Passing-Bablok Regression Basic Concepts

Motivation

Passing-Bablok regression is a non-parametric technique for comparing two methods (especially two measurement techniques) to see whether or not they yield similar results. This is also the motivation for Deming regression and Bland-Altman. Often you are comparing an existing method with some new method that has some advantages (less expensive, less-invasive, easier to apply, etc.), but you still want to make sure that the new method will produce similar results.

Ordinary linear or total least squares regression can be used for this purpose in which data from one method is regressed on data from the other to produce a linear regression model of the form y = βx + α. You then need to test the null hypotheses that the slope β = 1 and the intercept α = 0, in which case y = x, as desired.

Passing-Bablok regression is based on a similar motivation except that the normality assumption is dropped.

Steps

Passing-Bablok regression is performed on the data X = {x₁, …, x_n} and Y = {y₁, …, y_n} using the following steps:

Step 1: Calculate the slope of all possible pairs of XY points. There are N = C(n, 2) such pairs. More specifically, for the pairs (x_i, y_i) and (x_j, y_j) where j > i calculate the slope s_ij as follows:

We will then calculate the median of S = {s_ij: j > i} and use it as an estimate of the slope coefficient in the regression, although, as we will see shortly, we need to make a few modifications to this approach.

The first modification required is what to do when x_i = x_j, in which case s_ij is undefined. We handle this in the following ways:

1. If x_i = x_j and y_i = y_j (in which case s_ij = 0/0), then exclude that s_ij from the set S.
2. If x_i = x_j and y_i < y_j (in which case s_ij = ∞), set s_ij  = L where L is a large positive number (the exact value won’t figure into the calculation of the median of S)
3. If x_i = x_j and y_i > y_j (in which case s_ij = -∞), set s_ij  = –L where L is a large positive number (again, the exact value won’t figure into the calculation of the median of S)

In addition, we make the following modification:

• If s_ij = -1, then exclude that s_ij from the set S.

Step 2: We next set k equal to the number of elements in S that are less than -1. Instead of setting the regression slope coefficient to the median of S we set it to the median shifted k places to the right.

More specifically, if S contains M elements (i.e. N elements minus the elements that have been removed), then if M is odd, namely M = 2m+1, then the median is the m+1^th smallest element in S, and so we set the regression slope coefficient b = the m+1+k^th smallest element in S.

If instead, M is even, namely M = 2m, then the median is the average between the mth smallest element in S and the m+1^th smallest element in S, and so we set the regression slope coefficient b equal to the mean of the m+k^th and m+1+k^th smallest elements in S.

Note that in Excel, we will use the SMALL function to carry out these calculations.

The intercept coefficient a is now equal to the median of the set {y_i – bx_i: 1 ≤ i ≤ n}.

Step 3: We calculate a confidence interval for the regression coefficients as follows. Define

where z_crit = NORM.S.INV(1–α/2) and define

m₁ = (N – c)/2 rounded off to the nearest integer

m₂ = N – m₁ + 1

The  confidence interval for the slope coefficient is (b_lower, b_upper) where

b_lower is the m₁+k^th smallest element in S

b_upper is the m₂+k^th smallest element in S

The 1–α confidence interval for the intercept coefficient is (a_lower, a_upper) where

a_lower is the median of {y_i – b_upperx_i: 1 ≤ i ≤ n}

a_upper is the median of {y_i – b_lowerx_i: 1 ≤ i ≤ n}

Step 4: If 1 is contained in the confidence interval for the slope and 0 is contained in the confidence interval for the intercept, then we have confidence in the similarity between the two methods. Note that since we have two tests, it would be prudent to use a Bonferroni correction (e.g. by replacing alpha = .05 by alpha = .05/2 = .025). Keep in mind that some would accept 90% confidence (or even higher) rather than 95% confidence.

Example

Example 1: An existing measurement technique was used to obtain the measurements for 18 subjects shown in column B of Figure 1, while the corresponding measurements for these subjects based on a new technique are shown in column C. Using Passing-Boblok regression 1, determine whether the measurements from the new technique are sufficiently similar to those from the existing technique.

Note that the data are the same as those used for Example 1 of Lin’s CCC.

Figure 1 – Passing-Boblok Regression (part 1)

We start by creating an 18 × 18 array containing the s_ij values. To do this, we first place the array formula =TRANSPOSE(B4:C22) in range D2:V3. We next insert the following formula in cell E5

=IF($D5>E$4,IF($B5<>E$2,IF($B5+$C5<>E$2+E$3,($C5-E$3)/($B5-E$2),””), IF($C5>E$3,1000,IF($C5<E$3,-1000,””))),””)

and then highlight range E5:V22 and press Ctrl-R and Ctrl-D. In this formula, we have used 1000 as a large positive value (the L in the description of the P-B regression procedure).

We continue building the spreadsheet as shown in Figure 2. Here the formulas in cells Y9, Y20, and Y21 are array formulas.

Figure 2 – Passing-Boblok Regression (part 2)

We see that the slope coefficient is b = 1.127 with 95% confidence interval (.92, 1.49), while the intercept coefficient is a = -33.62 with 95% confidence interval (-142.8, 32.8). Since 1 is included in the confidence interval for the slope and 0 is included in the confidence interval for the intercept, we conclude that the new technique yields sufficiently similar measurements to the existing technique.

The one proviso is that we must also check to make sure that the linearity assumption holds.

Examples Workbook

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

References

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

Passing, H. and Bablok, W. (1983, 1984) Comparison of several regression procedures for method comparison studies and determination of sample sizes. Application of linear regression procedures for method comparison studies in Clinical Chemistry. J. Clin. Chem. Clin. Biochem
https://pubmed.ncbi.nlm.nih.gov/6481307/