Estimating Equations For Marginal Rate Models

Why estimating equations matter

A marginal rate model describes how covariates relate to a subject’s average recurrent-event rate over time. Its estimating equation uses each observed event and the covariate composition of the subjects still under observation at that time. Understanding how the equation is centered is essential: a marginal mean-rate assumption alone does not say that an event increment has the same mean after conditioning on the realized risk set, which can depend on event history.

The derivation below uses independent subjects and independent censoring conditional on baseline covariates. It permits arbitrary dependence among recurrences within a subject, subject to the stated marginal mean model and the regularity conditions needed for large-sample estimation. The key centering argument is at the population, asymptotic level—not a claim that each observed risk set produces a conditionally unbiased score.

Model, observed data, and assumptions

Definition: For subject $i$, let $N_i^*(t)$ be the event-counting process that would be observed in the absence of censoring, and let $Z_i$ be a vector of baseline covariates. The marginal rate model is $$ E\{dN_i^*(t)\mid Z_i\}=\exp(\beta_0^\mathsf{T}Z_i)\,d\Lambda_0(t), $$ where $\beta_0$ is the regression parameter and $\Lambda_0(t)$ is an unspecified baseline cumulative rate. Equivalently, $E\{N_i^*(t)\mid Z_i\}=\exp(\beta_0^\mathsf{T}Z_i)\Lambda_0(t)$. This is a conditional mean given $Z_i$; it does not specify the event increment conditional on the subject’s event history.

Let $C_i$ be the censoring time, $Y_i(t)=I(C_i\ge t)$ the indicator that subject $i$ remains observed and at risk just before $t$, and $N_i(t)=\int_0^t Y_i(u)\,dN_i^*(u)$ the observed event process. The data are independent across subjects (or independent sampling units), though recurrences for the same subject need not be independent.

For the estimating equation to target $\beta_0$, the following conditions are important.

Correct marginal mean model: the conditional mean rate given $Z_i$ has the stated proportional form over the analysis interval. No assumption of independent recurrences is needed for this mean statement.
Independent censoring given covariates: conditional on $Z_i$, censoring is independent of the subject’s full event process. Thus, with $G(t\mid Z_i)=P(C_i\ge t\mid Z_i)$, observed events are selected by the same censoring probability as risk-set membership.
Positivity: covariate patterns relevant to the comparison have a nonzero probability of remaining observed at times that contribute information.
Identification and regularity: covariates provide variation, the estimating equation has an identifiable root, the relevant event and risk-set quantities are finite, and large-sample averages converge. In practice, the number of independent subjects—not the number of recurrences—drives the independent information units.

Censoring may vary with $Z_i$; the condition does not require the same censoring distribution in every covariate group. If censoring depends on unobserved event history even after conditioning on $Z_i$, the selection argument below may fail. Likewise, a correctly specified marginal mean does not automatically imply that a history-conditional intensity model is correct.

From risk-set sums to the estimating equation

For $k=0,1$, define the empirical risk-set sums

S_n^{(0)}(\beta,t)=\frac{1}{n}\sum_{i=1}^nY_i(t)\exp(\beta^\mathsf{T}Z_i), \qquad S_n^{(1)}(\beta,t)=\frac{1}{n}\sum_{i=1}^nY_i(t)\exp(\beta^\mathsf{T}Z_i)Z_i.

When $S_n^{(0)}(\beta,t)>0$, their ratio is the risk-set weighted covariate mean

\bar Z_n(\beta,t)=\frac{S_n^{(1)}(\beta,t)}{S_n^{(0)}(\beta,t)}.

The marginal rate estimating equation for $\beta$ is

U_n(\beta)=\sum_{i=1}^n\int_0^\tau \{Z_i-\bar Z_n(\beta,t)\}\,dN_i(t)=0.

Here $\tau$ is the end of follow-up. Each event contributes the subject’s covariate vector minus the weighted mean covariate in the observed risk set at that event time. The equation resembles a partial-likelihood score, but it is an estimating equation for a marginal rate model; it need not be derived from a full event-history likelihood.

After estimating $\beta$, a corresponding baseline cumulative-rate estimator is

\widehat{\Lambda}_0(t)= \int_0^t\frac{\sum_{i=1}^n dN_i(u)} {\sum_{i=1}^nY_i(u)\exp(\widehat\beta^\mathsf{T}Z_i)}.

This is a cumulative rate on the model’s time scale. Its increments are event counts divided by covariate-weighted at-risk exposure at the event times. The covariate rate ratio for a one-unit contrast in a scalar $Z$ is $\exp(\beta_0)$, a ratio of marginal mean rates, not necessarily a ratio of event hazards conditional on recurrence history.

