Multiplicative Delta Adjustments for Positive Outcomes

Why Multiply a Positive Outcome?

For some outcomes, a departure from the missing at random (MAR) reference is more naturally described as a proportional change than as a fixed number of units. For example, a sensitivity assumption might be that the unobserved values of a positive laboratory measurement are, on average, 20% lower than their MAR reference values. Multiplication expresses that assumption directly: it lowers a reference value of 100 by more units than a reference value of 10, while applying the same proportion to both.

A multiplicative delta adjustment changes imputations for missing values, not the observed values. It is a sensitivity assumption: the data generally do not identify the departure from MAR, so an analyst examines how conclusions change across specified multipliers. The MAR imputation model supplies a reference; the multiplier describes a departure from that reference.

The ratio interpretation is individual to each reference draw: \(Y_{\delta}/Y_{\mathrm{MAR}}=\delta\). For instance, \(\delta=0.8\) means each adjusted imputation is 80% of its MAR reference, or 20% lower. It does not mean “0.8 outcome units lower.” The adjusted value retains the outcome's units, while \(\delta\) is dimensionless.

The adjustment is applied to missing values only. Observed outcomes remain as recorded, and a value imputed under one sensitivity scenario should not be treated as observed data. In multiple imputation, the adjustment is made to the relevant missing-value draws in each completed data set, and the analysis is repeated for each chosen \(\delta\).

Assumptions and a Log-Scale Justification

Before applying the adjustment, make its assumptions explicit. The outcome must be positive on the scale being multiplied. The MAR imputation model must provide a suitable reference distribution conditional on the variables used in that model. The sensitivity assumption is that, for the missing cases being adjusted, the departure from that reference is proportional and represented by the specified multiplier. In the simplest version, the same \(\delta\) applies to all those cases.

This assumption does not follow from the observed data. It is a chosen scenario, not an estimated correction. It also does not claim that the observed cases should be rescaled: their outcomes are observed and are left unchanged. If zero or negative outcomes are possible, multiplication may not express the intended departure, and it may not preserve the required support.

The connection with a log transformation explains why multiplication is convenient for positive outcomes. Taking logarithms of the adjustment gives an additive shift on the log scale:

So a fixed multiplier corresponds to the same additive shift, \(\log(\delta)\), for every missing value on the log scale. This is a mathematical representation of the adjustment, not a requirement that the original imputation model was fitted on the log scale. If \(Y_{\mathrm{MAR}}\) is positive, multiplication also preserves positivity. For any positive quantile \(q\), the adjusted quantile is \(\delta q\); in particular, the conditional mean is multiplied by \(\delta\) when it exists.

The log identity provides a useful check on interpretation. A multiplier of 0.8 gives a log-scale shift of \(\log(0.8)\), which is negative. A multiplier of 1.25 gives a log-scale shift of \(\log(1.25)\), which is positive. These shifts are not equal in magnitude: multiplying by 0.8 and then by 1.25 returns to the original value, but the percentages “20% lower” and “25% higher” are not symmetric descriptions around the original value.

Consequences for Means and Contrasts

The adjustment's effect on a completed-data mean depends on how many values are missing and on the sizes of their MAR draws. Suppose a data set has \(n\) individuals, \(r\) observed outcomes with total \(S_{\mathrm{obs}}\), and \(n-r\) missing outcomes with MAR draws \(y_1,\ldots,y_{n-r}\). The MAR-reference mean and the adjusted mean are:

Subtracting the first mean from the second shows exactly what determines the change:

Because the MAR draws are positive, their sum is positive. Therefore, \(\delta<1\) lowers this completed-data mean, \(\delta>1\) raises it, and \(\delta=1\) leaves it unchanged. The change grows with the total of the missing draws and shrinks as the total sample size grows. This calculation concerns one completed data set; with multiple imputations, the analysis and combination are repeated across the adjusted completed data sets.

For an inferential analysis, each chosen multiplier defines a separate sensitivity scenario. For each scenario, generate the adjusted imputations, calculate the analysis estimate and its complete-data variance in each imputed data set, and combine the results using the usual multiple-imputation rules. The resulting interval and estimate describe uncertainty under that particular scenario and its imputation model. Conventional within-scenario uncertainty does not account for uncertainty about which multiplier is appropriate; that uncertainty is explored by comparing scenarios.

Multiplicative and Additive Adjustments

An additive delta adjustment instead adds a fixed quantity to each missing MAR draw: \(Y_{\mathrm{add}}=Y_{\mathrm{MAR}}+a\), where \(a\) is measured in the outcome's units. Multiplication and addition encode different assumptions. A fixed additive shift changes every value by the same number of units; a fixed multiplier changes every value by the same proportion. Consequently, the two adjustments generally produce different ordering of changes across small and large reference values.

Diagnostics, Failure Modes, and Limitations

A multiplicative adjustment is simple to calculate, but its simplicity makes checking the implications important. Useful checks include:

• Verify the scale and support. Confirm that the outcome values being multiplied are positive and that the multiplier is positive. If zero or negative values are meaningful possibilities, this adjustment may not be suitable as written.
• Inspect adjusted values and summaries. Compare the MAR-reference and adjusted distributions of missing values, including their means, quantiles, and upper tails. Confirm that the intended ratio is applied and that the resulting values remain scientifically plausible.
• Check completed-data analyses across scenarios. Track the estimate and its uncertainty for each multiplier. A rapid change, a change in the substantive conclusion, or an unexpected direction can reveal that conclusions depend strongly on the unobserved outcomes.
• Check the model reference. A multiplier cannot repair a poor MAR imputation model. If the reference draws are implausible or the model omits important predictors, the adjusted results inherit those problems.

A common mistake is to say that \(\delta=0.8\) means “subtract 0.8 units from each imputation.” That is an additive interpretation and is wrong: the adjustment is multiplication, so a reference draw of 10 becomes 8, while a draw of 100 becomes 80. Another mistake is to multiply observed values as well as missing imputations. That changes the observed data and no longer represents the stated sensitivity departure for missing outcomes.

The constant-ratio assumption is also a limitation. It imposes the same proportional departure for all adjusted missing cases, even though the actual departure could differ across people or parts of the outcome distribution. A chosen multiplier is not validated simply because the adjusted values are positive or plausible. Nor does one multiplier capture every possible departure from MAR. Results should therefore be reported as a sensitivity analysis across clearly described scenarios, not as a data-estimated correction.

Finally, do not confuse the ratio for each missing draw with a ratio between group means, or between an adjusted estimate and the MAR estimate. Observed values are unchanged, and the proportion of missing outcomes may differ across groups. Those features make the effect on a full-sample mean or contrast depend on the data, as the worked calculations demonstrate.