Combining Current Value and Slope

Why Combine Marker Value and Slope?

A subject’s longitudinal marker at time $t$ describes the current level of the marker. Its slope describes how quickly that level is changing at that time. These features can convey different information: two subjects may have the same marker value but different rates of change, or different current values but similar slopes. A joint model can allow both features to contribute to the event hazard.

The central task is to define the hazard link precisely and derive the comparison it implies. In particular, the two association terms act together on the log hazard. Their contributions add on that scale; on the hazard-ratio scale, the corresponding multipliers combine multiplicatively.

Definition: Let $m_i(t)$ be subject $i$’s latent marker trajectory, and let $m_i'(t)=\frac{d}{dt}m_i(t)$ be its instantaneous slope. A current-value-and-slope association model specifies the conditional hazard as $$ h_i(t\mid m_i,w_i)=h_0(t)\exp\left\{\beta^\top w_i+\alpha_v m_i(t)+\alpha_s m_i'(t)\right\}. $$ Here $h_0(t)$ is a common baseline hazard, $w_i$ is a vector of covariates, $\beta$ is its coefficient vector, $\alpha_v$ is the current-value association parameter, and $\alpha_s$ is the slope association parameter. The hazard is conditional on the latent trajectory and covariates, not simply on a single noisy observed marker value.

This specification has several substantive assumptions. Conditional on the trajectory and covariates, the hazard has the proportional-hazards form: the baseline hazard is multiplied by an exponential linear predictor. The value and slope enter additively and linearly on the log-hazard scale, with coefficients that do not vary with time. The trajectory must be differentiable at the times where its slope is used. Unless an interaction is included, the value association does not depend on the slope, and the slope association does not depend on the current value.

The model is a statement about a conditional association, not a causal effect. Its interpretation also depends on the units and scale of the marker, the time unit used for the slope, and the assumptions used to represent the latent trajectory.

Deriving the Hazard Contrast

Compare subjects $A$ and $B$ at the same time $t$. They share the same baseline hazard $h_0(t)$, as specified in the model. Dividing their hazards cancels that baseline term. Taking the logarithm of the resulting ratio shows exactly how each difference contributes.

\begin{aligned} \frac{h_A(t)}{h_B(t)} &= \exp\Bigl\{\beta^\top(w_A-w_B) +\alpha_v\bigl[m_A(t)-m_B(t)\bigr] +\alpha_s\bigl[m_A'(t)-m_B'(t)\bigr]\Bigr\},\\ \log\left\{\frac{h_A(t)}{h_B(t)}\right\} &= \beta^\top(w_A-w_B) +\alpha_v\bigl[m_A(t)-m_B(t)\bigr] +\alpha_s\bigl[m_A'(t)-m_B'(t)\bigr]. \end{aligned}

If the subjects have the same covariates, the covariate term is zero. A difference in current marker value alone, $\Delta m$, then corresponds to a hazard ratio $\exp(\alpha_v\Delta m)$. A difference in slope alone, $\Delta m'$, corresponds to $\exp(\alpha_s\Delta m')$. When both differ, the hazard ratio is $\exp(\alpha_v\Delta m+\alpha_s\Delta m')$, equivalently the product $\exp(\alpha_v\Delta m)\exp(\alpha_s\Delta m')$.

These are comparisons of instantaneous hazards at a given time among subjects with the stated covariates and latent trajectory features. A hazard ratio is not a ratio of event probabilities over a follow-up period. If covariates differ, their contribution must be retained; it is not valid to attribute the whole contrast to marker value and slope.

Units and interpretation: If the marker is measured in units $U$, then $\alpha_v$ has units $1/U$, so $\alpha_v m_i(t)$ is dimensionless. If time is measured in units $T$, the slope has units $U/T$, and $\alpha_s$ has units $T/U$. Thus $\alpha_s m_i'(t)$ is also dimensionless. Changing the marker or time units changes the numerical coefficient needed to express the same association.

