MAR and Ignorability for Likelihood Inference

Why the Missingness Mechanism Matters

With incomplete multivariate data, the observed values do not by themselves describe how the data became incomplete. A subject’s probability of having a measurement recorded may depend on values that were observed, values that were not observed, or both. The likelihood must initially account for this recording process as well as for the model of the full data.

The important simplification is that, under suitable conditions, the likelihood for the observed data separates into a data-model factor and a missingness-mechanism factor. The first factor contains the parameter of interest for the full-data distribution. If the two parameter components are distinct, likelihood inference about that data-model parameter can use the first factor alone. This result is called ignorability of the missingness mechanism for likelihood inference.

Observed Data and MAR

Let $Y$ denote the complete data for one sampling unit, and let $R$ denote its missingness pattern. For a given pattern $r$, write $Y_{\mathrm{obs}}(r)$ for the components observed under that pattern and $Y_{\mathrm{mis}}(r)$ for the components missing. Let $f(y;\theta)$ be the full-data density or mass function, indexed by parameter $\theta$. Let $g(r\mid y;\psi)$ be the probability of pattern $r$ given the complete data, indexed by parameter $\psi$.

Definition: The missingness mechanism is missing at random (MAR) if, for every possible realized pattern $r$, the probability $g(r\mid y;\psi)$ depends on the complete data only through the components observed under $r$. That is, for all relevant $y_{\mathrm{mis}}$ and $\psi$, $$ g(r\mid y_{\mathrm{obs}},y_{\mathrm{mis}};\psi) = g(r\mid y_{\mathrm{obs}};\psi). $$ The mechanism is missing completely at random (MCAR) if this probability does not depend on $y$ at all. Thus MCAR is a special case of MAR.

MAR is pattern-specific: the set of observed components depends on the realized pattern. For example, if $Y_1$ is observed and $Y_2$ is missing, MAR permits the probability of that pattern to depend on $Y_1$, but not on the unobserved $Y_2$. It does not require missingness to be unrelated to every full-data value. The condition is understood on the support of the data model, for the parameter values under consideration.

Parameter Distinctness

MAR concerns how the mechanism depends on data values. A separate condition concerns how the model’s parameters are organized. Let $\Theta$ be the parameter space for the full-data model and $\Psi$ the parameter space for the missingness mechanism.

Definition: The data-model parameter $\theta$ and mechanism parameter $\psi$ are distinct when the joint parameter space is the Cartesian product $\Theta\times\Psi$. In particular, restrictions on $\theta$ do not constrain $\psi$, and restrictions on $\psi$ do not constrain $\theta$.

Parameter distinctness is a property of the model, not a statement that the parameters have different names or different numerical values. If the mechanism is required to use a parameter equal to a component of $\theta$, the parameters are not distinct, even if that equality is scientifically plausible. MAR and parameter distinctness are separate requirements for the likelihood factorization to justify ignoring the mechanism.

Observed-Data Likelihood and the Ignorability Result

For observed values $y_{\mathrm{obs}}$ and pattern $r$, the likelihood contribution is obtained by summing or integrating over all possible missing values. The appropriate measure may be counting measure, Lebesgue measure, or a product measure for mixed data.

L_{\mathrm{obs}}(\theta,\psi;y_{\mathrm{obs}},r) = \int f(y_{\mathrm{obs}},y_{\mathrm{mis}};\theta) g(r\mid y_{\mathrm{obs}},y_{\mathrm{mis}};\psi) \,d y_{\mathrm{mis}}.

Theorem: Suppose the full-data model and the missingness mechanism are dominated by suitable measures, the displayed observed-data likelihood is well-defined, the mechanism is MAR, and $\theta$ and $\psi$ are distinct. Then $$ L_{\mathrm{obs}}(\theta,\psi;y_{\mathrm{obs}},r) = L_{\mathrm{data}}(\theta;y_{\mathrm{obs}}) L_{\mathrm{mech}}(\psi;r,y_{\mathrm{obs}}), $$ where $$ L_{\mathrm{data}}(\theta;y_{\mathrm{obs}}) = \int f(y_{\mathrm{obs}},y_{\mathrm{mis}};\theta)\,d y_{\mathrm{mis}}, \qquad L_{\mathrm{mech}}(\psi;r,y_{\mathrm{obs}}) = g(r\mid y_{\mathrm{obs}};\psi). $$ Consequently, the missingness mechanism can be omitted for likelihood inference about $\theta$.

The theorem concerns the likelihood for the observed data, not the full-data likelihood evaluated as if the missing values had been observed. The data-model factor is the marginal density or mass of what was actually observed, obtained by integrating out the missing components.

Start with the joint model.
The complete data have density $f(y;\theta)$, and, conditional on those data, the pattern has probability $g(r\mid y;\psi)$.

Integrate out the missing values.
The observed-data likelihood is the integral of the joint density over $y_{\mathrm{mis}}$.

