Units and Scaling of Cumulative Effects

Why Units Matter for Cumulative Effects

A cumulative association uses the history of a marker, not just its value at one visit. That history is commonly represented by the area under a marker trajectory. The area’s units combine the marker’s measurement units with the time units used to measure follow-up. Consequently, the coefficient multiplying cumulative exposure has reciprocal units. Tracking both quantities is essential: changing a marker or time scale changes the numerical coefficient, even when the modeled association is unchanged.

Here we consider a proportional-hazards model with a cumulative marker term. Other covariates and association terms, if present, are held fixed. The goal is to interpret this term and translate it consistently between scales—not to change the definition of the exposure.

Definition: Let $m_i(t)$ be the marker trajectory for individual $i$, measured in marker units $U_m$, and let $t$ be measured in time units $U_t$. The cumulative exposure through time $t$ is $$ A_i(t)=\int_0^t m_i(s)\,ds. $$ It has units $U_m\cdot U_t$. In the model $$ h_i(t\mid b_i)=h_0(t)\exp\{\gamma^\top w_i+\alpha A_i(t)\}, $$ $h_0(t)$ is the baseline hazard, $w_i$ denotes covariates, and $b_i$ denotes any conditioning random effects. Since the exponent must be dimensionless, $\alpha$ has units $1/(U_m\cdot U_t)$.

At a fixed time $t$, for two otherwise identical individuals whose cumulative exposures differ by $\Delta A$, the hazard ratio is $$ \mathrm{HR}=\exp(\alpha\Delta A). $$ Thus, $\exp(\alpha)$ is the hazard ratio for a one-unit increase in cumulative exposure, where “one unit” means one unit of $U_m\cdot U_t$. It does not mean a one-unit increase in the marker at a single visit. The interpretation is conditional on the model’s other terms and assumptions; it is an association, not by itself a causal effect.

Calculating the Exposure and Checking Its Units

The integral accumulates marker values over time. For a constant marker $m_i(t)=c$, cumulative exposure through $t$ is $ct$. For a changing marker, the integral accounts for the full trajectory. Its units follow from multiplying marker units by time units; this remains true whether the integral is evaluated analytically or numerically.

Worked Example: Integrating a Linear Marker Trajectory

Suppose a hypothetical marker trajectory is $m(t)=2+0.5t$ mg/L, with $t$ measured in years. Compare its cumulative exposure through year 4 with a reference trajectory having zero cumulative exposure. Assume the cumulative term is the only difference in the hazard model and $\alpha=0.03$ per (mg/L-year).

Calculate the exposure. Integrating the trajectory gives

A(4)=\int_0^4(2+0.5s)\,ds =\left[2s+0.25s^2\right]_0^4 =8+4=12\ \text{mg/L-year}.

The units are mg/L multiplied by years. The exposure contrast is therefore $\Delta A=12$ mg/L-year.

Calculate the hazard ratio. The coefficient’s units are the reciprocal, L/(mg-year), so $\alpha\Delta A$ is dimensionless:

\mathrm{HR}=\exp(0.03\times12)=\exp(0.36)\approx1.433.

Under the specified model, at the same time and with the other terms held fixed, this 12 mg/L-year exposure contrast corresponds to a hazard 1.433 times as large. The calculation does not say that a 1 mg/L difference at a particular visit has this hazard ratio.

Rescaling the Marker

Suppose the same marker is recorded in a different unit, so that its numerical values are multiplied by a positive conversion factor $c$. Write $m_i^*(t)=c\,m_i(t)$. The transformed cumulative exposure is also multiplied by $c$, because integration preserves constant multiplication:

A_i^*(t)=\int_0^t c\,m_i(s)\,ds=cA_i(t).

To preserve the linear predictor, the coefficient must be divided by $c$. In other words, the coefficient’s numerical value changes inversely with the marker’s numerical scale. The hazard model is unchanged because the product of coefficient and exposure remains the same.

Worked Example: Converting a Marker from mg/L to µg/L

Suppose a cumulative exposure contrast is 3 mg/L-year and its coefficient is $\alpha=0.04$ per (mg/L-year). Convert the marker to µg/L, for which numerical marker values are 1,000 times as large.

Convert the exposure. Since $1$ mg $=1{,}000$ µg,

\Delta A^*=1{,}000(3)=3{,}000\ \text{µg/L-year}.

Convert the coefficient. To keep the product unchanged, divide its numerical value by 1,000. The new coefficient is $0.00004$ per (µg/L-year). The linear predictor contribution is

0.04(3)=0.12 \qquad\text{and}\qquad 0.00004(3{,}000)=0.12.

Both representations give $\mathrm{HR}=\exp(0.12)\approx1.127$ for this same physical exposure contrast. The coefficient is numerically smaller in µg/L units because one µg/L-year is a smaller exposure unit than one mg/L-year.

Rescaling Time

Changing the time unit also changes the numerical value of the integral. Let the new time coordinate be $\tau=rt$, where $r$ is the number of new time units per old time unit. For example, converting years to months uses $r=12$. The same physical trajectory, expressed on the new coordinate, has cumulative exposure

A_{i,\tau}(\tau)=\int_0^\tau m_i(v/r)\,dv =r\int_0^t m_i(s)\,ds =rA_{i,t}(t).