Worked Contrasts

Worked Example: Combining the Terms for One Subject

Suppose, hypothetically, that at a particular time the common baseline hazard is $0.0200$ per month, and the covariate contribution $\beta^\top w_i$ is zero. Let $\alpha_v=0.300$ per marker unit and $\alpha_s=-0.0800$ per marker unit per month. A subject’s latent marker value is $6.00$ units, and its slope is $0.500$ marker units per month. Calculate the subject’s hazard and the multiplier relative to a subject with marker value and slope both equal to zero, with the same covariates.

The combined association contribution to the log hazard is $$ 0.300(6.00)+(-0.0800)(0.500)=1.800-0.0400=1.760. $$

The hazard multiplier relative to the stated reference subject is therefore $$ \exp(1.760)=5.812. $$ The hazard is $$ 0.0200\ \text{per month}\times 5.812=0.1162\ \text{per month}. $$

Under this hypothetical model, the subject’s conditional instantaneous hazard is $5.812$ times the reference hazard at that time. The negative slope coefficient slightly reduces the multiplier: the value term alone gives $\exp(1.800)=6.050$, while the slope term multiplies it by $\exp(-0.0400)=0.9608$. This is a hazard comparison, not a statement that the subject is $5.812$ times as likely to experience the event over a specified period.

Worked Example: Subjects Differ Only in Slope

Consider two hypothetical subjects with the same covariates and the same latent marker value at time $t$. Their slopes are $0.200$ and $0.600$ marker units per month. Suppose $\alpha_s=0.250$ per marker unit per month. Find the conditional hazard ratio for the subject with the larger slope relative to the other.

Because the values and covariates match, those terms cancel in the contrast. The slope difference is $0.600-0.200=0.400$ marker units per month, so $$ \log(HR)=0.250(0.400)=0.100, \qquad HR=\exp(0.100)=1.105. $$

At that time, the subject with the larger slope has a conditional instantaneous hazard estimated to be $1.105$ times that of the other subject, or about $10.5\%$ higher, under the specified model. The interpretation is conditional on equal current marker value and equal covariates; it does not say that a larger slope causes the event.

Worked Example: Contrasting Subjects Who Differ in Both Features

Suppose time is measured in years and two hypothetical subjects have latent trajectories $$ m_A(t)=6+1.4t-0.10t^2, \qquad m_B(t)=5+0.6t-0.05t^2. $$ At $t=2$ years, calculate the hazard ratio for $A$ relative to $B$, assuming equal covariates, $\alpha_v=0.200$ per marker unit, and $\alpha_s=-0.300$ per marker unit per year.

First evaluate the values and their derivatives. Since $m_A'(t)=1.4-0.20t$ and $m_B'(t)=0.6-0.10t$, at $t=2$ the values are $m_A(2)=8.4$ and $m_B(2)=6.0$, while the slopes are $m_A'(2)=1.0$ and $m_B'(2)=0.4$, in marker units per year. Thus $\Delta m=2.4$ marker units and $\Delta m'=0.6$ marker units per year.

The log hazard ratio and hazard ratio are $$ \log(HR)=0.200(2.4)+(-0.300)(0.6)=0.480-0.180=0.300, \qquad HR=\exp(0.300)=1.350. $$

The model therefore gives subject $A$ a conditional instantaneous hazard $1.350$ times that of subject $B$ at year 2, or $35.0\%$ higher. The positive value contribution is partly offset by the negative slope contribution. Calculating only the value contrast would give $\exp(0.480)=1.616$, which does not represent the specified combined model.

Assumptions, Diagnostics, and Alternatives

The formula for the contrast follows directly from the model, but its usefulness depends on whether the model describes the data adequately. The linear value-and-slope link assumes that equal changes in marker value have equal effects on log hazard across the range considered, and likewise for equal changes in slope. It also assumes that these associations are constant over time on the log-hazard scale. These restrictions may be poor approximations in some applications.

