Cumulative Total Versus Average Marker Exposure

Why Distinguish Total Exposure from Average Exposure?

A cumulative marker association can describe how a participant’s accumulated marker history relates to the event hazard. There are two closely related summaries of that history: the integral of the marker trajectory up to time $t$, and that integral divided by the elapsed time. The first is a cumulative total; the second is a time-normalized average. Although one is obtained from the other by dividing by time, their hazard associations do not generally have the same interpretation.

The distinction matters because a total increases with the amount of exposure accumulated over time, whereas an average describes the mean marker level over the interval. The introductory material on cumulative association defines the total-exposure link and its hazard-ratio interpretation. Here the focus is how substituting the average changes the coefficient, its units, and the comparison between trajectories.

Definition: Let $m_i(s)$ be the latent marker trajectory for participant $i$, measured in marker units, at time $s$. For an integrable trajectory and $t>0$, define cumulative exposure and average exposure through time $t$ by $$ A_i(t)=\int_0^t m_i(s)\,ds, \qquad \bar m_i(t)=\frac{1}{t}\int_0^t m_i(s)\,ds=\frac{A_i(t)}{t}. $$ If time is measured in years, $A_i(t)$ has units of marker-years and $\bar m_i(t)$ has units of marker.

These definitions concern the underlying trajectory used in the joint model, not merely a sum of observed measurements. The integral must exist over the interval of interest. At $t=0$, the average as written is undefined; the comparisons below concern $t>0$. If the trajectory is continuous at zero, a limiting average may be defined, but that boundary detail does not change the distinction at positive times.

Two Hazard Links and Their Coefficients

For clarity, consider two alternative conditional hazard models. Let $h_0(t)$ be the baseline hazard and let $w_i$ denote other covariates, with coefficient vector $\gamma$. The models have the same baseline and covariates; they differ only in which summary of the marker trajectory enters the log hazard.

h_i(t\mid b_i)=h_0(t)\exp\left\{\gamma^{\mathsf T}w_i+\alpha_A A_i(t)\right\}

h_i(t\mid b_i)=h_0(t)\exp\left\{\gamma^{\mathsf T}w_i+\alpha_{\bar m}\bar m_i(t)\right\}

Here $b_i$ denotes the participant-specific latent effects used to define the marker trajectory, and the hazard is conditional on that trajectory and the included covariates. Each model assumes a linear contribution from its chosen exposure summary on the log-hazard scale. The models need not describe the same association: their coefficients are attached to quantities with different time behavior and different units.

Interpretation: In the total-exposure model, $\alpha_A$ is the change in log hazard per one-unit difference in cumulative exposure, holding the comparison time and other covariates fixed. In the average-exposure model, $\alpha_{\bar m}$ is the change in log hazard per one-unit difference in average marker level, under the same conditions. Exponentiating either coefficient gives the corresponding hazard ratio for a one-unit difference in its own exposure summary.

If marker values are measured in marker units and time in years, then $\alpha_A$ has units of inverse marker-years, while $\alpha_{\bar m}$ has units of inverse marker. These units are not interchangeable. A coefficient’s numerical value also depends on the chosen units: measuring time in months rather than years changes the numerical cumulative total and therefore the coefficient needed to describe the same hazard association.

Why the Coefficients Do Not Usually Match

The definitions imply $A_i(t)=t\bar m_i(t)$. At a particular fixed time $t$, the total-exposure model’s marker term can therefore be written as $\alpha_A t\bar m_i(t)$. This gives the precise relationship between the two links at that time.

\alpha_A A_i(t)=\alpha_A t\bar m_i(t)

Thus, at one fixed $t$, the same log-hazard contribution is obtained from an average-exposure term if $\alpha_{\bar m}=t\alpha_A$. This is a time-specific conversion, not generally one constant coefficient that makes the two models equivalent across all event times. If $t$ varies, the conversion factor $t$ varies too. The models represent different assumptions unless the setting or parameterization supplies a reason for that time dependence.

Justification by hazard ratios: Compare two participants with the same covariates at the same time $t>0$. Write their differences as $\Delta A(t)=A_1(t)-A_0(t)$ and $\Delta\bar m(t)=\bar m_1(t)-\bar m_0(t)$. Since $\Delta A(t)=t\Delta\bar m(t)$, their hazard ratios are $$ \mathrm{HR}_A(t)=\exp\{\alpha_A t\Delta\bar m(t)\}, \qquad \mathrm{HR}_{\bar m}(t)=\exp\{\alpha_{\bar m}\Delta\bar m(t)\}. $$ They agree at that time when $\alpha_{\bar m}=t\alpha_A$, provided the comparison in average exposure is nonzero. A fixed $\alpha_{\bar m}$ cannot satisfy this relation at every $t$ when $\alpha_A\ne 0$.

