Why Conditional Equality Can Look Like Marginal Inequality
A restriction on an unobserved outcome often says that two patterns have the same outcome distribution at a given observed history. It does not necessarily say that the patterns have the same outcome distribution overall. The distinction matters whenever the distribution of that history differs across patterns: each pattern averages over its own mix of histories.
This tutorial uses a two-visit setting to make that distinction explicit. Let \(Y_1\) be an outcome observed at visit 1, \(Y_2\) the visit-2 outcome, and \(S\) the follow-up pattern, where \(S=1\) denotes participants observed at both visits and \(S=0\) denotes participants observed at visit 1 but not visit 2. The observed history for the visit-2 outcome is \(H=Y_1\). All numerical examples below are hypothetical.
Under this restriction, the conditional distribution of \(Y_2\) for participants with \(S=0\) is borrowed from participants with \(S=1\) who have the same history. This identifies a distribution for the missing outcomes only if the restriction is accepted and the relevant history values have support in the reference pattern. It does not by itself equate the two patterns’ marginal distributions of \(Y_2\).
How the History Distribution Enters
Write \(F_s(y\mid h)=P(Y_2\leq y\mid H=h,S=s)\), and let \(G_s(h)=P(H\leq h\mid S=s)\). For discrete histories, the law of total probability gives the marginal distribution in pattern \(s\) as a weighted sum of its conditional distributions. For continuous histories, the corresponding expression is an integral.
If the conditional restriction holds, \(F_0(y\mid h)=F_1(y\mid h)\). Substituting this equality into the two marginal formulas gives
The integrand is the same, but the weights may differ. Thus, marginal equality follows if the history distributions are also equal, or if the conditional outcome distribution does not vary with history in a way that affects the marginal. In general, neither condition is guaranteed. For means, the same argument gives \(E(Y_2\mid S=s)=\int E(Y_2\mid H=h,S=s)\,dG_s(h)\). Equal conditional means do not imply equal marginal means when the history distributions differ.
Worked Example: Equal Conditional Distributions, Different Margins
Suppose \(H\) is binary, with values 0 and 1, and \(Y_2\) is a binary outcome. The assumed conditional probabilities of \(Y_2=1\) are 0.20 for \(H=0\) and 0.80 for \(H=1\), in both patterns. Among completers (\(S=1\)), 80% have \(H=1\); among participants who drop out (\(S=0\)), only 20% have \(H=1\). These are hypothetical probabilities.
The assumed restriction holds at both history values: \(P(Y_2=1\mid H=0,S=0)=P(Y_2=1\mid H=0,S=1)=0.20\), and \(P(Y_2=1\mid H=1,S=0)=P(Y_2=1\mid H=1,S=1)=0.80\). Now average separately using each pattern’s history proportions.
For completers, \(P(Y_2=1\mid S=1)=(0.20)(0.20)+(0.80)(0.80)=0.04+0.64=0.68\). For participants who drop out, \(P(Y_2=1\mid S=0)=(0.80)(0.20)+(0.20)(0.80)=0.16+0.16=0.32\).
Therefore, the marginal probability of the outcome is 0.68 among completers and 0.32 among those who drop out under the conditional restriction. The difference is 0.36, or 36 percentage points. It does not contradict conditional equality: the patterns have different proportions of participants with \(H=1\), and \(H=1\) has a higher conditional probability of \(Y_2=1\).
Conditional Equality Is Not Established by Marginal Equality
The converse error is also possible. Equal marginal distributions do not show that conditional distributions are equal within histories. Differences across histories can cancel when averaged. This matters when interpreting a marginal comparison: it cannot, on its own, confirm a history-specific restriction.
Worked Example: Equal Margins Despite Conditional Differences
Again let \(H\) and \(Y_2\) be binary. Suppose each pattern has half its participants at \(H=0\) and half at \(H=1\). The conditional probabilities of \(Y_2=1\) are as follows.
| History | \(P(Y_2=1\mid H,S=1)\) | \(P(Y_2=1\mid H,S=0)\) |
|---|---|---|
| \(H=0\) | 0.20 | 0.40 |
| \(H=1\) | 0.80 | 0.60 |
In pattern \(S=1\), the marginal probability is \((0.50)(0.20)+(0.50)(0.80)=0.10+0.40=0.50\). In pattern \(S=0\), it is \((0.50)(0.40)+(0.50)(0.60)=0.20+0.30=0.50\). The margins are equal, but the conditional probabilities differ by 0.20 at each history value. Marginal equality therefore does not establish the conditional restriction.
Computing a Marginal Mean Under a Conditional Mean Restriction
A restriction may concern only conditional means rather than full distributions. In that case, it says that \(E(Y_2\mid H=h,S=0)=E(Y_2\mid H=h,S=1)\) at each relevant \(h\); it does not assert equality of conditional variances or of the full conditional distributions. To obtain the dropout-pattern mean under this restriction, average the borrowed conditional means over the dropout pattern’s history distribution, not the completer pattern’s distribution.
