Why Missing Omics Layers Require a Joint Likelihood
An integrated molecular study may plan to measure several assay layers on each participant, yet some samples fail quality control, some assays are unavailable, and some measurements are not collected. Restricting analysis to participants with every layer observed is one response, but it discards usable measurements and can change the composition of the analyzed sample. Full-information maximum likelihood (FIML) offers another approach: specify a joint model for the layers and use each participant’s observed components in the likelihood.
The method does not recover unmeasured values as if they had been observed. Rather, it uses the probability distribution implied by the joint model for the components that are observed. Its validity depends on a correctly specified model and an appropriate assumption about why measurements are missing.
Observed-Data Likelihood and Missing at Random
Suppose participant \(i\) has \(q\) planned molecular outcomes, collected into the vector \(Y_i=(Y_{i1},\ldots,Y_{iq})^\mathsf{T}\). Let \(X_i\) denote observed predictors, such as genotype and prespecified covariates. Let \(R_i\) describe which outcomes are observed, and let \(S_i\) be the set of observed outcome indices. For example, \(S_i=\{1,3\}\) means that layers 1 and 3 are observed for participant \(i\), but the other layers are missing.
A stronger condition, missing completely at random (MCAR), requires the observation pattern to be independent of the outcomes and modeled predictors. Thus, a pattern can be independent of assay errors yet fail to be MCAR if it depends on genotype. In that case it may still be MAR when genotype is observed and included in the conditioning set.
The standard likelihood-based treatment also assumes that the parameters governing the outcome model are distinct from parameters governing the missingness process. Under MAR and this parameter distinctness, the missingness-process factor can be ignored for likelihood inference about the outcome-model parameters. This is called an ignorable missingness mechanism. MAR alone does not make a misspecified outcome model valid, nor does it guarantee that every parameter is identifiable from the observed patterns.
Gaussian Joint Models and FIML
A commonly used model for continuous molecular traits is a multivariate normal regression. Write
where \(B^\mathsf{T}X_i\) is the vector of conditional means and \(\Sigma\) is the \(q\times q\) residual covariance matrix. The model assumes independent participants conditional on their predictors, a correctly specified mean structure, and a residual distribution adequately represented by the multivariate normal model. The covariance matrix must be positive definite for the usual density to be defined.
For participant \(i\), let \(\mu_i=B^\mathsf{T}X_i\). The observed components have the marginal distribution
Here \(\mu_{i,S_i}\) selects the corresponding entries of the mean vector, and \(\Sigma_{S_i,S_i}\) selects the matching rows and columns of the covariance matrix. FIML multiplies these observed-data densities across participants:
A complete record contributes a \(q\)-dimensional density; a record with one observed layer contributes that layer’s marginal density. If no outcome is observed for a participant, the outcome likelihood contribution is 1, so that record supplies no direct information about \(B\) or \(\Sigma\). Under an ignorable MAR mechanism, the likelihood for the observed outcome data is the relevant likelihood for estimating the outcome-model parameters.
Worked Example: A Complete Record and a Single-Layer Record
Suppose two centered molecular outcomes have the illustrative model \(Y_i\sim N_2((0,0)^\mathsf{T},\Sigma)\), with known covariance matrix
For a hypothetical participant with both outcomes observed as \((1,2)\), the determinant is \(1-(0.6)^2=0.64\), and the quadratic form is
The bivariate normal density contribution is therefore
For a different hypothetical participant with only the first outcome observed, \(Y_{i1}=1\), the contribution is the marginal density \(f_1(1)=\exp(-1/2)/\sqrt{2\pi}\approx0.2420\). It is not the conditional density of the first outcome given a missing second outcome. The unobserved value is integrated out, leaving the first outcome’s marginal distribution.
Why Marginalization Gives the Correct Contribution
The observed-data likelihood follows directly from the joint model. For a participant with observed coordinates \(Y_{i,S_i}\) and missing coordinates \(Y_{i,M_i}\), where \(M_i\) is the complement of \(S_i\), the probability density of what was observed is obtained by integrating over every possible value of the missing coordinates:
For a multivariate normal model, this integral is the multivariate normal marginal with the selected mean entries and covariance submatrix. It is not necessary to fill in one guessed value for each missing assay. In fact, treating a single imputed value as known would fail to account for the uncertainty about the missing value.
The full-data likelihood can be written as an outcome-model factor and a missingness-pattern factor. Under MAR, the latter depends on observed outcomes and predictors, but not on the missing outcomes. If its parameters are distinct from the outcome-model parameters, it does not alter likelihood maximization for the outcome model. This justification does not apply when missingness depends on unobserved outcomes after conditioning on observed information, or when the outcome and missingness parameters are not distinct.
Worked Comparison With Complete-Case Analysis
Worked Example: Estimating Two Layer Means
Consider five hypothetical participants with two molecular outcomes, RNA and protein. Assume independent normal outcomes with known variance 1 for each layer, and unknown means \(\mu_1\) and \(\mu_2\). The protein assay is observed with probability that can vary with the observed RNA value; for example, its observation probability is 0.8 when RNA is at most 2 and 0.4 otherwise. This mechanism depends on an observed outcome, so it is MAR, not MCAR.
| Participant | RNA | Protein |
|---|---|---|
| 1 | 1 | 2 |
| 2 | 2 | 4 |
| 3 | 3 | Missing |
| 4 | 4 | Missing |
| 5 | Missing | Missing |
Under this independent, known-variance model, the likelihood separates by layer. For a normal sample with known variance, the maximum-likelihood estimate of a mean is the average of the observed values for that outcome. FIML therefore gives
Complete-case analysis retains only participants 1 and 2. Its RNA mean is \((1+2)/2=1.5\), and its protein mean is \((2+4)/2=3\). Thus, in this example the complete-case RNA estimate differs from the FIML estimate, while the protein estimates happen to agree. That agreement is not a general property: it results from these particular observed protein values and the specified independent-outcome model. If residual covariance were unknown or nonzero, the joint likelihood would no longer separate this way.
