values used while subjects remain at risk.

What the association parameter does and does not say

The value of \(\alpha\) alone does not describe a universal hazard ratio for a nonlinear link. The relevant quantity is \(\alpha[f(m_B)-f(m_A)]\), which depends on both marker values and the chosen transformation. In particular, a positive \(\alpha\) means higher transformed values correspond to a higher conditional hazard; it does not necessarily mean that every increase in the original marker raises the hazard.

A short derivative makes this distinction explicit. At fixed time and covariates, the marker-dependent part of the log hazard is \(\alpha f(m)\). For a small marker change \(\Delta m\), its first-order change is approximately \(\alpha f'(m)\Delta m\). Thus the local log-hazard change per marker unit is \(\alpha f'(m)\). For the square link it is \(2\alpha m\); for the log link it is \(\alpha/m\). These derivatives describe local sensitivity, while the finite hazard-ratio formula is the appropriate calculation for a specified, non-small contrast.

This identity follows by subtracting the two log hazards: terms held fixed, including \(\log h_0(t)\) and the covariate contribution, cancel. Exponentiating the remaining difference gives the hazard ratio. The derivation also shows why the contrast must keep the comparison order consistent: reversing \(A\) and \(B\) reciprocates the ratio.

Assumptions, diagnostics, and alternatives

A nonlinear current-value link is useful only if its assumptions are plausible. The longitudinal model must provide a meaningful latent trajectory on the scale being transformed; the hazard must depend on that current value through the specified \(f\); and the conditional hazard specification must be adequate given its baseline hazard, covariates, and random-effects structure. The marker contrast formula additionally assumes that the comparison holds those other terms fixed. Joint-model estimation accounts for uncertainty in the latent trajectory under the specified model, but it cannot make an incorrect transformation correct.

Useful checks focus on the restrictions the transformation imposes. Examine whether observed longitudinal measurements and fitted latent trajectories support the scale and range used in the hazard link. Assess whether the assumed transformed association is compatible with the event data, and whether conclusions depend heavily on a few subjects or on extrapolated trajectory values. For a log link, verify that trajectories are positive over the relevant at-risk times; values at or near zero are a warning, and negative values invalidate the link. For a square link, ask whether equal-magnitude positive and negative marker values should plausibly have equal transformed effects. For either link, inspect uncertainty in the association estimate and in model-implied contrasts rather than relying only on a point estimate.

A linear current-value link is simpler and imposes a constant log-hazard change per absolute marker unit. A square link permits curvature but imposes symmetry around zero. A log link is suited to relative contrasts but requires positive values. These forms answer different scientific questions; none is automatically preferable. Compare candidate specifications using appropriate model fit and predictive checks, and consider whether the observed marker range can distinguish their shapes. A complicated transformation can be weakly identified when there are few events or little information at the extremes.

A common mistake is to use a measured value \(y_{ij}\) in place of the latent value \(m_i(t)\), then interpret the resulting contrast as the joint model's current-value association. That reasoning ignores measurement error and may give a different, attenuated or otherwise distorted association. The correction is to define the hazard link in terms of the latent trajectory and use the observed measurements to inform that trajectory through the longitudinal model. Another mistake is to say that a positive square-link coefficient makes every marker increase raise the hazard: the local slope is \(2\alpha m\), so its sign depends on \(m\) as well as \(\alpha\).