Pattern-Mixture Models For MNAR

Representing Data by Missingness Pattern

A pattern-mixture model starts by grouping individuals according to which variables are observed. This is a useful alternative way to describe the joint distribution of outcomes and missingness: rather than first modeling the complete-data distribution and then how response depends on the data, it describes the probability of each pattern and the distribution of data within that pattern. Its key limitation is also central to its use: values missing in a pattern cannot be learned directly from that pattern's observed values.

Let \(Y=(Y_1,\ldots,Y_p)\) denote the complete data for an individual, and let \(R\) denote the missingness pattern. Write \(r\) for a particular pattern, \(O(r)\) for the set of variables observed in that pattern, and \(M(r)\) for the set that are missing. Thus \(Y_{\mathrm{obs}}(r)\) and \(Y_{\mathrm{mis}}(r)\) are the corresponding observed and unobserved components. For example, with two variables, the pattern in which \(Y_1\) is observed and \(Y_2\) is missing has \(O(r)=\{1\}\) and \(M(r)=\{2\}\).

The factorization is an identity, not an assumption that patterns arise randomly or that their distributions are the same. Pattern-mixture models become substantive models when restrictions or probability models are placed on these factors. The pattern probabilities \(P(R=r)\) and the distributions of observed components within each pattern can be estimated from the observed data, subject to ordinary sampling and model assumptions. The conditional distribution of missing components generally cannot be estimated from those same observations without additional assumptions.

For the model to describe the target population, the observed patterns must be represented in the data, and the pattern-specific quantities being estimated must be defined on suitable support. Modeling choices also need to be adequate for the observed data. Most importantly, any conclusion involving unobserved values depends on the chosen assumptions for their distributions. A pattern-mixture parameterization does not, by itself, make missingness ignorable or identify those distributions.

Where the Unobserved-Value Assumption Enters

For a person in pattern \(r\), the observed-data contribution to the joint distribution is $$ P(R=r)f\bigl(y_{\mathrm{obs}}(r)\mid R=r\bigr). $$ This follows by integrating over the missing components:

The second line uses the defining property of a marginal density: integrating the joint conditional density over the missing components gives the marginal density of the observed components. The conditional distribution \(f(y_{\mathrm{mis}}\mid y_{\mathrm{obs}},R=r)\) disappears from this observed-data expression. Consequently, it cannot be recovered just by fitting that expression. Estimating an overall quantity that depends on missing values requires specifying or restricting that conditional distribution.

Under a missing-at-random (MAR) restriction, the conditional distribution of missing values given the observed values does not additionally depend on the missingness pattern: \(f(y_{\mathrm{mis}}\mid y_{\mathrm{obs}},R=r)=f(y_{\mathrm{mis}}\mid y_{\mathrm{obs}})\), where the variables observed in each pattern are understood. A pattern-mixture model can impose such a restriction, but it need not. Pattern-specific departures from it can represent MNAR assumptions. The observed data alone cannot establish which unobserved-value assumption is correct.

Combining Patterns and Interpreting Sensitivity

For a variable \(Y_j\), if every pattern has a well-defined pattern-specific mean \(\mu_{jr}=E(Y_j\mid R=r)\), then the population mean is a mixture of those means:

This identity follows by partitioning the population into its mutually exclusive missingness patterns and adding the contribution from each pattern. It does not say all \(\mu_{jr}\) are observed or estimable. If \(Y_j\) is missing in pattern \(r\), its pattern-specific mean must be supplied by a model or an assumption. In more complex analyses, the same principle applies to distributions, not just means: an overall distribution is a mixture of the pattern-specific distributions, weighted by pattern proportions.

Comparison, Diagnostics, and Limitations

A selection model uses the alternative factorization \(f(y)P(R=r\mid y)\): it models the complete-data distribution and then models pattern probabilities given complete data. The pattern-mixture factorization instead models pattern probabilities and complete-data distributions within patterns. These are alternative factorizations of the same joint distribution, not competing claims that the data themselves have changed. Both require assumptions about unobserved values when the target includes them. A model that relies on MAR restrictions is another possible specification; it is not automatically justified merely because it is easier to estimate.

Since missing values are unavailable for direct checking, diagnostics cannot verify the MNAR assumptions. They can still identify problems in the parts of the model that are observed: examine whether the fitted distributions reproduce observed values within each pattern, whether pattern proportions are represented adequately, and whether conclusions are driven by sparse patterns or implausible extrapolation. For MNAR questions, report results under multiple scientifically plausible assumptions for the missing-pattern distributions. In MICE analyses, this can motivate sensitivity adjustments to imputation models, but the factorization here does not prescribe a particular imputation algorithm.

A common mistake is to treat an estimated distribution in the observed pattern as if it described people whose outcomes are missing. That step silently assumes the distributions are linked in a particular way. The correction is to state the link explicitly, calculate what it implies, and show how conclusions change under reasonable alternatives. Pattern-mixture models make this assumption visible; they do not remove the fundamental limitation that observed data alone cannot determine the distribution of values never observed.