Common Error: Confusing Pattern-Specific and Population Summaries

Why the Same Outcome Has Different Summaries

A mean reported for one dropout pattern answers a conditional question: what is the average outcome among people in that pattern? A mean for an entire treatment arm answers a different question: what is the average outcome across everyone in the arm’s target population? The two summaries can differ even when both are calculated correctly. Confusing them can lead to incorrect conclusions about the population, the treatment contrast, or the role of dropout.

The distinction matters whenever pattern-specific estimates are used to construct an arm-level summary. The calculation requires the pattern proportions in the target population, not an arbitrary average of the pattern means. It also requires care when a pattern has probability zero in an arm: its conditional mean is not defined, and it must not be included as though it were an observed summary.

Definition: Let $A=a$ denote membership in treatment arm $a$, and let $G$ be a dropout-pattern variable with possible categories $g=1,\ldots,K$. Let $Y$ be an outcome defined for members of the target population, whether or not it is observed for each person. The arm-specific pattern proportion is $\pi_{ag}=P(G=g\mid A=a)$. Define the pattern-specific mean $\mu_{ag}=E(Y\mid A=a,G=g)$ only when $\pi_{ag}>0$. The arm-level population mean is $\mu_a=E(Y\mid A=a)$. If the pattern categories partition the arm’s population, then $$ \mu_a=\sum_{g:\,\pi_{ag}>0}\pi_{ag}\mu_{ag}. $$ Patterns with zero probability contribute no term to this sum; their conditional means are not defined.

This identity describes a population quantity. In an analysis, $\pi_{ag}$ and $\mu_{ag}$ may be estimated, and a weighted estimate of $\mu_a$ is only as appropriate as the target population and the estimates used to construct it. A pattern-specific mean is not automatically the mean for the whole arm, nor is it automatically an estimate of that mean.

Why the Weighted Identity Holds

The pattern categories are mutually exclusive and collectively exhaustive within each arm. Therefore, for any person in arm $a$, exactly one of the indicators $I(G=g)$ equals one. The outcome can be written as a sum across patterns, with each term present only for members of that pattern. Taking the conditional expectation given $A=a$ gives the mixture identity.

\begin{aligned} E(Y\mid A=a) &=E\left(\sum_{g=1}^{K} I(G=g)Y\mid A=a\right)\\ &=\sum_{g:\,\pi_{ag}>0}P(G=g\mid A=a)\, E(Y\mid A=a,G=g)\\ &=\sum_{g:\,\pi_{ag}>0}\pi_{ag}\mu_{ag}. \end{aligned}

For a pattern with $\pi_{ag}=0$, $P(G=g\mid A=a)=0$, and conditioning on $A=a,G=g$ does not define an ordinary conditional mean in that arm. The first line of the derivation remains valid because the indicator for that pattern is zero with probability one. The conditional-mean term for that pattern is therefore omitted rather than assigned an estimated value and multiplied by zero.

The partition and the definition of $Y$ for the target population are important. The identity concerns an outcome defined for everyone in that population, even if some outcomes are missing from the recorded data. It does not, by itself, say that the pattern-specific means can be estimated from observed outcomes without additional modeling or assumptions.

Worked Example: Reconstructing an Arm-Level Mean

Suppose a hypothetical treatment arm has 100 people. Their dropout patterns are complete follow-up, later dropout, and early dropout. Assume the pattern-specific means of a specified outcome, in outcome units, are 12, 9, and 5, respectively, and the corresponding proportions are 0.60, 0.30, and 0.10. The proportions are nonnegative and sum to one, and each pattern has positive probability, so all three conditional means are defined.

The arm-level mean is the weighted sum:

\mu_a=(0.60)(12)+(0.30)(9)+(0.10)(5) =7.20+2.70+0.50=10.40.

Thus, the population mean is 10.40 outcome units. The unweighted average of the three pattern means is $(12+9+5)/3=8.667$ outcome units. That quantity gives each pattern equal weight, even though the patterns have different proportions, so it is not the arm-level population mean in this example.

Pattern Means and Treatment Contrasts

A treatment contrast can also be conditional or marginal. For a pattern present with positive probability in both arms, the within-pattern contrast is $\mu_{1g}-\mu_{0g}$. The arm-level marginal contrast is $\mu_1-\mu_0$, where each arm mean is constructed using that arm’s own pattern proportions:

\mu_1-\mu_0 = \sum_{g:\,\pi_{1g}>0}\pi_{1g}\mu_{1g} - \sum_{g:\,\pi_{0g}>0}\pi_{0g}\mu_{0g}.

These are different summaries. If pattern proportions differ between arms, the marginal contrast reflects both the pattern-specific means and how much population weight each arm assigns to its patterns. Consequently, averaging within-pattern treatment contrasts with arbitrary weights generally does not recover the marginal contrast. A within-pattern contrast is undefined in an arm where that pattern has zero probability.

Worked Example: Why the Marginal Contrast Is Not the Average Pattern Contrast

Consider a hypothetical two-arm comparison with two patterns, complete follow-up and dropout. Suppose both patterns have positive probability in both arms. In arm 0, their proportions are 0.80 and 0.20, and their respective means are 10 and 6 outcome units. In arm 1, the proportions are 0.50 and 0.50, and the means are 12 and 7 outcome units. Assume the means refer to the same specified outcome and target population within each arm.

First compute each arm’s population mean using its own pattern proportions:

\mu_0=(0.80)(10)+(0.20)(6)=8.00+1.20=9.20, \qquad \mu_1=(0.50)(12)+(0.50)(7)=6.00+3.50=9.50.

