Why use a count model in MICE?
When an incomplete variable records a count, its values are nonnegative integers: 0, 1, 2, and so on. A linear regression conditional model does not naturally respect that support. It can predict a negative value or a fraction of an event, and rounding those predictions does not represent the uncertainty in an individual’s unobserved count. A Poisson regression model provides a conditional distribution on nonnegative integers, so it can be used to generate plausible count imputations.
In MICE, a conditional model is fitted for the variable currently being imputed, using the other variables in the data as predictors. For a count variable, Poisson regression is a useful starting point when its assumptions are reasonable. The model describes the conditional distribution of the count given those predictors; it does not, by itself, resolve every issue about missingness or guarantee that the imputation model is appropriate.
Model specification and interpretation
The vector \(X_j\) usually includes an intercept and suitable predictors from the imputation data set. It should include variables needed to make the imputation model informative for the planned analysis, as well as observed variables associated with the count or its missingness. Categorical predictors can be represented with indicator variables; continuous predictors may require transformations or nonlinear terms if a straight-line relationship on the log-mean scale is implausible. Interactions can also be included when they are substantively justified.
The log link keeps the fitted mean positive for every finite linear predictor. A coefficient is additive on the log-mean scale but has a multiplicative interpretation on the mean scale. If \(X_{jk}\) increases by one unit, holding the other predictors fixed, the conditional mean is multiplied by \(\exp(\beta_k)\). For a binary predictor, \(\exp(\beta_k)\) is the ratio of the conditional means for the value 1 versus the reference value 0, conditional on the other predictors. It is not a difference in counts and does not establish a causal effect.
If cases have different observation times or other known exposure amounts, a rate model can include the logarithm of exposure as an offset. For exposure \(t_j>0\), the specification is \(\log(\lambda_j)=\log(t_j)+X_j^\mathsf{T}\beta\), making \(\lambda_j/t_j=\exp(X_j^\mathsf{T}\beta)\) the modeled rate per exposure unit. Use this only when exposure is meaningful and measured; a predictor coefficient is otherwise interpreted as a change in the expected count over the modeled observation period.
Worked Example: Interpreting a Log-Link Coefficient
Suppose a hypothetical study records the number of clinic visits per person during one year. A Poisson conditional model includes an indicator \(T\) for a program (1 if enrolled, 0 otherwise) and age \(A\), measured in decades above age 50. Its fitted equation is \(\log(\lambda)=\log(2)+0.4055T+0.1000A\). Interpret the program coefficient and calculate the fitted mean for a 60-year-old in the program.
For two otherwise identical people who differ only in program status, the fitted mean ratio is \(\exp(0.4055)\approx1.500\). Thus, the model associates program enrollment with a conditional expected count 1.5 times as high, or 50% higher, over one year, at a fixed age. This is a model-based association, not evidence that enrollment causes more visits.
For a 60-year-old in the program, \(A=1\) and \(T=1\). Substituting into the equation gives \(\log(\lambda)=\log(2)+0.4055+0.1000\approx1.1986\), so \(\lambda=\exp(1.1986)\approx3.316\) visits per year. This is a conditional mean, not a prediction that this person must have exactly 3.316 visits.
Assumptions for using the conditional model
The Poisson specification has assumptions about the count distribution and its relationship to predictors. These are modeling assumptions, not consequences of the data being integer-valued.
- Count support: the variable represents a nonnegative count of events over a defined unit or period. A bounded score, category code, or measurement recorded with decimals is not automatically a Poisson count.
- Conditional Poisson variation: given the predictors, the model has conditional mean and variance both equal to \(\lambda_j\). Substantial extra variation, clustering, or an excess of zeros relative to this model can indicate lack of fit.
- Correct conditional mean structure: the log of the conditional mean is adequately represented by the specified predictors and terms. Omitting a nonlinear pattern or an important interaction can distort fitted means and imputed values.
- Conditional independence for the basic likelihood: observations are treated as independent given the predictors. Repeated observations or clustered cases may require a model that accounts for their dependence.
- Missing-data and information conditions: the imputation model needs relevant observed predictors and enough observed count cases to estimate its parameters. The usual MICE justification relies on an appropriate missing-at-random assumption conditional on variables included in the imputation system, along with adequate overlap in the predictor patterns. Missing-at-random cannot be verified from observed data alone.
A Poisson model is therefore a candidate conditional model, not a rule to apply to every nonnegative integer variable. The substantive meaning of the count, the observed distribution, the predictor structure, and the amount of information available among observed cases all matter.
Worked Example: Calculating a Fitted Conditional Mean
Consider a hypothetical incomplete count of calls received by a service desk during a fixed one-week period. The model, fitted to cases with observed counts, is \(\log(\lambda_j)=0.6931+0.4055T_j+0.1000A_j\), where \(T_j\) indicates training (1=yes, 0=no) and \(A_j\) is age in decades above 50. For a case with a missing count, the known predictors are \(T_j=1\) and \(A_j=0\). Calculate the fitted mean.
