To evaluate the effect of a treatment or intervention on a clinical outcome, such as blood pressure, many researchers test the correlation between the change in the outcome and its initial value to see if the treatment is more effective in subjects with more severe disease. The conventional approach is to use correlation or regression analysis to test the relation between the baseline values and change in baseline values that have already been shown to be invalid, since the changes in baseline values is derived from the baseline value, giving rise to a statistical artifact known as mathematical coupling. Previously, Dr. Yu-Kang Tu in the Institute of Epidemiology & Preventive Medicine, College of Public Health, the National Taiwan University, has compared several approaches to correctly testing the relation between the baseline values and the change in baseline values. In his recent research published by Scientific Reports in March, he looks into a similar but more complex problem of how to correctly test the relation between baseline values and the percentage change in baseline values. He proposes a simple method for correctly testing the correlation between percentage change and baseline values.
“Back in 1897, the great British Statistician Karl Pearson derived a formula for approximate correlation between two ratio variables” said by Dr Tu. Suppose the baseline value and the follow-up value after treatment is denoted x and y, respectively, and then according to the Pearson’s formula, Dr. Tu shows that the expected correlation between x and y/x, rx,y/x, is determined by two parameters: (1) k, the ratio between the coefficients of variation for x and y, and (2) rxy, the correlation between x and y. The Pearson’s formula can then be re-arranged as:
The range of possible values for rx,y/x depends on k and rxy and does not always contain zero (see figure 1). Thus, testing against zero is potentially misleading.
Dr. Tu further shows that, when the correlation between percentage change and baseline values is null, the value of k should be close to unity. Therefore, to appropriately test the relation between change-baseline correlation, one should compare the observed correlation between x and y/x (rx,y/x) to , which is the expected correlation between x and y/x under k = 1. After applying Fisher’s z transformation to achieve normality, the final statistic z is shown to be where zr(t) is the Fisher’s z transformation , n is the sample size. The statistic z is then compared to the standard normal distribution, i.e. there is a significant correlation between baseline values and percentage change if |z| > 1.96. Moreover, the true direction of correlation between x and y/x is represented by the sign of instead of r.
Dr. Tu said “simulations based on real data shows that the conventional tests for correlation yield seriously misleading results in terms of both the statistical inference and direction of the relation, while my new approach maintains adequate type I error rates and statistical power as long as the measurement error in x and y is not relatively large.”
In many experiments, percentage change is used as the primary endpoint, as the response to the treatment or intervention is believed in proportion to baseline values. Dr Tu’s new approach provides a simple and more appropriate method to test the relation between baseline values and percentage change in baseline values.