Trace Plots of Dispersion Summaries

Why Trace Dispersion?

A chain’s imputed values can keep roughly the same mean while their spread changes substantially. For example, one iteration may place imputed values close to the center, while later iterations produce values that are much more dispersed. A trace of imputed means may not reveal this change: the center can remain nearly constant as the spread expands or contracts. Dispersion traces add a complementary diagnostic view.

Here the target is the spread of the current imputations for one continuous variable, summarized separately at each iteration of each chain. This is a descriptive summary of the imputed values in that iteration. It is not the variance of an eventual estimate, nor a standard error for the chain summary. The earlier material on forming chain summaries establishes the general idea of calculating a summary at each iteration and tracing it over iterations.

Definition: For one variable, let \(M\) be its set of missing-data cases and let \(r=|M|\) be the number of those cases. In chain \(c\) at iteration \(t\), write the imputed values as \(y^{(c,t)}_{i,\mathrm{imp}}\), for \(i\in M\). When \(r\geq 2\) and the values are finite, the iteration-level imputed mean, sample variance, and sample standard deviation are

\bar{y}^{(c,t)}_{\mathrm{imp}} = \frac{1}{r}\sum_{i\in M}y^{(c,t)}_{i,\mathrm{imp}}, \qquad s^{2(c,t)}_{\mathrm{imp}} = \frac{1}{r-1}\sum_{i\in M} \left(y^{(c,t)}_{i,\mathrm{imp}}-\bar{y}^{(c,t)}_{\mathrm{imp}}\right)^2, \qquad s^{(c,t)}_{\mathrm{imp}}=\sqrt{s^{2(c,t)}_{\mathrm{imp}}}.

The variance is in squared units of the variable; the standard deviation is in the variable’s original units. These summaries describe only the imputed values, not the observed and imputed values pooled together.

The definition presumes that the same cases are summarized at each iteration and that the variable is represented on a consistent scale. If there is only one missing value, its dispersion cannot be summarized by a sample variance; if there are no missing values, there is no imputed-value trace for that variable. When values are tied, as can occur with predictive mean matching, the variance is still defined, although it may be zero or change in discrete steps.

How to Read the Variance and Standard-Deviation Traces

Plot iteration on the horizontal axis and either \(s^{2(c,t)}_{\mathrm{imp}}\) or \(s^{(c,t)}_{\mathrm{imp}}\) on the vertical axis. Use a separate trace for each chain, and label the summary and its units. Variance makes squared-scale differences explicit; standard deviation is often easier to interpret in the measurement units. Since standard deviation is the square root of variance, the two traces contain the same ordering information when all values are nonnegative, but their vertical scales differ.

A trace that stays within a fairly stable band suggests that the spread of the imputed values is not visibly changing over the plotted iterations. Sustained expansion or contraction, a change in level, or repeated large excursions can signal that the distribution of imputed values is changing in a way a mean trace would miss. A stable-looking dispersion trace is not proof that the imputation process has converged: it checks one feature of the imputations, not every feature or the adequacy of the imputation model.

The sample variance formula centers each value on the iteration’s own mean. That is exactly why it can detect a change in spread without requiring a change in center. The identity below makes the distinction explicit: two sets of values can have the same mean but different sums of squared deviations from that mean, and hence different variances.

Justification: If every value in an iteration is shifted by the same constant \(a\), the mean shifts by \(a\), but each centered deviation is unchanged:

(y_i+a)-(\bar{y}+a)=y_i-\bar{y}.

Thus the variance is unchanged by a common shift. By contrast, multiplying every value by \(b\) multiplies each centered deviation by \(b\), so the variance is multiplied by \(b^2\):

\frac{1}{r-1}\sum_{i=1}^{r}\bigl(b(y_i-\bar{y})\bigr)^2 = b^2s^2.

A mean trace can therefore remain steady while the variance changes, and a change of units can greatly alter the numerical scale of a variance trace.

Worked Example: Stable Mean, Changing Spread

Suppose a hypothetical variable has four missing cases. At one iteration their imputed values are 8, 9, 11, and 12 units; at a later iteration they are 5, 7, 13, and 15 units. Calculate the mean and dispersion at both iterations.