The fixed-time qualification is important. Hazard ratios compare hazards at a specified time among participants with the stated exposure histories and covariates; they are not comparisons of event probabilities over an entire follow-up period. Neither coefficient, by itself, establishes a causal effect. The interpretation is conditional on the model and its assumptions.

Worked Example: Computing Both Summaries from a Hypothetical Trajectory

Suppose a hypothetical participant has marker values measured in marker units. The marker is 4 units from year 0 to year 2, then 6 units from year 2 to year 5. Compare this trajectory with a hypothetical participant whose marker remains at 4 units through year 5. Assume both are at risk at year 5 and have the same other covariates. The piecewise values make the integrals exact, and both trajectories are integrable.

For the first participant, the cumulative total at year 5 is $A_1(5)=(4\ \text{units})(2\ \text{years})+(6\ \text{units})(3\ \text{years})=8+18=26\ \text{unit-years}$. The time-normalized average is $\bar m_1(5)=26\ \text{unit-years}/5\ \text{years}=5.2\ \text{units}$.

For the comparison participant, $A_0(5)=(4\ \text{units})(5\ \text{years})=20\ \text{unit-years}$, and $\bar m_0(5)=20/5=4\ \text{units}$. The differences are $\Delta A(5)=6\ \text{unit-years}$ and $\Delta\bar m(5)=1.2\ \text{units}$; indeed, $5(1.2)=6$.

Suppose the fitted total-exposure coefficient is $\alpha_A=0.0300$ per unit-year. The total-link hazard ratio is $\exp\{0.0300(6)\}=\exp(0.180)=1.197$. Under that model, the first participant’s hazard at year 5 is estimated to be about 1.20 times the comparison participant’s hazard, conditional on the stated covariates and model.

If instead an average-link model has $\alpha_{\bar m}=0.100$ per unit, its hazard ratio for this comparison is $\exp\{0.100(1.2)\}=\exp(0.120)=1.127$. The different results do not indicate an arithmetic contradiction: the two coefficients specify different links. At year 5, matching the total-link term would require $\alpha_{\bar m}=5(0.0300)=0.150$ per unit, not 0.100.

What the Definitions Emphasize

The total and average can rank histories differently in their practical meaning, even though at the same fixed time one is a positive constant multiple of the other. A total directly retains duration: a sustained marker level accumulates exposure as time passes. An average divides away elapsed time and summarizes the mean level over the observed interval. For two histories evaluated at the same $t$, their totals and averages have the same ordering; for histories with different durations, the total also reflects that duration.

Worked Example: Same Average, Different Durations

Consider two hypothetical marker histories, both measured in marker units. Participant A has a constant marker value of 3 units through year 2; participant B has the same constant value through year 6. Assume each is at risk at the end of their stated interval. The trajectory is integrable in both cases.

For A, $A_A(2)=3(2)=6$ unit-years and $\bar m_A(2)=6/2=3$ units. For B, $A_B(6)=3(6)=18$ unit-years and $\bar m_B(6)=18/6=3$ units.

An average-exposure link assigns the same marker term, $\alpha_{\bar m}(3)$, to each history at its respective endpoint. A total-exposure link assigns terms $6\alpha_A$ and $18\alpha_A$. For illustration, if $\alpha_A=0.0400$ per unit-year, the terms are 0.240 and 0.720. Their hazard multipliers relative to a zero-total reference with the same other covariates are $\exp(0.240)=1.271$ and $\exp(0.720)=2.054$, respectively.

These are comparisons at different times, so the multipliers should not be interpreted as a hazard ratio between A and B at one common time. The example instead shows what each definition retains: the average records the common level, while the cumulative total records both level and elapsed duration.

Coefficient Conversion at a Fixed Time

A useful conversion can be made for a specific time point, but it should not be mistaken for a universal conversion over follow-up. The following example keeps the average contrast fixed while changing the comparison time. Both trajectories are defined over the intervals used, their integrals exist, and the participants are compared at the same time within each calculation.

