Mathematical coupling in medical research: lessons from studies of oxygen kinetics.

It is common practice in medical research to investigate the relation between two physiological variables by obtaining pairs of measurements over a range of values and then plotting one variable against the other. The nature of the relation may be examined using a technique such as regression analysis, and the strength of the relation assessed by calculating the correlation coefficient. It is often assumed that when a correlation is found between the variables they are physiologically related in some way. In some situations this may not be correct. Two variables, each of which is dependent on a third variable which may not be known, can correlate even if not causally linked. For example, unrelated variables that both change with time will correlate. Another cause of spurious correlation is illustrated in figure 1. A random number table was used to generate 50 pairs of numbers, x (n) and y (n) , between 1 and 10. When these numbers are plotted against one another a random scatter of points is produced and, as expected, the correlation coefficient is 0 (fig. 1A). If the same pairs of numbers are used to calculate a third variable x.y (n) , the product of each data pair, x (n) re-plotted against x.y (n) is no longer random. Instead, there is a linear relation with a correlation coefficient of 0.7 and a slope that is statistically different from 0 (fig. 1B). This surprising result occurs because the values of x (n) are present in both variables (x (n) and x y (n)), which are then linked mathematically. A similar result occurs when variables are derived by addition, subtraction, or division. This is mathematical coupling. There are many situations in medical research, and in the clinical management of patients, when physiological variables that are difficult to determine directly are calculated from other more easily measured variables. The formulas required are derived from basic physiological principles. Examples in anaesthesia include the deadspace ratio derived from arterial and mixed expired carbon dioxide partial pressures using the Bohr equation, and systemic vas-cular resistance derived from cardiac output and mean arterial pressure. If the relation is investigated between a derived variable and another variable that is either a component of the derived variable or shares a component with it, the relation may be affected by mathematical coupling. As a result, part or all of the relation may be spurious. Mathematical coupling can …