How Many Imputations Do You Need? From “Five Is Enough” to the Quadratic Rule

The honest answer is “more than you probably think, and it depends on how much information is missing” — not the flat m=5 that many textbooks and software defaults still imply. The number of imputations, m, is one of the few multiple-imputation decisions researchers routinely under-think: it is easy to accept a software default, run the analysis once, and move on, without asking whether that default was ever adequate for the amount of missingness actually present in the data. This guide works through why the old “5 is enough” convention existed, why it stopped being good advice, the percentage-of-missing-data rule that replaced it, the more accurate quadratic rule that replaced that, and the relative-efficiency-loss calculation that makes the cost of under-imputing concrete rather than abstract. It assumes you already know why your data are missing and have chosen multiple imputation as the method — see attrition bias and non-response bias for two common sources of missingness, full information maximum likelihood for an alternative that skips imputation entirely, and Rubin’s rules for the pooling step this guide’s m feeds into; see the research methods hub for the broader set.

The Old Answer: “3 to 10 Imputations Usually Suffice”

Rubin’s original 1987 treatment of multiple imputation included an efficiency argument showing that, for many practical purposes, a small number of imputations recovers most of the achievable precision on a point estimate. That result got compressed, over three decades of textbooks, tutorials and software documentation, into a much simpler and much more durable piece of folk wisdom: “m=5 is enough.” Several widely-used defaults still reflect it — R’s mice() function, for instance, defaults to m=5 imputed datasets unless you override it. The advice was not wrong when it was published; it was calibrated to the computing constraints and the specific quantity (point-estimate precision) Rubin was analyzing. It became misleading once researchers kept using m=5 for quantities the original argument never covered.

What m Actually Controls: Rubin’s Relative-Efficiency Formula

Rubin (1987) gives a closed-form expression for how much precision you lose by using a finite number of imputations m instead of an infinite number. Relative efficiency (RE) — the ratio of your actual estimator’s precision to the precision you’d get with infinite imputations — is:

RE = 1 / (1 + λ/m)

where λ is the fraction of missing information (FMI) for the specific estimate you care about, not the raw percentage of missing cells in your dataset — FMI accounts for how much the missing values actually affect that particular estimate, which is why two variables with identical percent-missing can have very different λ. The formula makes explicit what the old m=5 rule obscures: efficiency loss depends on both m and λ jointly, so a fixed m that’s adequate at low λ is inadequate at high λ.

The table below (independently computed, formula and method below) shows the efficiency loss — (1−RE)×100, i.e. how much precision you’re giving up relative to infinite imputations — across a realistic range of FMI and m:

Read one row: at λ=30% (a plausible FMI for a variable with roughly a third of its information missing), m=5 leaves 5.66% of achievable precision on the table; m=20 cuts that to 1.48%; m=50 to 0.60%. At λ=50%, m=5 costs 9.09% — almost a full order of magnitude more than the 0.99% that m=50 costs at the same λ. The equivalent standard-error inflation (relative to infinite m) is √(1+λ/m): at λ=30%, m=5 inflates your standard error by a factor of 1.0296 versus 1.0075 at m=20 — a small-looking number that can still be the difference between a confidence interval that excludes a null value and one that doesn’t, especially for an estimate already close to a significance threshold.

The Percentage-of-Missing-Data Rule

As computing power stopped being the binding constraint, methodological guidance moved toward a simple linear heuristic that is easy to state and easy to apply: use roughly as many imputations as the percentage of incomplete cases (or the percentage fraction of missing information) in your data — commonly written m ≈ 100 × FMI. Under this rule, 20% missing information calls for m≈20, 40% calls for m≈40, and so on. This rule is a real improvement on flat m=5: it at least ties m to the actual severity of missingness in a given dataset, rather than treating every analysis identically regardless of how much is missing. Its weakness is that it is linear where the underlying efficiency problem, once you also account for the stability of estimated quantities like the standard error and degrees of freedom (not just the point estimate Rubin’s original argument focused on), is not.

Von Hippel’s Quadratic Rule: A More Accurate Two-Stage Calculation

Paul von Hippel’s “How many imputations do you need? A two-stage calculation using a quadratic rule” (Sociological Methods & Research) argues that the number of imputations needed for a replicable standard-error estimate — one that won’t visibly change if you rerun the imputation with a different random seed — grows quadratically with FMI, not linearly. The two-stage calculation: first estimate FMI from a small pilot run (e.g. m=20), then plug it into

M ≈ 1 + 0.5 × (FMI / CV)²

where CV is the acceptable coefficient of variation of your standard-error estimate — how much run-to-run wobble in the SE you’re willing to tolerate, expressed as a fraction (CV=0.05 means you want the SE stable to within about 5% across reruns). Three worked values, computed directly from this formula and cross-checked against the published explainer of the same paper:

The pattern that makes this a genuinely different recommendation from the linear rule: at FMI=30%, the linear percentage rule says m≈30 regardless of how precise you need the SE to be. The quadratic rule says the honest answer is “it depends” — 6 imputations if a 10%-wobble SE is acceptable, 19 if you need it stable to 5%. For a small, exploratory analysis the linear rule can genuinely over-imput relative to what’s needed; for a confirmatory analysis with an estimate sitting near a significance threshold, where SE stability across reruns matters for reviewers and replication, the quadratic rule can call for more imputations than the linear rule would, once your target CV is tight enough. Neither the older m=5 convention nor the flat percentage rule captures that trade-off; the quadratic rule is the first of the three to make the target precision an explicit input rather than leaving it implicit.

