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Reference interval

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A Reference interval (Reference range, Normal range) for a parameter is the interval in which the central 95% values of apparently healthy subjects lie. For a double sided reference interval, where both low and high values are suspicious, there are 2 limits of normality: a lower limit (with 2.5% of healthy subjects below), and upper limit of normality (with 2.5% of subjects above that value).

When only low values are suspicious, then there is only a lower limit of normality (with 5% of healthy subjects beloww that value) and no upper limit of normality. This is a left sided reference interval.

On the other hand, when only high values are suspicious, then there is only a higher limit of normality (with 5% of healthy subjects above that value) and no upper limit of normality. This defines a right sided reference interval.

A 90% confidence interval (as recommended by the CLSI Guidelines C28-A3) is calculated for both limits of normality. As always, the confidence interval will be more narrow with higher number of subjects in the study, meaning more certainty about the reference limits.

The reference interval can be calculated using the following 3 methods: (a) using the Normal distribution (b) using a non-parametrical percentile method, and (c) optionally the robust method as described in the CLSI Guidelines C28-A3.

Normal distribution method

In the Normal distribution method, the mean, variance, and standard deviation of the sample data are calculated.

For a 2-sided reference interval, the limits of normality are :

$$ Lower\ limit = Mean - 1.96 \times SD $$ $$ Upper\ limit = Mean + 1.96 \times SD $$

The 90% confidence interval for each limit is given by (Bland, 2000):

$$ limit \pm 1.64 \times \sqrt{ Variance \times \Biggl( \frac{1}{n} + \frac{2}{n-1} \Biggr) } $$

The normal distribution method requires that the data present a normal distribution, possibly after logarithmic or Box-Cox transformation.

This method does not require a minimum number of subjects, but a minimum sample size of 40 is recommended (Le Boedic, 2019).

Percentile method

In the percentile method, the lower and upper limits of normality are given by the 2.5th and 97.5th percentiles for a double sided reference interval. The 5th percentile for a left sided reference interval, and 95th percentile for a right sided reference interval.

Following the CLSI guidelines, 90% confidence intervals are defined using the method of Reed et al. (1971).

For the calculation of a 90% confidence interval in the percentile method, a minimal sample size of 120 is required (CLSI, 2008).

Robust method

The calculation of a reference interval using the robust method (Horn & Pesce, 2005) involves an iterative process, in which the initial central value is estimated by the median and the initial spread by the median absolute deviation about the median (MAD). In the iterative process, actual observations are downweighted according to their distance from the central tendency of the sample. In each iteration, a quantity Tbi, representing the updated estimate of central tendency, is calculated, until the change in consecutive iterative values is negligible. Weighted estimators of variability and spread are calculated to establish the reference limits. See Horn & Pesce (2005) or CLSI (2008) for computational details.

90% confidence intervals for the reference limits are estimated using bootstrapping (percentile interval method, Efron & Tibshirani, 1993)

The robust method can be used as an alternative to the percentile method when sample size is less than 120.

Required input

Input for reference interval

In the dialog box you identify the variable with the measurements. You can click the Drop-down button button to obtain a list of variables. In this list you can select a variable by clicking the variable's name. You can also enter or select a filter in order to include only a selected subgroup of measurements in the statistical procedure, as described in the Introduction part of this manual.



Reference interval

Summary statistics

Logarithmic transformation

If the option Logarithmic transformation was selected, the program will display the back-transformed results. The back-transformed mean is named the Geometric mean. The Standard deviation cannot be back-transformed meaningfully and is not reported.

Suspected outliers

The program produces a list of possible outliers, detected by the methods based on Reed et al. (1971) or Tukey (1977). The method by Reed et al. tests only the minimum and maximum observations; the Tukey test can identify more values as outliers. Note that this does not automatically exclude any values from the analysis. The observations should be further inspected by the investigator who can decide to exclude the values. Click on the listed values (which are displayed as hyperlinks) to show the corresponding data in the spreadsheet (see Exclude & Include).

Reference interval

The program will give the 90, 95, 99, 99.9 or 99.99% Reference interval, double sided or left or right sided only, as selected in the dialog box.

The reference interval is calculated using 3 different methods: (a) using the Normal distribution (Bland, 2000; CLSI 2008), (b) using a non-parametrical percentile method, and (c) optionally a "robust method" as described in the CLSI Guidelines C28-A3.

90% Confidence Intervals are given for the reference limits.

For the robust method the confidence intervals are estimated with the bootstrap method (percentile interval method, Efron & Tibshirani, 1993). When sample size is very small and/or the sample contains too many equal values, it may be impossible to calculate the CIs.

The results from the Normal distribution method are not appropriate when the Test for Normal distribution (see above) fails. If the sample size is large (120 or more) the CLSI C28-A3 guideline recommends the percentile method, and for smaller sample sizes the "robust method".

The minimal sample size of 120 for the percentile method is the minimum number required to calculate 90% Confidence Intervals for the reference limits. A higher number of cases is required to achieve more reliable reference limits with narrower 90% Confidence Intervals.


Click the Graph button in the dialog box shown above to obtain the following Reference Interval Graph box:

Reference interval graph dialog box

This results in the following graph:

Reference interval graph


See also

External links

Recommended book

Book cover

An Introduction to Medical Statistics
Martin Bland

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This textbook is intended for medical researchers and includes the design of clinical trials and epidemiological studies, data collection, summarizing and presenting data, probability, standard error, confidence intervals and significance tests, techniques of data analysis including multifactorial methods and the choice of statistical method, problems of medical measurement and diagnosis, vital statistics, and calculation of sample size.