Why Use a Threshold or Piecewise-Linear Link?
A linear current-value link assigns the same change in log hazard to each one-unit increase in the latent biomarker, whatever its starting value. That can be clinically unappealing when risk is thought to change most after a threshold is crossed, or when the association differs across clinically defined ranges. A threshold link represents a jump at a cut point; a piecewise-linear link allows the rate of change in log hazard to differ between ranges.
These are alternative ways to describe the association between a patient's latent biomarker trajectory and the event hazard. They do not change the longitudinal model itself. To keep the comparison focused, hold the longitudinal model, covariates, baseline hazard specification, and other event-model components fixed while changing the biomarker link.
where \(h_0(t)\) is the baseline hazard, \(w_i\) is a vector of event-model covariates, \(\gamma\) is its coefficient vector, and \(\mathcal{M}_i\) represents the individual's longitudinal information used to infer the latent trajectory. The hazard is conditional on the modeled information; the association is not, by itself, evidence that changing the biomarker would cause a change in event risk.
Two Different Meanings of a Threshold
A step threshold separates values into two groups. For a cut point \(c\), define \(I\{m_i(t)>c\}\) to be 1 when the biomarker is above \(c\) and 0 otherwise. The link can be written
The coefficient \(\alpha\) is a log hazard contrast for being above rather than at or below \(c\), with other modeled quantities held fixed. The corresponding hazard ratio is \(\exp(\alpha)\). This model makes no distinction between values within either group: for example, \(c+0.1\) and \(c+5\) receive the same contribution to the log hazard. The convention at the boundary matters; here, a value exactly equal to \(c\) belongs to the lower group.
A continuous threshold, often written as a hinge or positive-part link, instead permits the association to begin or change after the cut point without an abrupt jump in the hazard. Define \((x)_+=\max(x,0)\). One simple link is
Below or at \(c\), this link contributes zero. Above \(c\), each additional biomarker unit changes the log hazard by \(\alpha\). The link is continuous at \(c\), but its slope changes there. A step threshold and a hinge therefore answer different scientific questions: does risk differ between groups, or does the rate of association change after a boundary?
Worked Example: Interpreting a Step Threshold
Suppose a hypothetical joint model uses a biomarker measured in mmol/L and the step link \(f(m)=\alpha I\{m>10\}\), with an estimated \(\alpha=\log(1.80)=0.588\). Compare the modeled hazards at values of 9.4 and 10.6 mmol/L for otherwise identical individuals at the same time.
At 9.4 mmol/L, the indicator is 0; at 10.6 mmol/L, it is 1. The log hazard contrast is therefore \(0.588(1-0)=0.588\), and the hazard ratio is
Under this model, the hazard at 10.6 mmol/L is 1.80 times the hazard at 9.4 mmol/L, conditional on the other modeled quantities. The same ratio would apply to any pair of values on opposite sides of 10 mmol/L. That is a strong structural restriction, not a claim that the data show a sharp biological jump exactly at 10.
Continuous Piecewise-Linear Links
A continuous piecewise-linear link allows a different slope in each biomarker range while avoiding a jump in the link at the breakpoints. With one breakpoint \(c\), use
For \(m\le c\), the slope of \(f(m)\) is \(s_1\). For \(m>c\), the slope is \(s_1+\delta\). The coefficient \(\delta\) is the change in slope at \(c\), not the slope above \(c\) itself. Since the positive-part term is zero at \(c\), both sides meet at \(f(c)=\beta_0\).
For a clinically meaningful middle range bounded by \(c_L\) and \(c_U\), a convenient two-knot specification is
The slopes below \(c_L\), between \(c_L\) and \(c_U\), and above \(c_U\) are, respectively, \(s_1\), \(s_1+\delta_L\), and \(s_1+\delta_L+\delta_U\). Thus, \(\delta_L\) and \(\delta_U\) describe changes in slope at the lower and upper breakpoint. The units of each slope are log hazard per biomarker unit. These coefficients describe the shape of the link; a hazard ratio for a particular contrast is obtained from the difference in the link values, not by exponentiating a slope without specifying a biomarker change.