Worked Example: Borrowed Conditional Means and Dropout Weights
Suppose \(H\) has three categories: low, middle, and high. In a hypothetical dataset, the observed visit-2 means among completers are 10, 15, and 20 points in those categories. Assume the conditional mean restriction assigns the same respective means to participants who drop out. The history proportions are 0.50, 0.30, and 0.20 among dropouts, and 0.20, 0.30, and 0.50 among completers.
Under the restriction, the implied mean visit-2 outcome among dropouts is \(E(Y_2\mid S=0)=(0.50)(10)+(0.30)(15)+(0.20)(20)=5+4.5+4=13.5\) points.
The completer mean is \(E(Y_2\mid S=1)=(0.20)(10)+(0.30)(15)+(0.50)(20)=2+4.5+10=16.5\) points. The means differ by 3.0 points, entirely because the history weights differ in this example. Using the completer weights to compute the dropout mean would instead give 16.5 points and would answer a different question: the mean under the completer history distribution, not the dropout history distribution.
Assumptions, Overlap, and Diagnostics
A conditional restriction is meaningful only after specifying what counts as the relevant history. It also requires that the reference pattern contain information at the histories where the restriction is applied. In the example above, the completer pattern must include participants with the dropout pattern’s history values; otherwise, its conditional distribution cannot directly supply the missing conditional distribution there.
- State the restriction at the intended level. Specify whether it equates full conditional distributions or only conditional means, and name the history variables being conditioned on.
- Check overlap. Compare the history values or ranges represented in each pattern. A history category found only among dropouts has no observed completer outcome distribution to borrow. Extrapolation then requires additional modeling assumptions.
- Inspect history imbalance. Tabulate or summarize the history distribution separately by pattern. Large differences signal that conditional equality can produce noticeably different marginal outcomes.
- Check stratum size and history specification. Sparse strata make conditional distributions or means unstable. Coarsening history can improve precision but may combine participants whose outcome distributions differ; adding many history variables can create sparse or empty strata.
These checks diagnose whether the restriction can be applied and how the averaging works; they do not prove that the restriction is scientifically correct. In particular, an observed conditional distribution in the reference pattern cannot reveal the true visit-2 distribution among participants whose visit-2 outcomes are missing. The restriction remains an assumption about that unobserved distribution.
How This Differs From Alternative Restrictions
| Approach | What it equates or uses | What determines the marginal result? |
|---|---|---|
| Conditional distribution restriction | Full outcome distributions within each history | The shared conditional distributions and each pattern’s history weights |
| Conditional mean restriction | Means within each history | The conditional means and each pattern’s history weights; not the full distribution |
| Marginal distribution restriction | Outcome distributions overall, without conditioning on history | The direct marginal equality assumption; it does not follow from conditional equality |
| Standardization to a common history distribution | Conditional outcomes averaged over a chosen common history mix | The chosen target weights, which need not be either pattern’s own weights |
Standardization can be useful for a fair comparison at a common history mix, but it changes the estimand: it is not the raw marginal distribution in either pattern unless the chosen weights match that pattern’s history distribution. Similarly, replacing a conditional restriction with marginal equality is not a harmless simplification; it is a different assumption. The earlier material on Observable-Data Compatibility Checks for Restrictions concerns whether proposed constraints conflict with observed information. Compatibility does not make conditional and marginal statements interchangeable.
Common Mistakes and Limitations
- Wrong: “The conditional distributions are equal, so the marginal distributions must be equal.” Correction: Marginal distributions average over pattern-specific history distributions, which may differ.
- Wrong: “The marginal distributions match, so the conditional restriction holds.” Correction: As the cancellation example shows, different conditional distributions can average to the same margin.
- Wrong: “The completer marginal mean is the dropout mean under a conditional mean restriction.” Correction: Borrow conditional means, then average them using the dropout history weights.
- Wrong: “A conditional mean restriction equates the entire outcome distribution.” Correction: Equality of means alone says nothing sufficient about variances, tails, or other distributional features.
Even with good overlap, a restriction can be sensitive to which histories are included and how finely they are represented. Results should therefore make the conditioning history and the target pattern’s history weights clear. If histories do not overlap, the restriction alone supplies no data-based conditional distribution for the unsupported region; reporting a marginal result without that limitation can conceal important extrapolation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- If conditional outcome distributions are equal across patterns and the history distributions are also equal, what follows about the marginal outcome distributions?
- Two history categories have conditional means 4 and 10 units in both patterns. The dropout pattern has history weights 0.75 and 0.25. What is its mean under the conditional mean restriction?
- In Question 2, the completer pattern has weights 0.25 and 0.75. Calculate its mean and explain whether the conditional mean restriction requires the two pattern means to match.
- Why can equal marginal probabilities fail to establish conditional equality?
Solutions. (1) The marginal outcome distributions are equal: each pattern averages the same conditional distributions using the same history weights. (2) The dropout mean is \((0.75)(4)+(0.25)(10)=3+2.5=5.5\) units. (3) The completer mean is \((0.25)(4)+(0.75)(10)=1+7.5=8.5\) units. The restriction equates conditional means, not pattern-specific marginal means; the 3.0-unit difference results from different history weights. (4) Conditional differences can cancel in the weighted averages. Therefore, equality of margins alone does not show equality within each history.