Accordingly, the coefficient on the new time scale is $\alpha_\tau=\alpha_t/r$. This conversion preserves the cumulative association contribution. The baseline hazard’s numerical scale also changes when time units change: a hazard is a rate per unit of time. A correctly transformed model accounts for that change as well as the coefficient conversion.

Worked Example: Converting a Cumulative Effect from Years to Months

Suppose a contrast in cumulative marker exposure is 2.5 mg/L-year and its coefficient is $0.08$ per (mg/L-year). Express the same contrast and association coefficient using months instead of years.

Convert the exposure. Twelve months equal one year, so

\Delta A_{\text{months}}=12(2.5)=30\ \text{mg/L-month}.

Convert the coefficient. Divide by 12:

\alpha_{\text{months}}=\frac{0.08}{12} \approx0.006667\ \text{per (mg/L-month)}.

The association contribution is $0.08(2.5)=0.20$ on the year scale and $0.006667(30)\approx0.20$ on the month scale. Thus both give $\mathrm{HR}=\exp(0.20)\approx1.221$ for the same physical contrast. A coefficient reported per mg/L-month describes a different-sized exposure increment from one reported per mg/L-year.

Why the Conversion Works

The invariance follows directly from the model’s linear predictor. If a scale conversion multiplies the numerical exposure by $q$, then multiplying the coefficient by $1/q$ leaves their product unchanged. For simultaneous marker and time conversions, $q=cr$, where $c$ multiplies marker values and $r$ multiplies the time coordinate.

A_i^*=crA_i,\qquad \alpha^*=\frac{\alpha}{cr}, \qquad \alpha^*A_i^* =\frac{\alpha}{cr}(crA_i) =\alpha A_i.

This is a units conversion, not a new association structure. If the exposure itself is redefined—for example, by changing its time window or replacing a cumulative total with an average—then it may no longer be a simple change of units. The coefficient’s interpretation and potentially the fitted association must then be reconsidered.

Write the exposure definition.
Record the marker units and the time unit used in the integral defining $A_i(t)$.

Find the numerical conversion.
If marker values multiply by $c$ and the time coordinate by $r$, the integrated exposure multiplies by $cr$.

Transform the coefficient and verify.
Divide the coefficient by $cr$, then check that $\alpha A_i(t)$ is unchanged for a stated exposure contrast.

Diagnostics, Alternatives, and Limitations

A practical diagnostic is to recompute a sample linear predictor after a unit conversion. If the marker and time units were converted consistently, $\alpha A_i(t)$ should agree before and after conversion, apart from rounding. A disagreement points to a conversion-factor error, a mismatched time unit, or an altered exposure definition. Check the units of the integral separately from the units of the coefficient; checking only one can conceal a mistake.

Researchers can report $\alpha$ per one unit of cumulative exposure, or report a hazard ratio for a prespecified, more interpretable exposure contrast. The latter can be easier to compare across settings if the contrast is stated with its units. Standardizing a marker is another possible presentation choice, but it changes what a one-unit marker increment means; for a cumulative term, the resulting integrated scale and coefficient must still be stated. These choices should not be confused with a unit conversion that preserves the same physical contrast.

The model also has substantive limitations. A single coefficient assumes that the log hazard changes linearly with cumulative exposure and that the association parameter is constant over the modeled range. Correct unit arithmetic cannot establish that these assumptions are appropriate, correct a misspecified trajectory, or remove uncertainty in the estimated coefficient. In addition, if two analyses use different integration origins, windows, or exposure definitions, their coefficients need not be directly comparable just because their units look alike.

Common Mistake

Incorrect reasoning: “The marker is now in µg/L, so the exposure is 1,000 times larger, but the coefficient can remain unchanged because the marker describes the same biology.”

Correction: The numerical exposure is 1,000 times larger, so the coefficient must be divided by 1,000 to preserve the same model contribution. Otherwise the exponent, and therefore the implied hazard ratio, changes by a factor that is an artifact of the units. A coefficient has meaning only together with the scale and units of its exposure.

Key takeaway: Integrated marker exposure has marker-times-time units; its cumulative association coefficient has reciprocal units. Convert both consistently, and verify that their product—and hence the modeled hazard contrast—remains unchanged.

Check Your Understanding

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

A marker is measured in mmol/L and integrated over years. What are the units of its cumulative exposure and of the coefficient multiplying it in a log-hazard model?
A cumulative contrast is 4 mg/L-year with coefficient $0.05$ per (mg/L-year). What is the hazard ratio for this contrast?
Marker values are multiplied by 10 under a new measurement scale. If the original coefficient is $0.05$, what is the transformed coefficient needed to preserve the model contribution?
A cumulative contrast is 2 mg/L-year. Express it in mg/L-month and convert a coefficient of $0.12$ per (mg/L-year) to the corresponding monthly scale.

Solutions. (1) Exposure is mmol/L-year; the coefficient is L/(mmol-year). (2) The log-hazard contrast is $0.05(4)=0.20$, so $\mathrm{HR}=\exp(0.20)\approx1.221$. (3) Divide by 10, giving $0.005$ per unit of the new cumulative exposure. (4) The exposure is $12(2)=24$ mg/L-month, and the coefficient is $0.12/12=0.010$ per (mg/L-month). Both products equal $0.24$, so both scales give $\mathrm{HR}=\exp(0.24)\approx1.271$.