Applying Complete-Case Restrictions to Regression Models

From a Mean Restriction to a Regression Restriction

A complete-case restriction can be applied not only to an outcome mean but to a conditional regression distribution. This distinction matters when patterns of missingness contain people with different covariate distributions. A shared regression relationship says how the outcome varies with covariates within each pattern; it does not say that the patterns have the same overall outcome distribution.

Let $G$ denote the missing-data pattern, let $C$ denote the complete-case pattern, let $Y$ be the outcome of interest, and let $X$ be a vector of covariates observed for the patterns being compared. The covariates might include baseline variables or observed history. For a pattern $g$, the relevant regression distribution is the conditional distribution of $Y$ given $X=x$ and $G=g$.

Definition: A complete-case restriction on the conditional outcome distribution equates the distribution of $Y$ in pattern $g$ to its distribution among complete cases at covariate values shared by the two patterns: $$ F_{Y\mid X,G=g}(y\mid x,g)=F_{Y\mid X,G=C}(y\mid x,C), \qquad x\in\mathcal S_g\cap\mathcal S_C. $$ Here $\mathcal S_g$ and $\mathcal S_C$ are the covariate supports in patterns $g$ and $C$. The restriction is conditional: it compares outcomes at the same $x$, rather than comparing pattern averages.

The support qualification is essential. If pattern $g$ has no observations near a covariate value represented among complete cases, the restriction does not supply an empirical comparison at that value. Any extension beyond the shared support is an extrapolation imposed by the model, not a consequence of the restriction itself.

What Equality of Regression Distributions Implies

Suppose a normal linear regression is used within each pattern:

Y\mid X=x,G=g\sim N(\alpha_g+\beta_g x,\sigma_g^2).

For a vector of covariates, replace $\alpha_g+\beta_g x$ by $z(x)^\mathsf{T}\theta_g$, where $z(x)$ contains the intercept and covariates. The regression formulation assumes the specified normal conditional distribution, the stated mean function, and (in this version) a variance that is constant within each pattern. These assumptions should be assessed using the pattern-specific regression diagnostics appropriate to the data, including residual and distributional checks. A convenient shared family does not itself establish the complete-case restriction.

Under the normal model, two conditional distributions at a given $x$ are equal exactly when their means and variances are equal at that $x$. Thus, the restriction requires equal conditional means throughout the common support and equal conditional variances. It does not automatically require equal coefficient vectors: that conclusion depends on whether the common support identifies the regression coefficients.

Key result: For the normal linear models above, equality of the conditional distributions on the common support requires $\sigma_g^2=\sigma_C^2$ and $\alpha_g+\beta_g x=\alpha_C+\beta_C x$ at every $x\in\mathcal S_g\cap\mathcal S_C$. The intercepts and slopes must be equal only if that support has sufficient affine rank to identify them.

For several covariates, write $z(x)^\mathsf{T}\theta_g=z(x)^\mathsf{T}\theta_C$ at every common-support value. Stack the corresponding design rows $z(x)^\mathsf{T}$ into a matrix $Z$. Equality of the fitted means says $Z(\theta_g-\theta_C)=0$. If $Z$ has full column rank, its null space contains only zero, so $\theta_g=\theta_C$. If it is rank deficient, different coefficient vectors can produce the same fitted means on the common support.

Worked Example: Checking the Support Before Equating Slopes

Suppose hypothetical complete cases and pattern $g$ both have covariate values $x=0$ and $x=1$. Their fitted models are $Y\mid x,G=C\sim N(10+2x,9)$ and $Y\mid x,G=g\sim N(10+2x,9)$. At $x=0$, each conditional mean is $10$; at $x=1$, each is $12$. The conditional variances are both $9$. The conditional distributions therefore agree at both shared values. The two design rows, $(1,0)$ and $(1,1)$, have rank 2, so this support identifies an intercept and slope; in this example the coefficients are indeed equal.

Now suppose the only shared covariate value is $x=0$, and the fitted models are $N(10+2x,9)$ for complete cases and $N(10+5x,9)$ for pattern $g$. At the only common value, both means are $10$, and both variances are $9$. Thus the conditional distributions agree on the common support even though the slopes differ. The single design row $(1,0)$ has rank 1, so it cannot identify two coefficients. Claiming that the restriction forces equal slopes would be incorrect.

Why Equal Conditional Regressions Do Not Equalize Margins

A pattern-specific marginal outcome distribution averages over that pattern’s own covariate distribution. In general, for any set $A$ of outcome values,

P(Y\in A\mid G=g)=\int P(Y\in A\mid X=x,G=g)\,dF_{X\mid G=g}(x).

Even if the conditional outcome distribution is shared across patterns, the distributions $F_{X\mid G=g}$ can differ. Consequently, averaging the same conditional regression over different covariate distributions can yield different marginal means, variances, and shapes. Under a linear conditional mean, the marginal mean is

E(Y\mid G=g)=\alpha+\beta E(X\mid G=g).

For a vector of covariates, the corresponding expression is $E(Y\mid G=g)=E\{z(X)^\mathsf{T}\theta\mid G=g\}$. Equality of conditional regressions gives a shared rule for each covariate value; equality of marginal outcomes additionally depends on how covariate values are distributed across patterns.

Worked Example: A Shared Regression, Different Pattern Means

Consider hypothetical data with $X$ taking values 0 and 4 in both patterns. In complete cases, the probabilities at $X=0,4$ are $0.75,0.25$; in pattern $g$, they are $0.25,0.75$. Assume the complete-case restriction holds and the shared model is $Y\mid X=x,G\sim N(50+4x,9)$ at these covariate values.