The linear predictor is \(0.6931+0.4055(1)+0.1000(0)=1.0986\). Exponentiating gives \(\lambda_j=\exp(1.0986)\approx3.000\) calls per week. This value is the model’s conditional mean for the case, provided the log-linear mean structure and Poisson assumptions are adequate.
The mean is not itself an imputed count: 3.000 happens to be an integer here, but a different predictor pattern could give, for example, a mean of 3.7. In either case, an individual imputation should be a draw from a count distribution, not the fitted mean rounded to an integer. For use in multiple imputation, parameter uncertainty also matters; a procedure that treats estimated coefficients as known can understate imputation uncertainty.
From a fitted mean to an integer-valued draw
Once a positive conditional mean has been obtained for a case, the Poisson distribution assigns probabilities to every nonnegative integer. For a mean \(\lambda\), the probability of a count \(y\) is \(P(Y=y)=\exp(-\lambda)\lambda^y/y!\), for \(y=0,1,2,\ldots\). Thus, a count imputation is generated by sampling an integer from the fitted conditional distribution, rather than by rounding its mean. A complete MICE implementation must also account for uncertainty in the fitted model parameters; the following calculation isolates the count-draw step to show why its output is integer-valued and variable.
Worked Example: Drawing a Count Given a Fitted Mean
Suppose a hypothetical missing count has fitted Poisson mean \(\lambda=2.4\). To illustrate an inverse-cumulative-probability draw, let \(U=0.70\) be a uniform random number between 0 and 1, and select the smallest count whose cumulative probability is at least \(U\). Calculate enough probabilities to identify the draw.
For zero, \(P(Y=0)=\exp(-2.4)\approx0.09072\). For one, \(P(Y=1)=\exp(-2.4)(2.4)\approx0.2177\), so the cumulative probability through one is \(0.09072+0.2177\approx0.3084\). For two, \(P(Y=2)=\exp(-2.4)(2.4^2/2)\approx0.2613\), giving a cumulative probability through two of approximately \(0.5697\).
For three, \(P(Y=3)=\exp(-2.4)(2.4^3/6)\approx0.2090\). Therefore, the cumulative probability through three is approximately \(0.5697+0.2090=0.7787\). Since \(0.5697<0.70\leq0.7787\), the draw is \(Y=3\). A different uniform draw could produce a different count; this randomness is the point of drawing rather than rounding the mean.
Diagnostics, alternatives, and limitations
Diagnostics should assess both the fitted mean structure and whether the Poisson distribution is credible for the observed counts. Inspect observed and fitted counts across important predictor patterns, compare the observed and model-implied frequency of zeros and large counts, and look for systematic patterns in residuals against fitted means and predictors. A substantial mismatch between observed variation and the Poisson model’s conditional variance is a warning, although simple summaries can be misleading when the mean varies across cases. Sparse predictor combinations and very small observed counts can also make coefficient estimates unstable.
Do not interpret a successful numerical fit as proof that the missing values are recoverable without bias. Imputed values depend on the predictors included, the assumed count distribution, the missingness assumptions, and the information in observed cases. Poor overlap—such as predictor combinations among missing cases with almost no comparable observed cases—makes the conditional predictions especially dependent on extrapolation. Sensitivity analyses may be needed if plausible departures from the missing-at-random assumption matter for the scientific conclusion.
The main alternatives address different problems. A linear regression conditional model may be convenient, but its unconstrained predictions can be negative or noninteger; truncating or rounding them does not create a coherent count distribution. A Bernoulli model is appropriate for a binary outcome, not an unrestricted count. Other count models may be more suitable when Poisson variation or the treatment of zeros is inadequate, but the choice and interpretation depend on the specific departure. Predictive mean matching is a donor-based alternative that selects observed values; it has different strengths and limitations and should not be confused with drawing from a Poisson conditional distribution.
Confirm that the target is a nonnegative count over a meaningful period or exposure, then choose relevant predictors and justified transformations.
Use observed values of the target to estimate the log-mean relation, then check fit, variation, zeros, sparse patterns, and plausible extrapolation.
Use its predictor values to obtain a positive conditional mean, allow for parameter uncertainty in the imputation procedure, and draw an integer-valued count.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A fitted coefficient for a binary predictor is \(0.6931\). What multiplicative change in the conditional mean does it represent?
- A fitted Poisson mean is 1.8. Is 1.8 a valid imputed count? What is the appropriate kind of imputation?
- Why can a linear regression model be problematic as a conditional model for an unrestricted count?
- What does a large discrepancy between observed and model-implied count variation suggest, and what should a practitioner do next?
Solutions. (1) \(\exp(0.6931)\approx2.000\): at fixed values of the other predictors, the conditional mean for predictor value 1 is twice that for the reference value. (2) No. It is a fitted mean, not an integer-valued observation; draw a nonnegative integer from the specified conditional count distribution, with parameter uncertainty represented in the multiple-imputation procedure. (3) It can predict negative or fractional values, and rounding such predictions does not model the count distribution or preserve its uncertainty. (4) It is evidence that the conditional Poisson distribution may not represent the data adequately. Examine fit across predictor patterns and consider whether another justified count model is needed rather than treating the Poisson specification as automatic.