Log-Rate Delta Adjustments for Count Outcomes

Why Adjust a Count Imputation on the Log-Rate Scale?

A count outcome records how many times an event occurred, such as hospital visits, relapses, or adverse events. For a missing count, a chained-equations model can use observed information to predict a conditional distribution and draw an integer from it. A sensitivity analysis asks what would happen if that distribution were systematically different for the missing cases than the reference model assumes.

For counts, a log-rate adjustment is useful because a change on the log scale corresponds to a multiplicative change in the expected count. A fixed delta therefore describes a proportional shift, rather than adding the same number of events to every prediction. The shift modifies the imputation distribution for missing values; it does not alter observed counts.

When \(\delta=0\), the adjustment is the reference model. When \(\delta>0\), missing counts have a higher conditional expected count than under that reference; when \(\delta<0\), they have a lower expected count. For example, \(\delta=\log(1.5)\) means that the conditional expected count is multiplied by 1.5, or increased by 50%. Delta is not itself a percentage.

Model, Assumptions, and the Adjustment

For a simple Poisson specification, the reference model is \(Y_i\mid X_i,t_i\sim\operatorname{Poisson}(\lambda_i)\), with \(\log(\lambda_i)=\log(t_i)+X_i^\mathsf{T}\beta\). The term \(\log(t_i)\) is an exposure offset: it has coefficient fixed at one, so doubling exposure doubles the mean count when other predictors are unchanged. This formulation requires a positive, correctly measured exposure and an appropriate log-linear relationship between predictors and the event rate.

The reference imputation also relies on a correctly specified conditional distribution and on a missing-at-random (MAR) assumption for its reference: conditional on the variables included in the imputation model, missingness does not further depend on the unobserved count. The delta adjustment is a sensitivity device for departures from that reference assumption; it does not establish which delta is true. Parameter uncertainty in the fitted model should be represented in the imputation procedure, and the chosen delta should be held fixed within a specified sensitivity analysis.

This model generates nonnegative integer counts directly. The adjustment changes the Poisson mean, then a count is drawn from the resulting distribution. It does not multiply an already drawn count by \(\exp(\delta)\), because that can produce fractional values and does not generally yield a draw from the intended conditional distribution.

The Poisson model entails equality of the conditional mean and variance. If the count distribution is overdispersed or has more zeros than the Poisson model can represent, a Poisson reference may generate unsuitable imputations. A model that better represents the conditional distribution can be used instead, with delta applied to its log mean while keeping its other distributional parameters appropriately specified. The mean multiplier interpretation remains the same; the full distribution and its variability need not be Poisson.

Choosing and Applying a Delta

A delta should express a defensible departure from the MAR reference in the units that matter for the count process. It can be chosen through subject-matter judgment or calibrated to external information. State the direction and scale in words, then convert the proposed multiplicative factor to the log scale using \(\delta=\log(r)\), where \(r\) is the intended ratio of conditional expected counts. For instance, a factor of 0.8 corresponds to \(\log(0.8)=-0.2231\), a 20% lower expected count.

The sensitivity analysis should compare a prespecified set of plausible values, including \(\delta=0\), rather than select a delta after seeing which value gives a preferred result. In each imputed data set, observed counts remain unchanged; missing counts are drawn under the selected adjusted model. The analysis model is then fitted to each completed data set and combined using the usual multiple-imputation pooling procedure. Conclusions should be compared across delta values, including changes in the substantive estimate and its uncertainty.

Diagnostics, Comparisons, and Limitations

A delta analysis is useful only if the reference model and the imposed shift are interpretable. Check whether the count model’s fitted values and residual behavior are compatible with the observed counts, and whether the chosen predictors capture important variation in the conditional rate. Compare observed and imputed count distributions, including the frequency of zeros and the upper tail. A large mismatch can indicate a poor count distribution, omitted predictors, or a poorly represented exposure process; changing delta alone may not fix any of these problems.

Also check the arithmetic and interpretation: a positive delta must increase the expected count, and the multiplier is \(\exp(\delta)\), not \(1+\delta\). Confirm that exposure is positive and measured on the intended scale. When exposure varies, an offset is important because a higher expected count may reflect more time at risk rather than a higher underlying rate. A common delta represents the same multiplicative shift in expected count at fixed predictors and exposure; it does not represent the same additive number of events for every case.

The principal alternative is to use the MAR reference without adjustment, which corresponds to \(\delta=0\). For positive outcomes that are not counts, a multiplicative adjustment may also be meaningful, but a count model must still produce a valid discrete distribution. An additive change to counts is a different assumption: it adds a fixed number of expected events rather than multiplying the expected count. Neither scale is universally preferable; the choice should reflect the scientific interpretation of departures from the reference.

The delta is not estimated from the observed data when the missing outcomes themselves provide no information about their unobserved values. It represents an assumption, so results can only show how conclusions vary over the values examined. A single delta also imposes the same log-rate shift on all missing cases to which it is applied. The analysis can be misleading if that assumption is implausible, if the reference model is misspecified, or if uncertainty in model parameters is ignored.