Measuring changes in logarithmic data, with special reference to bronchial responsiveness

Bronchial provocation tests with agents such as histamine and methacholine are commonly used in clinical and epidemiological studies of respiratory illness because bronchial hyperresponsiveness is a non-specific abnormality of the airways which is characteristic of asthma. However, measurements of bronchial responsiveness are log-normally distributed. As a result, special considerations need to be given to reporting within-subject changes in these measurements in longitudinal studies as, for example, in clinical trials or in any study in which a before-and-after experimental design is used. In these types of experiments, changes in bronchial responsiveness should not be simply expressed in the units of the measurement, such as dose of provoking agent, but must be expressed in units based on a logarithmic scale. The appropriate log-based units for measuring within-subject changes are doubling dose, fold difference or percent change. This paper explains the methods for calculating changes in these units in a statistically correct manner. All three units represent different ways of expressing the same change on a logarithmic scale. However, 'doubling dose' is only appropriate when it relates directly to the method of administering the provoking agent in doubling concentrations and 'fold difference' or 'percent change' are both appropriate for expressing any log-based changes. The methods for calculating changes in these units also apply to calculations of repeatability within test methods and to calculations of comparability and agreement between test methods. The methods are described solely for reporting changes in units of bronchial responsiveness but are applicable to other log-normally distributed measurements.