Detecting Multimodality and Artificial Smoothing

Why Shape Matters Beyond the Mean

Two distributions can have the same mean while differing in clinically important ways. One may contain two distinct clusters, while another concentrates values around the middle. Comparing only averages can conceal that distinction. For MICE diagnostics, this matters because a marginal distribution of imputed values that looks much smoother than the observed values may indicate that the imputation procedure is not reproducing important structure. Conversely, multiple peaks in an imputed distribution are not automatically an error: they may reflect real subgroups or a mixture of conditional distributions.

The diagnostic question is therefore not simply whether the imputed and observed means agree. It is whether their shapes are reasonably compatible with the variable, its predictors, and the imputation model. Histograms and density plots help reveal clusters, gaps, and unusual smoothing. They are exploratory tools, not tests that prove a model is correct or incorrect. The earlier material on density plots for continuous observed-imputed comparisons introduced their basic use; here the emphasis is on interpreting multiple modes and smoothing.

Definition: A mode of a distribution is a local maximum of its probability mass function or density. A distribution is multimodal when it has more than one mode. In a histogram, apparent modes are peaks formed by neighboring bins; in a density plot, they are local maxima of the estimated density. Whether a peak is visible can depend on the sample and on plotting choices, so an apparent mode is a diagnostic clue rather than a definitive finding about the population.

When comparing observed and imputed values, state which values are being plotted. A common display separates observed values from values imputed for missing entries. If values from several completed data sets are pooled for a descriptive plot, that plot can help summarize the imputed distribution, but pooled rows are not independent observations: the same records recur across imputations. The display is not a basis for ordinary independent-sample inference.

Reading Modes and Smoothness

A histogram groups values into intervals and displays the count or proportion in each interval. A density curve instead estimates a continuous distribution. For a sample \(x_1,\ldots,x_n\), the Gaussian-kernel density estimate at \(x\), using bandwidth \(h>0\), is

\hat f_h(x)=\frac{1}{nh\sqrt{2\pi}}\sum_{i=1}^{n} \exp\left[-\frac{1}{2}\left(\frac{x-x_i}{h}\right)^2\right].

The bandwidth \(h\) controls smoothing. A larger \(h\) spreads each observation's contribution over a wider range and tends to merge nearby peaks; a smaller \(h\) preserves local detail but can produce many bumps from sampling noise. The estimate has units reciprocal to the units of \(x\). It is a density, not the probability of observing exactly \(x\); probabilities are represented by areas under the curve.

A useful interpretation compares the observed and imputed shapes at more than one reasonable level of smoothing. If a suspected peak appears only under very narrow smoothing and disappears with a modest increase in bandwidth, it may be a sample fluctuation. If distinct peaks persist across reasonable displays and have a substantive explanation, they deserve investigation rather than automatic removal. If imputed values show a single broad, smooth mound while observed values have a clear gap or two clusters, consider whether the imputation model has represented the structure needed to generate those values.

Worked Example: Equal Means, Different Shapes

Suppose a hypothetical diagnostic comparison has the following six observed values and six imputed values, all in the same measurement units:

\text{Observed: } -2,\,-2,\,-1.8,\,1.8,\,2,\,2 \qquad \text{Imputed: } -0.4,\,-0.2,\,-0.1,\,0.1,\,0.2,\,0.4.

The observed mean is \((-2-2-1.8+1.8+2+2)/6=0\). The imputed mean is \((-0.4-0.2-0.1+0.1+0.2+0.4)/6=0\). Thus the means agree exactly.

But the shapes do not. Four observed values lie near \(-2\) or \(2\), with none between \(-1.8\) and \(1.8\). All six imputed values lie between \(-0.4\) and \(0.4\). A histogram or density plot would show two separated observed clusters and a concentrated imputed distribution near zero. In this hypothetical example, mean agreement alone would conceal a major distributional discrepancy. The difference prompts a review of the imputation model and the relevant predictors; it does not, by itself, establish which distribution is appropriate.

Bandwidth and Artificial Smoothing

The bandwidth dependence of a density plot can be made explicit. Consider the hypothetical observed values \(-2,-2,-1.8,1.8,2,2\), and evaluate their Gaussian-kernel estimate at zero using \(h=0.5\). Four observations have absolute value \(2\), and two have absolute value \(1.8\). Substitution gives

\hat f_{0.5}(0) =\frac{4e^{-\frac12(2/0.5)^2}+2e^{-\frac12(1.8/0.5)^2}} {6(0.5)\sqrt{2\pi}} =\frac{4e^{-8}+2e^{-6.48}}{7.5199} \approx 0.000586.

