Deriving Ignorability From MAR And Parameter Distinctness

Why the likelihood factorization matters

When follow-up outcomes are missing, a likelihood analysis may model both the outcomes and the process that determines which outcomes are recorded. Under particular conditions, the observed-data likelihood separates into an outcome term and a dropout term. The outcome term can then be used for likelihood-based inference about the outcome parameters without fitting the dropout model. The justification is a specific factorization, not simply the observation that missingness is related to recorded data.

The conditions have separate roles. Missing at random (MAR) lets the response-pattern probability be taken outside the sum or integral over unobserved outcomes. Distinct outcome and dropout parameters then ensure that this response term does not constrain the outcome parameter. If either condition fails, the factorization may not justify omission.

Definition: Let \(Y\) denote the full outcome vector for a participant, \(R\) its response pattern, and \(X\) observed covariates. For a given pattern \(r\), write \(Y=(Y_{\mathrm{obs}},Y_{\mathrm{mis}})\), with the observed and missing components determined by \(r\). An outcome model is \(f_\theta(y\mid x)\), and a response-pattern model is \(\pi_\psi(r\mid y,x)\). The parameters \(\theta\) govern the outcome distribution; \(\psi\) govern the response pattern.

The notation conditions on \(X\), which is treated as observed. In a sample of independent participants, the full-data likelihood contribution for participant \(i\) is the product \(f_\theta(y_i\mid x_i)\pi_\psi(r_i\mid y_i,x_i)\). The observed-data contribution is obtained by summing over missing outcomes for discrete variables or integrating over them for continuous variables.

Deriving the observed-data factorization

For a fixed observed pattern \(r\), the observed-data likelihood contribution is

L_{\mathrm{obs}}(\theta,\psi) = \int f_\theta(y_{\mathrm{obs}},y_{\mathrm{mis}}\mid x) \pi_\psi(r\mid y_{\mathrm{obs}},y_{\mathrm{mis}},x) \,d y_{\mathrm{mis}}.

For discrete outcomes, replace the integral by a sum over all possible values of \(Y_{\mathrm{mis}}\). MAR, for this pattern and the stated conditioning variables, means that the response-pattern probability does not depend on the missing outcomes once the observed outcomes and covariates are fixed: \(\pi_\psi(r\mid y_{\mathrm{obs}},y_{\mathrm{mis}},x)=\pi_\psi(r\mid y_{\mathrm{obs}},x)\). Consequently, that probability is constant with respect to the integration variable and can be moved outside the integral.

\begin{aligned} L_{\mathrm{obs}}(\theta,\psi) &= \int f_\theta(y_{\mathrm{obs}},y_{\mathrm{mis}}\mid x) \pi_\psi(r\mid y_{\mathrm{obs}},x)\,d y_{\mathrm{mis}}\\ &= \pi_\psi(r\mid y_{\mathrm{obs}},x) \int f_\theta(y_{\mathrm{obs}},y_{\mathrm{mis}}\mid x)\,d y_{\mathrm{mis}}\\ &= \pi_\psi(r\mid y_{\mathrm{obs}},x) f_{\theta,\mathrm{obs}}(y_{\mathrm{obs}}\mid x). \end{aligned}

The last integral is the marginal density of the observed outcomes under the outcome model. It sums or integrates over the missing components; it does not discard the participant’s observed outcomes.

Theorem: Suppose the full-data model factors as \(f_\theta(y\mid x)\pi_\psi(r\mid y,x)\), MAR holds for the observed pattern, and the parameter space is a Cartesian product \(\Theta\times\Psi\), with no constraints linking \(\theta\) and \(\psi\). Then the observed-data likelihood factors as \(L_{\mathrm{obs}}(\theta,\psi)=L_{\mathrm{out}}(\theta)L_{\mathrm{drop}}(\psi)\), where each factor may depend on the observed data. Likelihood-based inference for \(\theta\) can therefore use \(L_{\mathrm{out}}(\theta)\) without specifying or estimating the dropout model, subject to the usual model and regularity requirements.

For independent participants, the factorization applies to each contribution and hence to their product. The dropout factor can depend on observed histories and patterns; it is not necessarily constant across participants. What matters is that it depends on \(\psi\), not \(\theta\), and that the allowed parameter values for \(\theta\) do not depend on \(\psi\). The maximization then separates: maximizing the product over \((\theta,\psi)\in\Theta\times\Psi\) gives the same outcome-parameter maximizer as maximizing the outcome factor alone.