Why the estimating equation is centered

The distinction between population centering and conditioning on the realized risk set is fundamental. Define the population risk-set sums

s^{(0)}(\beta,t)=E\{G(t\mid Z)\exp(\beta^\mathsf{T}Z)\}, \qquad s^{(1)}(\beta,t)=E\{G(t\mid Z)\exp(\beta^\mathsf{T}Z)Z\}, \qquad \bar z(\beta,t)=\frac{s^{(1)}(\beta,t)}{s^{(0)}(\beta,t)}.

By independent censoring given $Z$, and the marginal mean-rate model,

E\{Y(t)\exp(\beta_0^\mathsf{T}Z)\} =s^{(0)}(\beta_0,t), \qquad E\{Z\,dN(t)\} =E\{G(t\mid Z)\exp(\beta_0^\mathsf{T}Z)Z\}\,d\Lambda_0(t).

The second identity follows by averaging over $Z$: conditional on $Z$, censoring retains an event at $t$ with probability $G(t\mid Z)$, while the uncensored event increment has mean $\exp(\beta_0^\mathsf{T}Z)d\Lambda_0(t)$. It does not require conditioning on the observed risk set.

As the number of independent subjects grows, the empirical risk-set sums converge to their population counterparts under the regularity conditions. The limiting mean score increment at $\beta_0$ is therefore

\begin{aligned} E\{G(t\mid Z)\exp(\beta_0^\mathsf{T}Z) [Z-\bar z(\beta_0,t)]\}\,d\Lambda_0(t) &=\{s^{(1)}(\beta_0,t)-\bar z(\beta_0,t)s^{(0)}(\beta_0,t)\}\,d\Lambda_0(t)\\ &=0. \end{aligned}

The last equality holds by the definition of $\bar z$. This establishes population centering of the limiting estimating equation. At finite sample size, the random risk-set ratio and the event increments can be dependent; the equation is not asserted to have zero conditional mean given that realized risk set. Arbitrary within-subject dependence is compatible with this argument because it uses the marginal first moment and independent censoring, not independent increments.

Worked Example: Solving a Binary-Covariate Rate Equation

Consider hypothetical data with a binary covariate $Z$, where $Z=1$ denotes the exposed group and $Z=0$ the unexposed group. At two event times, suppose the risk sets and event counts are as follows.

Time	At risk, $Z=0$	At risk, $Z=1$	Events, $Z=0$	Events, $Z=1$
$t_1$	8	4	1	2
$t_2$	5	3	1	1

For a scalar coefficient $\beta$, write $x=\exp(\beta)$. The exposed share of the weighted risk set is $4x/(8+4x)$ at $t_1$, and $3x/(5+3x)$ at $t_2$. The score is exposed events minus total events times the corresponding weighted share:

U(\beta)=3-3\frac{4x}{8+4x}-2\frac{3x}{5+3x}.

Set this score to zero. Dividing by 3 and simplifying gives

1=\frac{x}{2+x}+\frac{2x}{5+3x} \quad\Longrightarrow\quad 2x^2-2x-10=0.

The positive solution is $x=(1+\sqrt{21})/2=2.791$, so $\widehat\beta=\log(2.791)=1.026$. In this hypothetical analysis, the fitted marginal rate ratio for $Z=1$ versus $Z=0$ is 2.791. This calculation illustrates the estimating equation; it does not establish that the mean model or censoring assumptions hold for any real data.

Worked Example: Population Centering With Differential Censoring

Suppose, hypothetically, that half the population has $Z=0$, half has $Z=1$, and the true rate ratio is 2. At a particular time, let the conditional probabilities of remaining observed be $G(t\mid0)=0.8$ and $G(t\mid1)=0.5$. Use 100 subjects in each group to show how censoring enters the centering calculation.

The expected risk-set counts are $100(0.8)=80$ for $Z=0$ and $100(0.5)=50$ for $Z=1$. With rate weights $1$ and $2$, respectively, the risk-set weighted exposed share is

\bar z(\beta_0,t)=\frac{50(2)}{80(1)+50(2)} =\frac{100}{180}=0.5556.

For a small baseline increment $d\Lambda_0(t)$, the expected observed event contributions are $80d\Lambda_0(t)$ in the unexposed group and $50(2)d\Lambda_0(t)=100d\Lambda_0(t)$ in the exposed group. The expected exposed fraction among these contributions is $100/180=0.5556$, matching the weighted risk-set share. Hence the expected centered contribution is zero at the population level. This equality does not claim that the realized risk set has this composition or that its event increments are conditionally centered.

