Reference Interval

A reference interval defines the range of values expected in a healthy reference population. When a laboratory reports a result as "outside the reference range", the statistical meaning is precise: the value lies in the outer 5% of the distribution of that analyte in a well-defined healthy population — 2.5% at the lower tail and 2.5% at the upper tail. The interval therefore encloses the central 95% of the reference distribution. Understanding how reference intervals are constructed — and the assumptions behind each method — is essential for appreciating when they are reliable and when they should be treated with caution.

The Reference Sample

The statistical validity of a reference interval depends on the quality of the reference population as much as on the choice of method. The CLSI guideline C28-A3c defines the reference sample as a collection of individuals representative of the healthy population for whom the interval is intended — free from conditions or treatments that affect the analyte. Inclusion and exclusion criteria should be specified before sampling begins and should reflect the target clinical population in terms of sex, age, ethnicity, and any other known biological modifiers.

Sample size. The minimum requirements differ by method:

Parametric method: at least 40 observations.
Non-parametric method: at least 120 observations (required to compute confidence intervals for the reference limits).
Robust method: recommended when only 40–120 observations are available.

Larger samples always improve precision. Collecting 120 or more subjects gives access to all three methods and allows reliable confidence intervals regardless of which is chosen.

Assessing Normality

The choice of method depends on whether the reference sample follows a normal distribution. MedCalc uses the Shapiro-Francia test (Royston, 1993): if the $P$ value exceeds 0.05, the data are consistent with normality and the parametric method can be applied. If $P$ ≤ 0.05, normality is rejected.

When values are non-normal, a logarithmic transformation or a Box-Cox power transformation is applied before computing the reference interval. After transformation the normality test is repeated. If normality is achieved, the parametric method is applied on the transformed scale and the resulting limits are back-transformed to the original units — producing an asymmetric interval that correctly reflects the skewed shape of the original distribution. If no transformation achieves normality, the non-parametric or robust method is used.

The Parametric Method

When the reference sample is normally distributed — either directly or after transformation — the reference limits are:

$$ \text{Lower RL} = \bar{x} - 1.96 \, s \qquad \text{Upper RL} = \bar{x} + 1.96 \, s $$

where $\bar{x}$ is the sample mean and $s$ is the standard deviation. These limits enclose exactly 95% of a normal distribution.

The parametric method is the most statistically efficient: it extracts maximum information from the data and produces relatively narrow confidence intervals even at moderate sample sizes. The price is the normality assumption — if the distribution is genuinely skewed and no transformation corrects it, the parametric limits will be inaccurate, typically resulting in a lower limit that is too high or an upper limit that is too low.

The Non-parametric Method

When normality cannot be achieved even after transformation, the reference limits are estimated directly as the 2.5th and 97.5th sample percentiles. No distributional assumption is made; the limits are simply the values below which 2.5% and above which 2.5% of the reference sample fall.

The non-parametric method is robust to the shape of the distribution but requires substantially more data to achieve the same precision as the parametric method. A meaningful confidence interval for a percentile limit requires at least 5 observations in each tail — for a 95% reference interval this corresponds to n ≥ 120. With fewer observations the confidence intervals become very wide and the reference limits themselves become unstable. Below 120 observations, the robust method is preferable.

The Robust Method

The robust method provides a practical solution when the reference sample is smaller than 120 but the distribution departs from normality. It uses iterative re-weighting based on the median and median absolute deviation (MAD), producing estimates that are resistant to outliers and to mild non-normality — without requiring a separate outlier exclusion step.

The algorithm begins with initial estimates $T_1 = \text{median}(x_i)$ and $T_2 = \text{MAD}(x_i)\,/\,0.6745$, where the constant $0.6745$ scales the MAD to be a consistent estimator of the standard deviation under normality. At each iteration, each observation $x_i$ receives a weight that decreases as its distance from the current centre increases. Specifically, a transformed value $c_i$ is computed as:

$$ c_i = \frac{x_i - T_1}{3 \, T_2} $$

Observations with $|c_i| \geq 1$ receive zero weight; observations closer to the centre are upweighted smoothly. Updated estimates of $T_1$ and $T_2$ are derived from the weighted observations and the process repeats until convergence. The final reference limits are:

$$ \text{Lower RL} = T_1 - 1.96 \, T_2 \qquad \text{Upper RL} = T_1 + 1.96 \, T_2 $$

Because outlying values are downweighted rather than discarded, the robust method avoids the subjective step of deciding which observations to exclude. It can be applied to samples as small as 40 observations and handles skewed distributions better than the parametric method when no normalising transformation exists.

