What a Borrowed Conditional Distribution Implies
A restriction can specify more than the mean of an unobserved outcome. Suppose \(Y_1\) is observed at the first visit and \(Y_2\) is missing for a particular pattern \(G\). A distributional borrowing restriction assigns the conditional distribution of \(Y_2\) given \(Y_1\) in pattern \(G\) by borrowing it from a reference pattern \(C\), such as a pattern in which both visits are observed. That choice also constrains conditional variances and, through the distribution of \(Y_1\) in pattern \(G\), the marginal variance and cross-visit association implied for that pattern.
The key distinction is between a conditional distribution and the distribution of \(Y_1\) itself. Even if the conditional law of \(Y_2\) given \(Y_1\) is borrowed exactly, the distribution of \(Y_1\) can differ between patterns. Thus the marginal variance and correlation implied in \(G\) need not equal those in \(C\).
All variance and covariance calculations below assume finite second moments. The target pattern must also have appropriate support: if its observed \(Y_1\) values fall in regions where the reference conditional law is unavailable, the borrowed law does not by itself specify what is needed there. Equality of conditional distributions is a modeling restriction, not a consequence of the observed data.
General Moment Consequences
Write \(m(x)=E(Y_2\mid Y_1=x,C)\) and \(v(x)=\operatorname{Var}(Y_2\mid Y_1=x,C)\). Under the full conditional-distribution restriction, these are also the conditional mean and variance in pattern \(G\). Applying iterated expectation gives the target-pattern mean. Applying the law of total variance gives its marginal variance:
The two terms in the variance have distinct meanings. The first averages the borrowed conditional variance over the target pattern's \(Y_1\) distribution. The second measures variation in the conditional mean as \(Y_1\) varies in that pattern. Therefore, even when \(m\) and \(v\) are borrowed unchanged, changing the distribution of \(Y_1\) can change the marginal variance of \(Y_2\).
The same restriction implies a covariance identity. Since \(E(Y_2\mid Y_1,G)=m(Y_1)\), the conditional residual \(Y_2-m(Y_1)\) has conditional mean zero. Its covariance with \(Y_1\) is therefore zero, giving:
This identity depends on the conditional mean restriction, not on borrowing the entire conditional distribution. Full distributional borrowing additionally specifies \(v(x)\) and, beyond moments, the rest of the conditional law.
Linear Conditional Means and Correlation
A common special case has a linear conditional mean and constant conditional variance: \(E(Y_2\mid Y_1=x,G)=a+bx\) and \(\operatorname{Var}(Y_2\mid Y_1=x,G)=\tau^2\). These can follow, for example, from borrowing a reference conditional normal distribution with that mean and variance. For the moment calculations, normality itself is not required; the stated conditional moments and finite target moments suffice. Let \(\mu_{1G}=E(Y_1\mid G)\) and \(V_{1G}=\operatorname{Var}(Y_1\mid G)\). Substitution into the general identities yields:
The intercept affects the mean but not the variance or covariance. The slope determines the direction of the implied covariance: it is positive when \(b>0\), negative when \(b<0\), and zero when \(b=0\). Correlation requires both marginal variances to be positive. In this setup, the exact conditions are \(V_{1G}>0\) and \(\tau^2+b^2V_{1G}>0\). Only under both conditions is the correlation defined, with value:
Worked Calculations
Worked Example: Full Conditional Borrowing Determines a Correlation
Suppose, hypothetically, the reference pattern has conditional distribution \(Y_2\mid Y_1=x,C\sim N(2+0.6x,4)\), where the second argument is the variance. Pattern \(G\) has \(E(Y_1\mid G)=5\) and \(\operatorname{Var}(Y_1\mid G)=9\). Assume the reference conditional distribution is defined over the target pattern's \(Y_1\) support, and impose full conditional-distribution borrowing. Both outcomes are measured in the same units.
The borrowed conditional mean has \(a=2\) and \(b=0.6\), and the borrowed conditional variance is \(\tau^2=4\). The target-pattern mean is \(E(Y_2\mid G)=2+0.6(5)=5\).
The target-pattern variance and covariance are \(\operatorname{Var}(Y_2\mid G)=4+(0.6)^2(9)=4+3.24=7.24\) outcome-units squared, and \(\operatorname{Cov}(Y_1,Y_2\mid G)=0.6(9)=5.4\) outcome-units squared.
