Why Form Chain Summaries?
A MICE run produces successive imputed data states, usually in more than one chain. To describe how a particular variable’s imputations change across those states, first reduce the imputed values for that variable to a numerical summary at each iteration in each chain. The resulting iteration-by-chain summaries are the quantities that can be plotted or examined diagnostically.
The summary is not the entire imputed data set. A mean, for example, describes the center of the imputed values but does not show their full distribution or which records received particular values. Choosing and calculating summaries consistently is therefore essential: an apparent difference between chains is interpretable only if the same target cells, coding, and calculation are used throughout.
The fixed set \(M_j\) is important. In ordinary chained imputation, the observed/missing pattern is fixed, while imputed values are updated. Thus the number of imputed cells \(q_j\) for a variable stays the same across iterations and chains. If there are no missing values for \(j\), there is no set of imputed values to summarize. If \(q_j=1\), a mean is defined, but a sample variance across the imputed cells is not.
The arithmetic definitions below require no normality assumption. For the summaries to be comparable, however, the variable must have the same meaning, units, and coding in every chain; the summaries must use the same missing cells; and each summary must be calculated from the same point in the iteration. These are conditions for a meaningful comparison, not guarantees that the imputation models are appropriate or that a chain has stabilized.
Definitions for Numeric Imputations
For a quantitative variable, a basic summary is the mean of its imputed values in a chain at one iteration. Its sample variance records dispersion around that mean. The sample variance uses \(q_j-1\) in the denominator, so it is defined when \(q_j\geq 2\). It describes the spread among the imputed cells; it is not the variance of an estimator or a standard error.
Quantiles can add information about location and tails that a mean and variance do not preserve. To make hand calculations unambiguous here, use the empirical inverse-CDF convention: sort the \(q_j\) values as \(z_{(1)}\leq\cdots\leq z_{(q_j)}\), and define the \(u\) quantile, for \(0<u\leq1\), as \(z_{(\lceil q_j u\rceil)}\). This convention selects an observed value; software packages may use other interpolation conventions, so the convention should be recorded when exact numerical agreement matters.
These calculations describe the distribution of imputations for the missing cells, not the distribution of all observed and imputed values combined. The distinction matters: observed values may make up most of a variable’s completed data, so a completed-data mean can conceal changes among the imputed values.
Worked Example: Mean, Variance, and Quantiles Across Two Chains
Suppose a hypothetical continuous variable is missing for five records, so \(q_j=5\). At iteration 1, the imputed values are \((2,4,4,6,9)\) in chain 1 and \((3,3,5,8,11)\) in chain 2. At iteration 2, they are \((1,3,5,7,9)\) in chain 1 and \((2,4,5,6,8)\) in chain 2. Values are in hypothetical units.
For chain 1 at iteration 1, the mean is \((2+4+4+6+9)/5=25/5=5\). The squared deviations from 5 sum to \(9+1+1+1+16=28\), so the sample variance is \(28/(5-1)=7\) squared units. The sorted values are already \(2,4,4,6,9\). Under the stated convention, \(Q_{0.25}=z_{(2)}=4\), \(Q_{0.50}=z_{(3)}=4\), and \(Q_{0.75}=z_{(4)}=6\).
Applying the same definitions to all four chain-iteration pairs gives the following summary table. The median column is \(Q_{0.50}\); the quantile convention is unchanged across pairs.
| Chain | Iteration | Imputed mean | Sample variance | Imputed median |
|---|---|---|---|---|
| 1 | 1 | 5.000 | 7.000 | 4 |
| 1 | 2 | 5.000 | 10.00 | 5 |
| 2 | 1 | 6.000 | 12.00 | 5 |
| 2 | 2 | 5.000 | 5.000 | 5 |
For example, chain 2 at iteration 1 has mean \(30/5=6\); its squared deviations sum to \(48\), giving variance \(48/4=12\). These summaries show that the four states need not have the same spread even when their means are relatively close. They do not, by themselves, explain why the values differ or establish that the chains are adequately exploring their targets.
Category Proportions for Categorical Imputations
For a categorical variable with \(K\) possible levels, calculate the proportion of imputed cells assigned to each level at each chain-iteration pair. The category levels must be defined consistently across the run. For level \(k\), let \(I(z_{ij}^{(c,t)}=k)\) equal 1 when the imputed value is in level \(k\), and 0 otherwise. The denominator is the number of imputed cells, \(q_j\), not the number of records in the completed data set.
For a single-level, exhaustive coding with no missing or unrecognized category values, the proportions across all \(K\) levels sum to 1. This is a useful arithmetic check. A category with a zero proportion is absent from that chain’s current imputed values; it does not establish that the category is impossible in the population.
Worked Example: Binary Imputed Proportions
Suppose a hypothetical binary variable has eight missing records. At a particular iteration, chain 1 imputes three “Yes” values and five “No” values. Chain 2 imputes five “Yes” values and three “No” values. The denominator for each proportion is \(q_j=8\).