Genotype Effects and Observation Patterns
FIML also applies to regression coefficients. A key distinction is whether incomplete records are simply less precise or whether their selection changes which genotype values contribute. Complete-case analysis discards all partially observed participants, even when their observed layer is informative about its own genotype association.
Worked Example: Genotype-Dependent Assay Availability
Suppose \(G_i\) is an observed genotype dosage and RNA and protein follow separate linear regressions with independent normal errors of known variance. For RNA, the hypothetical data are \((G,Y_{\mathrm{RNA}})=(0,1),(1,4),(2,4),(3,8)\). Protein is observed only for the first two participants, with values 2 and 5. Imagine the protein observation probabilities depend on genotype, with positive probability at each genotype value. The realized pattern can occur under that mechanism.
Because genotype is observed and the pattern depends on genotype rather than on unobserved assay values after conditioning on genotype, the mechanism is MAR conditional on \(G\). It is not MCAR: the probability of observing protein varies with genotype. If the assay errors are independent of that pattern, this still does not make the pattern MCAR.
With known error variance and the stated independent-layer model, the FIML estimate of the RNA regression is the ordinary least-squares fit using all four observed RNA values. The genotype mean is 1.5, the RNA mean is 4.25, and the centered cross-product and genotype sum of squares are
Consequently, the slope is \(10.5/5=2.1\) RNA units per genotype-dose unit, and the intercept is \(4.25-(2.1)(1.5)=1.10\). Complete-case analysis uses only the first two participants, producing slope \((4-1)/(1-0)=3.00\) RNA units per genotype-dose unit and intercept 1.00. The two estimates differ because the complete-case subset has a different genotype range and different RNA values. This numerical difference does not prove that either estimate is biased; it shows why the observation process and the target model must be considered.
In this example, the protein regression has only two observed genotype values, so its fitted line can be calculated but is weakly supported by data. A fitted association based on limited genotype range should not be mistaken for strong empirical evidence across the full range. In general, parameters for a layer or cross-layer covariance may be poorly estimated or unidentifiable if the observed combinations do not provide enough information.
FIML and Complete Cases: What the Comparison Means
Complete-case analysis is easy to define: retain only participants observed on every variable required by the analysis, then fit the intended model to those participants. FIML instead evaluates the joint model for each participant’s observed subset. Under a correctly specified model and ignorable MAR, FIML can use incomplete records without treating missing values as observed. It often retains more information, but it is not automatically unbiased or more precise in every finite sample.
| Feature | Full-information maximum likelihood | Complete-case analysis |
|---|---|---|
| Records used | Each record contributes its observed outcomes | Only records complete for all required variables |
| Missingness requirement for standard validity | MAR given observed outcomes and predictors, plus distinct parameters and a correct outcome model | MCAR is a common sufficient condition for retaining the original marginal composition; other settings require analysis-specific justification |
| Model dependence | Uses the specified joint outcome distribution, including its covariance structure | Uses the model fitted to the selected complete subset |
| Main limitation | Can be sensitive to misspecification, weak identification, or nonignorable missingness | Can lose precision and can be biased when complete-case inclusion is related to modeled outcomes |
A complete-case estimate may still be appropriate under particular conditions, including suitable selection mechanisms or a model whose target parameter is unaffected by selection. Conversely, MAR does not guarantee that complete-case estimates are valid. The relevant question is whether the retained records support the estimand and model being used.
Practical Checks and Common Pitfalls
Before interpreting an FIML result, identify the planned layers, the observed pattern for each participant, the predictors included in the model, and the assumptions connecting the observation process to those observed variables. Then check whether the data provide enough observed combinations to estimate the parameters of interest. For example, estimating residual covariance between two layers requires participants with both layers observed; singleton records alone cannot directly reveal their residual covariance.
- Pitfall: Calling genotype-related missingness MCAR because assay errors are independent of it. Independence from errors alone is insufficient. If the observation pattern depends on genotype, it is not MCAR unless genotype is fixed or conditioned on under an explicitly stated convention. It may be MAR conditional on the observed genotype.
- Pitfall: Treating MAR as “missingness is harmless.” MAR is a conditional statement about the observation process. Estimates still rely on a suitable outcome model, parameter distinctness, and enough information to identify the target parameters.
- Pitfall: Filling in a missing value and then treating it as known. FIML does not require a single completed value. It integrates over missing outcomes through their model distribution.
- Pitfall: Assuming FIML guarantees a better estimate than complete-case analysis. FIML can use more observed data, but misspecification, sparse overlap between layers, or nonignorable missingness can undermine its results.
Check Your Understanding
Use the model assumptions as well as the observed pattern when answering.
- A participant has RNA and metabolite measurements but no protein measurement. Which distribution supplies that participant’s FIML contribution?
- Protein availability depends on observed genotype, while genotype is included in the outcome model. Is the pattern necessarily MCAR? State the relevant missingness condition.
- Why can a record with one observed layer inform that layer’s mean but provide no direct information about the residual covariance between two layers?
- In the two-layer mean example, why did the FIML estimate of the RNA mean use four records while the complete-case estimate used only two?
- Name two assumptions, beyond observing some data for each layer, that are needed for the standard FIML interpretation.