A reference interval, often called the reference range or loosely the normal range, is the central 95% of results from a suitable reference population. A clinician reads a patient’s result against it to decide whether the result is unremarkable or worth acting on. Two reference limits bound it: the 2.5th and 97.5th percentiles of the reference distribution.
Estimating those two percentiles sounds simple, but CLSI EP28-A3c offers several ways to do it, and the methods are not interchangeable. The right choice depends on two questions: how many reference individuals you have, and whether their values follow a Gaussian distribution. Get the pairing wrong and you either throw away information or report limits your data cannot support.
EP28-A3c sets 120 reference individuals per partition as the threshold for the recommended non-parametric method. The number is not arbitrary. About 120 is the smallest sample from which the non-parametric method can estimate the 2.5th and 97.5th percentiles with a 90% confidence interval on each limit.
Below 120 you have not failed. You have simply changed which methods are appropriate. With fewer individuals the non-parametric estimate becomes unstable in the tails. You move instead to a parametric or robust approach, which extracts more from a smaller sample by making (and checking) an assumption. Do not report a bare non-parametric interval from 40 samples as though it carried the same weight as one from 120.
Non-parametric quantile is the EP28 default at n ≥ 120. It ranks the data and reads the percentiles off directly, assuming nothing about the shape of the distribution. The method is robust and simple, but it needs that large sample, and its confidence intervals on the limits are wide. Analyse-it offers three standard computation approaches: (N+1)p, Np+½ and (N+⅓)p+⅓. At n = 120 the three place the 2.5th percentile at ranks 3.03, 3.50 and 3.34, a difference that is small beside the width of the confidence interval.
Parametric quantile assumes the values are Gaussian, if necessary after a transformation, and computes the limits from the mean and standard deviation (SD). Where that assumption holds, the parametric method is markedly more efficient than the non-parametric one: it works at smaller sample sizes and gives tighter confidence intervals. Where the assumption does not hold, and the data were not transformed to normality first, the limits are simply wrong. The normality check and the transformation step exist to prevent this.
Robust (bi-weight) is EP28’s small-sample option, workable with as few as around 20 reference individuals. The method estimates the limits iteratively, down-weighting values far from the centre, which makes it resistant to the occasional outlier a small sample cannot absorb without distortion. The robust method assumes the distribution is roughly symmetric, so transform a skewed sample before applying it. Robust estimation is the pragmatic choice when recruiting 120 healthy individuals per partition is not realistic, and for a partitioned analyte it often is not.
Harrell–Davis estimates each quantile as a weighted combination of all the order statistics rather than one or two ranked values. Using every observation lowers the variance of the estimate, including in the tails where reference limits sit. Harrell–Davis is a strong choice at moderate sample sizes, when you want more efficiency than the simple non-parametric method without committing to normality.
Bootstrap resamples the data many times, estimating each limit from the resamples and taking its confidence interval from their spread, with no distributional assumption. Bootstrapping is most useful when the analytic confidence intervals are questionable, or when you want a distribution-free interval on a limit estimated by another method.
Before you commit to a method, do two things. Screen for outliers with a Tukey box plot. Then check normality, using Shapiro–Wilk or Anderson–Darling together with a normal Q-Q plot. After that:
| Situation | Method |
|---|---|
| n ≥ 120, distribution non-Gaussian or unknown | Non-parametric quantile — the EP28 default |
| Distribution Gaussian or transformable to Gaussian, including n below 120 | Parametric — tighter CIs, once normality is confirmed |
| n below 120 (down to about 20), more individuals not obtainable, distribution symmetric or transformed to symmetry | Robust bi-weight |
| Moderate n, want efficiency and good tail behaviour without assuming normality | Harrell–Davis |
| Analytic CIs doubtful, or you want distribution-free CIs on a limit | Bootstrap |
| Adopting an interval established elsewhere | Transfer and verify — see below |
A skewed distribution does not force you onto the non-parametric method. If the skew is transformable, normalise the data first, estimate the limits parametrically, and back-transform them to the original scale. The transformations available for this are log, square root, cube root, reciprocal, Box–Cox, Manly exponential and two-stage exponential or modulus.
Many analytes need separate intervals for subgroups. Alkaline phosphatase by sex and age group is the classic example. Partition when the subgroups differ enough that a combined interval would misclassify results at the margins. The complication is arithmetic. Each partition needs its own adequate sample. Two sex partitions at the non-parametric threshold need 240 individuals, not 120. The cost is often why a careful laboratory uses the robust or parametric method in each partition.
You do not always have to establish an interval from scratch. EP28-A3c allows you to transfer an interval established elsewhere and verify that it holds in your population. The interval can come from a manufacturer or a published study, or from another method by way of a method-comparison regression. Verification is a small study, typically 20 reference individuals per partition.
EP28-A3c accepts the transferred interval if no more than two of those 20 fall outside it. A count that low is consistent, by a binomial test, with an interval that fits. Verification is a fraction of the work of establishment, and for most laboratories adopting a well-characterised assay it is the right route. The mercury transfer example below walks through one.
A worked example for each approach, ready to open in the Analyse-it trial:
The example workbook is downloading.
It opens in Excel on its own — the data and the finished results are both in it. Analyse-it is what lets you change the analysis and re-run it, try the same study on your own data, or work through it to see how the software handles it.
Every feature from all five editions for 15 days.
Reporting a non-parametric interval from too small a sample. Below 120 the tails are unstable. The number you print will not be reproducible. Move to a robust or parametric method rather than pretending the sample is larger than it is.
Applying the parametric method without checking normality. The efficiency of the parametric approach is real, but it is conditional. An untested normality assumption on a skewed analyte produces limits that are confidently wrong. Check normality first, and transform if needed.
Ignoring the confidence intervals on the limits. A reference limit is an estimate with uncertainty, and at realistic sample sizes that uncertainty is not small. Quoting the limits without their confidence intervals hides how much room there is around them.
Partitioning without the sample to support it. Splitting by sex and age can be right, but it doubles or quadruples the sample you need. Partition deliberately, and match the method to the sample each partition actually has.
Establishing when you could verify. If a well-established interval already exists for your method, a 20-sample verification may be all you need. Recruiting 120 individuals to re-establish it from scratch is effort spent where a transfer would have done.
Analyse-it covers the whole EP28-A3c workflow on your own reference data, inside Excel:
Every feature from all five editions for 15 days. Reference intervals are in the Medical, Method Validation and Ultimate editions, from US$ 340 a year. Transfer and verification of an existing interval are in Method Validation and Ultimate only, from US$ 475 a year. Validated against NIST and CLSI reference datasets. Full detail in the reference interval reference guide.