Confidence Intervals for Reference Limits

A reference interval estimated from a sample carries statistical uncertainty. Reporting 90% confidence intervals for the reference limits — the convention recommended by CLSI — communicates how precisely the limits have been pinned down. Narrow CIs reflect a large, well-behaved reference sample; wide CIs signal that the reference interval should be used with caution.

Parametric method. The 90% CI for each reference limit is:

$$ \text{RL} \pm 1.645 \times s \times \sqrt{\frac{1}{n} + \frac{1.96^2}{2(n-1)}} $$

The factor $1.645$ is the 95th percentile of the standard normal distribution (giving a one-sided 95% bound, equivalent to a 90% two-sided CI), and the term $1.96^2/[2(n-1)]$ accounts for uncertainty in the estimated standard deviation.

Non-parametric method. The CI for the 2.5th percentile lower limit uses the binomial distribution to identify the ranks surrounding the percentile (Reed et al., 1971). With sample size $n$ and tail probability $p = 0.025$, the 90% confidence bounds correspond to ranks:

$$ r = n \cdot p \pm 1.645 \times \sqrt{n \cdot p \cdot (1 - p)} $$

rounded to the nearest integers. The values at those ranks bracket the confidence interval for the reference limit.

Robust method. No closed-form variance formula exists for the robust estimates, so confidence intervals are obtained by bootstrap resampling (Efron & Tibshirani, 1993): the reference sample is drawn with replacement many times, the reference limits are recalculated for each resample, and the 5th and 95th percentiles of the bootstrap distribution form the 90% CI.

Choosing the Right Method

The decision follows a straightforward sequence:

Test normality (Shapiro-Francia). If the data are normal, or become normal after a log or Box-Cox transformation, apply the parametric method. This is the preferred route when applicable — it is the most efficient and gives the narrowest confidence intervals.
If no transformation achieves normality and $n \geq 120$, use the non-parametric method.
If no transformation achieves normality and $n < 120$, use the robust method.

The robust method is not merely a fallback for small samples. Because it downweights rather than excludes outlying values, it is the most defensible choice whenever the reference sample may contain extreme observations that cannot be verified or removed on clinical grounds.

Partitioning Reference Intervals

A single reference interval is inappropriate if the analyte differs systematically between subgroups. The most common biological partitioning factors are sex and age. Before establishing separate intervals, the cost in precision must be weighed against the clinical gain: splitting a sample of 150 into two groups of 75 reduces confidence interval width for a single pooled interval while producing wider, less reliable limits for each subgroup.

The decision to partition should be grounded in clinical knowledge and supported by statistical testing. The criterion recommended by Harris & Boyd (1990) evaluates the degree of overlap between the two candidate subgroup intervals: if the overlap is substantial — specifically, if more than 87.5% of values from both subgroups would fall within a common pooled interval — partitioning offers little clinical advantage. If the overlap is below this threshold, separate intervals are warranted. CLSI C28-A3c recommends a minimum of 120 observations per subgroup when partitioning.

Example: Serum creatinine increases with muscle mass and differs substantially between adult males and females — typically by more than half a standard deviation. Separate reference intervals by sex are clinically necessary. Serum sodium, on the other hand, varies little between the sexes; a single pooled interval is appropriate and more precise.

When the analyte varies $continuously$ with age rather than changing abruptly at a threshold, establishing separate reference intervals for discrete age classes is unsatisfactory — the limits at the boundaries of each class may be discontinuous and the intervals within each class will be imprecise if the groups are small. The preferred approach is to model the reference limits as smooth functions of age using weighted polynomial regression. This is described in the Age-related reference interval chapter.

Transferring and Validating Reference Intervals

Establishing a reference interval from scratch requires substantial resources — typically 120 or more carefully selected subjects and thorough pre-analytical standardisation. In practice, many laboratories verify whether a previously published or manufacturer-supplied reference interval is applicable to their local population.

CLSI EP28-A3c describes a formal validation procedure: a minimum of 20 reference subjects from the local population are tested. If no more than 2 of the 20 fall outside the candidate interval, the interval is accepted as transferable. This 20-subject approach is a pragmatic minimum, not a guarantee of accuracy — it detects gross incompatibilities between the source and local populations but cannot detect moderate systematic differences. When there is substantive reason to believe the local population differs from the source population (different ethnicity, different pre-analytical conditions, different analyser platform), a full de novo study is preferable.

MedCalc's Reference interval procedure applies all three methods — parametric, non-parametric, and robust — with automatic normality testing, optional log or Box-Cox transformation, and 90% confidence intervals for the reference limits computed by the appropriate method for each approach.