Testing the relation between percentage change and baseline value

Abstract

Testing the relation between percentage change and baseline value has been controversial, but it is not clear why this practice may yield spurious results. In this paper, we first explained why the usual testing of the relation between percentage change and baseline value is inappropriate and then demonstrated how the appropriate null hypothesis could be formulated. We also proposed a simple procedure for testing the appropriate null hypothesis based on the assumption that when there is no relation between percentage change and baseline value, the coefficients of variation for repeated measurements of a random variable should remain unchanged. Two examples were used to demonstrate how the usual testing gave rise to misleading results, whilst results from our simple test were in general consistent with those from simulations. We also undertook simulations to investigate the impact of measurement errors on the performance of the proposed test. Results suggested the type-I error rates increased with the magnitude of measurement errors, whilst the statistical power to detect a genuine relation decreased. The usual approach to testing the relation between percentage change and baseline value tended to yield misleading results and should be avoided.

Figure 1. The range of the correlation between x and y/x (r_x,y/x) for different values of k and k is the ratio of the two coefficients of variations for x and y, conditional on the correlation between x and y (r_xy).

Figure 2. Distribution of correlation coefficient between baseline pocket depth and percentage change in pocket depth based on 100,000 simulations of a group of 47 periodontal infrabony lesions treated with guided tissue regeneration to reduce pocket depth.

The median value of the correlation coefficients is 0.64, which is very close to 0.63 derived by Eq. 6.

Figure 3. Distribution of correlation coefficient between baseline CD4 count and percentage change in CD4 count based on 100,000 simulations of a group of 100 HIV infected subjects on antiretroviral therapy.

The aim of antiretroviral therapy was to increase CD4 count. The median value of the correlation coefficient is 0.479, which is slightly smaller than 0.508 derived by Eq. 6.

Figure 4

Results of simulations for type-I error rates and statistical power of the proposed test under different assumptions for the ratio of standard deviation of measurement error to the standard deviation of true baseline value for samples of varying sizes: (a) the type-I errors of the proposed test (in red lines) and the usual approach to testing the relation between percentage change and baseline value, when the follow-up value Y and baseline value X is Y = 0.8X; (b) the statistical power for the proposed test when Y = 0.8X − 0.001X2; (b) the statistical power for the proposed test when Y = 0.8X − 0.002X2; (b) the statistical power for the proposed test when Y = 0.8X − 0.003X2.