Why Examine Autocorrelation?
A trace plot shows the order of iteration-level summaries, but it can be hard to judge how strongly one iteration depends on the preceding ones. A chain may look stable in its overall range while moving only gradually from one region to another. Autocorrelation quantifies this serial dependence: it describes how closely a summary at one iteration is associated with the summary a specified number of iterations later.
In MICE, the object being diagnosed is a sequence of summaries, not an individual imputed value considered in isolation. For example, one might record the mean of the imputed values for a particular variable at every iteration. Large positive autocorrelation at short lags means successive summary values tend to remain similar. That pattern can indicate slow movement through the chain, but it does not, by itself, establish nonconvergence or show that imputations are inappropriate.
What a Chain-Level Summary Represents
A summary must be defined consistently across iterations. For instance, if \(Y\) is a variable with missing entries, \(S_{m,t}\) could be the mean of the values imputed for those entries in chain \(m\) at iteration \(t\). It could instead be a standard deviation or a chosen quantile. The same set of missing cells and the same summary rule should be used at every iteration; otherwise, changes in the summary may reflect a changing definition rather than movement of the chain.
Calculate autocorrelation separately for each chain. Concatenating chains can create artificial serial patterns at the points where one chain is joined to another, and can obscure differences among chains. The autocorrelation plot places lag \(k\) on the horizontal axis and \(\widehat{\rho}_m(k)\) on the vertical axis. Lag zero is conventionally one; diagnostic attention is generally on positive lags.
Assumptions Behind Interpretation
The formula is computable for any nonconstant finite sequence, but its diagnostic interpretation requires more care. The sequence should be approximately stable over the iterations being examined. In particular, the summary should have roughly constant mean and variance, and its lag dependence should be reasonably consistent over that period. These are forms of weak stationarity. If the summary is still drifting, one overall mean and one set of lag correlations can mix together distinct phases of the run.
This is an operating condition for interpreting the plot, not a guarantee that a MICE algorithm has a well-behaved stationary distribution. Conditional models in chained equations need not define a compatible joint model, and an autocorrelation plot cannot establish that the procedure targets a desired distribution. It also cannot verify assumptions about the missingness mechanism. The earlier material on diagnostic targets in chained imputation distinguishes computational behavior from other questions about imputation quality.
Worked Example: Calculating Lag-1 Autocorrelation
Suppose a hypothetical MICE chain produces the following sequence of means for a variable's imputed values, in arbitrary measurement units:
\(9.9,\ 10.9,\ 11.9,\ 10.9,\ 9.9,\ 8.9,\ 7.9,\ 8.9,\ 9.9,\ 10.9\).
The mean is \(10.0\), so the centered values are \(-0.1,\ 0.9,\ 1.9,\ 0.9,\ -0.1,\ -1.1,\ -2.1,\ -1.1,\ -0.1,\ 0.9\). The sum of their squared deviations is \(0.01+0.81+3.61+0.81+0.01+1.21+4.41+1.21+0.01+0.81=12.90\).
For lag 1, the adjacent products sum to \((-0.1)(0.9)+(0.9)(1.9)+(1.9)(0.9)+(0.9)(-0.1)+(-0.1)(-1.1)+(-1.1)(-2.1)+(-2.1)(-1.1)+(-1.1)(-0.1)+(-0.1)(0.9)=7.99\). Therefore,
This is a moderately strong positive lag-1 association in this short sequence: values at adjacent iterations tend to be similar relative to the sequence's overall variation. It is a descriptive estimate, not a test result or proof that the chain has failed to converge.
Why Persistent Dependence Matters
If successive summaries are strongly positively associated, a long run can contain fewer meaningfully distinct movements than its iteration count suggests. For example, a chain may spend many iterations near one value before moving gradually to another region. In an autocorrelation plot, this often appears as positive values that decline slowly as lag increases. A plot with substantial negative autocorrelation at lag 1 may instead reflect alternating movement, such as frequent movement from one side of the mean to the other. Either pattern is useful evidence about the chain's behavior, but its interpretation should be checked against the trace plot.
Autocorrelation describes dependence, not the level of a chain. In particular, centering each chain on its own mean means a constant shift in all values of a chain leaves its autocorrelation unchanged. This is why the plot cannot answer whether separate chains occupy similar regions.
Worked Example: Alternation and Negative Autocorrelation
Consider the hypothetical summary sequence \(0,1,0,1,0,1,0,1\). Its mean is \(0.5\), so each centered value is either \(-0.5\) or \(0.5\). The sum of squared deviations is \(8(0.25)=2.00\). Each of the seven adjacent pairs has product \(-0.25\), giving a lag-1 numerator of \(-1.75\). Thus
This strong negative estimate reflects the sequence's regular alternation, not persistent residence at one level. For comparison, at lag 2 there are six products, each \(0.25\), so \(\widehat{\rho}(2)=1.50/2.00=0.750\). The alternating pattern therefore produces negative association at odd short lags and positive association at even lags. In practice, a trace plot can help reveal whether an oscillating autocorrelation pattern corresponds to such structured movement.
