MCAR: Factorization and Complete-Case Validity

Why MCAR Matters

A complete-case analysis discards every subject with at least one missing variable. It is tempting to regard the remaining subjects as a smaller, otherwise ordinary sample. That interpretation is justified only under conditions on how observations become incomplete. Under missing completely at random (MCAR), the missingness pattern is independent of the full data vector. This gives both a useful likelihood factorization and a direct route to consistency results for complete-case estimates.

These conclusions have limits. MCAR does not guarantee that a useful number of complete cases will be observed, and the likelihood for all observed data is not generally the likelihood based only on complete cases. The distinction between those claims is central.

MCAR and the Observed-Data Likelihood

For one subject, let \(X=(X_1,\ldots,X_p)\) be the full data vector and let \(R=(R_1,\ldots,R_p)\), where \(R_j=1\) if \(X_j\) is observed and \(R_j=0\) otherwise. A missingness pattern \(r\) determines the observed coordinate set \(O(r)=\{j:r_j=1\}\). Write \(X_{O(r)}\) for the observed subvector and \(X_{M(r)}\) for the missing subvector.

Definition: Missingness is MCAR if \(R\) is independent of the full data vector \(X\). Equivalently, for every pattern \(r\), \(P(R=r\mid X=x)=P(R=r)\) for almost every \(x\). This condition concerns the full data, including values that are not observed.

Suppose \(X\) has density \(f(x;\theta)\) with respect to a product measure. For pattern \(r\), define the observed-data marginal density \(f_r(x_{O(r)};\theta)\) by integrating or summing over the missing coordinates. For example, with continuous coordinates,

f_r(x_{O(r)};\theta) = \int f(x_{O(r)},x_{M(r)};\theta)\,dx_{M(r)}.

For the pattern with no observed coordinates, set \(f_r=1\). Let \(g(r;\psi)=P(R=r)\), where \(\psi\) parameterizes the missingness distribution if it is modeled. Under MCAR, the joint density of the observed subvector and its pattern is the product of these two factors.

Theorem: Suppose the full-data density is \(f(x;\theta)\), the missingness pattern has probabilities \(g(r;\psi)\), and \(R\) is independent of \(X\). Then the joint observed-data likelihood for independent subjects is

L(\theta,\psi) = \prod_{i=1}^n g(r_i;\psi)\, f_{r_i}(x_{i,O(r_i)};\theta).

If \(\theta\) and \(\psi\) are distinct parameters, this factors into a missingness part and an observed-data part: \(L(\theta,\psi)=L_R(\psi)L_X(\theta)\).

To derive the result for one subject, the joint density of \(X\) and \(R=r\) is \(f(x;\theta)g(r;\psi)\), by independence. The recorded data retain \(x_{O(r)}\), so integrate this joint density over \(x_{M(r)}\):

\int f(x_{O(r)},x_{M(r)};\theta)g(r;\psi)\,dx_{M(r)} = g(r;\psi) f_r(x_{O(r)};\theta).

Independence of subjects gives the product over \(i\). Since the factor \(g(r_i;\psi)\) depends only on \(\psi\) and \(f_{r_i}\) only on \(\theta\), the likelihood separates when those parameters are distinct. The factorization does not say that every subject contributes a full-data density: a subject with missing coordinates contributes the appropriate marginal density.

Worked Example: Factoring a Bivariate Normal Likelihood

Let \(X=(X_1,X_2)\) have a bivariate normal distribution with mean \((\mu_1,\mu_2)\) and covariance matrix \(\Sigma=\begin{pmatrix}\sigma_{11}&\sigma_{12}\\ \sigma_{12}&\sigma_{22}\end{pmatrix}\). Assume MCAR, with probabilities \(0.5\) for observing both variables, \(0.2\) for observing only \(X_1\), \(0.2\) for observing only \(X_2\), and \(0.1\) for observing neither. These probabilities sum to \(1\).

