Passing-Bablok Regression Linearity Assumption

Linearity Test

The key assumption of Passing-Bablok regression is linearity. This can be tested using the following steps:

Step 1: Define ŷ_i = bx_i + a for each i and define n⁺ = # of i such that y_i > ŷ_i and n^– = # of i such that y_i < ŷ_i. Now define the r_i by

Step 2: For each i, define the distance d_i by

Step 3: Sort the r_i in the order of the d_i.

Step 4: Define the c_i and c_max by

Step 5: Test the null hypothesis that there is a linear relationship between the x_i and y_i using the test statistic

For any significance level α, we get a significant result (and so there isn’t a linear relationship between the x_i and y_i) when h ≥ h_crit where h_crit is the critical value from the Kolmogorov distribution at α, which can be calculated using the Real Statistics formula KINV(α). Figure 1 displays h_crit values for key values of alpha.

Figure 1 – h-crit values

Note that instead of steps 4 and 5 we could use a runs test to check for randomness, which is essentially what the KS test in steps 4 and 5 is doing.

Example

Example 1: Determine whether the linearity assumption holds for Example 1 of Passing-Bablok Regression Basic Concepts.

The test is carried out as shown in Figure 1. Since h < h-crit (AJ8 and AJ9), or p-value = .92 > .05 = α (AJ10), we conclude that the linearity assumption is likely to hold.

Figure 1 – Testing linearity

Some representative formulas from Figure 1 are shown in Figure 2.

Figure 2 – Key formulas from Figure 1

Note that the formulas in range AL2:AM20 and cell AN21 are array formulas.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

Hintze, J. L. (2020) Passing-Bablok regression for method comparison. NCSS
https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Passing-Bablok_Regression_for_Method_Comparison.pdf