Apply MAR.
For the realized pattern, $g(r\mid y_{\mathrm{obs}},y_{\mathrm{mis}};\psi)$ is constant as a function of $y_{\mathrm{mis}}$.

Factor the integral.
Pull that constant mechanism probability outside the integral. The remaining integral is the marginal data-model likelihood for $y_{\mathrm{obs}}$.

Explicitly, MAR gives

\begin{aligned} L_{\mathrm{obs}}(\theta,\psi;y_{\mathrm{obs}},r) &= \int f(y_{\mathrm{obs}},y_{\mathrm{mis}};\theta) g(r\mid y_{\mathrm{obs}};\psi)\,d y_{\mathrm{mis}}\\ &= g(r\mid y_{\mathrm{obs}};\psi) \int f(y_{\mathrm{obs}},y_{\mathrm{mis}};\theta)\,d y_{\mathrm{mis}}. \end{aligned}

The factorization follows because the mechanism term is constant over the integration variable; no interchange of a limit and an integral is involved. With independent sampling units, the likelihood is the product of these contributions, so the same separation holds across the sample. For each unit, the data-model factor uses that unit’s observed components and the mechanism factor uses its observed values and pattern. $\square$

The likelihood consequence can also be stated through profiling. Under distinctness, the maximization over $\psi$ ranges over the same $\Psi$ for every $\theta$. Therefore

\sup_{\psi\in\Psi}L_{\mathrm{obs}}(\theta,\psi) = L_{\mathrm{data}}(\theta) \sup_{\psi\in\Psi}L_{\mathrm{mech}}(\psi).

The second factor is constant in $\theta$. Thus the profile likelihood for $\theta$, up to a multiplicative constant, is the likelihood from the observed data model alone. Likelihood ratios comparing two values of $\theta$ are unchanged by the mechanism factor, and maximizing the observed-data likelihood over $\theta$ gives the same result with or without that factor. This argument assumes the mechanism factor is well-defined and has a positive supremum; as usual, likelihood inference also requires that the data-model parameter be identifiable from the observed-data distribution for the question at hand.

Worked Examples

Worked Example: MAR That Is Not MCAR

Suppose $X$ and $Y$ are independent Bernoulli variables with $P(X=1)=p$ and $P(Y=1)=q$. $X$ is always observed. Let $R=1$ mean $Y$ is observed, and suppose $P(R=1\mid X=x,Y)=a_x$. The mechanism depends on $X$, which is observed, but not on $Y$, which may be missing. It is therefore MAR, but it is not MCAR unless $a_0=a_1$.

Consider six independent units. Among the three with $X=0$, two have observed $Y$, with values 0 and 1, and one has missing $Y$. Among the three with $X=1$, one has observed $Y=1$ and two have missing $Y$. The observed data-model likelihood is

L_{\mathrm{data}}(p,q) = p^3(1-p)^3q^2(1-q).

There are three observed $X=1$ values and three observed $X=0$ values, giving $p^3(1-p)^3$. Among the three observed $Y$ values, two are 1 and one is 0, giving $q^2(1-q)$. The missing $Y$ values contribute no additional factor to this marginal likelihood: summing their Bernoulli probabilities gives $q+(1-q)=1$ for each.

The mechanism contribution is

L_{\mathrm{mech}}(a_0,a_1) = a_0^2(1-a_0)a_1(1-a_1)^2.

There are two observed and one missing $Y$ among the $X=0$ units, and one observed and two missing among the $X=1$ units. Thus the full observed-data likelihood is the product of these two factors. Maximizing the data-model factor gives $\hat p=3/6=1/2$ and $\hat q=2/3$. Maximizing the mechanism factor separately gives $\hat a_0=2/3$ and $\hat a_1=1/3$. The mechanism is not MCAR because its observation probabilities differ by $X$, yet it can be omitted for likelihood inference about $p$ and $q$, since it is MAR and the parameter spaces are distinct.

Worked Example: Factoring a Conditional Data Model

Suppose $X\sim\operatorname{Bernoulli}(p)$, and $Y\mid X=x\sim\operatorname{Bernoulli}(q_x)$. Again, $X$ is always observed and $R=1$ indicates that $Y$ is observed. Let $P(R=1\mid X=x,Y)=a_x$. Consider three records: $(X=0,R=1,Y=1)$, $(X=1,R=0)$, and $(X=1,R=1,Y=0)$.

For the first record, the data-model probability is $(1-p)q_0$, and the mechanism probability is $a_0$. For the second record, the observed value is $X=1$, while $Y$ is missing. Summing over its two possible values gives

p\sum_{y=0}^1P(Y=y\mid X=1)(1-a_1) = p\{q_1+(1-q_1)\}(1-a_1) = p(1-a_1).