At the first iteration, the mean is \((8+9+11+12)/4=10\) units. The deviations from 10 are \(-2,-1,1,2\), so

s^2_{\mathrm{imp}} = \frac{(-2)^2+(-1)^2+1^2+2^2}{4-1} = \frac{10}{3} = 3.333\ \text{units}^2, \qquad s_{\mathrm{imp}}=\sqrt{3.333}=1.826\ \text{units}.

At the later iteration, the mean is also \((5+7+13+15)/4=10\) units. The deviations are \(-5,-3,3,5\), giving

s^2_{\mathrm{imp}} = \frac{25+9+9+25}{3} = \frac{68}{3} = 22.67\ \text{units}^2, \qquad s_{\mathrm{imp}}=\sqrt{22.67}=4.761\ \text{units}.

The mean trace alone would show no change between these two iterations. The standard deviation, however, increased from 1.826 to 4.761 units, and the variance increased from 3.333 to 22.67 squared units. The imputed values are much more dispersed at the later iteration, despite the identical mean.

Comparing Chains With Different Dispersion Scales

When chains for the same variable have very different dispersion levels, a shared linear vertical axis can compress the smaller trace until its fluctuations are difficult to see. Begin by checking that the chains really are comparable: they must summarize the same variable, the same set of missing cases, and the same measurement units and transformation. If those conditions hold, a difference in dispersion is itself important and should not be hidden.

Useful displays include separate panels with identical labeled units, or a logarithmic vertical axis when variances are positive and span several orders of magnitude. A supplementary within-chain relative trace can show changes relative to a chain’s own typical level. For example, divide each variance by that chain’s median variance over the plotted iterations. This relative view can make proportional movements visible, but it removes differences in absolute levels. It must therefore accompany, not replace, a view of the raw variance or standard deviation.

Worked Example: Two Chains on Very Different Scales

Consider two hypothetical chains for the same variable and missing cases. Their iteration-level standard deviations, in units of the variable, are:

Chain	Iteration 1	Iteration 2	Iteration 3
A: standard deviation	2	2.2	1.8
B: standard deviation	10	11	9

The corresponding variances are the squared standard deviations. For Chain A they are \(2^2=4\), \(2.2^2=4.84\), and \(1.8^2=3.24\) squared units. For Chain B they are \(10^2=100\), \(11^2=121\), and \(9^2=81\) squared units. Chain B’s standard deviations are about five times Chain A’s at each iteration, while its variances are about 25 times as large.

To inspect proportional movement, divide each chain’s standard deviations by its own median standard deviation. The medians are 2 units for Chain A and 10 units for Chain B. The relative sequences are therefore \(1,1.1,0.9\) for both chains. This shows similar proportional fluctuations, but it does not make their spreads equal: the raw scales still differ sharply. If these are truly chains for the same variable under comparable conditions, the difference in level warrants investigation rather than being dismissed because the relative traces match. A clear display would show the raw values in labeled panels and, if helpful, the relative traces separately.

Calculation, Diagnostics, and Common Failure Modes

A disciplined workflow keeps the summary interpretable. Use the same definition at every iteration, calculate dispersion from the imputed entries only, and retain the measurement units. If the goal is to compare chains, use a consistent axis or clearly label separate scales. A log axis can help reveal relative variation when all variances are strictly positive, but it cannot display a zero variance; do not add an arbitrary offset without stating and justifying it.

Identify the target values.
For one variable, select its missing cases and extract their imputed values at each chain iteration. Confirm that the set of cases and variable coding stay consistent.

Calculate the summaries.
Compute the sample variance using deviations from that iteration’s imputed mean; take its square root if a standard-deviation trace is more interpretable.

Plot and compare carefully.
Label units, chain, and summary. Inspect sustained changes and differences between chains in both the raw dispersion and, when useful, a clearly identified relative display.