The marginal contrast, arm 1 minus arm 0, is $9.50-9.20=0.30$ outcome units. The within-pattern contrasts are $12-10=2$ outcome units for complete follow-up and $7-6=1$ outcome unit for dropout. Their simple average is 1.50 outcome units, not 0.30.

The discrepancy is not an arithmetic error. The simple average gives both patterns equal weight and ignores that the pattern proportions differ across arms. To see the components explicitly, use arm 1’s weights for the within-pattern contrasts and add a composition term based on arm 0’s means:

\begin{aligned} \mu_1-\mu_0 &=\sum_g \pi_{1g}(\mu_{1g}-\mu_{0g}) +\sum_g(\pi_{1g}-\pi_{0g})\mu_{0g}\\ &=(0.50)(2)+(0.50)(1) +\big[(0.50-0.80)(10)+(0.50-0.20)(6)\big]\\ &=1.50+(-3.00+1.80)=0.30. \end{aligned}

This algebraic decomposition is valid here because both pattern-specific means are defined in both arms. It does not turn the within-pattern contrasts into a marginal contrast by themselves: the composition term is also needed. The marginal contrast is descriptive unless the study design and analysis justify a causal interpretation.

Zero-Probability Patterns and Sparse Data

A true zero pattern proportion and an observed sample count of zero are not the same statement. If a pattern truly has probability zero in an arm, its conditional mean in that arm is undefined and its term is omitted from the mixture. If a finite sample contains no members of a pattern, that alone does not establish that the population probability is zero. A model that estimates a conditional mean for an unobserved pattern is making an extrapolation, not calculating a directly observed within-pattern mean.

Worked Example: Omitting a Zero-Probability Pattern

Suppose a hypothetical target population in arm 0 has two possible pattern labels. Its pattern proportions are $\pi_{01}=1.00$ and $\pi_{02}=0.00$. For the first pattern, the mean outcome is $\mu_{01}=8$ units. Since the second pattern has zero probability, $\mu_{02}=E(Y\mid A=0,G=2)$ is not defined by conditioning in this arm. Assume the outcome is defined for the arm’s target population and the two labels partition it.

Only the positive-probability pattern enters the mixture:

\mu_0=\sum_{g:\,\pi_{0g}>0}\pi_{0g}\mu_{0g} =(1.00)(8)=8\text{ units}.

It would be misleading to report a pattern-2 mean as though it had been calculated from arm-0 members. Writing $(1.00)(8)+(0.00)\mu_{02}$ can disguise the fact that $\mu_{02}$ is undefined. A zero weight does not make an undefined conditional mean into an estimated quantity.

Practical Checks and Common Errors

Before interpreting a pattern-weighted summary, specify the target population, the outcome, the treatment arm, and the pattern definition. Then check whether the proportions refer to that same arm and target population. If proportions are estimated, their sum should be one (apart from rounding), and the patterns should be mutually exclusive and collectively exhaustive for the population being summarized.

Common warning signs include:

Equal weighting without justification. Averaging pattern means equally answers a question about an equally weighted set of pattern means, not generally about the arm population.
Using the wrong proportions. Weights from the other arm, from a different population, or from a selected subset do not generally reconstruct the intended arm-level mean.
Calling a conditional mean a population mean. A mean among completers describes the completer pattern. It is not automatically the mean for all people in the arm.
Comparing within-pattern effects as if they were marginal. Different pattern proportions can make the marginal contrast differ substantially from within-pattern contrasts.
Reporting a mean in an empty pattern. A zero sample count does not provide a sample pattern mean. A modeled value may be available under further assumptions, but it must be identified as model-based.

Estimated pattern proportions and estimated pattern means both contribute uncertainty to a weighted summary. Treating the weights as fixed when they were estimated from the same data can understate uncertainty. Standard errors for pattern-weighted summaries should reflect the estimation procedure and relevant dependence; the earlier material on standard errors for pattern-weighted summaries addresses that calculation.

Pattern-specific summaries are useful for describing heterogeneity and for building an arm-level summary when the target weights are appropriate. They should not be substituted for population summaries merely because they are easier to interpret or estimate. Conversely, a marginal summary can conceal important differences across patterns. Reporting both may be informative, provided each is labelled with its conditioning set and target.

Key takeaway: A pattern-specific mean is conditional on arm and pattern. An arm-level mean is a mixture of the conditional means, weighted by the arm-specific pattern proportions and summed only over patterns with positive probability. Within-pattern treatment contrasts and marginal treatment contrasts are not interchangeable.

Check Your Understanding

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

An arm has two patterns with proportions 0.25 and 0.75 and means 4 and 10 units. What is the arm-level mean? Why is the equal-weighted average not the answer?
In one arm, a pattern has probability zero. Is its conditional mean defined, and should it appear in the mixture identity?
In two arms, pattern-specific means are both available, but pattern proportions differ. Is the simple average of the within-pattern treatment contrasts necessarily the marginal contrast?
A sample has no members in one pattern. Does that prove the population pattern proportion is zero? What should be reported about the within-pattern sample mean?

Solutions. (1) The arm mean is $(0.25)(4)+(0.75)(10)=1+7.5=8.5$ units. The equal-weighted average, 7 units, gives the patterns equal weights rather than their population proportions. (2) No. Conditioning on a zero-probability pattern does not define the conditional mean; omit that pattern from the sum. (3) No. The marginal contrast uses each arm’s own pattern proportions, so unequal pattern proportions can change the result; a simple average need not match it. (4) No. A zero sample count does not establish a zero population probability. There is no observed within-pattern sample mean to report; any model-based estimate should be identified as such and supported by its assumptions.