Worked Example: Why a Fixed Coefficient Conversion Depends on Time

Suppose the difference in average marker exposure between two hypothetical participants is 1.5 units at each comparison time. Let $\alpha_A=0.0250$ per unit-year. At year 4, the cumulative-exposure difference is $4(1.5)=6.0$ unit-years. The total-link log hazard ratio is $0.0250(6.0)=0.150$, so the hazard ratio is $\exp(0.150)=1.162$.

At year 2, the same average-exposure difference corresponds to a cumulative difference of $2(1.5)=3.0$ unit-years. The total-link log hazard ratio is $0.0250(3.0)=0.0750$, giving $\exp(0.0750)=1.078$.

To match the year-4 total-link comparison with an average link, one would use $\alpha_{\bar m}=4(0.0250)=0.100$ per unit. At year 2, matching instead requires $\alpha_{\bar m}=2(0.0250)=0.0500$ per unit. There is no single fixed average-link coefficient that performs both conversions in this example. The calculations are conditional hazard comparisons; they do not state that either marker history causes the observed hazard difference.

Assumptions, Diagnostics, and Common Errors

Before interpreting either association, check that the marker trajectory and time scale have been defined consistently, that the integral is meaningful on the interval, and that the chosen summary is the one intended scientifically. The hazard model also assumes the specified log-linear contribution: a one-unit change in the included summary has the same coefficient on the log-hazard scale, conditional on the other modeled terms. In the average link this is a constant coefficient per marker unit; in the total link it is a constant coefficient per marker-year. The conditional hazard model and the fitted longitudinal model must both be adequate for their intended roles.

Useful checks include verifying the marker and time units in the data and model code, confirming that averages are computed using elapsed time rather than a count of irregularly spaced visits, and examining whether the chosen link gives a plausible pattern of conditional hazard differences over follow-up. A coefficient with unexpected magnitude may reflect a unit mismatch or a scaling error rather than strong evidence of association. Model fit and residual diagnostics should be considered in the context of the joint model; a numerical comparison of coefficients alone cannot decide which association structure is appropriate.

Confusing the two coefficient units. With marker units and years, $\alpha_A$ is per marker-year, while $\alpha_{\bar m}$ is per marker. They cannot be compared numerically without specifying a time point and conversion.
Calling $A_i(t)$ a time average. The integral is a total, not an average; the average is $A_i(t)/t$.
Assuming a single conversion works at all times. The factor relating the coefficients is $t$, so a conversion at one follow-up time need not hold at another.
Reversing units when the marker rate is used. If marker values are measured in units per year and time in years, the integral has units and the time average has units per year. Units must follow from the actual marker scale.
Interpreting association as causation. A fitted hazard ratio describes a conditional model association; it is not, by itself, a causal effect.

The main alternative is not an algebraic shortcut but a different substantive summary: use the total when accumulated amount is the intended association, or the average when mean level over elapsed time is the intended association. Their adequacy depends on the scientific question and model fit. Neither summary captures every possible feature of a marker history; distinct trajectories can yield the same total or the same average. More elaborate association structures are separate modeling choices and should not be implied by either coefficient here.

Key takeaway: $A_i(t)=t\bar m_i(t)$, but a fixed total-exposure coefficient and a fixed average-exposure coefficient generally define different hazard models across follow-up. State the exposure definition, units, and comparison time before interpreting a coefficient or hazard ratio.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

If the marker is measured in marker units and time in years, what are the units of $A_i(t)$, $\bar m_i(t)$, $\alpha_A$, and $\alpha_{\bar m}$?
At $t=3$ years, two participants differ by 0.8 marker units in average exposure. If $\alpha_A=0.0500$ per marker-year, calculate the total-link hazard ratio for the higher-average participant.
A marker is measured in units per year and time in years. What are the units of its cumulative total and time-normalized average?
Why does $\alpha_{\bar m}=t\alpha_A$ not usually provide one fixed conversion across all event times?
What does an average-exposure link retain that a cumulative total does not directly distinguish, and what does the total retain that the average normalizes away?

Solutions. (1) $A_i(t)$ is in marker-years, $\bar m_i(t)$ is in marker units, $\alpha_A$ is per marker-year, and $\alpha_{\bar m}$ is per marker. (2) The cumulative difference is $3(0.8)=2.4$ marker-years, so $\mathrm{HR}=\exp\{0.0500(2.4)\}=\exp(0.120)=1.127$. (3) Integrating units per year over years gives units; dividing by years gives units per year. (4) The conversion factor is $t$, which changes with the comparison time. (5) The average expresses mean marker level over the interval, while the total retains accumulated amount and therefore reflects duration as well as level.