Here \(V_{1G}=9>0\), and \(\tau^2+b^2V_{1G}=7.24>0\), so both marginal variances are positive and correlation is defined. Its value is \(5.4/\sqrt{9(7.24)}=5.4/\sqrt{65.16}\approx0.669\). Thus the restriction implies a positive cross-visit correlation of about 0.669 in pattern \(G\); it does not establish that this correlation is an observed fact for that pattern.
Worked Example: A Mean-Only Restriction Leaves Variance Undetermined
Keep the hypothetical target pattern's \(Y_1\) mean and variance from the previous calculation, and impose only the conditional mean \(E(Y_2\mid Y_1=x,G)=2+0.6x\). Suppose, for illustration, that the target conditional variance is 16 rather than the borrowed value 4. This is one possible distribution consistent with the mean-only restriction, not a consequence of it.
The mean remains \(2+0.6(5)=5\). The covariance also remains \(0.6(9)=5.4\), because the conditional mean is unchanged. But the variance becomes \(\operatorname{Var}(Y_2\mid G)=16+(0.6)^2(9)=16+3.24=19.24\) outcome-units squared.
Both marginal variances are positive, so this particular choice gives correlation \(5.4/\sqrt{9(19.24)}=5.4/\sqrt{173.16}\approx0.410\). The mean-only restriction does not select this correlation: another allowable conditional variance would yield another marginal variance and correlation. Full distributional borrowing would instead fix the conditional variance at 4 and, in this example, imply the correlation calculated above.
Worked Example: Equal Conditional Laws, Different Pattern Correlations
Consider a hypothetical binary \(Y_1\), measured as 0 or 1. In both patterns, let \(Y_2\mid Y_1=x\) have a normal distribution with mean \(2x\) and variance 1. In pattern \(G\), suppose \(P(Y_1=1\mid G)=0.25\); in reference pattern \(C\), suppose \(P(Y_1=1\mid C)=0.50\). Thus the full conditional law is the same, but the distribution of \(Y_1\) differs.
For \(G\), \(E(Y_1\mid G)=0.25\) and \(V_{1G}=0.25(0.75)=0.1875\). Therefore \(E(Y_2\mid G)=2(0.25)=0.5\), \(\operatorname{Var}(Y_2\mid G)=1+4(0.1875)=1.75\), and \(\operatorname{Cov}(Y_1,Y_2\mid G)=2(0.1875)=0.375\). The correlation is \(0.375/\sqrt{0.1875(1.75)}\approx0.655\).
For \(C\), \(E(Y_1\mid C)=0.50\) and \(V_{1C}=0.25\). Its corresponding moments are \(E(Y_2\mid C)=1\), \(\operatorname{Var}(Y_2\mid C)=1+4(0.25)=2\), and \(\operatorname{Cov}(Y_1,Y_2\mid C)=2(0.25)=0.5\). The correlation is \(0.5/\sqrt{0.25(2)}\approx0.707\). The conditional distributions match exactly, yet the marginal correlations differ because the patterns have different \(Y_1\) distributions.
Derivation and Comparison With Mean-Only Borrowing
The variance identity follows by writing the deviation of \(Y_2\) from its target mean as two parts: \(Y_2-E(Y_2\mid G)=[Y_2-m(Y_1)]+[m(Y_1)-E\{m(Y_1)\mid G\}]\). When this expression is squared and averaged within \(G\), the cross-product has expectation zero: conditional on \(Y_1\), the first bracket has mean zero, while the second is fixed. The remaining terms are the average conditional variance and the variance of the conditional mean. This gives the stated law of total variance.
For the covariance identity, write \(Y_2=m(Y_1)+\epsilon\), where \(E(\epsilon\mid Y_1,G)=0\). Then \(\operatorname{Cov}(Y_1,\epsilon\mid G)=E[(Y_1-E(Y_1\mid G))\epsilon\mid G]=0\), by conditioning on \(Y_1\). Hence the covariance reduces to \(\operatorname{Cov}(Y_1,m(Y_1)\mid G)\). If \(m(x)=a+bx\), this is \(bV_{1G}\). The variance becomes \(\tau^2+b^2V_{1G}\) when the conditional variance is the constant \(\tau^2\).
