Pattern-Specific Quantiles and Tail Behavior

Why Means Do Not Describe Every Pattern

Within a dropout pattern, two outcome distributions can have the same mean and still differ substantially in their spread, skewness, or upper tail. Those differences matter when the scientific question concerns a typical value, an unusually high outcome, or the proportion exceeding a clinically meaningful threshold. Pattern-specific quantiles and tail probabilities provide direct summaries for such questions.

The conditioning is important. A pattern-specific summary describes the outcome distribution among individuals in a specified dropout pattern; it is not automatically a summary for the full study population. In particular, if a pattern’s members have already dropped out by a visit, their outcomes at that visit are unobserved. A distributional summary for those outcomes then depends on the model or assumptions used to describe them, rather than being directly calculated from observed values.

The infimum definition works for both continuous and discrete outcomes. For a continuous distribution, the \(p\)th quantile usually satisfies \(F_{t\mid r}(Q_{t\mid r}(p))=p\). For a discrete distribution, the cumulative probability can jump over \(p\), so the quantile is the smallest outcome value at which the cumulative probability reaches or exceeds \(p\). For integer-valued outcomes, distinguish \(P(Y_t>c\mid R=r)\) from \(P(Y_t\geq c\mid R=r)\); they need not be equal.

Calculating Quantiles and Tails From a Model

If the outcome at visit \(t\) is modeled as normal within pattern \(r\), with mean \(\mu_{t\mid r}\) and standard deviation \(\sigma_{t\mid r}>0\), then the quantile and upper-tail probability follow by standardizing the outcome. Write \(\Phi\) for the standard normal cumulative distribution function and \(z_p=\Phi^{-1}(p)\).

These formulas rely on the normal model being an adequate description of the conditional distribution, not merely on having calculated a mean and standard deviation. In particular, a symmetric normal model cannot represent strong skewness or a very heavy tail. When a different distribution is specified, calculate its quantiles and tails from that distribution instead.

Skewed Distributions and Tail Behavior

When outcomes are positive and right-skewed, a transformed model may be more appropriate than a normal model on the original scale. For example, if \(\log(Y_t)\mid R=r\sim N(\eta_{t\mid r},\tau_{t\mid r}^2)\), then the conditional outcome is lognormal. Its \(p\)th quantile is \(\exp(\eta_{t\mid r}+\tau_{t\mid r}z_p)\). For a positive threshold \(c\), its upper-tail probability is \(1-\Phi((\log(c)-\eta_{t\mid r})/\tau_{t\mid r})\). The logarithm is taken on the same outcome scale as the model.

Discrete Outcomes and Threshold Definitions

For binary, ordinal, or count outcomes, a quantile is located from the cumulative probabilities rather than from a continuous quantile formula. The probability table must cover all possible outcome values, and its probabilities must be nonnegative and sum to one. A tail probability is then the sum of probabilities over the values satisfying the chosen threshold definition.

Pattern-Specific Versus Pattern-Weighted Summaries

A pattern-specific quantile conditions on \(R=r\). A population-level distribution instead combines patterns using their population pattern probabilities, say \(\pi_r=P(R=r)\). Its cumulative distribution function is

To find a population-level quantile, first form this mixture distribution and then find where its cumulative probability reaches the desired level. In general, the result is not the weighted average of the pattern-specific quantiles. Quantiles are nonlinear summaries.

For instance, consider a hypothetical mixture of two normal pattern distributions with standard deviation \(1\), means \(0\) and \(5\), and pattern weights \(0.8\) and \(0.2\). Their pattern-specific medians are \(0\) and \(5\), whose weighted average is \(0.8(0)+0.2(5)=1\). But the mixture median solves \(0.8\Phi(q)+0.2\Phi(q-5)=0.5\). Since \(\Phi(q-5)\) is essentially zero near the solution, \(q\) is approximately \(\Phi^{-1}(0.625)=0.319\), not \(1\). This illustrates why a mixture quantile must be calculated from the mixture CDF.

By contrast, when the pattern-specific means exist, the mean of a mixture is the weighted average of the pattern-specific means: \(\mu_t=\sum_r\pi_r\mu_{t\mid r}\). That convenient identity for means should not be transferred to quantiles or tail probabilities. For a threshold \(c\), the population tail probability is the weighted average of the pattern-specific tail probabilities, \(\sum_r\pi_rP(Y_t>c\mid R=r)\), provided the pattern weights and conditional distributions refer to the same target population.

Assumptions, Diagnostics, and Limitations

The calculations above are descriptive only to the extent that the conditional distributions and pattern weights are known or estimated appropriately. Before interpreting a result, check that the visit, outcome scale, pattern definition, and threshold match the scientific question. For model-based summaries, check whether the chosen distribution can represent the observed shape: a normal model may miss skewness or tail weight, while a lognormal model requires positive outcomes and a plausible log-scale normal distribution. For discrete outcomes, confirm that the outcome support and cumulative probabilities have been specified correctly.

When using observed data within a pattern, quantiles and tail proportions can be unstable in small strata, especially near extreme quantiles or when few observations exceed the threshold. A few unusual values can strongly affect an estimated tail proportion. Sparse outcome categories also make discrete quantiles jump abruptly as data change. These are reasons to report the pattern sample size and relevant uncertainty, not reasons to present a modeled result as certain.

For a visit before or at dropout, the relevant outcome may be observed for members of the pattern, so summaries can be calculated from observed outcomes subject to the usual sampling and measurement considerations. For a visit after dropout, those outcomes are missing by the pattern’s definition. Their pattern-specific quantiles and tails cannot be recovered from observed data alone without assumptions or a model for the unobserved outcomes. Clearly label such summaries as model-based, and avoid implying that observed completers directly reveal the missing pattern’s post-dropout distribution.

The main alternatives answer different questions. A mean describes the distribution’s center in the original measurement scale but may be insensitive to how probability is arranged in the tails. A median is a robust measure of the 50th percentile but does not describe a specified extreme threshold. A tail probability directly answers a threshold question, while a quantile identifies a threshold for a chosen cumulative probability. Reporting a small set of complementary summaries can be more informative than selecting one summary for every purpose.