Worked Example: A Contrast Across One Breakpoint
Consider a hypothetical continuous link with biomarker units in mmol/L, breakpoint \(c=8\), lower-range slope \(s_1=0.08\), and slope change \(\delta=0.17\). Compare values \(m=7\) and \(m=10\), holding time and other modeled covariates fixed.
From 7 to 8, the link increases by \(0.08(8-7)=0.08\). Above 8, the slope is \(0.08+0.17=0.25\), so from 8 to 10 it increases by \(0.25(10-8)=0.50\). The total log hazard contrast is \(0.08+0.50=0.58\), giving
The modeled hazard at 10 mmol/L is about 1.79 times that at 7 mmol/L. The hazard ratio from 8 to 10 alone is \(\exp\{0.25(2)\}=\exp(0.50)=1.649\). The different contrasts illustrate why it is not enough to report the slope above the breakpoint: the biomarker distance and the path across any other segments also matter.
Deriving Hazard Contrasts Across Ranges
A hazard ratio between two biomarker values follows directly from the hazard model. Consider two otherwise identical modeled states at the same time, with biomarker values \(m_a\) and \(m_b\). The common baseline hazard and covariate contribution cancel in their ratio:
For a piecewise-linear link, divide the interval from \(m_a\) to \(m_b\) at each breakpoint it crosses. On each resulting segment the change in \(f\) equals that segment's slope multiplied by the biomarker distance within it. Adding those changes gives \(f(m_b)-f(m_a)\); exponentiating gives the hazard ratio. This calculation also makes the interpretation's scope explicit: it is a modeled, conditional contrast, not an absolute event probability.
Worked Example: A Contrast Across Two Clinical Breakpoints
Suppose a hypothetical two-knot link has \(c_L=6\) and \(c_U=10\) mmol/L. The slopes below, within, and above the clinical range are \(0.05\), \(0.20\), and \(0.10\) per mmol/L, respectively. Compare \(m_a=5\) with \(m_b=12\) mmol/L.
The contrast crosses all three segments. Its log hazard difference is
Therefore, the hazard ratio is \(\exp(1.05)=2.858\). Under this fitted link, the hazard at 12 mmol/L is about 2.86 times that at 5 mmol/L, conditional on the other modeled quantities. The calculation uses each segment's slope only over the biomarker distance that lies in that segment.
Assessing Dependence on the Cut Point
A cut point can be chosen from clinical knowledge, a measurement convention, or exploratory analysis. If it is not firmly established in advance, a substantive conclusion should not rest on one convenient value. Fit the same model with a small set of clinically plausible cut points, changing only the cut point and its associated link specification. Then compare a clearly defined contrast that is meaningful across fits, such as the modeled hazard ratio between two fixed biomarker values. For a step link, also report the estimated cut-point coefficient and its uncertainty; for a piecewise-linear link, inspect the segment slopes and contrasts that cross the knots.
Worked Example: Checking a Step-Threshold Cut Point
Suppose a hypothetical analysis considers step thresholds of 8, 9, and 10 mmol/L. The fitted coefficients are \(0.47\), \(0.55\), and \(0.38\), respectively. Each coefficient is the log hazard contrast for being above that model's cut point. The estimated hazard ratios are
All three estimates point toward a higher modeled hazard above the selected threshold, but their sizes vary: the estimates range from 1.46 to 1.73. This supports a directionally similar association across these specifications, while showing that its estimated magnitude depends on the cut point. It does not establish that the thresholds are interchangeable or that any one is biologically correct. In an actual analysis, uncertainty intervals and the clinical plausibility of each threshold are also needed to judge how informative these differences are.
A practical sensitivity check asks whether the conclusion changes in direction, magnitude, or clinical interpretation over plausible specifications. A sign that reverses, a large change in a clinically important hazard contrast, or a breakpoint that moves the apparent association into a different range deserves attention. Small differences in estimates alone do not prove sensitivity, particularly when estimates are imprecise.