Within-Chain Dependence Is Not Between-Chain Separation
A chain-level autocorrelation plot describes each chain's own serial pattern. It does not compare the location of one chain with another. To see why, add a constant \(c\) to every value in a chain. The centered values do not change:
The numerator and denominator in the autocorrelation formula are therefore unchanged. Two chains can have identical autocorrelation plots but substantially different average levels. Conversely, chains with similar levels can have different within-chain autocorrelation patterns.
Worked Example: Identical Autocorrelation, Separated Chains
Suppose two hypothetical chains have these summary sequences:
Chain A: \(0,1,0,1,0,1,0,1\). Chain B: \(10,11,10,11,10,11,10,11\).
For Chain A, the mean is \(0.5\) and the lag-1 autocorrelation is \(-0.875\), as calculated above. Chain B is Chain A shifted upward by 10 units; its mean is \(10.5\), and its centered values are still \(-0.5,0.5,-0.5,0.5,-0.5,0.5,-0.5,0.5\). It consequently has the same lag-1 autocorrelation, \(-0.875\).
The chains' means differ by 10 units, despite their matching autocorrelation plots. The example separates two diagnostic questions: do values persist across iterations within each chain, and do different chains explore comparable regions? Trace plots overlaid across chains address the latter more directly; autocorrelation addresses the former.
Reading Plots and Recognizing Failure Modes
Use an autocorrelation plot alongside the chain's trace plot and the earlier diagnostics for drift, stabilization, sticky behavior, and between-chain mixing. A practical reading sequence is:
Look for a sustained change in level or spread in the trace. If the sequence is drifting, interpret its autocorrelations cautiously because a single stationary pattern may not describe the whole run.
A large positive value at lag 1 that tapers slowly suggests persistence over multiple iterations. Alternating signs can suggest oscillation; an isolated bar should not be interpreted without the neighboring lags and trace.
Look for chain-specific patterns, then use overlaid traces to assess whether chains occupy similar ranges. Do not substitute within-chain autocorrelation for a between-chain comparison.
Confirm that the same variable, imputed cells, and calculation were used at every iteration. A changing set of cells or inconsistent calculation can make the sequence difficult to interpret.
A common rough reference for autocorrelations under independent sampling is a band of approximately \(\pm 1.96/\sqrt{T}\). It is not a universal pass/fail boundary: it relies on a simplified reference setting, and repeated inspection across many lags can produce bars outside the band by chance. MICE chains are serial processes, so the pattern and context matter more than mechanically counting bars beyond such a band.
Comparison and Limitations
| Diagnostic | Main question | What it does not establish |
|---|---|---|
| Autocorrelation plot | How strongly are summaries associated across a specified number of iterations within a chain? | Whether different chains occupy similar regions, or whether the imputation model is appropriate. |
| Trace plot | How does the summary move over iteration, including drift, abrupt changes, or sticky stretches? | A numerical measure of dependence at each lag. |
| Overlaid chains | Do chains appear to explore similar ranges and regions? | Whether each chain has weak serial dependence. |
Autocorrelation estimates from short runs can be noisy, especially at larger lags, where fewer pairs contribute to the numerator. A smooth-looking plot is not proof of convergence, and low autocorrelation is not proof that the chains represent the intended imputations. Strong autocorrelation is a reason to investigate chain behavior and run adequacy, not a stand-alone verdict. Converting autocorrelation into effective sample size or Monte Carlo error is a separate calculation.
Common Mistakes
- Calling a high lag-1 value proof of nonconvergence. It indicates adjacent summaries are associated; convergence and stability require broader assessment.
- Reading a negative value as automatically desirable. Strong negative autocorrelation can indicate regular alternation, which may itself reflect unusual or problematic chain movement.
- Combining chains before calculating autocorrelation. Chain boundaries and differing chain locations can distort the serial pattern; calculate within each chain first.
- Using autocorrelation to judge model validity. It diagnoses dependence in the recorded summary sequence, not whether the imputation models or missingness assumptions are suitable.
- Interpreting a drifting sequence with one stationary summary. First inspect the trace for changes over time; a single mean-centered ACF may conceal distinct phases.
Check Your Understanding
Answer each question, then check the brief solutions below.
- A chain has lag-1 autocorrelation \(0.82\), with positive autocorrelations that decline slowly across subsequent lags. What pattern does this suggest, and what additional plot should you inspect?
- For the sequence \(2,4,2,4,2,4\), calculate the lag-1 sample autocorrelation using the stated formula.
- If every summary in a second chain is exactly 5 units larger than the corresponding summary in a first chain, what happens to their autocorrelation plots? What important feature might still differ?
- Why should an autocorrelation plot be interpreted cautiously if the trace plot shows sustained drift?
Solutions. (1) The pattern suggests persistence across iterations and potentially slow movement; inspect the trace plot for drift, sticky stretches, or changing behavior, and compare chains separately. (2) The mean is 3 and the deviations are \(-1,1,-1,1,-1,1\). The denominator is \(6\), and the five adjacent products are all \(-1\), so \(\widehat{\rho}(1)=-5/6=-0.8333\). (3) The plots are unchanged because adding a constant leaves centered values unchanged. The chain locations can still differ by 5 units. (4) Sustained drift conflicts with the stable-sequence interpretation: a single overall mean can combine different phases, making the estimated lag pattern hard to interpret as stationary dependence.