Worked Example: Estimating Baseline Rate Increments

Use the hypothetical risk sets and events from the first example, with $\widehat\beta=1.026$ and $\exp(\widehat\beta)=2.791$. At $t_1$, the weighted risk-set total is $8+4(2.791)=19.16$, and there are 3 events. At $t_2$, it is $5+3(2.791)=13.37$, and there are 2 events. The baseline cumulative-rate increments are therefore

\Delta\widehat\Lambda_0(t_1)=\frac{3}{19.16}=0.1566, \qquad \Delta\widehat\Lambda_0(t_2)=\frac{2}{13.37}=0.1496.

The estimated cumulative baseline rate through $t_2$ is $0.1566+0.1496=0.3062$ events per unit of the time scale, for a subject with $Z=0$. For $Z=1$, the model multiplies this cumulative rate by 2.791. The estimates depend on the assumed rate model and on adequate censoring-adjusted observation of the relevant risk sets.

Diagnostics, alternatives, and limitations

A numerical root is not evidence that the assumptions are adequate. Before interpreting estimates, inspect the data and the fitted equation for signs of failure.

Empty or sparse risk sets: If a covariate group has few subjects remaining under observation, the weighted risk-set sums can be unstable or nearly zero. This threatens positivity and can make the coefficient estimate highly sensitive to a few events.
No covariate overlap or separation: If events occur almost exclusively in one group while the other group remains in the risk set, the score may have no finite root or may be very flat. Check whether the score changes sign over a sensible range of $\beta$, and inspect the information available across event times.
Questionable censoring: Compare follow-up and loss-to-follow-up patterns across covariate and measured history groups. Such checks cannot prove independent censoring. If censoring depends on event history after conditioning on included covariates, the stated centering derivation is not justified without an appropriate adjustment or stronger model.
Mean-model mismatch: A single proportional rate ratio may not represent the data across time or across recurrence experience. Examine whether estimates are sensitive to analysis interval and scientifically justified covariate specifications. Such checks diagnose tension with the model but do not prove the mean relationship.
Dependence and uncertainty: Recurrences from one subject are not independent observations. Treating every event as an independent subject can understate uncertainty. Variance estimation must respect the subject as the independent sampling unit; the estimating equation alone does not supply a valid standard error.

The marginal rate equation is useful when the target is an average rate contrast over the population, rather than an event-order-specific or history-conditional effect. In contrast, PWP models explicitly condition on prior event order and use event-order risk sets. Andersen–Gill models are commonly expressed through a conditional-intensity formulation. These approaches answer different questions and rely on different modeling statements; the marginal rate equation should not be interpreted as establishing either a conditional hazard ratio or a causal effect.

The equation also has limitations. It specifies a proportional covariate effect on the marginal mean rate and leaves the baseline rate unspecified; misspecification of that proportional form can make the fitted coefficient an inadequate summary. Informative censoring can change which events are observed relative to the risk-set weights. Finally, a rate ratio compares mean event rates over the modeled time scale; it does not by itself describe the distribution of event counts, the chance of any recurrence, or individual trajectories.

Common mistake

Common mistake: “Given the observed risk set, each subject’s mean event increment is $Y_i(t)\exp(\beta_0^\mathsf{T}Z_i)d\Lambda_0(t)$, so the score is conditionally centered.” This does not follow from the marginal mean-rate model. Conditioning on risk-set membership can alter the event-increment mean when that membership is related to event history. The correction is to use the marginal model with independent censoring to derive the population event and risk-set limits, then show that their limiting centered score is zero. A conditional-on-risk-set argument requires an additional conditional-intensity assumption and must state it explicitly.

Check Your Understanding

Try each question, then compare with the solutions below.

What does the marginal rate model condition on, and what does it leave unspecified about event history?
In the population-centering argument, why does $G(t\mid Z)$ appear in both the event contribution and the risk-set sums?
For a binary covariate with risk-set counts 10 unexposed and 5 exposed, and $\exp(\beta)=3$, calculate the weighted exposed share.
Name one assumption whose failure can invalidate the stated centering derivation even if the marginal mean-rate model is correct.

Solutions. (1) It conditions on baseline covariates $Z$; it specifies a marginal mean rate, not the event increment conditional on prior event history. (2) Under censoring independent of the full event process conditional on $Z$, an event at time $t$ is observed with the same conditional probability $G(t\mid Z)$ that the subject remains in the risk set. (3) The share is $5(3)/\{10+5(3)\}=15/25=0.6000$. (4) For example, censoring that depends on event history after conditioning on $Z$ can make observed event contributions and risk-set membership differ in the required way.

Time	At risk, \(Z=0\)	At risk, \(Z=1\)	Events, \(Z=0\)	Events, \(Z=1\)
\(t_1\)	8	4	1	2
\(t_2\)	5	3	1	1