Suppose three subjects have records \((X_1,X_2)=(1,2)\), \(X_1=0\) only, and \(X_2=-1\) only, and a fourth subject has neither variable observed. Let \(\phi_2(\,\cdot\,;\mu,\Sigma)\) denote the bivariate normal density and \(\phi(\,\cdot\,;\mu_j,\sigma_{jj})\) the univariate marginal density of \(X_j\). The joint likelihood is

L = (0.5)(0.2)(0.2)(0.1) \phi_2((1,2);(\mu_1,\mu_2),\Sigma) \phi(0;\mu_1,\sigma_{11}) \phi(-1;\mu_2,\sigma_{22}).

The missingness factor is \(0.5\times0.2\times0.2\times0.1=0.002\). The fully missing subject contributes no data-density factor, because its observed-data density is \(1\), but its pattern still contributes probability \(0.1\). The data-density factor uses a bivariate density for the complete record and univariate marginal densities for the two partial records.

When Complete-Case Estimates Are Consistent

Let \(C=1\) indicate a complete case, meaning \(R=(1,\ldots,1)\), and write \(q=P(C=1)\). MCAR implies \(C\) is independent of \(X\). If \(q>0\), the distribution of \(X\) among complete cases is therefore the population distribution:

P(X\in A\mid C=1) = \frac{P(X\in A,C=1)}{P(C=1)} = \frac{P(X\in A)q}{q} = P(X\in A).

This equality is the distributional reason complete-case estimates can work. It does not, by itself, establish consistency: the number of complete cases must grow, and the estimator must consistently estimate its target under the population distribution.

Theorem: Suppose \((X_i,C_i)\), \(i=1,\ldots,n\), are independent and identically distributed, \(C_i\) is independent of \(X_i\), and \(q=P(C_i=1)>0\). If \(h(X)\) is integrable, then

\frac{\sum_{i=1}^n C_i h(X_i)} {\sum_{i=1}^n C_i} \xrightarrow{a.s.} E[h(X)].

The ratio is considered when its denominator is positive; under the stated assumptions, that holds eventually with probability one.

For the proof, apply the strong law of large numbers separately to the numerator and denominator after dividing each by \(n\). Since \(E[|C h(X)|]\leq E[|h(X)|]<\infty\),

\frac{1}{n}\sum_{i=1}^n C_i h(X_i) \xrightarrow{a.s.} E[C h(X)] = qE[h(X)], \qquad \frac{1}{n}\sum_{i=1}^n C_i \xrightarrow{a.s.}q.

The equality \(E[C h(X)]=qE[h(X)]\) follows from independence. Because \(q>0\), the denominator limit is nonzero, so the ratio converges almost surely to \(qE[h(X)]/q=E[h(X)]\). \(\square\)

Taking \(h(X)=X_j\) gives consistency of each complete-case sample mean when \(E[|X_j|]<\infty\). Taking \(h(X)=X_jX_k\), when that product is integrable, gives consistency of complete-case raw second moments. Combining these convergent moments gives consistency of the usual sample covariance entries when the required second moments are finite.

More generally, a complete-case maximum likelihood estimator can be consistent when the complete-case model is identifiable and the usual MLE consistency conditions hold. One sufficient set of conditions is a compact parameter space, an almost surely continuous log-density \(m_\theta(X)=\log f(X;\theta)\), an integrable envelope for \(\sup_\theta |m_\theta(X)|\), and a unique maximizer of \(E[m_\theta(X)]\) at the true parameter. Under MCAR with \(q>0\), the complete-case observations have the same distribution as \(X\), and their number tends to infinity almost surely. The uniform law of large numbers then makes the complete-case average log-likelihood converge uniformly to \(E[m_\theta(X)]\); uniqueness of its maximizer implies consistency. Identifiability matters: representing the same population distribution by multiple parameter values does not permit consistent estimation of one unique parameter value.

Worked Example: Complete-Case Mean and Variance

Suppose \(X\sim\operatorname{Bernoulli}(0.3)\), so \(E[X]=0.3\) and \(\operatorname{Var}(X)=0.3(0.7)=0.21\). Each subject is complete with probability \(q=0.6\), independently of \(X\) and of other subjects. For the mean, the theorem applies with \(h(X)=X\), so the complete-case mean converges almost surely to \(0.3\).

