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CDF for PROs
A Note on Cumulative Distribution Functions for Patient-Reported Outcomes

Andrew G. Bushmakin, MS
Joseph C. Cappelleri, PhD, MPH


Pfizer Inc., 445 Eastern Point Road, MS 8260-2502 Groton, CT 06340

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Keywords: cumulative distribution function, percent change, transformed scores, original scores, patient-reported outcomes, label claim




Purpose: To quantify differences between cumulative distribution function (CDF) plots using percent change and simple change from baseline with original scores versus their transformed scores (0-100 scale) on patient-reported outcomes.
Method and Results: Derivations show that percent change on transformed scores can give more extreme values than percent change on original scores. Unlike simple change, percent change based on transformed scores and original scores can give different p-values in the comparison of CDF plots between treatments.
Conclusions: The choice of which type of score (original or transformed) and type of metric (percent or simple) to use should be based on how a patientreported outcome was developed, analyzed, interpreted, and reported before CDF plots are later considered for the purpose to enhance interpretation on a patient-reported outcome.


In terms of patient-reported outcomes, a cumulative distribution function (CDF) gives the probability that, in the data under consideration, the individual scores on a patient-reported outcomes measure are less than or equal to a given score on the same measure.1-2 A CDF is a useful graphical and descriptive display because it captures all available data. In doing so it provides a comprehensive profile on the distribution of individual scores for each treatment group. Treatment groups can then be compared in terms of their cumulative distributions to determine whether their probability distributions differ; the corresponding p-value between independent groups can be obtained from the Kolmogorov-Smirnov test.1-2


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