For chain 1, the “Yes” proportion is \(3/8=0.3750\), and the “No” proportion is \(5/8=0.6250\). For chain 2, the corresponding proportions are \(5/8=0.6250\) and \(3/8=0.3750\). In each chain, the two proportions add to 1. The difference describes the current imputed allocations; it is not a test of a difference in population proportions.
If the variable has more than two categories, calculate each level’s count divided by the same denominator. For example, counts of 2, 3, and 3 among eight imputations give proportions \(0.2500\), \(0.3750\), and \(0.3750\). Their sum is \(1.000\), as required when the three levels exhaust the coding.
Matching a Summary to the Variable
No single summary captures every feature of a chain’s imputed values. The mean is sensitive to large values; a median or other quantile describes a location in the ordered values; variance summarizes squared spread; and category proportions describe allocation among levels. A sensible set of summaries is guided by the variable’s scale, support, and the feature that needs monitoring. For skewed or bounded variables, a mean alone can miss changes in the distribution’s center or tails.
Worked Example: Why a Tail Quantile Can Add Information
Consider six hypothetical imputed values for a nonnegative, skewed count variable at one chain-iteration pair: \(0,0,1,1,2,12\). The mean is \((0+0+1+1+2+12)/6=16/6=2.667\) counts. The sorted values are already in order.
Under the empirical inverse-CDF convention, \(Q_{0.25}=z_{(\lceil1.5\rceil)}=z_{(2)}=0\), \(Q_{0.50}=z_{(3)}=1\), \(Q_{0.75}=z_{(\lceil4.5\rceil)}=z_{(5)}=2\), and \(Q_{0.90}=z_{(\lceil5.4\rceil)}=z_{(6)}=12\). The mean of 2.667 is substantially above the median of 1 because the value 12 raises the mean. The upper quantile makes that tail value visible in a way the median does not. With only six values, these quantiles are coarse summaries; they should not be read as precise estimates of population quantiles.
Construction Checks and Common Pitfalls
Before comparing or storing summaries, verify the data being summarized and the calculation itself. A concise construction sequence is:
Specify the variable, its units or category levels, and the fixed set \(M_j\) of records imputed for that variable.
Use the current imputed values for \(M_j\) from one chain and one iteration. Do not combine chains or iterations before calculating the summary.
Use the same formula, denominator, quantile convention, and coding rule for every state being compared.
Confirm that the number of values is \(q_j\), units are correct, and category proportions sum to 1 when levels are exhaustive.
A common mistake is to calculate a mean from all values in a completed data set and label it the “mean of the imputations.” That reasoning is wrong because the completed-data mean includes observed values as well as imputed values. The correction is to subset to \(M_j\) first and calculate the summary from those imputed cells alone. A completed-data summary can answer a different question, but it should be named accordingly.
Other failures include changing the denominator across iterations, treating category codes as quantitative measurements, silently dropping a category, mixing measurement units, and using different quantile conventions in different chains. A mislabeled or inconsistent summary can create an apparent difference that comes from data handling rather than from the imputation process. If the missing-cell set or coding genuinely differs, the summaries are not directly comparable until that difference is resolved or explicitly accounted for.
How These Summaries Differ From Alternatives
An observed-data summary uses only records with observed values; it can provide context but does not summarize the current imputed values. A completed-data summary combines observed and imputed values and may be useful for describing a completed data set, but it can dilute changes in the imputed subset. A chain summary instead isolates a chosen feature of the current imputations for one variable, chain, and iteration.
A single pooled summary across chains or iterations also loses the separate chain-by-iteration structure. It may conceal that different states produced different summaries. Conversely, one scalar summary cannot replace examination of the underlying values: equal means can arise from different distributions, and category proportions do not retain which record received which category. These are complementary summaries, not interchangeable representations of the full imputed state.
The earlier material on diagnostic targets in chained imputation motivates checking features of the imputed distributions. Here the task is to construct those features consistently. Whether their behavior indicates stabilization, drift, or other chain behavior requires additional diagnostics; a correctly calculated summary alone does not establish convergence or validate the imputation models.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A continuous variable has three imputed values, 2, 4, and 6, in one chain at one iteration. Find the imputed mean and sample variance.
- For a binary variable, six of ten imputed values are “Yes.” What are the “Yes” and “No” proportions, and what denominator is used?
- Under the empirical inverse-CDF convention, find \(Q_{0.75}\) for the sorted values \(1,2,4,7\).
- Why is a completed-data mean not generally the same as the mean of the imputations?
Solutions.
- The mean is \((2+4+6)/3=4\). The squared deviations sum to \((2-4)^2+(4-4)^2+(6-4)^2=4+0+4=8\), so the sample variance is \(8/(3-1)=4\) squared units.
- The “Yes” proportion is \(6/10=0.6000\), and the “No” proportion is \(4/10=0.4000\). The denominator is the ten imputed cells for that variable, not the number of all records in the completed data set.
- Here \(q_j=4\), so \(\lceil4(0.75)\rceil=3\). Thus \(Q_{0.75}=z_{(3)}=4\).
- A completed-data mean includes observed values along with imputed values. The mean of the imputations is calculated only from records whose values were missing, so the two means use different sets of values and can differ.