Worked examples of the factorization

Worked Example: A Continuous Outcome With a Missing Follow-Up

Suppose a hypothetical participant has baseline outcome \(Y_1\) recorded as 2.1 units and a later outcome \(Y_2\) missing. For illustration, assume the outcome model has \(Y_1\sim N(\mu,1)\), and the dropout model is MAR with probability 0.30 of this response pattern conditional on the recorded \(Y_1\). Let the distinct parameter values be \(\mu=2.0\) and a dropout parameter \(\psi\) that gives this probability.

The observed-data contribution is \(\int f_\mu(2.1,y_2)\pi_\psi(R=(1,0)\mid 2.1,y_2)\,d y_2\). By MAR the response probability is 0.30 for every \(y_2\), so

\begin{aligned} L_{\mathrm{obs}}(\mu,\psi) &=0.30\int f_\mu(2.1,y_2)\,d y_2\\ &=0.30 f_\mu(2.1)\\ &=0.30\left[\frac{1}{\sqrt{2\pi}} \exp\left\{-\frac{(2.1-2.0)^2}{2}\right\}\right]\\ &\approx 0.30(0.39695)=0.1191. \end{aligned}

The density is approximately 0.39695 per outcome unit, and the likelihood contribution is approximately 0.1191 per outcome unit, multiplied by the response-pattern probability. Because the factor 0.30 contains no \(\mu\), maximizing this contribution over \(\mu\) gives the same result as maximizing \(f_\mu(2.1)\). The missing \(Y_2\) has been integrated out; it was not replaced by a value or ignored in the full-data model.

Worked Example: Summing Over a Missing Binary Outcome

Consider a hypothetical two-visit study. Let \(P(Y_1=1)=0.60\), and let \(P(Y_2=1\mid Y_1=1)=0.70\). A participant has \(Y_1=1\), while \(Y_2\) is missing. Suppose the probability of this response pattern, conditional on \(Y_1=1\), is 0.25 and does not depend on \(Y_2\). The response mechanism is MAR for this pattern, and suppose its parameter is distinct from the outcome parameters.

The contribution sums the joint outcome probability over the two possible values of \(Y_2\), then multiplies by the response probability:

\begin{aligned} L_{\mathrm{obs}} &=0.25\{P(Y_1=1,Y_2=0)+P(Y_1=1,Y_2=1)\}\\ &=0.25\{0.60(0.30)+0.60(0.70)\}\\ &=0.25(0.18+0.42)\\ &=0.25(0.60)=0.150. \end{aligned}

The outcome factor is 0.60, the marginal probability of the recorded \(Y_1=1\), and the dropout factor is 0.25. The sum \(0.30+0.70=1\) uses the conditional distribution of the missing \(Y_2\); it demonstrates why the outcome factor is the marginal probability of what was observed.

Where the conditions enter—and what fails without them

The derivation identifies the exact contribution of each assumption. MAR is used only in the step that takes \(\pi_\psi\) outside the sum or integral over \(Y_{\mathrm{mis}}\). Parameter distinctness is used after factorization: it makes the response-pattern factor irrelevant to likelihood maximization over \(\theta\). More precisely, distinctness requires a parameter space that permits every \(\theta\in\Theta\) with every \(\psi\in\Psi\), rather than a constraint linking the two.

Worked Example: MAR Does Not Rescue a Shared Parameter

Let a hypothetical binary outcome satisfy \(Y\sim\mathrm{Bernoulli}(\theta)\), and let \(R=0\) indicate that \(Y\) is missing. Suppose \(P(R=0\mid Y;\theta)=\theta\) for either value of \(Y\). This response probability is independent of \(Y\), so it is MAR. But the same \(\theta\) governs both outcome and response probabilities.

For an observation with \(R=0\), the observed-data likelihood contribution is

\begin{aligned} P(R=0;\theta) &=\sum_{y=0}^1 P(Y=y;\theta)P(R=0\mid Y=y;\theta)\\ &=\{(1-\theta)+\theta\}\theta\\ &=\theta. \end{aligned}

The outcome distribution alone contributes \(\sum_yP(Y=y;\theta)=1\), which contains no information about \(\theta\) for this missing outcome. Omitting the response factor would therefore change the likelihood from \(\theta\) to 1. MAR allowed the response probability to be moved outside the sum, but it did not make that factor irrelevant to \(\theta\). The required parameter distinctness fails.

