Why Model Choice Begins With the Variable
In MICE, each incomplete variable is updated using a conditional model: the variable being imputed is treated as the response, and other variables provide predictors. The first model-choice question is not which regression procedure is most familiar. It is what values can this response take, and what scale do those values represent? A model that assigns substantial probability to impossible values cannot be repaired merely by running more iterations.
Let \(X_j\) denote the variable currently being updated, and let \(W_j\) denote the predictors used in its conditional model. For records with \(X_j\) observed, the model describes the conditional distribution of \(X_j\) given \(W_j\). That fitted conditional distribution is then used to generate imputations for records where \(X_j\) is missing. The predictor set and the model family are separate choices: a useful predictor set cannot make an unsuitable response distribution appropriate.
This matching is a necessary modeling check, not a guarantee that the entire imputation procedure is valid. The conditional relationships still need to be plausible, the predictors need to be chosen with the analysis in mind, and the missing-data assumptions require separate consideration. In particular, choosing the right distribution family does not establish the missingness assumptions discussed in Missing Data.
Match the Response Scale to a Conditional Distribution
A variable’s support is the set of values it can take. The support immediately rules out some candidate models. A binary variable has support \(\{0,1\}\); a count has support \(\{0,1,2,\ldots\}\); an ordinal variable has ordered categories; and a nominal variable has categories without an inherent order. A continuous variable may have a wide or restricted range, depending on how it is defined.
| Response type | Candidate conditional model | Key matching question |
|---|---|---|
| Continuous | Normal linear regression is a common candidate | Are values roughly compatible with a conditionally normal distribution and its variance assumptions? |
| Binary | Bernoulli model with logistic regression | Does the model keep conditional probabilities between 0 and 1? |
| Ordinal | Ordinal regression, often a cumulative-link model | Is category order meaningful, and is the model’s shared-effect assumption defensible? |
| Nominal | Multinomial regression | Are categories genuinely unordered, and is there enough information to estimate category probabilities? |
| Count | Poisson or negative-binomial regression; binomial regression for fixed-trial counts | Are counts nonnegative integers, and is dispersion consistent with the proposed distribution? |
These are families of models, not automatic decisions. A continuous variable may be strongly skewed or bounded, a count may be overdispersed relative to a Poisson model, and a categorical variable may have sparse levels. Such features can make a basic model inadequate even when its support is technically correct.
Worked Choices for Common Variable Types
Worked Example: A Binary Clinical Indicator
Suppose a hypothetical dataset contains a binary indicator for whether a patient experienced a particular symptom during follow-up: \(1\) means yes and \(0\) means no. A candidate model is Bernoulli logistic regression, with \(\operatorname{logit}\{P(X_j=1\mid W_j)\}=\eta\). Assume, only for illustration, that \(\eta=-2+0.05(\text{age})+0.8(\text{group indicator})\). For a 50-year-old in group 1, the linear predictor is \(\eta=-2+0.05(50)+0.8(1)=1.3\).
The corresponding probability is \(P(X_j=1\mid W_j)=1/(1+\exp(-1.3))\approx0.786\). Thus the conditional distribution assigns probability about 0.786 to 1 and 0.214 to 0. Both possible imputations are within the response’s support. This makes the Bernoulli-logistic family a natural candidate; it does not establish that the predictors or fitted relationship are adequate. For example, very sparse outcomes or predictions near 0 and 1 warrant additional scrutiny.
Worked Example: Ordered Categories Without Equal Spacing
Suppose a hypothetical pain variable has four categories: none, mild, moderate, and severe. Their order is meaningful, but the difference between none and mild is not necessarily equal to the difference between moderate and severe. An ordinal model preserves the ordering without treating category labels as equally spaced numbers.
For illustration, consider a cumulative-logit model \(\operatorname{logit}\{P(X_j\leq k\mid W_j)\}=\theta_k-\eta\), where \(k\) indexes the first three cut points. Suppose the cut points are \(-1, 0.5, 2\), and a record has \(\eta=0.4\). Applying the logistic function to each cut point minus \(\eta\) gives cumulative probabilities approximately \(0.198,\ 0.525,\ 0.832\).
The probabilities of the four categories are the successive differences, including the end categories: \(0.198,\ 0.525-0.198=0.327,\ 0.832-0.525=0.307,\ 1-0.832=0.168\). They are nonnegative and sum to 1. The ordinal family is appropriate only if the ordering is real and its shared slope, or proportional-odds, assumption is reasonably tenable. If category effects vary substantially across cut points, that assumption may be too restrictive.
Worked Example: Unordered Categories
Imagine a hypothetical variable recording one of three mutually exclusive discharge destinations: home, rehabilitation, or another facility. If there is no scientifically meaningful ranking among these labels, the response is nominal rather than ordinal. A multinomial regression is a natural candidate because it models probabilities for the categories without imposing an order.
Suppose a fitted model for one record gives probabilities \(0.60\) for home, \(0.25\) for rehabilitation, and \(0.15\) for another facility. These are valid category probabilities because each is between 0 and 1 and their sum is \(0.60+0.25+0.15=1.00\). An imputation must be one of the three labels, selected according to those probabilities; it should not be a numerical average of category codes. If one category is rare, its conditional probabilities may be difficult to estimate, so category counts and fitting warnings matter alongside the support check.
