Measurement error proportional to the mean

摘要

We often need to know the error with which measurements are made—for example, so that we can decide whether the change in a clinical observation represents a real change in a patient's condition. We have discussed previously the within-subject standard deviation as a practical index of measurement error. We said that this approach should be used when the measurement error was not related to the magnitude of the measurement and recommended that we plot the subject standard deviation against the subject mean to check this. Table 1 shows some duplicate salivary cotinine measurements taken from a larger study. Figure 1 shows absolute subject difference against subject mean, which is equivalent to a standard deviation versus mean plot when we have only two measurements per subject. If we are to use the within-subject standard deviation as an index of measurement error we need the subject standard deviation to be independent of the subject mean. Here, there is a clear relationship, the variability increasing with the magnitude. We can test this using a rank correlation coefficient if we wish; here Kendall's τ = 0.62, P = 0.0001. Under these circumstances a logarithmic transformation of the data almost always solves the problem, but we can check by plotting log standard deviation against log mean. For these data the slope is 0.9; as this is very close to 1 the subject standard deviation is roughly proportional to the subject mean and a log transformation is indicated. Figure 2 shows the plot of absolute difference versus subject mean for the log transformed data. There is now no evidence of a relationship (Kendall's τ = 0.07, P = 0.7).