The estimate at the center is very small because all observations are far from zero relative to this bandwidth. In contrast, at \(x=1.9\), three observations are only \(0.1\) away, so their kernel contributions dominate; the estimate is approximately \(0.391\). This is the signature of a pronounced valley between clusters under this relatively narrow smoothing.

Worked Example: A Broader Bandwidth Can Hide a Valley

Use the same six observations, but set \(h=2\). At zero, the standardized distances are \(2/2=1\) and \(1.8/2=0.9\), giving

\hat f_2(0) =\frac{4e^{-1/2}+2e^{-0.9^2/2}}{6(2)\sqrt{2\pi}} =\frac{4(0.6065)+2(0.6670)}{30.0795} \approx 0.1250.

At \(x=1.9\), three observations are \(0.1\) away, one is \(3.7\) away, and two are \(3.9\) away. Therefore

\hat f_2(1.9) =\frac{3e^{-\frac12(0.1/2)^2} +e^{-\frac12(3.7/2)^2} +2e^{-\frac12(3.9/2)^2}}{30.0795} \approx\frac{3(0.9988)+0.1806+2(0.1493)}{30.0795} \approx 0.1155.

Now the density at the center is slightly higher than at \(1.9\), unlike the narrow-bandwidth comparison. Because the data are symmetric, zero is a stationary point; here the second derivative of the estimate at zero is negative, so zero is a local maximum. The two clusters that produced a deep valley with \(h=0.5\) have merged into a central peak with \(h=2\). This calculation shows why smoothness in a plotted curve is not, by itself, evidence that the imputed values are naturally unimodal.

The mechanism follows by differentiating the Gaussian estimate twice. For one observation \(x_i\), its kernel contribution is proportional to \(\exp[-(x-x_i)^2/(2h^2)]\). Its second derivative is that contribution multiplied by \((x-x_i)^2/h^4-1/h^2\). Summing these terms gives the curvature of the estimated density. At a symmetric center, the first derivative is zero; the sign of the second derivative distinguishes a local maximum from a local minimum. Thus bandwidth changes not only visual smoothness but can change the number and locations of estimated modes.

Diagnostic Workflow and Assumptions

A useful diagnosis combines the plot with assumptions about what the variable represents and how it was imputed. Histograms and density estimates describe the values supplied to them; they do not establish that the values are representative of the missing-data distribution. Interpreting a discrepancy requires that the observed and imputed values refer to the same variable and scale, that the plotted samples are identified correctly, and that the diagnostic is relevant to the intended imputation model. The missing values are unobserved, so an observed-imputed comparison cannot directly verify their true distribution.

Locate the shape feature.
Note apparent peaks, valleys, gaps, and unusual smoothness in the observed and imputed distributions. Check that the feature is not simply a difference in plotting scale or sample size.

Check sensitivity to smoothing.
Compare a histogram with a density curve, and assess whether a suspected mode persists under reasonable smoothing. A single narrow-bandwidth bump is weaker evidence than a stable, interpretable cluster.

Relate shape to the model.
Ask whether predictors or known subgroups could produce multiple clusters, and whether the imputation model can represent those conditional distributions. A simple normal conditional model, for example, can smooth away distinct structure if it is not represented in the predictors or model form.

Investigate, then qualify the conclusion.
Use complementary summaries and substantive knowledge to decide whether the discrepancy merits model revision. Treat plots as diagnostics, not as a formal proof of model failure or validity.

A particularly important failure mode is excessive smoothing in the imputation model. A model that predicts values close to a conditional mean, with too little variation or too little representation of subgroups, can generate an imputed distribution that is narrower and more regular than the observed distribution. This is not the same as choosing a large bandwidth for the diagnostic curve: one is a property of the imputation process, the other is a plotting choice. A broad plotting bandwidth can hide a discrepancy, but changing the plot cannot repair the imputation model.

Multimodality also requires caution. Several peaks can arise from a genuine mixture of populations, from predictors associated with different conditional distributions, or from a small sample. A marginal plot can combine records with different predictor values and thereby show several modes even when each conditional distribution is simpler. Conversely, marginal imputed values can look smooth while important subgroup-specific discrepancies remain. The plot is therefore a prompt to ask a more specific question, not a substitute for understanding the data-generating context.

