Why a Trace Can Look Like It Is Moving
A trace plot records a summary of the imputed data at successive MICE iterations. Even after the initial transient period has subsided, a summary may continue to move. That movement has at least two importantly different explanations: the chain may be undergoing drift, or it may be fluctuating slowly because successive values are strongly dependent. Both can produce long runs of values that rise or fall. A short segment alone may not distinguish them.
The distinction matters because a chain that is still drifting has not settled into a stable regime for the summary being inspected. By contrast, a slowly fluctuating chain may be stable in the long-run sense but explore that stable regime inefficiently. Neither a visually smooth trace nor a trace with no obvious slope proves convergence. The useful question is whether the whole long trace supports a stable level, while allowing for substantial serial dependence.
The summary might be an imputed mean, a dispersion measure, a quantile, or another quantity defined for the variable being monitored. Its units and meaning matter: a slope in an imputed mean measured in mmol/L is not directly comparable to a slope in an imputed proportion. Also, stability of one summary does not guarantee stability of every feature of the imputed data.
Reading Long Traces in Blocks
A practical way to avoid overreacting to individual iterations is to divide a sufficiently long trace into consecutive blocks and calculate the average summary within each block. Plotting or tabulating these block means reduces point-to-point noise while preserving broad movement. Blocks should be long enough to smooth short fluctuations, but not so long that a trend is averaged away. It is useful to examine more than one reasonable block length.
Block means are descriptive aids, not independent observations. When values are strongly dependent, adjacent blocks may also be dependent. Therefore, a visually increasing sequence of block means is a warning to investigate, not a formal test result by itself. Likewise, a sequence that appears flat at one block length can conceal a change that appears at another.
State what is being traced, its units, and what changes in that summary would matter for the analysis.
Look for sustained changes across the run, not just a local rise or fall. Retain the iteration order; a histogram alone discards evidence about movement.
Calculate block means or another suitable block summary, and compare early, middle, and late portions. Repeat with a different reasonable block size if the conclusion depends on the choice.
Extend the run or examine additional chains and summaries when a sustained shift remains plausible. Do not convert visual impressions into a claim of convergence or nonconvergence without considering dependence and model behavior.
Worked Examples: Drift and Slow Fluctuation
Worked Example: A Persistent Increase Across Blocks
In a hypothetical MICE run, a researcher tracks the mean of the imputed values for a continuous clinical measurement, in mmol/L. After the previously identified burn-in period, six consecutive blocks of 100 iterations have means 4.05, 4.18, 4.32, 4.47, 4.61, and 4.76 mmol/L. Assess the pattern descriptively.
The change from the first to the last block mean is
There are five block-to-block intervals, so the average change per block interval is
Since each block represents 100 iterations, this endpoint calculation corresponds to an average change of \(0.142/100=0.00142\) mmol/L per iteration across the span. More important than the numerical slope alone is that every successive block mean is higher. This persistent directional pattern is evidence of possible drift in this summary and deserves investigation. It does not prove drift: the block means may themselves be dependent, and a sufficiently persistent stationary fluctuation can temporarily rise. The appropriate conclusion is that this trace does not yet provide reassuring evidence of a stable level for the imputed mean.
Worked Example: A Slow, Wavelike Fluctuation
Now suppose a different hypothetical run has 12 consecutive block means, in mmol/L: 9.7, 10.3, 10.6, 10.4, 9.9, 9.5, 9.4, 9.8, 10.2, 10.5, 10.3, and 9.8. The values move in long waves rather than alternating rapidly. Calculate the overall, first-half, and second-half averages.
The sum of all 12 block means is \(120.4\) mmol/L, so their average is
The first six sum to \(60.4\), giving \(60.4/6=10.067\) mmol/L. The final six sum to \(60.0\), giving \(60.0/6=10.000\) mmol/L. The two half-run averages differ by only \(10.000-10.067=-0.067\) mmol/L, while the block means have a range of \(10.6-9.4=1.2\) mmol/L.
The relatively similar half-run averages and the rise-and-fall pattern are more consistent with slow fluctuation around a stable level than with a persistent increase or decrease across this trace. They are not proof of stationarity. The trace should still be checked at other block sizes and alongside relevant summaries. In this example, the modest difference between the halves is not enough to justify calling the process drift simply because several neighboring blocks move in the same direction.
Why Slow Mixing Can Resemble Drift
A short derivation explains why a stationary but persistent process can produce misleading stretches. Let \(S_t\) be a scalar chain summary at iteration \(t\), and suppose, for this justification, that it is weakly stationary after burn-in: its mean is a constant \(\mu_S\), and the covariance between two values depends only on their lag. Write \(\gamma(h)=\operatorname{Cov}(S_t,S_{t+h})\). For the average of \(n\) consecutive values, \(\bar{S}_n\), linearity of expectation gives
But the variance of that average is
To see the expression, expand the variance of the sum: there are \(n\) variance terms \(\gamma(0)\), and for each lag \(h\), there are \(n-h\) pairs in each of the two orders, contributing \(2(n-h)\gamma(h)\). Divide by \(n^2\) to obtain the variance of the average. When the covariances at many lags are positive, the terms in the sum remain positive, so the average can vary more than it would if successive values were independent. A long, smooth-looking excursion can therefore occur even when the expected summary is constant.