The covariate means are $E(X\mid G=C)=0(0.75)+4(0.25)=1$ and $E(X\mid G=g)=0(0.25)+4(0.75)=3$. Therefore, the marginal outcome means are $50+4(1)=54$ outcome units for complete cases and $50+4(3)=62$ outcome units for pattern $g$. The conditional regression is the same, but the marginal means differ by $8$ outcome units because the covariate distributions differ.

For either pattern, the covariate variance is $3$: for complete cases it is $0.75(0-1)^2+0.25(4-1)^2=3$, and for pattern $g$ it is $0.25(0-3)^2+0.75(4-3)^2=3$. The marginal outcome variance is $9+4^2(3)=57$ outcome-units squared in each pattern. This variance calculation uses the law of total variance: the conditional variance is 9 and the variance of the conditional mean $50+4X$ is $16\operatorname{Var}(X\mid G)$. Equal marginal variances here do not make the full marginal distributions equal: their component means are 50 and 66 in complete cases, versus 50 and 66 with the probabilities reversed in pattern $g$.

Applying the Restriction and Checking Its Limits

A disciplined application starts by naming the outcome, pattern, and conditioning variables. Then specify the conditional model and its assumptions before imposing a restriction. A useful sequence is:

Define the comparison.
Identify complete cases $C$, the pattern $g$, the outcome $Y$, and covariates $X$ observed in both groups.

Check overlap and rank.
Identify common-support values and check whether the associated design rows have sufficient rank for the coefficient claims you intend to make.

State the restricted distribution.
Specify which conditional distribution features are equated, such as the conditional mean function and variance under a normal model.

Interpret the result conditionally.
Keep pattern-specific covariate distributions when reconstructing or describing marginal outcomes; do not replace them with the complete-case covariate distribution unless that is an explicit standardization target.

If the model is identifiable and the complete-case restriction is represented as equality of regression parameters, an unrestricted pattern-specific fit can be compared with a constrained fit. A likelihood-ratio test has its usual large-sample chi-squared reference distribution only under the regularity conditions for that model, including identifiable parameters and a correctly specified likelihood. A test of parameter equality is not a test of the restriction on a rank-deficient common support: in that case some coefficient contrasts are not determined by the shared-support conditional distributions.

Diagnostics can reveal problems with the regression family, influential observations, sparse covariate regions, and poor overlap. They cannot, by themselves, establish equality of an outcome distribution that is missing for one pattern. When the outcome is unobserved for members of $g$, the key complete-case restriction is generally an identifying assumption, not an empirically verified fact. Observed history and covariates can inform plausibility, but they do not reveal the missing outcomes.

Compared with a restriction only on means, a conditional-distribution restriction is stronger: it can constrain variance and other distributional features as well as the conditional mean. Compared with an unrestricted pattern-specific regression, it reduces pattern-specific flexibility by borrowing the complete-case relationship. That can stabilize estimates, but it can also impose a wrong relationship. The introductory material on complete-case restrictions for means concerns mean-level constraints; a regression restriction must additionally address covariate conditioning, support, and coefficient identifiability.

Common Mistakes and Failure Modes

Equating coefficients without checking support. Equality of fitted conditional means on observed shared covariate values implies coefficient equality only when the design has sufficient rank. With rank deficiency, only certain linear combinations of coefficients are constrained.
Calling a shared regression a shared marginal distribution. Margins also depend on each pattern’s covariate distribution. Report which covariate distribution is being averaged over.
Extrapolating silently. A restriction stated on common support does not identify predictions outside that support without additional model assumptions. Sparse overlap makes those predictions especially fragile.
Treating the restriction as testable from complete cases alone. Complete cases reveal their own outcome distribution, not the unobserved outcomes in another pattern. Diagnostics can assess model fit among observed outcomes but cannot prove the missing-outcome restriction.
Assuming a mean constraint equates full distributions. Equal conditional means do not imply equal conditional variances or shapes. State which distributional features the model and restriction actually constrain.

Key takeaway: A complete-case regression restriction equates conditional outcome distributions at shared covariate values. Parameter equality follows only with adequate support and design rank; marginal outcome equality requires additional conditions on the covariate distributions.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Two patterns have a shared normal linear conditional model with equal variances. Their only common covariate value is $x=0$, where the fitted means agree. Must their slopes be equal?
Under the shared model $E(Y\mid X,G)=20+3X$, pattern A has covariate mean 2 and pattern B has covariate mean 5. What are their marginal outcome means?
For two covariates and an intercept, the shared-support design matrix has rank 2 although there are 3 coefficients. Can equality of conditional means on that support identify all three coefficients?
If conditional means agree across patterns, what further condition is needed to conclude that the conditional distributions agree in a normal model?

Solutions. (1) No. Agreement at $x=0$ constrains the intercept there, but the slope is not identified by a design row $(1,0)$. (2) Pattern A has mean $20+3(2)=26$ outcome units; pattern B has mean $20+3(5)=35$ outcome units. (3) No. Rank 2 leaves a nonzero null space, so some coefficient differences produce identical fitted means on the shared support. (4) The conditional variances must also agree at the shared covariate values; under the stated homoskedastic model, this means the pattern-specific variances are equal.

Applying Complete-Case Restrictions to Regression Models

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