Worked Example: Checking a Count Model
Consider a hypothetical count of visits in a fixed follow-up period, with six observed values \(0,1,1,2,3,5\). The sample mean is \(\bar{x}=(0+1+1+2+3+5)/6=12/6=2\) visits. The sum of squared deviations from 2 is \(4+1+1+0+1+9=16\), so the sample variance is \(s^2=16/(6-1)=3.20\) visits squared.
A basic Poisson model has conditional variance equal to its conditional mean. Here the sample variance exceeds the sample mean, which is a warning that a simple Poisson model may not describe the data well; it is not, with only six values, decisive evidence against it. A negative-binomial model allows conditional variance to exceed the mean and may be a more suitable candidate if overdispersion persists after accounting for predictors. If the variable instead counted successes out of a fixed number of trials, a binomial model could be more appropriate. A count model must also respect nonnegative integer support.
A Support Check and Why It Matters
The support argument can be made directly. If a model assigns positive probability to values the variable cannot take, then it can generate invalid imputations. For example, a normal distribution has support over all real numbers. If a count were modeled as \(X_j\sim N(2,2^2)\), the probability of a negative value would be \(P(X_j<0)=P\{(X_j-2)/2<-1\}=\Phi(-1)\approx0.1587\). That is about a 15.9% probability of an impossible negative count under this illustrative model.
This calculation does not mean that every continuous response must be normally distributed, nor that a normal model is always inappropriate for a variable whose observed values are positive. It shows why the response scale and distributional assumptions must be examined rather than assumed. In practice, inspect the observed values and compare the model’s possible predictions and draws with the variable’s actual definition.
Assumptions, Diagnostics, and Limits
A model choice should be accompanied by an explicit account of what it assumes. A normal linear model assumes a conditionally normal response around a linear predictor, with a variance structure appropriate to the model. A Bernoulli-logistic model represents a binary response with a conditional probability described by the logistic regression structure. An ordinal cumulative-link model uses ordered categories and, in a proportional-odds version, a common predictor effect across cumulative cut points. A multinomial model treats categories as unordered and models their conditional probabilities. A Poisson model assumes a conditional mean-variance relationship characteristic of the Poisson distribution; a negative-binomial alternative accommodates greater dispersion. These statements describe model assumptions, not properties guaranteed by the variable’s label.
Useful checks include comparing observed and imputed ranges, inspecting category frequencies, checking for extreme fitted probabilities, and examining whether count variability is much larger than the fitted mean. For an ordinal response, check whether the ordering is substantively meaningful and whether the common-effect assumption seems plausible. For any family, investigate convergence warnings, sparse predictor patterns, and predictions that are implausible in context. A model can run without an error message and still be a poor match.
The main alternatives differ in what structure they retain. Treating an ordinal variable as nominal avoids imposing a shared ordinal structure but discards its ordering. Treating it as a numeric continuous variable imposes spacing and arithmetic interpretations that may not be meaningful. A Poisson model is simpler than a negative-binomial model but can underrepresent variability when counts are overdispersed. A normal model is a useful baseline for many continuous responses, but a poor choice when its conditional distribution or support conflicts with the data. There is no universally best family independent of the variable definition and observed conditional patterns.
Finally, MICE uses a collection of conditional models. Matching each model to its response type does not ensure that all conditionals correspond to one coherent joint distribution. Nor does it guarantee adequate exploration, an issue considered in the earlier material on FCS diagnostics. Model selection is one important part of imputation design, not a substitute for evaluating the resulting chains and imputations.
Common Mistakes
- Using a normal model for every response. This can produce impossible values for binary or count variables. Choose a family whose support matches the response.
- Calling every category variable nominal. If the categories have a genuine order, an ordinal model can use that information; do not invent an order when none exists.
- Treating ordinal codes as equally spaced measurements. Codes such as 1, 2, 3, and 4 label ordered categories but do not establish equal distances between them.
- Assuming a Poisson model fits because the response is a count. Counts can be overdispersed or generated from a fixed number of trials; check the sampling structure and variability.
- Assuming the right family solves every problem. Sparse categories, inappropriate predictors, or incompatible conditional relationships can remain even when support is respected.
Check Your Understanding
For each item, identify a reasonable starting family and state the main check that could change your choice.
- A variable records whether a hypothetical participant completed a program: yes or no.
- A survey response is rated as poor, fair, good, or excellent.
- A variable records the number of emergency visits in a year, with no fixed maximum number.
- A variable records the number of correct answers on a test containing exactly 20 questions.
Solutions. (1) A Bernoulli model with logistic regression is a reasonable starting point because the response has two possible values. Check for sparse outcomes or unstable fitted probabilities. (2) An ordinal model is a reasonable starting point because the four labels have a natural order; assess whether its ordering and, for a proportional-odds model, shared-effect assumption are plausible. (3) A Poisson model is a possible starting point because the response is a nonnegative count with no stated fixed upper limit; check for overdispersion and consider a negative-binomial model if it is substantial. (4) A binomial model may be appropriate because the count is the number of successes among 20 trials, provided the trial structure and binomial assumptions are reasonable. A generic Poisson model would ignore the known upper bound of 20.