Several patterns deserve follow-up. A smooth increase or decrease can indicate changing spread even when the mean is nearly flat. Abrupt jumps may reflect a change in the imputed values or a coding or processing error; check the underlying values and the iteration records before assigning a statistical explanation. A trace with frequent large fluctuations may be hard to interpret if the number of missing cases is small: a few individual values can move the variance substantially. A flat trace can also be uninformative if all imputed values are tied or if a coding mistake has accidentally reused the same values at every iteration.

Do not treat the \(r\) imputed values as an independent random sample in order to attach a conventional standard error to the variance trace. Imputations within an iteration may be dependent through shared fitted models, predictors, or donor selection. The trace is a diagnostic sequence of descriptive summaries, not a sequence of independent replicate estimates. Likewise, pooling observed and imputed values changes the target: fixed observed values can dominate the total spread and mask changes in the imputed portion.

Worked Example: A Small Set of Imputations

Suppose only three values are missing. In a hypothetical iteration, their imputations are 4, 4, and 10 units. The mean is \((4+4+10)/3=6\) units. The sample variance and standard deviation are

s^2_{\mathrm{imp}} = \frac{(4-6)^2+(4-6)^2+(10-6)^2}{3-1} = \frac{4+4+16}{2} = 12\ \text{units}^2, \qquad s_{\mathrm{imp}}=\sqrt{12}=3.464\ \text{units}.

This calculation is valid, but it is based on only three values. If one imputation changed from 10 to 8 units, the mean would become \(16/3=5.333\) units and the variance would become

\frac{(4-5.333)^2+(4-5.333)^2+(8-5.333)^2}{2} \approx 5.333\ \text{units}^2.

A single changed value has more influence here than it would in a much larger set. Thus, pronounced iteration-to-iteration changes in a dispersion trace with few missing cases require inspection of the individual imputations; they are not automatically evidence of a broad change in the imputation process.

What Dispersion Traces Add—and What They Cannot Show

A mean trace and a dispersion trace answer different descriptive questions: where are the imputed values centered, and how spread out are they? Neither alone describes the entire distribution. For instance, two sets can share the same mean and variance while differing in other distributional features. Quantile and tail summaries are useful complementary views, but their specific interpretation is a separate topic.

Other approaches also answer different questions. Summarizing the observed and imputed values together assesses the completed variable’s overall spread, but can obscure iteration changes among imputations. A variance trace isolates imputed-value spread, while the mean trace isolates location. Comparing chains on raw scales preserves differences in actual dispersion; within-chain normalization helps compare proportional movement but conceals differences in level. These displays are complementary rather than interchangeable.

A dispersion trace is therefore a focused diagnostic, not a pass/fail test. Its interpretation depends on a consistent target, adequate data to make the summary informative, and attention to the observed values and imputation procedure. It cannot establish that missingness assumptions are appropriate or that the imputation model is scientifically adequate. Its practical value is narrower and important: it can expose changes in spread that a center-only trace would leave unseen.

Common mistake: “The imputed means look stable, so the imputed distributions must be stable too.” This is incorrect because the mean does not determine the variance. Calculate and inspect a dispersion summary separately; then interpret it alongside the mean trace and the values that generated it.

Check Your Understanding

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

Four hypothetical imputations are 2, 4, 6, and 8 units. Calculate their sample variance and sample standard deviation.
Can two iterations have equal imputed means but unequal variances? State why.
One chain’s standard deviations are near 2 units and another’s near 10 units. What does a within-chain relative plot help show, and what does it hide?
Why can pooling observed values with imputations conceal an iteration-level change in imputed spread?
What happens to the sample variance if all imputed values are multiplied by 3?

Solutions. (1) The mean is 5 units. The squared deviations sum to \(9+1+1+9=20\), so \(s^2=20/(4-1)=6.667\) squared units and \(s=\sqrt{6.667}=2.582\) units. (2) Yes. Equal means do not require equal sums of squared deviations from those means, so the variances can differ. (3) A relative plot helps compare proportional movement within each chain; it hides the large difference in absolute dispersion, which must remain visible in a raw-scale display. (4) Observed values are fixed across iterations and may dominate the pooled spread, diluting changes among the imputed values. (5) Each deviation is multiplied by 3, so each squared deviation is multiplied by \(3^2=9\); the sample variance becomes nine times as large.