A simple linear relationship between the results of a medical device and the true concentration of an analyte is important to health care providers. The assessment of linearity is rather straightforward. The results of the medical device are regressed on the reference values and the regression parameters are evaluated statistically. The following regression models are fitted:
Where Y = the device's test results, X = the reference values and a, b, c and d are the regression parameters.
Equations (1), (2) and (3) above represent linear, quadratic and cubic models, respectively. A minimum of five different levels of reference values are needed to perform the regressions shown above and to estimate the residuals. Linearity over the test range is established when the non-linear parameters (c and d) are shown to be statistically insignificant. The value of each non-linear parameter and its standard error are used to perform the statistical test of significance. Do not panic if any of the nonlinear terms is found to be statistically significant. One way to deal with non-linearity is to reduce the testing range (the range of the levels of the reference values.) Another way to deal with significant non-linearity is to assess the degree of deviation from linearity and evaluate the deviation from linearity in terms of clinical relevance and the repeatability of the device. Statistically significant non-linear terms may not be clinically relevant.
The linear model provides a simple relationship between the bias and the reference values. In terms of the X and Y as defined above, the bias is given as:
Using equation (1), we can rewrite equation (4) as:
Equation (5) reduces to:
As shown in equation (6), the bias has two components: a fixed component (given by the first term on the right-hand side, a, and is known as the fixed bias,) and a proportional component (given by the term (b-1) and is known as the proportional bias.) Equation (6) can be used to estimate the bias at certain values of X (the concentration of the analyte) that are clinically important.