The distinction between restrictions can be summarized as follows.
| Restriction in pattern \(G\) | What it determines, given the \(Y_1\) distribution | What remains unspecified |
|---|---|---|
| Full conditional distribution borrowed | Conditional moments and law; consequently the implied marginal distribution and its moments | Whether the restriction is substantively appropriate; the law outside supported \(Y_1\) values |
| Conditional mean borrowed | Marginal mean of \(Y_2\) and covariance with \(Y_1\), if the needed moments exist | Conditional variance and shape; generally the marginal variance and correlation |
A mean-only restriction can still determine variance in a special case where additional assumptions supply the conditional variance. That determination then comes from those extra assumptions, not from equality of conditional means. Likewise, a full conditional law does not force the two patterns to have equal marginal variances or correlations unless their \(Y_1\) distributions also make those quantities equal.
Diagnostics, Failure Modes, and Limitations
These calculations provide a coherence check on the implications of a restriction. Before reporting an implied correlation, verify that the target \(Y_1\) variance and implied \(Y_2\) variance are both strictly positive. For the linear constant-variance case, check \(V_{1G}>0\) and \(\tau^2+b^2V_{1G}>0\) separately. A zero denominator is not a correlation of zero; it means at least one variable is constant within the pattern and the correlation is undefined.
Also check that the borrowed conditional law applies over the target pattern's \(Y_1\) support. A formula borrowed from the reference pattern cannot determine the target conditional distribution where the reference provides no basis for it without an additional extrapolation rule. Numerical estimates can create a further warning: a negative estimated conditional variance is incompatible with a variance model and may indicate estimation or specification problems. These checks assess mathematical coherence and support; they do not prove that the borrowed law is correct for unobserved outcomes.
- Confusing conditional and marginal equality. Equal \(Y_2\mid Y_1\) distributions do not imply equal marginal \(Y_2\) distributions when the distribution of \(Y_1\) differs.
- Using a mean restriction to claim a variance. A common conditional mean does not fix conditional residual variability. State the extra variance assumption if one is used.
- Computing correlation with a zero variance. Check both marginal variances before dividing; a zero denominator leaves correlation undefined.
- Treating an implication as an empirical finding. If \(Y_2\) is unobserved in \(G\), the implied target-pattern association comes from the restriction and observed \(Y_1\) distribution, not from directly observed pairs in \(G\).
The practical value of this distinction is that a distributional restriction carries consequences beyond a mean trajectory. It can shape uncertainty and cross-visit dependence used in downstream summaries or imputation. A mean-only restriction is weaker and may be preferable when the mean is the intended identifying commitment, but additional modeling choices are then required to specify variability and dependence. Neither approach removes sensitivity to the assumptions used for unobserved outcomes.
Check Your Understanding
Answer each question using the stated assumptions. Solutions follow in the same box.
- Under a borrowed conditional law with mean \(1+0.5Y_1\), constant conditional variance 3, and \(\operatorname{Var}(Y_1\mid G)=4\), find the implied variance of \(Y_2\) and covariance of \(Y_1,Y_2\) in \(G\).
- If a conditional mean is borrowed but the conditional variance is unspecified, which of the marginal mean, covariance, and variance of \(Y_2\) are generally determined, assuming finite moments?
- In the linear constant-variance formula, when is the correlation undefined? Give both required checks.
- Two patterns share the same full conditional distribution of \(Y_2\mid Y_1\), but have different \(Y_1\) distributions. Must their marginal \(Y_2\) variances or correlations agree? Explain briefly.
Solutions. (1) The variance is \(3+(0.5)^2(4)=3+1=4\) outcome-units squared, and the covariance is \(0.5(4)=2\) outcome-units squared. (2) The marginal mean and covariance are determined by the conditional mean and the target \(Y_1\) distribution; the marginal \(Y_2\) variance is generally not determined because the conditional variance is unspecified. (3) Check \(V_{1G}>0\) and \(\tau^2+b^2V_{1G}>0\). If either fails, a marginal variance is zero and correlation is undefined. (4) No. The marginal variance averages conditional variance over the pattern-specific \(Y_1\) distribution and adds the variance of the conditional mean; the covariance and correlation can also change with that distribution.