Diagnosing Heaping and Rounding Artifacts

Why Digit Patterns Matter

Two sets of measurements can have similar centers and spreads while differing in how their values are recorded. Self-reported ages may cluster at multiples of five or ten; a device may record a laboratory measurement to the nearest tenth; and a fitted imputation model may instead generate smooth values at arbitrary decimal places. These patterns can reveal information about measurement and recording that is not obvious in a density plot or summary of means and standard deviations.

In MICE, the observed values and imputed values are not generally interchangeable samples. Observed values belong to people whose measurements were recorded, while imputed values fill in measurements missing for other people. Their marginal patterns may differ for legitimate reasons, including differences in predictors or in the missingness mechanism. A digit-pattern comparison is therefore a diagnostic question: are the imputed values compatible with the measurement process, and is there a discrepancy that the imputation model should explain?

Define the Pattern Before Counting It

A useful check begins with the variable’s meaning and measurement history, not with a convenient digit test. Ask whether the variable is a count, a self-report, a device measurement, or a value transformed from another measurement. Establish the units and the smallest meaningful or recorded increment. An integer-valued count has a different support from a continuous measurement rounded to the nearest integer. Likewise, an age recorded in whole years may reflect both the measurement convention and respondents’ tendency to report preferred ages.

For a specified event \(A\), such as “the value is a multiple of five,” calculate its frequency among observed values and among imputed values. If \(O\) denotes the indices with observed measurements and \(M\) the indices with missing measurements, one simple comparison is:

Here \(I(\cdot)\) equals one when the condition is true and zero otherwise, and \(l\) identifies an imputed data set. Calculate the imputed frequency separately in each completed data set, then summarize its variation across imputations. Do not pool all imputed cells across imputations and treat them as independent observations: each set of imputations represents uncertainty about the same missing entries.

The comparison is descriptive. Even if observed and imputed values are generated under a well-specified model, their marginal pattern need not match: the missing cases can differ from the observed cases. Under missing completely at random, marginal patterns might be expected to agree apart from sampling and imputation variability, but that expectation does not hold in general under missing at random. More directly, if the measurement process is the same for both groups, imputed values should normally respect its units, support, and recording resolution. That requirement is not the same as demanding identical frequencies of every digit.

Connect the Pattern to the Recording Rule

A fixed rounding rule creates a lattice of possible recorded values. If a continuous measurement \(X\) is reported to the nearest increment \(u\), the recorded value is \(R=u\,\mathrm{round}(X/u)\). Thus \(R\) can take values spaced \(u\) units apart. For example, values reported to the nearest tenth lie on a grid of tenths. An imputed value with additional nonzero digits may be incompatible with the recorded variable, even if it is a plausible value for an unobserved latent measurement.

This distinction depends on what is being imputed. If the analysis concerns the recorded measurement, the imputation should represent values at the recorded resolution. If the scientific target is the underlying unrounded quantity, imputing a latent value is a different modeling task: a recorded value \(R=r\) corresponds to an interval of possible latent values, approximately from \(r-u/2\) to \(r+u/2\), rather than revealing the exact \(X\). Silently treating the recorded number as exact or rounding arbitrary model draws can each misrepresent the target.

Quantify and Investigate the Discrepancy

A rate difference is easy to communicate, but its interpretation depends on the pattern selected. For terminal digits, one may compare the full distribution of last digits rather than only grouping values into “round” and “not round.” For measurements rounded to a known increment, count values on the permitted grid and values off it. For a suspected heaping point, compare its frequency with frequencies at nearby values. State the rule in advance and report counts as well as percentages; a percentage based on a few observed values can be unstable.

The following justification explains why a grid check is appropriate when the recorded variable is rounded. If every recorded value is rounded to increment \(u\), then by the recording rule each value equals an integer multiple of \(u\). Therefore, an off-grid imputed value cannot be a possible value of that recorded variable under the same rule. This establishes a compatibility warning, not the cause of the warning: the model may be targeting the latent quantity, the recording rule may have been misstated, or the imputation implementation may be inappropriate.

Compare Diagnostics and Recognize Failure Modes

Common failure modes include assuming every multiple-of-five pattern is a defect, treating observed and imputed marginal proportions as if they must be equal, and examining just one completed data set. A digit pattern may be a genuine feature of self-report, a consequence of a device, or an artifact of data processing. Conversely, a smooth-looking imputed distribution can conceal an important mismatch in recording resolution. Review value frequencies alongside the data dictionary, units, and known measurement procedures.

Avoid using a uniform terminal-digit distribution as a default benchmark. Uniformity is justified only when the variable and its measurement process make that benchmark plausible. For example, genuine ages or work hours need not have equally frequent terminal digits. Likewise, an apparent excess of zeroes may come from a real mass at zero rather than rounding. Comparing nearby values and understanding the measurement process are more informative than mechanically declaring any unequal digit frequencies suspicious.

Choose a Response Proportional to the Evidence

When imputations lose a plausible heaping pattern, investigate whether the imputation model represents the process that generated the recorded values. When they contain unsupported precision, first clarify whether the target is the recorded or latent measurement. Depending on that answer, a model that respects the recording scale, an explicit measurement model, or a justified post-processing rule may be considered. Any change should be checked for its effects on distributions and on the substantive analyses; post hoc rounding is not a universal repair.

These checks complement rather than replace distributional comparisons. Density plots and empirical distribution functions can reveal broad shape differences, while targeted digit counts can reveal fine-scale patterns. A pattern discrepancy is a reason to investigate, not a standalone test of validity or a guarantee that a revised model is correct. Sparse observed values, uncertain recording histories, and differences between observed and missing cases all limit what can be concluded from the marginal comparison.