A Practical Decision Procedure

1. Run a pilot imputation at a moderate m (20 is a reasonable starting point) and inspect the reported FMI/λ for each estimate you actually care about — not just the dataset’s raw percent-missing, which is a different and less directly useful number.
2. Decide how much SE instability you can tolerate. An exploratory analysis can typically accept a looser CV (0.10); a pre-registered confirmatory analysis, or an estimate close to a decision threshold, should target something tighter (0.05 or lower).
3. Apply the quadratic rule to the highest-FMI estimate that matters for your conclusions — not the average FMI across all variables, since the estimate with the most missing information is the one whose SE will be least stable at low m.
4. Round up, and re-run to confirm stability — rerun the final imputation at the chosen m with a different random seed and check that key SEs and p-values don’t move enough to change a conclusion.
5. Record the chosen m and the reasoning in the statistical analysis plan before final analysis, not after — choosing m post hoc, especially if a smaller m happens to produce a more favorable p-value, is a form of analytic flexibility that undermines the entire point of pre-specifying an analysis plan. See CASRAI’s Statistical Analysis Plan (SAP) entry for what belongs in that document generally.

When the Choice of m Matters Less

None of this means every analysis needs m=50. When FMI is genuinely low (a variable with only a few percent missing, or missingness with little relationship to the quantity being estimated) the efficiency-loss table above shows the practical difference between m=5 and m=50 is often under one percent — not worth the added computation or reporting complexity. The relative-efficiency and quadratic-rule calculations exist precisely so this is a calculation rather than a guess: the point is not “always use more,” it’s “know your FMI and your precision target before picking m,” which sometimes means m=5 genuinely was enough, and sometimes means it wasn’t. Note also that m only addresses how many imputations to run under whatever missingness mechanism you’ve assumed; it says nothing about whether that assumption itself is safe. If your outcome data are plausibly missing not at random, pairing your chosen m with a tipping-point sensitivity analysis tests the more fundamental assumption a bigger m can’t fix.

Frequently Asked Questions

Is fraction of missing information (FMI) the same as the percentage of missing data?

No, and conflating them is the most common mistake in applying either rule of thumb. Percent-missing describes how many cells or cases are incomplete in the raw data. FMI describes how much statistical information about a specific estimate was lost because of that missingness, which depends on how strongly the missing values are related to the variables used to impute them and to the estimate itself. A variable can have 40% of its cases missing but a low FMI if the imputation model predicts it well from other observed variables, or a modest percent-missing but a high FMI if the missingness is hard to predict.

Does a higher m ever hurt?

Not statistically — relative efficiency is monotonically increasing in m, so more imputations never make an estimate less precise. The real costs of a larger m are computational (larger m means longer run times and more storage for large datasets or complex imputation models) and reporting complexity (some pooled diagnostics and sensitivity checks get more tedious to report at very large m). For most social-science and clinical datasets, m in the tens rather than the hundreds is a practical ceiling long before efficiency loss becomes the binding constraint.

My software defaults to m=5. Is that wrong?

It’s a starting point, not a stopping point. A default has to pick something, and m=5 is a reasonable low-cost first pass for inspecting whether your imputation model runs and produces sensible values. Treat the default run as the pilot step in the practical procedure above — check the reported FMI, decide your precision target, and increase m from there if FMI turns out to be non-trivial for the estimates that matter to your conclusions.

Do I need to recompute m separately for every variable in my dataset?

Only for the estimates that matter to your paper’s conclusions. Use whichever of those has the highest FMI to set m for the whole imputation — since all analysis-model estimates are computed from the same m completed datasets, you pick one m for the run, and it should be driven by your least-favorable (highest-FMI) estimate of interest, not by an average across every variable in the file.

Reproducibility Note

Every number in the two tables above was independently computed via a Node.js script, not looked up or estimated. The relative-efficiency table applies Rubin’s (1987) published closed-form formula RE = 1/(1+λ/m) directly for λ ∈ {0.10, 0.20, 0.30, 0.50} and m ∈ {3, 5, 10, 20, 50, 100} — a deterministic calculation, not a simulation, since the formula itself is exact. The quadratic-rule table applies M = 1 + 0.5×(FMI/CV)²; the 0.5 coefficient was not assumed but back-solved by testing candidate coefficients against the three worked examples published in von Hippel’s own explainer of this paper — a coefficient of 1 (i.e. M = 1 + (FMI/CV)²) reproduces 10, 37, and 17 for the three published examples, which does not match; 0.5 reproduces 5.5→6, 19, and 9 exactly, matching all three published results after rounding up to a whole number of imputations. The script and its output are available on request.