Limitations of calculating “true” regression slope: impact on estimates of minimal important difference

  1. J.W. Dodd and
  2. P. Jones
  1. Dept of Respiratory Medicine, Clinical Sciences, St George’s University of London, London, UK
  1. J.W. Dodd, Dept of Respiratory Medicine, Clinical Sciences, St George’s University of London, Cranmer Terrace, London SW17 0RE, UK. E-mail: jdodd{at}sgul.ac.uk

To the Editors:

We read with interest the recent article 1 that addresses the important question of minimal important difference (MID) for exercise tests in chronic obstructive pulmonary disease. Puhan et al. 1 use both distribution- and anchor-based methods, and it is the anchor-based methodology that we would like to consider here. Those authors use linear regression analysis between change in the dependent variable, in this case 6-min walk distance, and change in an anchor variable (St George’s Respiratory Questionnaire and University of California San Diego Shortness of Breath Questionnaire) as the independent variable. We would like to highlight limitations of this approach and demonstrate, with example data, the potential impact on estimates of MID. The problem is based on the assumption that the independent variable (X) is measured without error, but the slope of linear regression will be a measure of the “true” relationship only if all variance of X is due to the regression and not due to measurement error. Although no measurement is free from error, some signals are rather more “noisy” than others. The presence of high correlation between the variables will indicate the reduction in such a potential error. This point is made by the authors, who suggest an r-value of 0.5 as a cut-off and a linear relationship between the two variables.

Figures 1 and 2, and tables 1 and 2 show a worked example using fictional data of two linear and strongly correlated (r = 0.76) variables, A and B. The MID for B is known to be 2; this example will illustrate the impact of switching variables on the x- and y-axes of the scatterplot on estimating the MID of variable A.

Figure 1–
Figure 1–

Scatter plot of variable A versus variable B. Linear r2 = 0.582.

Figure 2–
Figure 2–

Scatter plot of variable B versus variable A. Linear r2 = 0.582.

View this table:
Table 1– Linear regression analysis with variable A as the dependent variable
View this table:
Table 2– Linear regression analysis with variable B as the dependent variable

This example demonstrates that estimates of MID vary considerably (-0.4 versus -1.5) despite a high correlation coefficient r = 0.76, well above the threshold suggested by Puhan et al. 1.

The problem of measurement error when comparing two variables has been discussed at length in work by Daubenspeck and Ogden 2, Kendall 3, and Kermack and Haldane 4. They propose that this may be resolved by either normalising one axis or dividing each observation by its standard deviation, estimated from the pooled observations. This approach would eliminate the dependency of the fitted line parameters on the scale of the measurement used; however, it would not readily allow estimation of an MID. The message is that mapping exercises such as this become less reliable with lower correlations between the two variables that are being compared and, even when those variables are well correlated, may lead to an unreliable estimate of the MID. MID will be estimated most reliably though direct measurement rather than statistical inference.

Footnotes

REFERENCES

    1. Puhan MA,
    2. Chandra D,
    3. Mosenifar Z,
    4. et al
    . The minimal important difference of exercise tests in severe COPD. Eur Respir J 2011; 37: 784–790.
    1. Daubenspeck JA,
    2. Ogden RD
    . Estimation of response slopes in respiratory control using directional statistics. J Appl Physiol 1978; 45: 823–829.
    1. Kendall MG
    A Course in Multivariate Analysis. London, Giffin 1957.
    1. Kermack KA,
    2. Haldane JBS
    . Organic correlation and allometry. Biometrika 1950; 37: 30–41.