This argument does not show that any observed trace is stationary. It shows why visual persistence is not by itself decisive: slow movement can be generated by dependence, while a genuine change in the underlying regime can generate similar-looking movement. A sustained pattern across a longer trace, across sensible block choices, and across relevant summaries is more concerning than one local ramp.
Worked Example: A Trend That Reverses
A third hypothetical trace has eight block means for an imputed mean, in mmol/L: 9.6, 9.8, 10.0, 10.2, 10.4, 10.2, 10.0, and 9.8. A reader notices a clear increase at first and suspects drift. Compare the two halves and the endpoints.
The first four block means sum to \(39.6\), so their average is \(39.6/4=9.90\) mmol/L. The last four sum to \(40.4\), so their average is \(40.4/4=10.10\) mmol/L. The latter half is \(0.20\) mmol/L higher. However, the overall trace rises from 9.6 to 10.4 mmol/L and then falls to 9.8 mmol/L; the endpoint change from the first to the last block is only \(9.8-9.6=0.2\) mmol/L.
This short trace supports neither a simple persistent upward drift nor a confident claim of stable mixing. The rise reverses, but the halves differ, and eight block means provide limited information about longer-run behavior. The correct response is to inspect a longer run and other relevant summaries, rather than treating the initial rise as decisive or treating the later reversal as proof that the trace has stabilized.
Diagnostics, Alternatives, and Limits
Several observations make drift more plausible: block means move in the same direction over a substantial portion of the trace; the shift is large relative to ordinary variation in the summary; and the pattern persists when reasonable block lengths or related summaries are examined. Repeated shifts in different parts of the run, or a late level that remains separated from the early level, are also reasons to investigate further. These are diagnostic clues, not a universal numerical threshold.
Slow mixing is more plausible when the trace makes broad, persistent excursions but repeatedly reverses, with no sustained directional change over the full run. However, a long excursion can occupy much of a finite trace, making slow mixing look like drift. Similarly, aggressive smoothing or very large blocks can erase a real change. A trace that appears stable after changing the vertical axis or smoothing is not thereby more trustworthy; inspect the unmodified values as well.
| Feature | More consistent with drift | More consistent with slow fluctuation |
|---|---|---|
| Direction across the full trace | Persistent increase or decrease across many blocks | Long excursions that reverse direction |
| Early-versus-late blocks | Late blocks remain shifted from early blocks | Block averages are broadly similar, despite local waves |
| What to do next | Investigate initialization, model specification, and whether a longer run changes the pattern | Collect a longer trace and assess whether the excursions continue around a stable level |
A formal test for a time trend is not a universal solution. Ordinary regression calculations that treat iteration values as independent can misrepresent uncertainty when the trace is serially dependent. A non-significant trend test would not establish stability, and a significant test can flag a pattern without identifying its cause. Visual inspection, block comparisons, and additional runs answer different diagnostic questions; none should be mistaken for a proof that all aspects of a MICE procedure have converged.
The main alternatives are to extend the run, inspect multiple chains where available, and examine more than one scientifically relevant summary. Extending the run can reveal whether an apparent shift reverses or persists. Different starting values can help show whether chains approach comparable regimes, though chains that appear similar on one summary can still differ elsewhere. If a concerning pattern remains, review the imputation models and their implementation rather than assuming that simply adding iterations will resolve it.
Common Mistake and Practical Takeaway
Common mistake: “The trace rises for 50 iterations, so the chain is drifting.” This treats a local sequence as if it were independent evidence of a persistent change. A highly dependent trace can rise for many successive iterations and then return. The correction is to assess the full trace, compare multiple blocks, and ask whether directional movement persists at longer scales. The reverse error is also common: “The first and last values are similar, so there is no drift.” A trace can move substantially between its endpoints or have a change that reverses. Inspect the path, not just two values.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A trace summary increases in six consecutive blocks, and the late block means remain above the early ones. What does this pattern suggest, and what can it not prove?
- For block means 6.2, 6.3, 6.1, and 6.2 units, calculate the first-half and second-half averages. Is there a persistent endpoint increase?
- Why can a stationary chain summary produce a long upward or downward stretch?
- Name two checks that could help investigate an ambiguous trace without treating a single visual feature as conclusive.
Solutions. (1) It suggests possible drift and warrants investigation, but does not prove nonstationarity because slow dependent fluctuation can produce a sustained-looking segment. (2) The first-half average is \((6.2+6.3)/2=6.25\) units; the second-half average is \((6.1+6.2)/2=6.15\) units. The final value is \(6.2-6.2=0\) units from the first, so this sequence does not show a persistent endpoint increase. (3) Positive dependence can keep neighboring values similar, producing a long excursion even when the stationary mean is constant. (4) Examples include extending the run, changing the block length, inspecting other relevant summaries, and examining additional chains or starting values.