Shifting Binary Imputations on the Log-Odds Scale

Why Shift Binary Imputations on the Log-Odds Scale?

When a binary outcome is missing, a logistic imputation model can predict the probability of outcome 1 from the available information. Under a missing-at-random (MAR) reference, that model supplies the conditional prediction for a missing case. But the unobserved outcomes might, after accounting for the predictors, be systematically more or less likely to equal 1 than the reference model predicts. A log-odds delta adjustment represents that uncertainty with a specified shift.

The delta is a sensitivity parameter: it is not estimated from the observed outcomes alone. Its role is to define alternative assumptions about missing outcomes, then assess whether the substantive conclusions change across plausible values. Here the adjustment is a constant added to the logistic linear predictor for missing cases. It is not a change to the observed outcomes or a model of the probability that an outcome is missing.

Definition: Let $Y_i$ be a binary outcome, coded 1 or 0, and let $X_i$ denote the predictors used in its logistic imputation model. Write $p_{i,0}=P(Y_i=1\mid X_i)$ for the MAR-reference prediction and $\operatorname{logit}(p)=\log\{p/(1-p)\}$. A log-odds delta adjustment specifies $$ \operatorname{logit}(p_{i,\delta})=\operatorname{logit}(p_{i,0})+\delta $$ for cases whose $Y_i$ is missing. The corresponding conditional odds multiplier, relative to the MAR reference for the same predictors, is $\exp(\delta)$.

Model, Assumptions, and the Odds Multiplier

Suppose the MAR-reference imputation model is

\operatorname{logit}(p_{i,0})=\eta_i=X_i^\mathsf{T}\beta,

where $X_i$ includes the chosen predictors and $\beta$ is the logistic regression coefficient vector. In chained equations, the available values of other incomplete variables are the current values for that iteration. The logistic model should be fitted to cases whose binary outcome is observed; for the MAR reference to extrapolate to missing outcomes, it must be reasonable to assume that, conditional on the included observed information, their outcome distribution follows the same conditional model. This is the MAR assumption for the reference model.

The model also requires an adequately specified relationship between predictors and the log odds, sufficient information to estimate its coefficients, and no serious lack of overlap or separation that makes predictions unstable. Delta does not repair a misspecified logistic model, sparse data, or an inadequate predictor set. It changes the reference prediction for missing outcomes according to an explicit sensitivity assumption.

The multiplier follows directly from the definition. The reference odds are $p_{i,0}/(1-p_{i,0})=\exp(\eta_i)$. Adding $\delta$ gives shifted odds $\exp(\eta_i+\delta)$, so their ratio is:

\frac{p_{i,\delta}/(1-p_{i,\delta})} {p_{i,0}/(1-p_{i,0})} = \frac{\exp(\eta_i+\delta)}{\exp(\eta_i)} = \exp(\delta).

Thus $\delta=0$ recovers the MAR reference, $\delta>0$ makes outcome 1 more likely in the missing cases in the conditional-odds sense, and $\delta<0$ makes it less likely. The multiplier applies at fixed $X_i$; it is not generally a marginal odds ratio comparing all missing cases with all observed cases. Because the odds-to-probability conversion is nonlinear, a fixed odds multiplier does not produce the same probability difference for everyone.

For a given reference probability, the adjusted probability is

p_{i,\delta} = \frac{\exp(\delta)p_{i,0}} {1-p_{i,0}+\exp(\delta)p_{i,0}}.

This identity is useful for checking the calculation. It also shows why the direction of the probability change agrees with the sign of $\delta$, while the size of that change depends on the initial probability.

Worked Example: Doubling the Conditional Odds

Suppose a hypothetical missing case has a MAR-reference probability of $0.30$ for outcome 1. A sensitivity scenario sets $\delta=\log(2)$. Calculate the adjusted probability and explain the odds multiplier.

The reference odds are $0.30/0.70=0.428571\ldots$. Since $\exp\{\log(2)\}=2$, the shifted odds are $(0.30/0.70)\times 2=0.857142\ldots\approx0.8571$. The corresponding probability is $0.857142\ldots/(1+0.857142\ldots)=0.461538\ldots\approx0.4615$.

Equivalently, the adjusted log odds are $\operatorname{logit}(0.30)+\log(2)$. The scenario doubles the odds of outcome 1 relative to the MAR prediction for a case with these predictors. It does not double the probability: the probability rises from $0.30$ to about $0.4615$, an increase of about $0.1615$, or 16.15 percentage points.

Applying the Shift During Imputation

A practical implementation retains the logistic model’s uncertainty and the randomness of a binary outcome draw. In each imputation, draw or estimate the logistic coefficients using the chosen multiple-imputation procedure, calculate the MAR-reference linear predictor for each missing case, add the prespecified $\delta$, and draw the missing outcome from a Bernoulli distribution with the resulting probability. Do not shift or redraw observed outcomes. Repeat the procedure across imputations, and repeat the full analysis for each delta scenario of interest.

Fit the reference model.
Use observed outcomes and the selected predictors to fit the logistic imputation model within the current chained-equations iteration.

Predict and shift.
For each missing outcome, calculate $\eta_i=X_i^\mathsf{T}\beta$ and use $\eta_i+\delta$ as its shifted linear predictor.

Draw the binary value.
Set $p_{i,\delta}=1/\{1+\exp[-(\eta_i+\delta)]\}$ and draw $Y_i$ as 1 with probability $p_{i,\delta}$, or 0 otherwise.

Repeat and compare scenarios.
Complete the chained-equations iterations and imputations for each prespecified delta. Summarize the substantive estimates and uncertainty separately by scenario.