Assumptions, Diagnostics, and Alternatives
These links are meaningful only under the assumptions of the joint model in which they are used. The longitudinal component must represent the latent biomarker trajectory adequately; otherwise, a threshold pattern in the event association can reflect a misspecified trajectory rather than a genuine change in the association. The event model must correctly specify how the link enters the hazard, and censoring must be handled under the model's assumptions. The hazard contrasts above additionally compare states at the same time and with the same other modeled covariates.
Useful checks include plotting the fitted link over the observed biomarker range, marking the cut points, and checking how much data support each segment. A breakpoint with few observations on one side can yield an unstable segment slope or threshold contrast. Also check whether the fitted shape is driven by a few extreme values and whether proposed cut points are clinically defensible. A plot of the link is a diagnostic aid, not proof that the chosen shape is correct.
| Link choice | What it represents | Main limitation |
|---|---|---|
| Step threshold | A constant contrast above versus at or below a cut point | Discards within-group differences and imposes a jump |
| Continuous hinge or piecewise-linear link | A continuous association with different slopes across ranges | Results depend on knot locations and may be weakly supported in sparse ranges |
| Linear current-value link | A single constant slope over the biomarker range | Cannot represent a change in slope or a threshold-specific contrast |
The linear link is a useful simpler alternative when a constant slope is scientifically plausible and the data do not support multiple ranges. A threshold or segmented link can make a clinical contrast easier to communicate, but it uses more structure and may be sensitive to the chosen cut point. More flexible nonlinear links are another alternative; they avoid fixing a small number of straight-line segments but can be harder to summarize and still require adequate data over the biomarker range. No link is automatically preferred simply because it looks more clinically familiar.
Common Mistakes and Limitations
- Confusing a step with a change in slope. A step link changes abruptly at \(c\); a hinge link remains continuous and changes slope. Choose the form that matches the scientific contrast being asked.
- Exponentiating a slope as if it were a hazard ratio. A slope is a log hazard change per biomarker unit. For a change of \(d\) units within one segment, the hazard ratio is \(\exp(sd)\), where \(s\) is that segment's slope.
- Assuming a cut point is discovered because it was fitted. A fitted threshold is conditional on the specified model and candidate cut point. Clinical justification and sensitivity to plausible alternatives remain important.
- Interpreting a conditional hazard ratio as a risk ratio or causal effect. The contrast concerns instantaneous hazard under the model, conditional on its covariates and latent trajectory; it is not a ratio of cumulative event probabilities and does not establish causation.
Threshold links also simplify a potentially gradual biological process. A genuine association may be smooth rather than discontinuous or segmented, and a clinically meaningful boundary may not correspond to a sharp change in risk. Sparse measurements or limited event information near a cut point can make apparent range-specific patterns difficult to distinguish from uncertainty. Report the link definition, cut point units, and the biomarker contrast used so the interpretation can be checked.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A step link has \(\alpha=\log(1.5)\). What is the modeled hazard ratio for a value above the cut point versus a value at or below it?
- A hinge link has breakpoint \(c=4\), lower slope \(0.10\), and slope change \(0.20\). What is the slope above the breakpoint?
- Using the link in Question 2, calculate the hazard ratio for a change from 3 to 6 biomarker units.
- In a sensitivity analysis, the threshold estimate is positive at three plausible cut points but its hazard ratio varies from 1.2 to 2.0. What can be concluded, and what cannot be concluded from this information alone?
Solutions. (1) The hazard ratio is \(\exp\{\log(1.5)\}=1.5\). (2) The slope above the breakpoint is \(0.10+0.20=0.30\) log hazard units per biomarker unit. (3) From 3 to 4, the link increases by \(0.10(1)=0.10\); from 4 to 6, it increases by \(0.30(2)=0.60\). The total is \(0.70\), so the hazard ratio is \(\exp(0.70)=2.014\). (4) The estimated direction is consistent across those fits, while the estimated magnitude varies. Without uncertainty intervals, sample support near the cut points, and clinical context, one cannot determine whether the differences are meaningful or identify a uniquely correct cut point.