Worked Example: MNAR Prevents the MAR Factorization

Now let \(Y\sim\mathrm{Bernoulli}(\theta)\), but suppose the probability of a missing outcome depends on its value: \(P(R=0\mid Y=0)=0.10\) and \(P(R=0\mid Y=1)=0.60\). For a participant with \(Y\) missing, take \(\theta=0.40\). The response mechanism is not MAR because the missing outcome changes the response probability.

The observed-data probability for \(R=0\) is

\begin{aligned} P(R=0;\theta) &=(1-\theta)(0.10)+\theta(0.60)\\ &=0.10+0.50\theta\\ &=0.10+0.50(0.40)=0.300. \end{aligned}

The outcome distribution sums to one when \(Y\) is unobserved, but the actual response-pattern contribution remains \(0.10+0.50\theta\), which depends on the outcome parameter. It cannot be pulled outside the sum as a factor independent of the missing outcome. Even with separately named parameters, an MNAR response probability can carry information about \(\theta\) through the unobserved outcome distribution.

Interpretation, diagnostics, and limits

Likelihood-based ignorability means that, under the stated model and assumptions, a separate response-pattern model need not be included to obtain the likelihood for \(\theta\). It does not mean that the data are complete, that participants with missing outcomes can be removed, or that any analysis using only observed outcomes is valid. The outcome likelihood must correctly represent the marginal distribution of the observed outcome components, including their dependence where relevant.

MAR is a condition on the response probability after conditioning on the specified observed outcomes and covariates. It is not established merely because dropout is predictable from recorded information. In general, observed data alone cannot distinguish MAR from all alternatives that depend on unobserved outcomes. Thus a factorization is justified by a modeling assumption, not certified simply by a successful fit. The earlier material on Ignorability Conditions For Likelihood-Based Outcome Analysis sets out the broader conditions; here the algebra shows how MAR and distinctness supply the factorization.

A useful diagnostic is to inspect the proposed joint model before omitting anything. Verify that the response probability is constant as the missing outcomes vary, given the pattern’s observed information. Then check that the outcome and dropout parameters are not shared or constrained. A dependence on a missing outcome, a shared parameter, or a joint parameter constraint invalidates the simple omission argument. Poor fit of the outcome model is a separate problem: factorization does not make a misspecified outcome distribution appropriate.

Situation	What the likelihood derivation gives	Can the dropout factor be omitted for inference on \(\theta\)?
MAR and distinct parameters	Outcome factor times a response factor in \(\psi\)	Yes, under the stated model and regularity conditions
MAR but shared or constrained parameters	Factors may separate algebraically, but the response factor can depend on \(\theta\)	No, not from this argument
MNAR	Response probability varies with missing outcomes inside the integral or sum	No, not from this argument
Complete-case analysis	Uses only participants with all outcomes observed	Not implied by likelihood ignorability

This result is about likelihood-based inference. Bayesian omission requires additional care: distinct likelihood parameters alone do not ensure that the prior links neither parameter. A common sufficient formulation is a prior that factors independently across the outcome and dropout parameters; without an appropriate prior factorization, the posterior need not separate.

Key takeaway: MAR makes the response-pattern probability constant over the missing outcomes in the observed-data sum or integral. Parameter distinctness makes the resulting response factor irrelevant to likelihood inference about the outcome parameter. Neither condition can substitute for the other.

Check Your Understanding

Answer each question, then compare with the solutions below.

In the observed-data likelihood integral, which assumption permits the response-pattern probability to be moved outside the integral?
Suppose MAR holds and the likelihood separates, but the parameter space imposes \(\psi=\theta\). Is the dropout factor necessarily irrelevant to likelihood inference for \(\theta\)?
A binary outcome is missing, and \(P(R=0\mid Y=0)=P(R=0\mid Y=1)=0.20\). Is this response probability MAR with respect to \(Y\)? If the dropout parameter is distinct, what is the response factor for this pattern?
Explain why likelihood ignorability does not imply that complete-case analysis is justified.

Solutions. (1) MAR: given the observed outcomes and covariates, the response-pattern probability does not vary with the missing outcomes being integrated over. (2) No. The constraint links the parameters, so the response factor may vary with \(\theta\) even if the algebraic factors separate. (3) Yes. The response probability is constant across the possible missing outcome values, and the response factor is 0.20; with a distinct parameter it does not alter likelihood inference for the outcome parameter. (4) Ignorability retains each participant’s observed outcome contribution and integrates over missing components under the outcome model. Complete-case analysis discards partial outcome information and requires its own justification.