Missing Measurements in a Nutrition Cohort

Why Missing Measurements Matter

A nutrition cohort may record participants’ usual diet, blood pressure, age, and body mass index, but some participants skip a diet questionnaire, miss a clinic visit, or leave a question blank. A regression fitted to the available records can still produce a coefficient, but the people contributing to that coefficient may differ systematically from those whose measurements are missing. The key issue is therefore not only how many values are absent, but also what process made them absent.

This tutorial considers a hypothetical cohort in which the research question is whether usual sodium intake is associated with systolic blood pressure after adjusting for age, sex, body mass index, and energy intake. The study is observational, so an estimated association does not by itself establish that changing sodium intake would change blood pressure.

What Complete-Case Analysis Does

A complete case is a participant with an observed value for the outcome and every variable included in the regression model. Complete-case analysis fits the model using only those participants. It is simple and transparent, but it can discard substantial data when missing values occur in different variables.

To reason about the consequences, let “observed” indicate that a value is recorded and “missing” indicate that it is not. Three standard descriptions of the missingness process are useful:

• Missing completely at random (MCAR): the chance a measurement is missing is unrelated to the observed data and to the unobserved value. For example, a randomly lost batch of forms could produce MCAR missingness.
• Missing at random (MAR): after conditioning on observed information, missingness does not additionally depend on the value that is missing. For example, if younger participants are less likely to complete a diet questionnaire, age may explain that pattern if age is observed.
• Missing not at random (MNAR): even after accounting for observed information, the chance a value is missing still depends on that unobserved value. For example, participants with especially high sodium intake might be less likely to report it, even among people with the same recorded age and health measures.

These labels describe assumptions about the process, not patterns that can always be diagnosed from the observed data. In particular, observed associations between missingness and recorded variables can challenge MCAR, but cannot prove MAR: the missing values themselves are unavailable for a direct check.

MCAR is a sufficient, relatively strong condition under which complete cases are not selected according to the study measurements. It is not the only possible condition under which a particular regression coefficient may remain unbiased. For example, selection can sometimes preserve the conditional relationship being estimated. But MAR alone does not guarantee that complete-case regression is unbiased: exclusion related to recorded age or health can change the composition of the analyzed sample and distort the fitted relationship. The mechanism and the model both matter.

Reading a Complete-Case Result

Before interpreting a complete-case estimate, state which participants contributed and what assumptions support its uncertainty calculation. In the example below, suppose checks of the hypothetical complete-case data support the chosen linear mean form and there is no evident dependence between participants. Assume the conventional model-based standard error is appropriate; if the variance pattern makes it unsuitable, the uncertainty calculation would need an appropriate alternative. These checks do not establish that complete cases represent participants with missing measurements.

Multiple Imputation as a Practical Alternative

Rather than deleting every incomplete record, multiple imputation creates several completed versions of the data. Each version fills in missing values with plausible draws that reflect uncertainty about those values. The regression is fitted separately to each completed data set, and the estimates and their uncertainty are combined. Unlike replacing every missing value once with a mean, proper multiple imputation carries information about imputation uncertainty into the final standard error.

For this regression, an imputation model should generally include the outcome, all variables in the analysis model, and useful observed auxiliary variables that help predict missingness or the missing measurements. It should respect variable types and preserve important features of the analysis, such as nonlinear terms or interactions if those are in the regression. Including systolic blood pressure when imputing missing sodium helps retain the sodium–blood pressure relationship; imputing a predictor without using the outcome can weaken that relationship.

A practical analysis commonly relies on MAR conditional on the information included in the imputation model, along with appropriate imputation models and adequate computation. Imputation does not make that assumption true. Implausible imputed values, poorly chosen predictors, or an imputation model inconsistent with the analysis can still produce misleading results.

Pooling an Imputation Analysis

The pooling calculation follows directly from separating two sources of uncertainty. There is sampling uncertainty within each completed data set, summarized by \(\bar{U}\), and there is variation among the completed-data estimates, summarized by \(B\). Because the average across \(m\) imputations estimates the missing-data uncertainty, its contribution is inflated by \(1+1/m\), giving \(T=\bar{U}+(1+1/m)B\). When a confidence interval is needed, a \(t\) critical value with an imputation-adjusted degrees of freedom is commonly used; with many imputations and ample data, it is often close to the normal critical value 1.96. The calculation below uses that close approximation for illustration.

Comparing Results and Checking Assumptions

The complete-case estimate, 1.60, and the pooled imputation estimate, 1.30, are not identical because they use information differently and need not target the same result under every missingness process. Their confidence intervals overlap substantially, and both include zero in this hypothetical example. That comparison is informative, but it is not a test proving that either missing-data assumption is correct. A report should state the sample sizes, missingness patterns, imputation variables and methods, and the assumptions on which interpretation depends.

Investigators can compare observed characteristics of participants with and without complete measurements, and tabulate missingness across relevant recorded groups. Such checks can identify patterns that make MCAR doubtful and guide the imputation model. They cannot determine whether missingness depends on the unobserved sodium value after conditioning on observed information. When MNAR is plausible, a sensitivity analysis can repeat the analysis under explicit alternative assumptions—for example, shifting imputed sodium values by a prespecified amount—and show whether the substantive conclusion changes. The shift and its rationale must be reported; sensitivity analysis measures dependence on assumptions rather than eliminating it.

Common Mistakes

• Wrong: “Only 15% of sodium values are missing, so missingness cannot matter.” Correction: Even a moderate missing fraction can bias a coefficient if missingness relates to the outcome or exposure. Examine joint patterns and the likely reasons for missingness.
• Wrong: “The imputed analysis uses everyone, so it is unbiased.” Correction: Imputation relies on a model and assumptions such as MAR conditional on included information; it is not a guarantee against bias.
• Wrong: “Fill each gap with the observed mean, then run ordinary regression.” Correction: Single mean filling understates variability and can distort associations. Multiple imputation represents uncertainty across plausible completions and pools that uncertainty.
• Wrong: “The two analyses differ, so one is proven correct.” Correction: Differences can signal sensitivity to missing-data handling, but observed data alone may not reveal the true missingness mechanism.