The binary draw matters: replacing each missing outcome with a probability would create fractional values and would not be a proper binary imputation. Also, uncertainty in fitted model parameters should be handled as required by the multiple-imputation procedure. The specified delta is held fixed within a scenario, so the resulting inference does not account for uncertainty about which delta is appropriate.

Worked Example: A Negative Shift

For a second hypothetical missing case, suppose the MAR-reference probability of outcome 1 is $0.70$. Consider $\delta=\log(0.5)$. Find the shifted odds and probability, then interpret them.

The reference odds are $0.70/0.30=2.3333\ldots$. The multiplier is $\exp\{\log(0.5)\}=0.5$, so the shifted odds are $(0.70/0.30)\times0.5=1.1666\ldots\approx1.167$. The adjusted probability is $1.1666\ldots/(1+1.1666\ldots)=0.538461\ldots\approx0.5385$.

This scenario halves the conditional odds relative to the MAR reference; it does not halve the probability. The probability falls from $0.70$ to about $0.5385$. A Bernoulli imputation draw for this case would therefore be 1 with probability about $0.5385$, and 0 with probability about $0.4615$.

Worked Example: Applying a Shift to a Logistic Prediction

In a hypothetical imputation model, suppose $\eta_i=-1.2+0.8X_i$, where $X_i$ is a binary predictor. For a missing case with $X_i=1$, use $\delta=\log(1.5)$. Calculate the reference and shifted predictions.

The reference linear predictor is $-1.2+0.8(1)=-0.4$. Thus $p_{i,0}=1/\{1+\exp(0.4)\}\approx0.4013$, and the reference odds are $\exp(-0.4)\approx0.6703$.

The offset is $\log(1.5)\approx0.4055$, giving the shifted linear predictor $-0.4+0.4055=0.0055$ (more precisely, $0.005465$). Therefore $p_{i,\delta}=1/\{1+\exp(-0.005465)\}\approx0.5014$. As a check, the shifted odds are $\exp(0.005465)\approx1.0055$, which are $1.5$ times the reference odds: $0.6703\times1.5\approx1.0055$, using the unrounded reference odds.

The factor $1.5$ describes the conditional odds for missing cases relative to the reference prediction at $X_i=1$. It is not the effect of $X_i$, which is represented by the model coefficient $0.8$, nor does this calculation establish a causal effect.

Choosing Scenarios, Diagnostics, and Limits

Observed data generally cannot determine the correct delta for outcomes that are missing. Scenario values therefore need substantive justification, for example from subject-matter knowledge or a transparent range of departures considered meaningful. Reporting only one convenient value can conceal how dependent the conclusion is on an untestable assumption. A useful sensitivity analysis shows results at the MAR reference and across a stated range of positive and negative offsets, while explaining the conditional odds multipliers.

Several checks can expose implementation or model problems:

Confirm that $\delta=0$ reproduces the MAR-reference predictions and that positive and negative values move predictions in the expected directions.
Check that the stated odds multiplier agrees with $\exp(\delta)$, and verify adjusted probabilities are between 0 and 1.
Inspect fitted logistic models for convergence problems, separation, sparse predictor combinations, or implausible predictions. An offset cannot fix these failures.
Compare imputed distributions with observed outcomes across relevant predictor values. A mismatch may indicate model inadequacy or a strong sensitivity assumption; it does not by itself identify the correct delta.
Check whether the substantive estimates and uncertainty change meaningfully across the reported scenarios. Large changes indicate that conclusions depend on assumptions about the missing outcomes.

The log-odds scale is convenient because the delta has a constant conditional odds-multiplier interpretation. It does not imply a constant probability increment. A probability-scale delta is an alternative scale for expressing sensitivity, but it has a different interpretation; the chosen scale should match the scientific question. The MAR reference, by contrast, makes no such departure. Each approach still requires a plausible reference model and cannot recover information that is absent from the observed data without assumptions.

Common pitfall: “A delta of $\log(2)$ doubles the probability for every missing case.” This is incorrect. It doubles the conditional odds relative to each case’s MAR prediction. For example, a reference probability of $0.30$ becomes about $0.4615$, not $0.60$. The probability change varies with the reference probability.

The conclusions from a delta analysis are conditional on both the imputation model and the chosen delta values. A constant offset may be too simple if the departure from the MAR reference differs across subgroups or missingness patterns; such complexity needs explicit justification rather than being inferred from the observed outcomes. The essential report is therefore not merely a delta number, but its odds-multiplier interpretation, the reference model, the scenarios examined, and how the substantive results respond.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

If $\delta=0$, what happens to the MAR-reference prediction?
What conditional odds multiplier corresponds to $\delta=\log(0.25)$?
A MAR-reference probability is $0.20$ and $\delta=\log(3)$. What is the adjusted probability?
Why does an odds multiplier of 2 not mean that every missing case has twice the reference probability?
Name one key assumption needed to use the logistic MAR-reference model for missing outcomes.

Solutions. (1) Since $\exp(0)=1$, the odds and probability remain at the MAR reference. (2) The multiplier is $\exp\{\log(0.25)\}=0.25$, so the conditional odds are one quarter of the reference odds. (3) Reference odds are $0.20/0.80=0.25$; shifted odds are $0.25\times3=0.75$, giving probability $0.75/(1+0.75)=0.4286$. (4) Odds and probabilities are different quantities, and converting shifted odds to probability depends on the starting probability. (5) For example, conditional on included observed information, the outcome distribution among missing cases follows the reference logistic model under MAR; the logistic functional form must also be adequate.

Shifting Binary Imputations on the Log-Odds Scale

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