Worked Example: Distinguishing a Model Concern From a Plotting Choice

In a hypothetical clinical variable measured in millimetres, the observed values show two clusters around 12 and 20, while the imputed values appear concentrated near 16. Investigators make two density curves using different bandwidths. With a narrow bandwidth the observed curve has two peaks and a low valley; with a broad bandwidth the peaks merge. The imputed curve remains centered near 16.

Step 1: Check the means. If both groups average about 16 mm, that agreement does not resolve the shape difference. A mean is a location summary, not a description of modality.

Step 2: Separate plotting from imputation. If the observed peaks disappear only after broadening the diagnostic bandwidth, that is evidence that the display is smoothing the observed shape. It is not evidence that the imputation process created the smooth imputed distribution.

Step 3: Examine a plausible explanation. If the two observed clusters correspond to clinically meaningful subgroups or predictor levels, a model that only generates values around a common conditional mean may be inadequate. If they arise from a few observations or arbitrary grouping, the apparent bimodality may not be stable.

Conclusion. The plots identify a question to investigate: whether the imputation model accounts for the structure associated with the two clusters. They do not establish that the missing values should belong to either cluster, because those values are not observed.

Comparisons, Limitations, and Common Pitfalls

Histograms and density estimates answer related but different descriptive questions. A histogram makes the frequency of observed intervals visible, but its appearance depends on how values are grouped. A density estimate gives a continuous curve, but its shape depends on bandwidth and kernel choice. A rug display can show individual observations and help assess whether a peak has data support, though it becomes crowded with many values. These displays are complementary: agreement among them can strengthen a descriptive impression, but none alone establishes the correct imputation distribution.

There is no universal numerical threshold for how many modes a plot must show or how much smoothness is “artificial.” Small samples can produce unstable peaks, and larger samples may reveal structure hidden by a coarse display. The appropriate interpretation depends on measurement scale, sample size, predictors, and the scientific meaning of the variable. Formal tests for modality, where used, require their own assumptions and are not replaced by visual inspection.

Mistake: “The means match, so the distributions match.” Equal means do not imply equal variance, tails, gaps, or modes. Compare the shape directly and use relevant context.
Mistake: “A smooth imputed curve proves the imputation model is good.” A curve can look smooth because of plotting bandwidth or because the model has suppressed variation. Check both the diagnostic display and the imputation model.
Mistake: “Every visible peak is a real population mode.” Sampling variation and smoothing choices can create or remove peaks. Look for stability and a plausible explanation before interpreting a mode substantively.
Mistake: “Pooled imputed values are independent data.” Records recur across completed data sets. A pooled plot is descriptive and should not be used as though it were a new independent sample for ordinary inference.

Key takeaway: Use modes and smoothness as clues about distributional compatibility, not as verdicts. Similar means can conceal strikingly different shapes, while bandwidth can either reveal or hide those differences. Interpret plots alongside the imputation model, predictors, and substantive knowledge.

Check Your Understanding

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

Two hypothetical samples each have mean 10 units. One has values clustered near 6 and 14; the other lies near 10. What does the mean comparison establish, and what does it miss?
For a Gaussian-kernel estimate, what is the effect of increasing the bandwidth \(h\) on nearby peaks? Why is this not a change to the imputation model?
In the worked kernel example, why is \(\hat f_{0.5}(0)\) much smaller than \(\hat f_{0.5}(1.9)\)?
An imputed density looks unimodal under one bandwidth, while the observed density looks bimodal under a narrower bandwidth. Name two checks before concluding that the imputation model has over-smoothed the data.

Solutions. (1) The equal means establish agreement in average location only; they do not establish agreement in spread, clustering, or modality. (2) A larger \(h\) spreads each observation's kernel contribution over a wider range and can merge nearby peaks. It changes the descriptive density estimate, not the values generated by the imputation procedure. (3) The observed values are all far from zero, so their kernel contributions at zero are small; three values are close to 1.9, making the estimate there much larger. (4) Check whether the observed bimodality persists under reasonable smoothing and whether the peaks have substantive or predictor-based support. Also distinguish a plotting choice from a model property, and do not treat the marginal plot as proof of the missing values' distribution.