Delta Adjustments for Counts and Rates

Why use a log-scale delta for counts and rates?

A count outcome might be the number of symptom episodes, hospital admissions, or adverse events during a specified observation period. When follow-up time differs among people, the event rate—the expected count per unit of exposure—can be more useful than the count alone. In a sensitivity analysis for missing outcomes, a delta adjustment can express how the missing-outcome rate is assumed to differ from its missing-at-random (MAR) reference rate.

For counts, a log-link model gives a particularly interpretable adjustment: adding delta to the log rate multiplies the rate by $e^\delta$. The delta is therefore dimensionless, while the adjusted rate and expected count retain their original units. This lets an analyst specify a proportional departure from the reference rather than an additive change that could imply a negative expected count.

Definition: Let $Y_i$ be a nonnegative count for person $i$, observed over exposure time $t_i>0$. Let $R_i=1$ indicate that the outcome is observed and $R_i=0$ that it is missing. Write $\lambda_i^{MAR}$ for the MAR-reference expected rate for a missing outcome, conditional on the covariates used in that reference model. A log-rate delta adjustment defines the missing-outcome rate and expected count as $$ \lambda_i(\delta)=\lambda_i^{MAR}e^\delta, \qquad \mu_i(\delta)=t_i\lambda_i^{MAR}e^\delta. $$ Here $\delta=0$ gives the MAR reference, $\delta>0$ increases the assumed missing-outcome rate, and $\delta<0$ decreases it.

From a log-link model to an adjusted expected count

Suppose the MAR reference model uses a log link and an exposure offset. For covariate vector $x_i$, it may be written as

\log \left(\lambda_i^{MAR}\right)=x_i^\mathsf{T}\beta, \qquad \log \left(\mu_i^{MAR}\right)=\log(t_i)+x_i^\mathsf{T}\beta.

The offset $\log(t_i)$ accounts for the length of time during which events could be counted. A delta adjustment for the missing outcome adds $\delta$ to the log rate, or equivalently to the log expected count when exposure is held fixed:

\log \left(\lambda_i(\delta)\right)=x_i^\mathsf{T}\beta+\delta, \qquad \log \left(\mu_i(\delta)\right)=\log(t_i)+x_i^\mathsf{T}\beta+\delta.

Exponentiating these expressions gives the definition above. In particular, the rate ratio comparing the adjusted assumption with the MAR reference is

\frac{\lambda_i(\delta)}{\lambda_i^{MAR}} = \frac{\mu_i(\delta)}{\mu_i^{MAR}} =e^\delta.

This ratio is the same for every person when a single common delta is used, but the absolute change in expected count depends on each person’s reference rate and exposure time. The calculation concerns conditional means. A log-link mean model does not by itself require that the conditional variance equal the mean; a Poisson model adds that distributional assumption, whereas an overdispersed count model can use a different variance structure.

Assumptions and interpretation: The MAR reference model must provide a defensible conditional expected rate for the missing outcomes, and the relevant exposure time must be known or specified. The sensitivity assumption is that the missing-outcome rate is the MAR rate multiplied by $e^\delta$, conditional on the reference-model covariates. The adjustment applies to missing outcomes, not to outcomes already observed. The observed data alone do not identify the appropriate nonzero value of $\delta$; it represents a sensitivity assumption.

Worked rate and count calculations

Worked Example: A 50% Higher Missing-Outcome Rate

Suppose a hypothetical MAR reference model gives a missing participant an expected event rate of 0.40 events per person-year. The participant has 1.5 years of exposure. Consider the sensitivity value $\delta=\log(1.5)$. Calculate the adjusted rate and expected count.

The multiplier is $e^\delta=e^{\log(1.5)}=1.5$. Thus the adjusted rate is

\lambda(\delta)=0.40\ \frac{\text{events}}{\text{person-year}}\times1.5 =0.60\ \frac{\text{events}}{\text{person-year}}.

The adjusted expected count over 1.5 person-years is

\mu(\delta)=1.5\ \text{person-years}\times0.60\ \frac{\text{events}}{\text{person-year}}=0.90\ \text{events}.

For comparison, the MAR-reference expected count is $1.5\times0.40=0.60$ events. The sensitivity assumption raises the expected count by $0.90-0.60=0.30$ events for this participant. An expected count need not be an integer; it is an average over the assumed outcome distribution, not a prediction that this person will experience a fractional event.

Worked Example: Aggregating Different Exposure Times

In a hypothetical study, three participants have missing outcomes. Their MAR-reference rates are 0.20, 0.50, and 0.80 events per person-year, and their exposure times are 1, 2, and 0.5 person-years, respectively. Use $\delta=\log(0.8)$, a 20% rate reduction, to calculate the group’s adjusted expected total count.

First calculate the MAR-reference expected counts: $1(0.20)=0.20$, $2(0.50)=1.00$, and $0.5(0.80)=0.40$ events. Their reference total is $0.20+1.00+0.40=1.60$ events.

Since $e^\delta=0.8$, each missing participant’s rate and expected count are multiplied by 0.8. The adjusted expected counts are $0.20(0.8)=0.16$, $1.00(0.8)=0.80$, and $0.40(0.8)=0.32$ events. Therefore,

\sum_{i=1}^{3}\mu_i(\delta) =0.16+0.80+0.32=1.28\ \text{events}.

Equivalently, because the same delta applies to all three people, $1.60e^{\log(0.8)}=1.60(0.8)=1.28$ events. The total expected count is 0.32 events lower than the MAR-reference total. This shortcut works for a common delta; calculating participant-level expectations first also makes the exposure and rate contributions visible.

