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Age-related reference interval

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An age-related (age-specific or age-dependent) reference interval is a reference interval that varies with the patients' age.

The methodology that MedCalc uses to model a continuous age-related reference interval is based on the methods described by Altman (1993), Altman & Chitty (1993), and Wright & Royston (1997a).

The method includes the following steps:

  1. If the distribution of the measurements (the variable for which to establish a reference interval) shows skewness at different levels of age, the measurements are transformed logarithmically or using a Box-Cox power transformation.
  2. The transformed measurements are modelled on age using weighted polynomial regression (Altman & Chitty, 1994). This regression model gives the mean of the (transformed) measurements as a function of age: mean(age).

  3. The residuals of this regression model are calculated.
  4. The absolute residuals, multiplied by $ \sqrt { \pi / 2 } $ are modelled on age using weighted polynomial regression (Altman, 1993). This second regression model gives the standard deviation of the (transformed) measurements as a function of age: SD(age).
  5. For every age in the observed range, the reference interval is calculated by taking mean(age) ± z SD(age). For a 95% reference interval z = 1.96. If the data were initially transformed in step 1, MedCalc back-transforms the results to the original scale.
  6. The model is evaluated by analyzing and plotting z-scores for all observations. The z-score for an observed value y is calculated by
    $$ z = \frac { y - \operatorname {mean}(age) } { \operatorname {SD}(age) } $$
    The z-scores should be normally distributed. If they are not, the model may not be appropriate and other powers for the polynomial model may be selected.

Required input

The example makes use of the data on biparietal diameter (outer-inner) from Chitty et al., 1994. Data downloaded from

Age-related reference interval - MedCalc dialog box

Measurements and age variables

Reference interval options

Powers for polynomial model

Options for Measurements variable



Age-related reference interval - MedCalc report

Suspected outliers

The program produces a list of possible outliers of the measurements, 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).

Model summary

This table gives a summary of the model.

The first row shows the outcome variable.

Next the regression equation is given for Mean and SD of the outcome variable.


This table lists the centiles at different ages (for about 6 to 12 values of age).

Below this table there is a hyperlink to get a more comprehensive table in Excel format, for about 60 to 120 values of age. This Excel file includes the formulae for the different centiles and therefore can easily be shortened or expanded to your needs.

Fitted equations for Mean and Standard Deviation

This table lists the details of the weighted regression for the Mean of the measurements and next for the Standard Deviation.

The different coefficients are listed with their standard error and P-value.

The P-values should not be given too much attention. Specifically, they must not be used to decide if a term should remain or should be removed from the model. It is the magnitude of the coefficient itself that is of interest.


The analysis of the z-scores is an important step in the evaluation of how well the model fits the data.


Scatter plot with centile curves

Age-related reference interval - centiles

This plot shows a scatter diagram of the measurements versus age with the calculated mean (central line) and centile curves.


This graph shows the z-scores plotted against age.

Age-related reference interval - z-scores

Horizontal lines are drawn at z-scores of −1.645 and 1.645.

The central line (red in the example) is a 80% smoothed LOESS (Local Regression Smoothing) trendline.

The z-scores should not display any pattern and must be randomly scattered about 0 at all ages (Altman & Chitty, 1993). It is expected that 5% of cases lie above the line corresponding to z=1.645 and 5% of cases are expected to lie below the line corresponding with z=−1.645; and these cases should be randomly distributed across the observed age range. Any deviation from this indicates that the model needs modification.

In the graph's Info panel, the exact number of observations below z=−1.645 and above z=1.645 is reported.

Confidence intervals

MedCalc allows to calculate confidence intervals for the reference limits using bootstrapping (Wright & Royston, 1997b).

Proceed as follows:


See also

Recommended book

Book cover

Handbook of Parametric and Nonparametric Statistical Procedures, Fifth Edition
David J. Sheskin

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The Handbook of Parametric and Nonparametric Statistical Procedures, provides researchers, teachers, and students with an all-inclusive reference on univariate, bivariate, and multivariate statistical procedures.