Ratios and Rates With Incomplete Components

Why Ratios Need Special Care

Ratios and rates are often calculated from two measured quantities: events per unit of exposure, expenditure per person, or a biomarker divided by body mass. If either component is incomplete, the ratio cannot be calculated from the observed components alone. In multiple imputation, however, the numerator and denominator can be imputed and the ratio can then be calculated from those completed values.

The important detail is that a ratio is a deterministic function of its components. It should agree exactly with them in every completed data set. That is the purpose of passive calculation here: after the components have been imputed, calculate the derived quantity from their current values rather than treating it as an unrelated variable. This preserves the arithmetic identity; it does not, by itself, guarantee that the imputation models are otherwise appropriate.

For a rate of events per person-time, for example, \(N_i\) might be an event count and \(D_i\) the corresponding follow-up time. A rate reported per 100 person-years is \(100N_i/D_i\), provided follow-up time is positive. A ratio may be mathematically defined for a negative denominator, but that may not make it scientifically meaningful. Many rates require a strictly positive denominator by definition.

Before deriving a ratio, verify that the numerator and denominator refer to the same unit or population, time window, and intended quantity. Check that their units make the quotient interpretable and state the denominator domain required by the measure. If the measure is defined only for positive denominators, the imputed values must respect that domain; a computed zero or negative denominator cannot be treated as an ordinary valid exposure.

Passive Calculation in Completed Data

Suppose one or both components are missing. In each completed data set, use the observed or imputed values of \(N_i\) and \(D_i\) to calculate \(R_i\). If \(D_i\) is nonzero, the result is defined; if the rate requires positive exposure, check that \(D_i>0\). The calculation is repeated separately in every completed data set. Do not calculate the derived value once from a single imputation and carry it over to the others.

Here \(m\) indexes a completed data set. Superscripts \(m\) on the components indicate that the values may differ across imputations. The derived value must be updated whenever a component changes during the chained-equations procedure or when a completed data set is assembled. The arithmetic is exact given the components: passive calculation does not add a separate random draw for the ratio.

Undefined Values and the Denominator Domain

A zero denominator is not a very large or very small ratio; division by zero is undefined. In particular, \(0/0\) is also undefined, not zero. Replacing an undefined result with zero, an arbitrary constant, or an infinite value changes the data and can distort subsequent summaries or models. First determine why the denominator is zero and whether the target quantity is meaningful for that unit.

For a rate with positive exposure as its definition, a zero or negative imputed exposure indicates that the completed value does not satisfy the rate's domain. It should be flagged for investigation rather than silently converted into a numerical rate. The appropriate response depends on the data-generating and measurement context: the exposure definition may need clarification, an impossible imputed value may point to a poorly specified component model, or the unit may not belong to the population for which the rate is defined.

A small positive denominator creates a different issue. The ratio is mathematically defined, but it may be highly sensitive to small changes in either component. For example, 1 event over 0.01 person-years gives 100 events per person-year. This value may be correct arithmetically but unstable or scientifically implausible for the intended setting. Do not mistake a defined ratio for a reliable or substantively sensible one.

Individual Ratios and Pooled Rates Are Different

The choice of calculation must match the target quantity. An average of unit-specific ratios gives equal weight to each unit, whereas a ratio of totals weights each unit's rate in proportion to its denominator, when denominators are positive. These answer different questions and generally produce different results.

The equality follows because \(D_i(N_i/D_i)=N_i\) for every unit with \(D_i>0\). Thus the pooled rate is a denominator-weighted average of the individual rates. It is not generally equal to their unweighted mean.

Why Calculate Passively?

Once the components have been completed, the ratio is determined. If the components are \(N_i=2\) and \(D_i=4\), their ratio must be \(0.5\). Drawing or imputing a separate value \(R_i=0.8\) would produce a completed record in which \(R_i\ne N_i/D_i\). Passive calculation prevents that arithmetic contradiction by setting \(R_i\) from the completed components every time.

Directly imputing the derived ratio instead can be useful only when the target is not truly a deterministic function of the available components, or when a different model is explicitly intended. For a deterministic ratio, separate imputation can break the identity and yield combinations of values that cannot all be true at once. Complete-case calculation avoids imputation but discards records with incomplete components and changes the analyzed set. Passive calculation retains the imputed components while ensuring that the derived ratio agrees with them; it does not repair unsuitable component models or establish that their assumptions hold.

This distinction matters in analyses that use the ratio as an outcome or predictor: a ratio that disagrees with its components can make the analysis refer to no coherent completed data set. At the same time, preserving the identity is only one requirement. The numerator and denominator must still be imputed in a way appropriate to the analysis and the data, and the resulting ratios must be examined for undefined or implausible values.

Diagnostics and Common Pitfalls

For each completed data set, check the denominator before calculating the ratio. Summarize how many denominators are zero, negative when positivity is required, or unusually small. Record the number of undefined ratios rather than dropping them without explanation. Check that the numerator and denominator have consistent units and definitions across records and imputations, and verify that every calculated ratio matches its components.

Compare the distribution of the derived ratio across completed data sets. Large changes can arise from imputed components, particularly when denominators are small. Such variation should prompt review of component draws and the scientific meaning of the measure; it is not a reason to replace the passive formula with a separate ratio imputation. If the rate is intended to be per 1,000 units, verify that the multiplier is applied consistently and report the unit explicitly.

• Wrong reasoning: “A zero numerator and zero denominator mean there were no events, so the rate is zero.” Correction: \(0/0\) is undefined; there is no positive exposure interval defining that rate.
• Wrong reasoning: “The mean of individual rates must equal the group rate.” Correction: an unweighted mean and a ratio of totals use different weights and usually target different quantities.
• Wrong reasoning: “A very large ratio proves an error.” Correction: small positive denominators can produce large, defined ratios. Check the source values and context before deciding whether the value is erroneous.
• Wrong reasoning: “A separate imputed ratio is acceptable if it looks plausible.” Correction: if the ratio is defined deterministically by the components, calculate it passively so the identity holds exactly.