For the third record, the data-model probability is $p(1-q_1)$, and the mechanism probability is $a_1$. Multiplying the three contributions gives

L_{\mathrm{obs}} = (1-p)q_0\,a_0 \cdot p(1-a_1) \cdot p(1-q_1)\,a_1 = \underbrace{p^2(1-p)q_0(1-q_1)}_{L_{\mathrm{data}}} \underbrace{a_0(1-a_1)a_1}_{L_{\mathrm{mech}}}.

For instance, at $p=0.6$, $q_0=0.7$, $q_1=0.4$, $a_0=0.8$, and $a_1=0.3$, the data-model factor is $0.6^2(0.4)(0.7)(0.6)=0.06048$, and the mechanism factor is $0.8(0.7)(0.3)=0.168$. Their product is $0.01016064$. The mechanism depends on the observed $X$, so this example is MAR but generally not MCAR. Its factor contributes to the joint likelihood, but it is constant in $p,q_0,q_1$ when the $a_x$ parameters are distinct.

Worked Example: Why Parameter Distinctness Is Necessary

Suppose four units have a fixed, fully observed covariate value $X=0$. Their responses are independent with $Y\sim\operatorname{Bernoulli}(\theta)$. Let $R=1$ mean the response is observed. Define the mechanism by $P(R=1\mid X=0,Y)=\theta$; for $X=1$, let the observation probability be $1/2$. This is MAR because the mechanism depends only on the observed $X$, not on $Y$. But the mechanism parameter at $X=0$ is constrained to equal the data-model parameter $\theta$, so the parameters are not distinct.

Suppose three responses are observed, with values $1,1,0$, and the fourth response is missing. The data-model factor is $\theta^2(1-\theta)$. The mechanism factor is $\theta^3(1-\theta)$, since there are three observed responses and one missing response at $X=0$. The full observed-data likelihood is therefore

L_{\mathrm{obs}}(\theta) = \theta^2(1-\theta)\theta^3(1-\theta) = \theta^5(1-\theta)^2.

Ignoring the mechanism would maximize $\theta^2(1-\theta)$, whose log-likelihood derivative is $2/\theta-1/(1-\theta)$. Setting it to zero gives $2(1-\theta)=\theta$, so $\hat\theta=2/3$. The full likelihood has log-likelihood derivative $5/\theta-2/(1-\theta)$, giving $5(1-\theta)=2\theta$ and $\hat\theta=5/7$. The estimates differ. MAR alone is not enough: the factorization is algebraically valid in terms of separate data and mechanism factors, but when the parameters are tied, the mechanism factor is not constant as $\theta$ varies and cannot be discarded for inference about $\theta$.

What Ignorability Does—and Does Not—Say

For likelihood inference, ignorability means that the observed-data likelihood for $\theta$ is proportional to the marginal data-model likelihood, with a proportionality factor that does not involve $\theta$. It does not mean the missingness mechanism is scientifically unimportant, nor does it mean that the missing values can be treated as observed. The missing values have already been accounted for by marginalizing the full-data model.

Nor does MAR guarantee that a particular summary, such as a complete-case estimate, is valid. The theorem justifies using the observed-data likelihood under the stated model and conditions; it does not turn every analysis of the observed subset into a likelihood analysis of that model. Whether the observed-data likelihood identifies the parameters also depends on which patterns occur and on the data model.

Finally, likelihood ignorability is not automatically Bayesian ignorability. A Bayesian analysis also specifies a joint prior for $(\theta,\psi)$. If the prior links these parameters, or otherwise fails to separate the relevant factors, the mechanism can affect the posterior for $\theta$ even when MAR holds. A common sufficient Bayesian setup uses distinct parameters and a prior that factors as $\pi(\theta)\pi(\psi)$; then the mechanism factor does not alter the posterior for $\theta$.

Common Mistake

Wrong reasoning: “The probability of missingness depends on $X$, so the mechanism is not ignorable.”

Correction: If $X$ is observed under the realized pattern, dependence on $X$ is permitted by MAR. Check whether the mechanism depends on any value that is missing under that pattern, and check separately whether the data and mechanism parameters are distinct. Both conditions matter for likelihood ignorability.

Key takeaway: Under MAR, the missingness probability for a realized pattern is constant over the missing values, so it factors out of the observed-data integral. If the data-model and mechanism parameters are also distinct, that factor is free of the data-model parameter and can be omitted for likelihood inference about it.

Check Your Understanding

Use the observed-data likelihood and the two ignorability conditions to answer these questions.

If missingness depends on an observed covariate but not on any missing value, which missingness condition is satisfied? Is MCAR necessarily satisfied?
State the parameter-distinctness condition in terms of the joint parameter space.
In the likelihood integral, which term can be taken outside the integral under MAR, and why?
Why can a MAR mechanism still matter for likelihood inference about the data-model parameter?
What additional prior condition is commonly used when seeking Bayesian ignorability?