Proportional scatter: per cent predicted is acceptable in children and adolescents!
The logarithms of the FVC (or other lung volumes) and standing height are linearly related;
after allowing for height a residual scatter remains. This
residual spread is also on a logarithmic scale. As in any
other regression equation with normally distributed residuals
we can now compute confidence intervals by adding to or subtracting
from the mean the residual standard deviations multiplied
by an appropriate number, for example +1.64·RSD for
the 5th percentile. As we obviously prefer to work with liters,
rather than with log liters, we need to retransform the prediction
equation by exponentiation. You are reminded that
log (a·b) = log (a) + log (b)
Let us assume that a = predicted FVC, and b = 1.64·RSD; it follows from the above that upon retransformation we should multiply the predicted FVC or other volume (a) by a factor (b).
An example. According to Schrader (ref. 1) the predicted FVC in an adolescent boy is:
loge FVC = 3.3073·loge(height) – 8.6846, RSD 0.1024
At a standing height of 170 cm the
predicted FVC = exp{3.3073·loge(170) –
8.6846} = 4.028 L. To estimate the 5-percentile,
assuming a normal distribution, we need to subtract 1.64·RSD
in the logarithmic scale. Hence, upon retransformation, we
must multiply the predicted FVC by exp(-1.64·0.1024)
= 0.845. In other words, the 5-percentile for the FVC is 84.5%
of the predicted value, for whatever age or height. This is
an example of a proportional scatter, where expressing the
observed value as a percentage of the predicted value is quite
legitimate.
As we shall see the above prediction equation in not satisfactory for the whole age range, because the relationship between the dimensions of the trunk to height changes in children, but this can be resolved by incorporating age-height interaction.
Children, adolescents and adults can be compared without difficulty if we employ standard deviation or Z-scores (ref. 2); to that end, in children and adolescents one should first transform the FVC and the FEV1 logarithmically, in adults this step is not required.
See also:
Advantage of using the natural logarithm
Reference values for adults
SpirXpert software
Ref. 1 - Schrader PC - cited in: Standardization of lung function tests in pediatrics. Working Group Paediatrics SEPCR. Ed. PhH Quanjer, P Helms, J Bjure, Cl Gaultier. Eur Respir J 1989; 2 suppl. 4, p. 201s
Ref. 2 - Standard deviation score = Z-score: The standard deviation score is the difference between observed and predicted value (the deviation) divided by the scatter about the predicted value (the residual standard deviation or RSD in prediction equations). Hence:
SDS = Z-score = (observed - predicted)/RSD predicted