The ratio calculation also displays the target directly: \(E[CX]=E[C]E[X]=(0.6)(0.3)=0.18\), while \(E[C]=0.6\). Their ratio is \(0.18/0.6=0.3\). For the second raw moment, \(X^2=X\), so the complete-case second moment also converges to \(0.3\); subtracting the squared mean gives \(0.3-(0.3)^2=0.21\).

In one possible finite sample, six complete cases might have values \(1,0,0,1,0,0\). Their mean is \(2/6=1/3\), not \(0.3\). Their variance with denominator \(6-1\) is \(\{2(1-1/3)^2+4(0-1/3)^2\}/5= (4/3)/5=4/15\), not \(0.21\). Consistency is a large-sample statement, not a claim of exact equality in each realized sample.

Conditions, Limitations, and Common Pitfalls

The positive-probability condition \(q>0\) is essential for complete-case estimation of the full vector. If \(q=0\), there are no complete cases, even though a missingness mechanism can still be independent of \(X\). If \(q\) is positive but small, the complete-case count grows asymptotically, but it may be too small for stable estimation in a particular study.

Worked Example: MCAR but Too Few Complete Cases in a Small Study

Suppose each of \(n=20\) subjects is a complete case independently with probability \(q=0.05\), and completeness is independent of the measurements. The mechanism is MCAR and \(q>0\), but the probability of observing no complete cases is

P\left(\sum_{i=1}^{20}C_i=0\right) = (1-0.05)^{20} = 0.95^{20} \approx 0.3585.

Thus there is about a \(35.85\%\) chance that complete-case estimates of full-vector quantities cannot even be calculated in this study. The asymptotic consistency result does not remove this finite-sample risk.

Common mistake: “The data are MCAR, so complete-case estimates must be consistent.” This omits the required positive complete-case probability and the assumptions on the estimator and target. The correction is to check that \(q>0\), that the number of complete cases grows, and that the complete-case estimator is consistent and identifiable under the population distribution.

A related error is to treat MCAR as a statement that missingness rates are merely unrelated to recorded values. MCAR requires independence from the full data. For example, suppose \(X\) is Bernoulli with \(P(X=1)=0.3\), and a case is complete with probability \(0.9\) when \(X=1\) and \(0.3\) when \(X=0\). Then \(P(C=1)=0.3(0.9)+0.7(0.3)=0.48\), and \(P(X=1\mid C=1)=0.3(0.9)/0.48=0.5625\), not \(0.3\). Complete cases are not representative in this example because completeness depends on \(X\); the MCAR argument does not apply.

Finally, likelihood factorization and complete-case validity are separate conclusions. Under MCAR, the observed-data likelihood uses every subject through a pattern-specific marginal density. A complete-case analysis discards the partial records and may still consistently estimate a target when the conditions above hold, but it can lose information and precision.

Key takeaway: Under MCAR, the observed-data likelihood separates into missingness probabilities and pattern-specific data marginals. Complete cases preserve the full-data distribution, so complete-case estimates can be consistent when the complete-case probability is positive and the estimator satisfies the relevant identification and regularity conditions.

Check Your Understanding

Use the conditions and distinctions in this tutorial to answer the following.

For a subject with pattern \(r\), why does the observed-data density involve \(f_r(x_{O(r)};\theta)\) rather than the full density \(f(x;\theta)\)?
State the independence condition in the definition of MCAR and explain why it implies \(P(X\in A\mid C=1)=P(X\in A)\) when \(q>0\).
Which two limits from the strong law establish consistency of the complete-case average of an integrable function \(h(X)\)?
Why does MCAR with \(q=0\) fail to justify complete-case estimation of a full-vector parameter?
In the Bernoulli example, calculate the complete-case probability when \(P(C=1\mid X=1)=0.9\) and \(P(C=1\mid X=0)=0.3\). Does this mechanism satisfy MCAR?

MCAR: Factorization and Complete-Case Validity

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