Worked Example: Combining Observed Counts and Adjusted Missing Counts

Suppose a hypothetical group has 12 observed participants with a total of 9 recorded events. Four other participants have missing outcomes. Their MAR-reference expected counts, calculated using their individual exposure times, are 0.30, 0.60, 0.90, and 1.20 events. Use $\delta=\log(1.25)$ to calculate the group’s expected total after adjustment, holding the observed counts fixed.

The four missing-outcome reference expectations sum to $0.30+0.60+0.90+1.20=3.00$ events. The multiplier is $e^\delta=1.25$, so their adjusted expected total is $3.00(1.25)=3.75$ events. The expected group total, combining the observed count with the adjusted expectation for missing outcomes, is

9+3.75=12.75\ \text{events}.

Under the MAR reference it would be $9+3.00=12.00$ events. The sensitivity adjustment adds 0.75 expected events to the group total. The observed total of 9 is not multiplied by 1.25: delta represents a departure assumption for missing outcomes, not a retrospective rescaling of observed counts.

Derivation and practical checks

The multiplicative interpretation follows directly from the log-link form. For one person, write the MAR log expected count as $\log(\mu_i^{MAR})$. Adding delta gives $\log(\mu_i(\delta))=\log(\mu_i^{MAR})+\delta$. Exponentiating both sides yields

\mu_i(\delta) =\exp\{\log(\mu_i^{MAR})+\delta\} =\exp\{\log(\mu_i^{MAR})\}\exp(\delta) =\mu_i^{MAR}e^\delta.

Because $\mu_i^{MAR}=t_i\lambda_i^{MAR}$, dividing the adjusted expected count by $t_i$ gives $\lambda_i(\delta)=\lambda_i^{MAR}e^\delta$. This derivation explains both why the same factor applies to rates and counts for fixed exposure and why delta has no count or time units.

For a common delta applied to a set $M$ of missing outcomes, the adjusted expected total is

\sum_{i\in M}\mu_i(\delta) =e^\delta\sum_{i\in M}t_i\lambda_i^{MAR}.

This is a total expected count, not necessarily an integer. If the goal is a rate for the group, the corresponding expected rate per total exposure is the adjusted expected total divided by $\sum_{i\in M}t_i$. It is not generally correct to treat a total count as a rate without accounting for exposure.

Several checks help catch implementation errors:

Check the sign and multiplier. A positive delta must give $e^\delta>1$, and therefore a higher assumed rate. A negative delta must give a multiplier between 0 and 1.
Check exposure units. If rates are per person-year, exposure must be in person-years. Converting exposure from months to years requires dividing months by 12.
Check which outcomes are adjusted. Apply the sensitivity factor to the missing-outcome expectations, not to observed counts.
Check the estimand. If the target is a rate, divide expected events by exposure. If it is an expected count, retain the exposure multiplication.
Check whether the reference mean is plausible. Delta scales the reference prediction; it does not fix a poorly specified reference model, incorrect exposure, or inappropriate covariate adjustment.

How this adjustment compares with alternatives

A raw additive shift would set $\mu_i(\delta)=\mu_i^{MAR}+\delta$. That can be meaningful for some continuous outcomes, but for counts it can produce a negative expected count and does not preserve a common proportional change across different baseline rates. A log-rate delta instead preserves nonnegative expected means and states a proportional change directly.

The adjustment is also distinct from changing a binary outcome’s log odds: a rate ratio $e^\delta$ is not an odds ratio, and an event rate is not automatically a probability. For a count observed over time, the rate’s units and the exposure window matter. Use a rate-based interpretation only when the outcome and exposure define a meaningful count-per-exposure quantity.

The main limitation is that a single delta is a transparent but restrictive sensitivity assumption. It imposes the same proportional shift for all missing outcomes covered by that adjustment, regardless of their different reference rates. It also changes the conditional mean without, by itself, specifying how the rest of the missing-outcome distribution should change. A delta sensitivity analysis should therefore report the reference model, exposure definition, delta values and resulting adjusted quantities. It should not present a chosen delta as something learned from the observed outcomes.

Common mistake

Common mistake: “A delta of 0.2 adds 0.2 events to every missing participant’s expected count.” That is incorrect for a log-rate delta. It adds 0.2 to the log rate, so the multiplier is $e^{0.2}\approx1.221$, or about a 22.1% increase. A MAR expected count of 0.5 becomes $0.5e^{0.2}\approx0.611$ events, while a reference expected count of 2.0 becomes $2.0e^{0.2}\approx2.44$ events. The absolute increases differ because the reference expectations differ.

Check Your Understanding

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

A missing outcome has a MAR-reference rate of 0.30 events per person-month. What adjusted rate corresponds to $\delta=\log(2)$?
A missing participant has a reference expected count of 1.2 events over the relevant exposure. Calculate the adjusted expectation for $\delta=-0.1$, to three significant figures.
Two missing participants have reference expected counts of 0.4 and 1.1 events. What is their adjusted expected total under a common $\delta=\log(1.5)$?
Explain why a delta adjustment should not multiply an already observed count in this sensitivity analysis.

Solutions. 1. The multiplier is $e^{\log(2)}=2$, so the rate is $0.30(2)=0.60$ events per person-month. 2. The multiplier is $e^{-0.1}\approx0.9048$, so the adjusted expectation is $1.2(0.9048)=1.09$ events. 3. The reference total is $0.4+1.1=1.5$ events; the adjusted total is $1.5(1.5)=2.25$ events. 4. Delta encodes the assumed departure from the MAR reference for missing outcomes. Observed outcomes are already known and are held fixed, so rescaling them would change the observed data rather than express a missing-outcome assumption.