Assessment should consider whether the fitted model captures the event-time pattern and whether its assumed trajectory gives plausible values and derivatives over the observed follow-up. In particular, slopes can be sensitive to how the trajectory is represented and to sparse or noisy measurements. A rapidly changing estimated slope based on little information should not automatically be treated as a precisely known subject feature. In a joint analysis, uncertainty in the latent trajectory and in the event model matters; treating estimated trajectories as error-free can understate uncertainty.

Useful warning signs include systematic mismatch between observed and model-implied event patterns, implausible fitted trajectories or derivatives, and evidence that the marker association changes over time in a way the constant coefficients cannot capture. A strong association between the value and slope features can also make their separate coefficients difficult to interpret: the model’s contrast remains defined, but the data may provide limited information for separating the two terms. Model checks can reveal concerns, but no single diagnostic proves the assumptions true.

Association specification	What it allows the hazard to depend on	Main limitation
Current-value only	The latent marker level at time $t$	Does not represent a separate association with the rate of change.
Slope only	The instantaneous rate of marker change at time $t$	Does not represent a separate association with the current level.
Current value and slope	Both features, with additive contributions to log hazard	Requires the linear, time-constant association assumptions for both terms.

The simpler specifications may be appropriate when supported by the scientific question and data, but omitting a relevant feature can leave the intended association unrepresented. Conversely, adding both terms does not guarantee a better-supported model: it uses more information from the data and may be difficult to estimate when the trajectory features are strongly related. This specification also does not represent an accumulated effect of the marker history; its terms describe the value and slope at the current time.

Common Pitfall

A common error is to add the two hazard ratios, or to interpret $\alpha_s$ as the hazard ratio for a one-unit increase in slope. Neither is correct. The coefficients are changes in the log hazard per unit change in their respective features. For changes $\Delta m$ and $\Delta m'$, the log-hazard contrast is $\alpha_v\Delta m+\alpha_s\Delta m'$, and exponentiating that sum gives the hazard ratio. For example, the separate multipliers $1.616$ and $\exp(-0.180)=0.8353$ in the third example combine as $1.616\times0.8353$, approximately $1.350$; they do not add.

A second practical caution is to keep the comparison time and units explicit. A slope of $0.5$ units per month is not numerically interchangeable with $0.5$ units per year. The subject contrast must use values and slopes evaluated at the same time and expressed on the same scales.

Key takeaway: In a current-value-and-slope link, the log hazard contains $\alpha_v m_i(t)+\alpha_s m_i'(t)$. For a same-time comparison, multiply each coefficient by the corresponding between-subject difference, add those contributions (and any covariate contrast), then exponentiate to obtain the conditional hazard ratio.

Check Your Understanding

Answer each question, then check the brief solutions below.

With equal covariates, $\alpha_v=0.10$ per unit, and a value difference of 3 units but no slope difference, what is the hazard ratio?
With $\alpha_s=0.20$ per unit per year, what is the log-hazard contrast for a slope difference of $-0.5$ units per year, if all other terms cancel?
If the value contribution to the log hazard is $0.4$ and the slope contribution is $-0.1$, what is the combined hazard multiplier?
Why must two subjects’ slopes be evaluated at the same time for the stated contrast?
State one limitation of the current-value-and-slope specification.

Solutions. (1) $HR=\exp(0.10\times3)=\exp(0.30)=1.350$. (2) The log-hazard contrast is $0.20(-0.5)=-0.100$; the corresponding ratio is $\exp(-0.100)=0.9048$. (3) Add on the log scale and exponentiate: $HR=\exp(0.4-0.1)=\exp(0.3)=1.350$. (4) The model compares the value and instantaneous slope features at a common time; slopes can change over time, so comparing slopes from different times would not be the specified same-time hazard contrast. (5) For example, it assumes constant, linear effects of value and slope on log hazard; this may not describe a time-varying or nonlinear association.

Association specification	What it allows the hazard to depend on	Main limitation
Current-value only	The latent marker level at time \(t\)	Does not represent a separate association with the rate of change.
Slope only	The instantaneous rate of marker change at time \(t\)	Does not represent a separate association with the current level.
Current value and slope	Both features, with additive contributions to log hazard	Requires the linear, time-constant association assumptions for both terms.